---
abstract: 'We study the effect of electron interaction in an electronic system with a high-order Van Hove singularity, where the density of states shows a power-law divergence. Owing to scale invariance, we perform a renormalization group (RG) analysis to find a nontrivial metallic behavior where various divergent susceptibilities coexist but no long-range order appears. We term such a metallic state as a *supermetal*. Our RG analysis reveals the Gaussian fixed point and a nontrivial interacting fixed point, which draws an analogy to the $\phi^4$ theory. We further present a finite anomalous dimension at the interacting fixed point by a controlled RG analysis, thus establishing an interacting supermetal as a non-Fermi liquid.'
author:
- Hiroki Isobe
- Liang Fu
title: Supermetal
---

Introduction
============

A Bloch electron in a crystal is described by the energy dispersion $E_{\bm{k}}$ that relates the energy with its wave vector $\bm{k}$. For metals, the energy dispersion determines the density of states (DOS) at the Fermi level, which to a large extent governs various thermodynamic properties such as charge compressibility, spin susceptibility and specific heat. Van Hove’s seminal work [@VanHove] revealed that the DOS exhibits non-analyticity at an extremum or a saddle point of the energy dispersion, where $\nabla_{\bm{k}} E_{\bm{k}}=0$. Importantly, Van Hove singularities (VHS) are guaranteed to exist in every energy band by the continuity and the periodicity of $E_{\bm{k}}$ over the Brillouin zone. The behavior of the DOS at a VHS depends on whether it is at an energy extremum or a saddle point, and also on the dimensionality of the system. For example, at a saddle point in two dimensions with $E_{\bm{k}} = k_x^2-k_y^2$, the DOS diverges logarithmically. As the chemical potential crosses the VHS, the topology of Fermi surface changes from electron to hole type, known as electronic topological transition.

Recently, we have extended the notion of VHS to high-order saddle points, where, besides $\nabla_{\bm{k}}E_{\bm{k}}=0$, the Hessian matrix $D_{ij} =\partial_{k_i} \partial_{k_j} E$ satisfies $\det D(\bm{k})=0$ [@high-order]. These high-order saddle points occur where two Fermi surfaces touch [*tangentially*]{}, or at the common intersection of three or more Fermi surfaces [@Fu2011; @multicritical1]. An example of the former is $E_{\bm{k}}= k_x^2 - k_y^4$, and of the latter is $E_{\bm{k}}=k_x^3 - 3 k_x k_y^2$. Generally speaking, high-order saddle points can be realized by tuning the energy dispersion with one or more control parameter. At high-order saddle points in two dimensions, the DOS shows a power-law divergence $|E|^{-\epsilon}$, much stronger than a logarithmic one at ordinary VHS [@high-order; @multicritical1].

The existence of high-order VHS has recently been identified in a variety of materials including twisted bilayer graphene near a magic angle, trilayer graphene-hexagonal BN heterostructure [@high-order], and Sr$_3$Ru$_2$O$_7$ [@multicritical2]. In particular, scanning tunneling spectroscopy measurements [@Pasupathy] on twisted bilayer graphene found power-law divergent DOS at the magic twist angle, providing direct evidence for high-order VHS [@high-order].

In the presence of electron-electron interaction, a large DOS near the Fermi level may have important consequences. On the one hand, it may trigger Stoner instability to ferromagnetism. On the other hand, a large DOS may result in strong screening of repulsive interaction, so that a Fermi liquid description remains valid at low energy.

In this work, we study interacting electron systems with a high-order saddle point near the Fermi level. Assuming that electron interaction is weak, dominant contributions to low-energy thermodynamic properties of the system come from those states in the vicinity of the saddle point, from which the DOS divergence originates. This allows us to formulate a [*continuum*]{} field theory of interacting fermions by taking the leading-order energy dispersion relation $E_{\bm{k}}$ near the saddle point and extending the range of momentum to infinity.

In this field theory, when the high-order VHS is right at the Fermi level, the Fermi surface in $\bm{k}$-space becomes [*scale-invariant*]{}. As the VHS approaches the Fermi level, charge and spin susceptibilities exhibit power-law divergence, reminiscent of critical phenomena. Motivated by these observations, we develop a renormalization group (RG) theory for interacting fermions near high-order VHS, which parallels Wilson–Fisher RG approach to $\phi^4$ theory [@Wilson-Fisher; @Wilson]. By introducing a small parameter $\epsilon$ associated with the DOS divergence, we present a [*controlled*]{} RG analysis and find that short-range repulsive interaction is relevant at the noninteracting fixed point and drives the system into a nontrivial $T=0$ interacting fixed point. The former is the analog of Gaussian fixed point in Fermi system, and the latter the analog of the Wilson–Fisher fixed point.

The metallic state at the interacting fixed point exhibits scale-invariance in space/time and power-law divergent charge and spin susceptibility, but finite pairing susceptibility. In other words, this is a metal on the verge of becoming charge-ordered and ferromagnetic. We call such a metallic state with various power-law divergent susceptibilities but without any long-range order, a *supermetal*. We further show by a two-loop RG calculation that the fermion field acquires a finite anomalous dimension. Hence the interacting supermetal we found is a non-Fermi liquid.

The outline of the paper is as follows: In Sec. \[sec:model\], we introduce a model with high-order VHS and calculate the power-law divergent DOS, whose exponent is determined from the scaling property of energy dispersion.

In Sec. \[sec:energy-shell\], we perform the energy-shell RG analysis step by step. We first define the energy shell as a region of momentum space. Then, the tree-level and one-loop RG equations for the chemical potential and interaction strength are derived in sequence, which resembles the case of $\phi^4$ theory. We identify the noninteracting Gaussian fixed point and the nontrivial interacting fixed point which is the analog of the Wilson–Fisher fixed point in Fermi system. We next consider other relevant perturbations to the system, including Zeeman and pairing field as well as additional symmetry-allowed terms in the energy dispersion. The discussion about higher-loop renormalization in the energy-shell RG analysis follows, though an actual two-loop calculation appears in the later section.

In Sec. \[sec:analysis\], we perform the scaling analysis for thermodynamic quantities and correlation functions. The generic formalism is first presented, followed by the one-loop result for various exponents of divergent susceptibilities. We also discuss the Ward identity in this section, which results from charge conservation and imposes a constraint on the field renormalization and the charge compressibility.

In Sec. \[sec:field\_theory\], another RG scheme, the field theory approach, is introduced. We briefly discuss the two RG schemes before presenting the RG analysis from the field theory approach. The field theory approach has an advantage in calculating higher-order perturbative corrections compared to the energy-shell RG analysis. The UV regulator with a soft energy cutoff is then introduced, which is confirmed to satisfy the Ward identity. The one-loop calculation reproduces the energy-shell RG analysis in Sec. \[sec:energy-shell\]. Furthermore, the two-loop calculation shows the finite anomalous dimension at a high-order VHS realized at a saddle point of an energy dispersion. It proves the non-Fermi liquid nature of interacting supermetal.

In Sec. \[sec:lifetime\], we evaluate the quasiparticle lifetime at finite temperature due to electron interaction. From a perturbative calculation at two-loop order, we find an unusual temperature dependence in the quasiparticle lifetime.

In Sec. \[sec:discussions\], we summarize the results and discuss their significance in the broad context of Van Hove physics, RG approach to Fermi systems, and non-Fermi liquids. We also discuss possible material realizations of supermetal.

Model {#sec:model}
=====

An example of high-order VHS in two dimensions {#sec:tight-binding}
----------------------------------------------

We consider a tight-binding model on an anisotropic square lattice $$\begin{aligned}
H = - \sum_j \left( t_x c_{j+\hat{x}}^\dagger c_j + t_y c_{j+\hat{y}}^\dagger c_j + t'_y c_{j+2\hat{y}}^\dagger c_j \right) + \text{H.c.}\end{aligned}$$ $t_x$ and $t_y$ are the nearest-neighbor hopping amplitudes along the $x$ and $y$ directions, respectively, and $t'_y$ is the second-nearest neighbor hopping along the $y$ direction. The energy dispersion is obtained as $$\label{eq:tb_dispersion_full}
E_{\bm{k}} = -2t_x \cos(k_xa) -2t_y \cos(k_ya) -2t'_y \cos(2k_ya),$$ with the lattice constant $a$.

For $|t_y|\geq|t'_y|/4$, there are four VHS points in the Brillouin zone at the high symmetry points: $\Gamma=(0,0)$, $X=(\pi/a,0)$, $Y=(0,\pi/a)$, and $M=(\pi/a,\pi/a)$. With $t_x$, $t_y$, $t'_y>0$, the energy minimum and maximum are located at $\Gamma$ and $M$ points, respectively, and $X$ and $Y$ points are the saddle points \[Fig. \[fig:tb\](a)\]. At the special value $t'_y=t_y /4$, the energy dispersions takes the form $$E_{\bm{k}} = k_x^2 - k_y^4 \label{x2y4}$$ near $Y$ and $k_x^2 + k_y^4$ near $M$, where coefficients have been eliminated by rescaling $k_x$ and $k_y$. The former is a high-order saddle point, while the latter is a high-order energy extremum.

![ Lattice model for a high-order VHS. (a) Energy contour plot with $t_y/t_x=0.8$ and $t'_y/t_x=0.2$. We find the energy minimum at $\Gamma$, the maximum at $M$, and the two saddle points at $X$ and $Y$. $Y$ and $M$ are high-order VHS points. At $Y$, we see that the two Fermi surfaces touch tangentially while they cross linearly at $X$. (b) DOS for the energy dispersion in (a). The four VHS points give rise to analytic singularities in the DOS, where the corresponding points are labeled in the figure. The two peaks at $Y$ and $M$ correspond to high-order VHS, fitted by the analytic formula for the continuum theory Eq. . []{data-label="fig:tb"}](tb_c.pdf){width="\hsize"}

A VHS manifests itself as an analytic singularity in the DOS $$\label{eq:DOS_formula}
D(E) = \int_{\bm{k}} \delta(E-E_{\bm{k}}),$$ where $\int_{\bm{k}}=\int\frac{d^dk}{(2\pi)^d}$ stands for the momentum integration in $d$ dimensions. The DOS for the present model $(d=2)$ is depicted in Fig. \[fig:tb\](b). We find four singularities in the DOS and each of them tied to the individual VHS of the model. The band bottom at $\Gamma$ gives rise to a discontinuity in the DOS and the saddle point at $X$ shows a logarithmic divergence in the DOS. Those two are conventional VHS, known since the original work of Van Hove [@VanHove]. Here we focus on the high-order VHS at $Y$ and $M$. They exhibit distinct behavior: the DOS has a power-law divergence as $|E|^{-1/4}$ instead of a logarithm. In addition, the divergence at $Y$ is stronger on the electron side by the factor $\sqrt{2}$ than on the hole side. Such an asymmetry is not seen for a conventional VHS with a logarithmic divergence at $X$. The two Fermi surfaces touch tangentially at $Y$ at the Van Hove energy \[Fig. \[fig:contour\](a)\]. When the chemical potential $\mu$ crosses the Van Hove energy, the Fermi surface topology changes from being closed to open in the $k_y$ direction.

In Fig. \[fig:contour\](b), the DOS peaks at the two high-order VHS in our tight-binding model are fitted by the analytical expressions of the DOS calculated from the continuum model (\[x2y4\]). The calculation will be shown in the next subsection. We can see a close fit within a finite energy range. Since the divergent DOS and hence susceptibilities originate from the vicinity of the high-order VHS, the continuum model is expected to capture universal features at low energy. Using the continuum model has the advantage of removing non-universal aspects associated with high-energy regions away from the high-order VHS in the tight-binding model. We will show that infrared (IR) scaling properties are not indeed affected by the UV cutoff in the continuum model.

Before proceeding, we briefly mention the Fermi surfaces in strained Sr$_2$RuO$_4$ [@ruthenate1; @ruthenate2; @ruthenate3]. It has a quasi-two-dimensional electronic structure with a layered perovskite structure. Under uniaxial pressure, a Lifshitz transition occurs on the Brillouin zone boundary [@ruthenate2]. At the transition point, there is one VHS in the Brillouin zone at the Fermi energy. The Fermi surface of the band of interest resembles the one obtained from Eq. .

Generalization {#sec:generalization}
--------------

From now on, we study a continuum model of fermions with a high-order energy dispersion. For the purpose of controlled RG analysis later, here we consider the generalized energy dispersion in the $d$-dimensional $k$-space $$\label{eq:model}
E_{\bm{k}} = A_+ k_+^{n_+}- A_- k_-^{n_-}.$$ The momentum is denoted by $$\bm{k} = (\bm{k}_+, \bm{k}_-),$$ where $\bm{k}_\pm$ are $d_\pm$-dimensional vectors with $d_++d_-=d$, and $k_\pm = |\bm{k}_\pm|$. Analyticity of the energy dispersion requires $n_\pm$ to be positive integers. We consider the case of even $n_\pm$, so that $E_{\bm{k}} = E_{-\bm{k}}$ satisfies time-reversal symmetry. When at least one of $n_\pm$ is greater than two, this energy dispersion has a high-order VHS at $k=0$, which is defined as a point where the Hessian matrix $D_{ij} =\partial_{k_i} \partial_{k_j} E_{\bm{k}}$ fulfills $\det D_{ij}=0$.

The energy dispersion Eq.  follows the scaling relation $$E_{\bm{k}} = b E_{\bm{k}'} \text{ with } \bm{k}'=(\bm{k}_+/b^{1/n_+},\bm{k}_-/b^{1/n_-}). \label{scalingE}$$ It then follows from Eq.  and Eq.  that the DOS satisfies $$\begin{gathered}
\label{eq:DOS}
D(E) =
\begin{cases}
D_+ E^{-\epsilon} & (E>0) \\
D_- (-E)^{-\epsilon} & (E<0),
\end{cases}\end{gathered}$$ where the DOS singularity exponent $\epsilon$ is $$\epsilon = 1-\frac{d_+}{n_+}-\frac{d_-}{n_-}.$$ Throughout this work, we consider the case $\epsilon>0$. For example, the high-order VHS introduced in the preceding section corresponds to the case of $d_+ = d_-=1, n_+=2, n_-=4$, so that $\epsilon = 1/4$.

We calculate the prefactors $D_\pm$ for the dispersion (\[eq:model\]) explicitly and find

\[eq:DOS\_prefactor\] $$\begin{gathered}
D_s = D_0 \sin\left(\frac{\pi d_s}{n_s}\right) \quad (s=\pm),\end{gathered}$$ with the common factor $$D_0 = \frac{4\Gamma(\epsilon)}{\pi(4\pi)^{d/2}} \prod_{s=\pm} \frac{\Gamma\left(d_s/n_s\right)}{n_s A_s^{d_s/n_s} \Gamma\left(d_s/2\right)}.$$

We note that in calculating DOS, the $d$-dimensional momentum integral over $\bm k\in (-\infty, \infty) $ is convergent for all $E \neq 0$.

Note that $D_+ \neq D_-$ for $d_+/n_+ \neq d_-/n_-$. It describes the asymmetry in the DOS above and below $E=0$. This is a feature of the high-order saddle points defined by Eq.(\[eq:model\]), distinct from conventional saddle points in two dimensions where the logarithmically divergent DOS peak is symmetric.

![ Energy contour plot for $E_{\bm{k}} = k_x^2 - k_y^4$. The thick line is the Fermi surface at the Van Hove energy, which is scale-invariant. The colored region has the energy inside the cutoff $\Lambda$, where the red (blue) area corresponds to $E>0$ ($E<0$). At every RG step of the energy-shell RG scheme, high-energy modes within the energy shell shown in darker colors are integrated out. In the field theory approach, all states below the cutoff $\Lambda$ are integrated over at once. []{data-label="fig:contour"}](contour_a_2.pdf){width="\hsize"}

The nontrivial fixed point to be shown later is controlled by the smallness of $\epsilon$. For the model defined by Eq. , the exponent can be any rational number between $0<\epsilon<1$. By choosing positive integers $n_\pm$ and $d_\pm$ judiciously, we can make $\epsilon$ arbitrarily small in high-dimensional crystals, while keeping the energy-momentum dispersion an analytic function.

We now introduce our model of interacting electrons near a high-order VHS: $$\begin{aligned}
\label{eq:action}
S =&\int_0^{1/T} d\tau \int d^dr
\Big[ \bar{\psi}_{\sigma} (\partial_\tau + E_{-i\partial_{\bm{r}}} -\mu) \psi_{\sigma}
+ g \bar{\psi}_{\uparrow} \bar{\psi}_{\downarrow} \psi_{\downarrow} \psi_{\uparrow} \Big].\end{aligned}$$ We set $k_B=\hbar=1$ throughout the paper. Here we formulate the model at temperature $T$. Temperature $T$ is regarded as the system size $L_\beta \equiv 1/T$ in the imaginary time direction. $\psi_\sigma$ is the fermionic field and the summation over the spin index $\sigma=\uparrow (+)$, $\downarrow (-)$ is implicit. We consider the local contact interaction between electrons with opposite spins. Later, we shall consider the effects of other interactions and external fields.

From the action, we define the noninteracting Green’s function $$G_0(\bm{k},\omega_n)= \frac{1}{i\omega_n - E_{\bm{k}}},$$ with the fermionic Matsubara frequency $\omega_n=(2n+1)\pi T$ ($n$: integer). The partition function $\mathcal{Z}$ is expressed as $$\mathcal{Z} = \int D\bar{\psi}D\psi e^{-S}. \label{Z}$$ We are interested in thermodynamic quantities such as specific heat. These are obtained from the free energy density $$\label{eq:free_energy}
F=-\frac{T}{V}\ln \mathcal{Z}$$ where $V$ is the volume of the system.

Energy-shell RG analysis {#sec:energy-shell}
========================

Formalism at zero temperature
-----------------------------

In this section, we adopt the Wilsonian approach to the RG equations for the action Eq. . For clarity, we consider first the action at $T=0$, where we will find fixed points. Then, the Matsubara frequency becomes continuous $\omega_n \to \omega$, and the action is written with frequency $\omega$ and momentum $\bm{k}$ as $$\begin{aligned}
S &= \int\frac{d\omega}{2\pi} \int_{\bm{k}} \bigg[ \bar{\psi}_\sigma(k) (-i\omega + E_{\bm{k}} -\mu) \psi_\sigma(k) \nonumber\\
&\quad + g \bigg( \prod_{j=1}^{4} \int\frac{d\omega_j}{2\pi} \int_{\bm{k}_j} \bigg) (2\pi)^{d+1} \delta (k_1+k_2-k_3-k_4) \nonumber\\
&\quad\ \times
\bar{\psi}_{\uparrow}(k_1) \bar{\psi}_{\downarrow}(k_2) \psi_{\downarrow}(k_3) \psi_{\uparrow}(k_4).\end{aligned}$$ We introduce the shorthand notation $k = (\bm{k},\omega)$.

We impose a UV energy cutoff $\Lambda$ on this action to remove unphysical UV divergences that appear in electron density of the ground state, etc. We note that the UV cutoff here is imposed on energy, but not on momentum directly. The region in $\bm{k}$-space with $|E_{\bm{k}}| \leq \Lambda$ still extends to infinity. Importantly, this UV cutoff does not affect universal scaling properties of IR fixed points in the analysis of high-order VHS, as we shall show. The UV cutoff merely appears in the prefactors of IR scaling functions.

We use two different energy cutoff schemes in this paper: an energy shell with a hard cutoff and a soft energy cutoff. The former scheme allows the Wilsonian RG approach, which offers a rather simple analysis and understanding. The latter requires a field theoretical analysis, which is apparently complicated, but high order perturbative corrections become more tractable.

This section focuses on the energy-shell RG scheme, which imposes a constraint on momentum integrals. By converting the momentum integral to an energy integral with the help of the DOS, we write the momentum integral with the cutoff $\Lambda$ as $$\begin{aligned}
\int_{\bm{k}}^\Lambda \mathcal{F}(E_{\bm{k}}) = \int_{-\Lambda}^{\Lambda} dE D(E) \mathcal{F}(E),\end{aligned}$$ for an arbitrary function $\mathcal{F}$. We denote the action with the energy cutoff $\Lambda$ as $S_\Lambda$, obtained by replacing the momentum integral $\int_{\bm{k}}$ by $\int_{\bm{k}}^\Lambda$. The UV energy cutoff designates an unbounded region in $\bm{k}$-space, reflecting the extended Fermi surface with scale invariance (Fig. \[fig:contour\]). Note that frequency integrals still range from $-\infty$ to $+\infty$.

We now sketch how an RG transformation works with the energy-shell RG scheme. To access the IR behavior, we progressively eliminate UV modes and focus more on remaining modes. In the energy-shell RG scheme, we first split the energy range into two parts; one corresponds to lower energies $E_{\bm{k}} \in [-\Lambda/b, \Lambda/b]$ and the other to higher energies $E_{\bm{k}} \in[-\Lambda,-\Lambda/b)$, $(\Lambda/b,\Lambda]$ $(b>1)$. Accordingly, the fermion field $\psi$ is decomposed as $$\begin{aligned}
\psi_\sigma(k) = \psi_\sigma^<(k) + \psi_\sigma^>(k),\end{aligned}$$ where $\psi_\sigma^<$ represents the low-energy modes and $\psi_\sigma^>$ the high-energy modes. We write a momentum integral in the same way: $$\int_{\bm{k}}^\Lambda = \int_{\bm{k}}^< + \int_{\bm{k}}^>.$$ Due to this division, the action is decomposed into the three parts as $$S_\Lambda[\psi] = S^<[\psi^<] + S^>[\psi^>] + S^{<>}[\psi^<,\psi^>].$$ The first term $S^<[\psi^<]$ consists only of the low-energy modes $\psi^>$ and the second term $S^>[\psi^>]$ of the high-energy modes $\psi^>$. The last term $S^{<>}[\psi^<,\psi^>]$ describes the coupling of the low- and high-energy modes, which arises when the interaction is finite $(g \neq 0)$. To obtain the effective action without the high-energy modes, we need to integrate them out: $$\begin{aligned}
\label{eq:effective}
&\quad S_{\Lambda/b}[\psi^<] \nonumber\\
&=
S^<[\psi^<] - \ln \left( \int D\bar{\psi}^> D\psi^> e^{-S^>[\psi^>] -S^{<>}[\psi^<,\psi^>]} \right) \nonumber\\
&=
S^<[\psi^<] - \ln \left( \int D\bar{\psi}^> D\psi^> e^{-S^{<>}[\psi^<,\psi^>]} \right) + \text{const}.\end{aligned}$$

Now the high-energy modes are eliminated and the new action has the smaller cutoff $\Lambda/b$. One may be tempted to compare $S_\Lambda[\psi]$ and $S_{\Lambda/b}[\psi^<]$ to look into low-energy properties. However, it is like “comparing apples to oranges” [@Shankar] as the two actions are defined in different domains. For a fair comparison, we should make a change of variables ($\bm{k}$, $\omega$ and $\psi$) to restore the cutoff $\Lambda$. This procedure, called rescaling, completes the RG step. It results in the change of parameters in the model, which is described by RG equations.

The RG equations describe the flow of the parameters under a scale transformation. When the parameters do not change under a scale transformation, the system reaches an RG fixed point and exhibits scale-invariant properties. Away from a fixed point, the parameters flow. If the flow converges to a fixed point in its vicinity, then the fixed point is called a stable fixed point. If the parameters flow away from a fixed point, then it is an unstable fixed point. The RG equations also tell us how various susceptibilities and correlation lengths diverge as the critical point is approached, and the scaling properties of correlation functions at the critical point.

Tree-level analysis
-------------------

The mixing term $S^{<>}$ can be calculated by expanding the logarithm in powers of the coupling constant $g$. We first consider the zeroth-order contribution in $g$. Since the remaining terms are described by tree diagrams without loops, the approximation is referred to as the tree-level analysis.

At tree-level, the effective action with the cutoff $\Lambda/b$ becomes $S_{\Lambda/b}[\psi^<] = S^<[\psi^<]$. To compare with $S_\Lambda[\psi]$, we need to change the variables to put the cutoff $\Lambda/b$ to $\Lambda$. Now we change the variables so that the energy satisfies the relation

$$\label{eq:rescaling_energy}
E_{\bm{k}'} = bE_{\bm{k}}.$$

For the energy dispersion given by Eq. , this immediately leads to rescaling of the momentum $$\label{eq:rescaling_momentum}
k_+' = b^{1/n_+} k_+, \quad
k_-' = b^{1/n_-} k_-,$$ while the coefficients do not change: $$\begin{gathered}
A'_+ = A_+, \quad A'_-=A_-.\end{gathered}$$

To retain the form of the action, we also need to rescale the field $\psi$, frequency $\omega$, chemical potential $\mu$, and coupling constant $g$ to be

$$\begin{gathered}
\psi' = b^{-(3-\epsilon)/2} \psi^<, \\
\label{eq:omega_tree}
\omega' = b \omega, \\
\label{eq:mu_tree}
\mu' = b \mu, \\
\label{eq:g_tree}
g' = b^{\epsilon} g.\end{gathered}$$

When we look at the parameters of the model, the chemical potential $\mu$ and the coupling constant $g$ change after an RG step, whereas the coefficients of the energy dispersion $A_\pm$ do not. The flow of an parameter under an infinitesimal scale transformation $(b\to 1)$ is described by a differential equation, namely the RG equation. For $\mu$ and $g$, the RG equations are obtained from Eqs.  and : $$\begin{gathered}
\label{eq:RG_tree}
\frac{d\mu}{dl} = \mu, \quad
\frac{dg}{dl} = \epsilon g.\end{gathered}$$ with $l = \ln b$.

In the present case, we find the Gaussian fixed point at $\mu=g=0$ in Eq. , where the system becomes noninteracting and the partition function takes a functional form of the Gaussian integral. If the parameters are away from the fixed point, they grow as $l$ increases i.e., in low energies, and flow away from the fixed point. Therefore, the fixed point at $\mu=g=0$ is unstable and both $\mu$ and $g$ are relevant perturbations to the unstable fixed point.

So far we have only considered the contact interaction. However, electron-electron interactions can take a more complicated form. Other types of interactions will be generated under RG even if not present initially, and thus their effects should be considered as well. In general, a finite-range interaction can be expanded in powers of spatial derivatives, with contact interaction being the lowest order term. The next leading term $g_- (\bar{\psi}_\uparrow \partial_{r_-} \bar{\psi}_\downarrow) (\psi_\downarrow \partial_{r_-} \psi_\uparrow)$ contains two spatial derivatives, and has a different scaling relation: $g_-' = b^{\epsilon-2/n_-} g_-$, which has a much smaller exponent than $\epsilon$ for the contact interaction. As an example, for the energy dispersion (\[x2y4\]) in two dimensions, we have $\epsilon=1/4$ and $n_-=4$, so that $g_-$ is irrelevant. It is therefore legitimate to retain only the contact interaction in RG analysis.

One-loop analysis {#sec:energy-shell_one-loop}
-----------------

In the presence of interaction, elimination of the high-energy modes gives rise to corrections in the effective action through the mixing of low- and high-energy modes in $S^{<>}[\psi^<,\psi^>]$. When depicted diagrammatically, $S^{<>}[\psi^<,\psi^>]$ involves diagrams with loops, corresponding to integrations of the high-energy modes. We here consider perturbative corrections to one-loop order.

The effective action Eq.  can be calculated perturbatively with respect to the coupling constant $g$ when it is small. Here we also treat the chemical potential $\mu$ as a perturbation as we are interested in critical phenomena where there is no characteristic scale in the system. Including perturbative corrections, we write down the action in the form $$\begin{aligned}
\label{eq:action_one-loop}
&\quad S_{\Lambda/b}[\psi^<] \nonumber\\
&= \int\frac{d\omega}{2\pi} \int_{\bm{k}}^< \bar{\psi}^<_\sigma(k) [-i\omega + E_{\bm{k}} -\mu +\Sigma] \psi^<_\sigma(k) \nonumber\\
&\quad + (g+\delta g) \bigg( \prod_{j} \int\frac{d\omega_j}{2\pi} \int_{\bm{k}_j}^< \bigg) \nonumber\\
&\quad\ \times (2\pi)^{d+1} \delta (k_1+k_2-k_3-k_4) \nonumber\\
&\quad\ \times
\bar{\psi}^<_{\uparrow}(k_1) \bar{\psi}^<_{\downarrow}(k_2) \psi^<_{\downarrow}(k_3) \psi^<_{\uparrow}(k_4) \nonumber\\
&\quad+ \cdots,\end{aligned}$$ where $\delta g$ is a correction to the coupling constant and $(\cdots)$ consists of interactions with derivatives that may be generated after integrating out the high-energy modes. As we have discussed above, finite-range interactions are irrelevant, so that we can safely neglect them.

![ Diagrammatic representation of perturbative corrections. The solid lines with arrows are the noninteracting electron propagators $G_0$. Each vertex corresponds to the contact interaction with the coupling constant $g$. (a) Self-energy $\Sigma$ to one-loop order. The first term represents the Hartree term and the second shows the one-loop correction linear in the chemical potential $\mu$. (b) Correction to the coupling constant $\delta g$. There are particle-particle (left) and particle-hole (right) contributions. (c) One-loop correction to the pairing field $\Delta$. (d) Two-loop correction to the self-energy, which gives rise to the finite field renormalization, and thus to the anomalous dimension. []{data-label="fig:loop"}](fig_loop_8-3.pdf){width="\hsize"}

Perturbative corrections to the lowest order, namely to one-loop order, are diagrammatically depicted in Fig. \[fig:loop\](a) and (b), corresponding to $\Sigma$ and $\delta g$, respectively. We find that the one-loop corrections to the self-energy $\Sigma$ and coupling constant $\delta g$ can be written as

$$\begin{gathered}
\label{eq:sigma_t0}
\Sigma = -g\Sigma_\text{H} + g\mu \Pi_\text{ph}, \\
\label{eq:delta_g_t0}
\delta g = -g^2 (\Pi_\text{pp}+\Pi_\text{ph}).\end{gathered}$$

We emphasize that the all loop corrections should be evaluated at zero external frequency and momentum. The one-loop corrections are obtained to $O(l)$ $(l = \ln b)$ as

\[eq:one-loop\_T0\] $$\begin{gathered}
\Sigma_\text{H} = \int\frac{d\omega}{2\pi} \int_{\bm{k}}^> G_0(\bm{k},\omega) \simeq l c_\text{H} \Lambda D(\Lambda) , \\
\label{eq:one-loop_T0_pp}
\Pi_{\text{pp}} = \int\frac{d\omega}{2\pi} \int_{\bm{k}}^> G_0(\bm{k},\omega) G_0(-\bm{k},-\omega) \simeq l c_\text{pp} D(\Lambda) , \\
\Pi_\text{ph} = \int\frac{d\omega}{2\pi} \int_{\bm{k}}^> G_0(\bm{k},\omega) G_0(\bm{k},\omega) =0,\end{gathered}$$

where $D(\Lambda)$ is the DOS at the cutoff energy and the dimensionless constants $c_\text{H}$ and $c_\text{pp}$ are

$$\begin{gathered}
\label{eq:c_H_0}
c_\text{H} =  \frac{1}{2} \left( 1-\frac{D_-}{D_+} \right), \\
\label{eq:c_pp_0}
c_\text{pp} = \frac{1}{2} \left( 1+\frac{D_-}{D_+} \right).\end{gathered}$$

We can see that the particle-hole contribution vanishes identically after the frequency integration, i.e., at $T=0$ there is no particle-hole screening coming from states near the cutoff energy $\Lambda$. On the other hand, the particle-particle loop has a finite contribution. The Hartree contribution $\Sigma_\text{H}$ can be finite only when the DOS is asymmetric on the electron and hole side: $D_+ \neq D_-$.

There is no frequency or momentum dependence in the self-energy to one-loop order, so that the self-energy only renormalizes the chemical potential $\mu$. The field renormalization or renormalization of the energy dispersion does not appear at one-loop order. They appear at two-loop order from the diagram shown in Fig. \[fig:loop\](d), which will be examined with the field theory approach in Sec. \[sec:field\_theory\].

With the one-loop corrections obtained, the new parameters $\mu'$ and $g'$ after rescaling are

$$\begin{gathered}
\mu' = b (\mu-\Sigma) \simeq b [\mu - lc_\text{H}\Lambda gD(\Lambda)], \\
g' = b^\epsilon (g + \delta g) \simeq b^\epsilon g [ g- l c_\text{pp} g^2 D(\Lambda) ],\end{gathered}$$

which lead to the RG equations for the chemical potential $\mu$ and the coupling constant $g$. It is convenient to define the dimensionless chemical potential $\bar{\mu}$ and coupling constant $\bar{g}$ as $$\begin{gathered}
\label{eq:dimensionless_mu_g}
\bar{\mu} = \frac{\mu}{\Lambda}, \quad
\bar{g} = g D(\Lambda).\end{gathered}$$ Then, we obtain RG equations for $\bar{\mu}$ and $\bar{g}$ as

\[eq:RG\_mu\_g\_0\] $$\begin{gathered}
\label{eq:RG_mu_0}
\frac{d\bar{\mu}}{dl} = \bar{\mu} + c_\text{H} \bar{g}, \\
\label{eq:RG_g_0}
\frac{d\bar{g}}{dl} = \epsilon \bar{g} - c_\text{pp} \bar{g}^2.\end{gathered}$$

![ RG flow of the coupling constant $\bar{g}$. There are two fixed points: $\bar{g}_1^*=0$ corresponds to the Gaussian fixed point and $\bar{g}_2^*$ to the nontrivial interacting fixed point. On the $\bar{g}$ line, the nontrivial fixed point $\bar{g}_2^*$ is stable whereas the Gaussian fixed point $\bar{g}_1^*$ is unstable. $\bar{g}_2^*$ is a positive number of order $\epsilon$, i.e., the stable fixed point has weak repulsive interaction with its strength controlled by the DOS singularity exponent $\epsilon$. []{data-label="fig:flow_g"}](flow_g.pdf){width="0.9\hsize"}

Since we are interested in the low-energy behavior, we consider the RG flow by increasing $l$. In the RG equation for the coupling constant $\bar{g}$, we find two fixed points $$\begin{gathered}
\label{eq:fixed_points}
\bar{g}_1^* = 0, \quad
\bar{g}_2^* = \frac{\epsilon}{c_\text{pp}}(>0).\end{gathered}$$ The RG flow is shown in Fig. \[fig:flow\_g\]. $\bar{g}_1^*$ corresponds to the Gaussian (noninteracting) fixed point as it has already been seen in the tree-level analysis. The new fixed point $\bar{g}_2^*$ is the nontrivial interacting fixed point with finite repulsive interaction, whose strength is of order $\epsilon$. The smallness of the coupling constant allows a controlled analysis by the DOS singularity exponent $\epsilon$ about the nontrivial fixed point.

We can find the similarity to the $\phi^4$ theory in the structure of the RG equation : the coefficient of the quadratic term $r \phi^2$ corresponds to the chemical potential $\bar{\mu}$ and the quartic interaction term $\phi^4$ to the coupling constant $\bar{g}$. From this viewpoint, our theory can be regarded as the fermionic analog of the $\phi^4$ theory.

The nontrivial fixed point in the $\phi^4$ theory is referred to as the Wilson–Fisher fixed point. One can perform a controlled analysis around the Wilson–Fisher fixed point because of the smallness of the coupling constant of order $\epsilon$. We emphasize that the small parameter $\epsilon$ has different meanings in the $\phi^4$ theory and the present model. In the $\phi^4$ theory, $\epsilon$ describes the dimension of the system measured from the upper critical dimension of four, i.e, the dimension of the system is $4-\epsilon$. On the other hand, $\epsilon$ is intrinsic to the energy dispersion for a high-order VHS and it is not restricted to an integer value but can take any rational value as discussed in Sec. \[sec:generalization\].

In the $\phi^4$ theory, the RG flow of $r$ describes the phase transition between ordered and disordered states: the RG flow to $r \gg 0$ corresponds to the disordered state and $r \ll 0$ to the ordered state, where the field $\phi$ has a finite expectation value associated with spontaneous symmetry breaking. The parameter $r$ is analogous to the chemical potential $\mu$ in the present fermionic model, where $\mu \gg 0$ yields the electron Fermi surface and $\mu \ll 0$ the hole Fermi surface. The sign change of $\mu$ describes the electronic topological transition at which the Fermi surface changes the topology. Note that the electron and hole Fermi liquids are indistinguishable by symmetry.

Let us now look into the RG equation for $\bar{\mu}$ \[Eq. \] in detail. We have $\bar{\mu} \neq 0$ at the nontrivial interacting fixed point with $\bar{g}_2^*\neq 0$, whereas $\bar{\mu}=0$ at the Gaussian fixed point at $\bar{g}_1^*=0$. The shift of the chemical potential is in parallel with the shift of $r$ at the Wilson–Fisher fixed point in the $\phi^4$ theory. This states that the Wilson–Fisher fixed point relocates $r$ from the mean-field value and that the critical properties are observed at the displaced $r$.

As we have mentioned, the contribution from the Hartree term $c_\text{H}$ is finite when the DOS is asymmetric on the electron and hole sides: $D_+ \neq D_-$. The RG equation for $\bar{\mu}$ asserts that a finite chemical potential is generated by the interaction when $c_\text{H}\neq 0$. This is consistent with $D_+ \neq D_-$, since a finite chemical potential $\bar{\mu}$ breaks electron-hole symmetry and hence the contact interaction itself cannot generate $\bar{\mu}$ unless the symmetry does not exist from the beginning.

Relevant perturbations
----------------------

We have identified the two fixed points: the Gaussian fixed point and the nontrivial interacting fixed point. With the chemical potential tuned at $\bar{\mu}=0$, the Gaussian fixed point $\bar{g}_1^*$ is an unstable fixed point and the nontrivial fixed point $\bar{g}_2^*$ is a stable fixed point. The chemical potential is a relevant perturbation around both fixed points. We have included the chemical potential even in the analysis of the simplest case above as it can be generated by interaction in the absence of particle-hole symmetry.

In addition to the chemical potential, we consider other relevant perturbations to the fixed points, including the magnetic field $h$ and the $s$-wave pairing field $\Delta$. Those relevant perturbations adds the following terms to the action at criticality: $$\begin{gathered}
-\mu \bar{\psi} \psi, \quad
h (\bar{\psi_\uparrow} \psi_\uparrow - \bar{\psi_\downarrow} \psi_\downarrow ), \quad
\Delta \bar{\psi}_\uparrow \bar{\psi}_\downarrow + \Delta^* \psi_\downarrow \psi_\uparrow.\end{gathered}$$ Finite temperature is also a relevant perturbation. Its effect is taken account of via Matsubara frequencies. We further consider other relevant perturbations. For an energy dispersion, e.g., $E_{\bm{k}}=k_x^2-k_y^4$, the fermion bilinear terms with derivatives $\partial_{k_x}$, $\partial_{k_y}$, $\partial_{k_y}^2$, $\partial_{k_y}^3$, $\partial_{k_x} \partial_{k_y}$ are also relevant perturbations.

Perturbations to the system are subject to symmetry constraints: Particle conservation forbids the pairing term, spin-rotational symmetry nonzero $h$, and reflection symmetry odd-derivative terms in $x$ or $y$. With all three symmetries present, only two terms $\mu\bar{\psi}\psi$ and $\bar{\psi}\partial_{k_y}^2\psi$ are allowed as perturbations to the system with $E_{\bm{k}}=k_x^2-k_y^4$. This means that we need to tune two parameters to reach the critical point.

To one-loop order, the term $\bar{\psi}\partial_{k_y}^2\psi$ does not receive a correction from the interaction since the self-energy $\Sigma$ is independent of momentum.

We assume that the pairing term is induced by a proximity effect, or it might be regarded as a test field for the $s$-wave pairing fluctuation, because it must vanish in the presence of particle number conservation. Likewise, the magnetic field $h$ can be thought of as an external field or a test field for the spin susceptibility. In this viewpoint, the chemical potential is conjugate to the particle number, and hence it is related to the charge compressibility.

Corrections to the perturbations $h$ and $\Delta$ are calculated similarly as those for $\mu$ and $g$ at $T=0$. To consider a correction to the pairing field $\Delta$, we include the particle-particle loop diagram, where the one-loop diagram is shown in Fig. \[fig:loop\](c). We include the corrections to write the magnetic field $h + \delta h$ and the pairing field $\Delta + \delta\Delta$.

Integrating out the high-energy modes is followed by rescaling. The parameters of the model should be rescaled at tree level as $h' = bh$ and $\Delta' = b\Delta$. Those parameters are relevant and thus their values increase as we proceed with RG steps. When the perturbative corrections are included, the new parameters after an RG step are $$\begin{gathered}
h' = b (h + \delta h), \quad
\Delta' = b (\Delta + \delta\Delta).\end{gathered}$$

To one-loop order, the correction terms are expressed as $$\begin{gathered}
\delta h = 0, \quad \delta\Delta = -g\Pi_\text{pp}.\end{gathered}$$ $\delta h=0$ follows from spin-rotational symmetry of the model. The one-loop correction $\Pi_\text{pp}$ is obtained in Eq. . Then, the parameters change as $$\begin{gathered}
h' = b h, \quad
\Delta' \simeq b [\Delta - lc_\text{pp}gD(\Lambda)].\end{gathered}$$ With the dimensionless quantities $$\begin{gathered}
\bar{h}=\frac{h}{\Lambda}, \quad
\bar{\Delta}=\frac{\Delta}{\Lambda},\end{gathered}$$ we reach the RG equations

\[eq:RG\_h\_delta\] $$\begin{gathered}
\label{eq:RG_h}
\frac{d\bar{h}}{dl} = \bar{h}, \\
\label{eq:RG_Delta}
\frac{d\bar{\Delta}}{dl} = ( 1- c_\text{pp}\bar{g} ) \bar{\Delta}. \end{gathered}$$

We confirm that the perturbations $h$ and $\Delta$ are relevant around the two fixed point, given in Eq. . Finite temperature is also a relevant perturbation, which scales in the same manner as energy and frequency. All low-energy fixed points are found at $T=0$, and thus we focus on zero temperature in the main part. The one-loop RG equations at finite temperature are presented in Appendix \[sec:finite-temperature\]. The physical consequences, i.e., scaling properties of thermodynamic quantities, are discussed in the next section.

Discussion for higher-order corrections {#sec:RG_Wilson_discussion}
---------------------------------------

So far, we have made the energy-shell RG analysis to one-loop order. We now illustrate how it works in the case with higher-order corrections. Again, we consider here the minimal case with $\mu$ and $g$ at $T=0$ for clarity. Inclusion of other relevant contributions such as $T$, $h$, and $\Delta$ is straightforward.

Higher-order perturbative corrections give rise to the frequency and momentum dependence in the self-energy $\Sigma$ in Eq. . We expand the self-energy with respect to the frequency and momentum to find corrections to the field, energy dispersion, and chemical potential. In the following discussion, we assume that the momentum-dependent part of the self-energy involves terms proportional to $k_+^{n_+}$ and $k_-^{n_-}$ and that other relevant terms are not generated or eliminated by symmetry or fine-tuning the system to criticality. Then, the expansion of the self-energy is given by $$\begin{aligned}
\label{eq:sigma_expansion}
\Sigma &= \Sigma_0 + (i\omega) \Sigma_\omega + \mu \Sigma_\mu + A_+ k_+^{n_+} \Sigma_+ - A_- k_-^{n_-} \Sigma_- \nonumber\\
&\quad + \text{(high order terms)},\end{aligned}$$ where irrelevant high order terms are safely neglected. After integrating out the high-energy modes within the energy shell, we obtain the effective action

$$\begin{aligned}
S_{\Lambda/b}[\psi^<]
&= \int\frac{d\omega}{2\pi} \int_{\bm{k}}^< \bar{\psi}^<_\sigma(k)
\{ -i\omega (1-\Sigma_\omega) + [A_+ k_+^{n_+}(1+\Sigma_+) -A_- k_-^{n_-}(1+\Sigma_-)] -\mu(1-\Sigma_\mu) \} \psi^<_\sigma(k) \nonumber\\
&\quad + (g+\delta g) \bigg( \prod_{j} \int\frac{d\omega_j}{2\pi} \int_{\bm{k}_j}^< \bigg)
(2\pi)^{d+1} \delta (k_1+k_2-k_3-k_4)
\bar{\psi}^<_{\uparrow}(k_1) \bar{\psi}^<_{\downarrow}(k_2) \psi^<_{\downarrow}(k_3) \psi^<_{\uparrow}(k_4).\end{aligned}$$

The next step in the energy-shell RG analysis is to rescale the momentum and restore the energy cutoff $\Lambda/b$ to $\Lambda$; see Eqs.  and . However, the effective action $S_{\Lambda/b}$ still evidently has a different form from $S_\Lambda$. To recover the form of the action, we rescale the other quantities as follows:

$$\begin{gathered}
\label{eq:scaling_omega}
\omega' = b\omega , \\
\label{eq:scaling_A}
A'_\pm = (1+\Sigma_\pm) (1-\Sigma_\omega)^{-1} A_\pm \equiv b^{\gamma_{A_\pm}} A_\pm, \\
\label{eq:scaling_mu}
\mu' = b (1-\Sigma_\mu) (1-\Sigma_\omega)^{-1} \mu \equiv b^{\gamma_\mu} \mu, \\
g' = b^\epsilon (1-\Sigma_\omega)^2 (g+\delta g), \\
\psi' = b^{-(3-\epsilon)/2} (1-\Sigma_\omega)^{-1/2} \psi^< \equiv b^{-(3-\epsilon)/2} b^{\gamma_\psi/2} \psi^<.\end{gathered}$$

Here we also introduce the scaling exponents $\gamma_{A_\pm}$, $\gamma_\mu$, and $\gamma_\psi$. Note that there is an ambiguity in defining $\omega'$ and $\psi'$ as the factor $(1-\Sigma_\omega)$ can be imposed on either $\omega'$ or $\psi'$. We choose to scale $\omega$ linearly in $b$ and hence the factor $(1-\Sigma_\omega)$ contributes to the field renormalization. We shall show later that the Ward identity requires $\gamma_\mu = 1$.

For $\gamma_{A_\pm} \neq 0$, if we continue to rescale momentum according to Eq.  and the coefficients $A_\pm$ according to Eq. , the cutoff energy $\Lambda/b$ is not mapped to $\Lambda$. To remedy this issue, we rescale momentum as $$\label{eq:scaling_k_tilde}
k'_\pm = b^{1/\tilde{n}_\pm} k_\pm \text{ with } \tilde{n}_\pm = \frac{n_\pm}{1+\gamma_{A_\pm}},$$ so that $E_{\bm{k}'} = bE_{\bm{k}}$ is satisfied. In this way, the coefficients $A_\pm$ do not change under rescaling.

Rescaling of the magnetic field $h$ and the pairing field $\Delta$ can be considered similarly. Including the field renormalization, we obtain

$$\begin{gathered}
\label{eq:scaling_h}
h' = b (h+\delta h) (1-\Sigma_\omega)^{-1} \equiv b^{\gamma_h} h, \\
\label{eq:scaling_Delta}
\Delta' = b (\Delta + \delta\Delta) (1-\Sigma_\omega)^{-1} \equiv b^{\gamma_\Delta} \Delta,\end{gathered}$$

where we define the exponents $\gamma_h$ and $\gamma_\Delta$.

Analysis {#sec:analysis}
========

Scaling analysis
----------------

### Generic case {#sec:scaling_generic}

Scale invariance at the fixed points enables us to extract various scaling relations. Since the partition function $\mathcal{Z}$ is invariant under the scale transformation, the free energy density $F$, defined in Eq. , reflects the scaling of the factor $T/V$: $$F' = b^{1 + d_+/\tilde{n}_+ + d_-/\tilde{n}_-} F,$$ where the volume $V$ scales according to Eq.  and temperature scales the same manner as energy and frequency. For convenience, we rewrite the exponent as $$\begin{aligned}
1 + \frac{d_+}{\tilde{n}_+} + \frac{d_-}{\tilde{n}_-} &= 2 - \left( \epsilon - \frac{d_+ \gamma_{A_+}}{\tilde{n}_+} - \frac{d_- \gamma_{A_-}}{\tilde{n}_-} \right) \nonumber\\
&\equiv 2-\tilde{\epsilon}.\end{aligned}$$ By explicitly showing the parameters of $F$, we obtain the scaling relation of the free energy density $$\label{eq:scaling}
F(\mu,h,\Delta;T) = b^{-2+\tilde{\epsilon}} F(b \mu , b^{\gamma_h}h, b^{\gamma_\Delta}\Delta; b T).$$ Here, the exponents $\gamma_h$ and $\gamma_\Delta$ correspond to the values at a fixed point, $\gamma_h(\bar{g}^*)$ and $\gamma_h(\bar{g}^*)$, respectively. In deriving this relation, we use the Ward identity $\gamma_\mu = 1$. The coupling constant $g$ itself does not appear in the scaling relation of the free energy density $F$, but the effect is imprinted on $\gamma_h$, $\gamma_\Delta$, and $\tilde{\epsilon}$ as the fixed point properties. We shall see that $\gamma_{A_\pm}$ are at most of order $\epsilon^2$ at the nontrivial interacting fixed point and thus $\tilde{\epsilon}$ is also a small positive quantity.

We then consider the critical exponents of the charge compressibility $\kappa$, magnetic susceptibility $\chi$, heat capacity per unit volume $C_V$, and $s$-wave pairing susceptibility $\chi_\text{BCS}$. From Eq. , we find $$\begin{gathered}
\label{eq:scaling_compressibility}
\kappa = \left(\frac{\partial n}{\partial\mu}\right)_T \sim
\begin{cases}
T^{-\tilde{\epsilon}} \\
|\mu|^{-\tilde{\epsilon}},
\end{cases} \\
\label{eq:scaling_susceptibility}
\chi = -\lim_{h\to0}\left(\frac{\partial^2 F}{\partial h^2}\right)_T \sim
\begin{cases}
T^{-(\tilde{\epsilon}+2\gamma_h-2)} \\
|\mu|^{-(\tilde{\epsilon}+2\gamma_h-2)},
\end{cases} \\
\label{eq:scaling_heat-capacity}
\frac{C_V}{T} = -\left(\frac{\partial^2 F}{\partial T^2}\right)_V \sim
\begin{cases}
T^{-\tilde{\epsilon}} \\
|\mu|^{-\tilde{\epsilon}} \\
|h|^{-\tilde{\epsilon}/\gamma_h} \\
|\Delta|^{-\tilde{\epsilon}/\gamma_\Delta},
\end{cases} \\
\label{eq:scaling_pairing}
\chi_\text{BCS} = \left(\frac{\partial^2 F}{\partial\Delta \partial\Delta^*}\right)_T
\sim
\begin{cases}
T^{-(\tilde{\epsilon}+2\gamma_\Delta-2)} \\
|\Delta|^{-2+(2-\tilde{\epsilon})/\gamma_\Delta}.
\end{cases}\end{gathered}$$

We also examine the pair correlation function $$\begin{aligned}
C(\bm{r},\tau) &= \langle (\psi_\uparrow \psi_\downarrow) (\bm{r},\tau) (\bar{\psi}_\downarrow \bar{\psi}_\uparrow) (0,0) \rangle \nonumber\\
&\quad - \langle (\psi_\uparrow \psi_\downarrow) (\bm{r},\tau) \rangle \langle (\bar{\psi}_\downarrow \bar{\psi}_\uparrow) (0,0)\rangle,\end{aligned}$$ with $\langle \mathcal{O} \rangle = \int D\bar{\psi} D\psi \mathcal{O} e^{-S}/\mathcal{Z}$. From the comparison between $\chi_\text{BCS}$ and $C(\bm{r},\tau)$, we obtain the scaling form $$\begin{aligned}
C(r_+,r_-,\tau) = \kappa^{2(2-\tilde{\epsilon}-\gamma_\Delta)} \hat{c}\left( r_+ \kappa^{1/\tilde{n}_+}, r_- \kappa^{1/\tilde{n}_-}, \tau \kappa \right),\end{aligned}$$ where $\kappa$ is an arbitrary energy scale and $\hat{c}$ is a scaling function.

The field renormalization with the exponent $\gamma_\psi$ appears in the two-point correlation function $G(\bm{r},\tau)$. We shall show the derivation later with the field theory approach. In the critical region, the exponent $\gamma_\psi$ can be replaced with a constant $\eta = \gamma_\psi(\bar{g}^*)$; the scaling form is given by

\[eq:correlation\_scaling\] $$\begin{aligned}
\label{eq:correlation_scaling_real}
G(r_+,r_-,\tau) = \kappa^{1-\tilde{\epsilon}+\eta} \hat{g}\left( r_+ \kappa^{1/\tilde{n}_+}, r_- \kappa^{1/\tilde{n}_-}, \tau \kappa \right),\end{aligned}$$ or its Fourier transform is $$\begin{aligned}
\label{eq:correlation_scaling_frequency}
G(k_+,k_-,\omega) = \kappa^{-(1-\eta)} \hat{g}' \left( \frac{k_+}{\kappa^{1/\tilde{n}_+}}, \frac{k_-}{\kappa^{1/\tilde{n}_-}}, \frac{\omega}{\kappa} \right),\end{aligned}$$

where $\hat{g}$ and $\hat{g}'$ are scaling functions. Particularly, we see the frequency dependence $G(\omega) \propto 1/|\omega|^{1-\eta}$, which differs from the noninteracting correlation function $G(\omega)\propto 1/|\omega|$ with finite $\eta$. $\eta$ corresponds to the anomalous dimension and specifies the non-Fermi liquid behavior.

### One-loop results

To one-loop order, we find from the RG equation the exponents at the fixed points $$\begin{gathered}
\label{eq:exponents_one-loop_1}
\gamma_h=1, \\
\gamma_\Delta=1-c_\text{pp}(0)\bar{g}_j^* =
\begin{cases}
1 & \text{(Gaussian)} \\
1-\epsilon & \text{(Nontrivial)}.
\end{cases}\end{gathered}$$ with $\tilde{\epsilon} = \epsilon$. Most exponents in Eq. – are the same at the Gaussian and nontrivial fixed points, which is identical to that of the DOS in the noninteracting state. The difference is found when the pairing field $\Delta$ is involved. The exponent for the pairing field $\gamma_\Delta$ renders different exponents for the pairing susceptibility $\chi_\text{BCS}$: $$\chi_\text{BCS} \sim
\begin{cases}
T^{-\epsilon},\ |\Delta|^{-\epsilon} & \text{(Gaussian)} \\
T^{+\epsilon},\ |\Delta|^{+\epsilon} & \text{(Nontrivial)}.
\end{cases}$$ The $s$-wave pairing susceptibility remains finite at the nontrivial fixed point whereas it diverges at the Gaussian fixed point. We also find a difference in the pair correlation function $$\begin{aligned}
&\quad C(r_+,r_-,\tau) \nonumber\\
&= \hat{c} \left( r_+ \kappa^{1/\tilde{n}_+}, r_- \kappa^{1/\tilde{n}_-}, \tau \kappa \right) \times
\begin{cases}
\kappa^{-2(1-\epsilon)} & \text{(Gaussian)} \\
\kappa^{-2} & \text{(Nontrivial)}.
\end{cases}\end{aligned}$$ It shows a faster decay at the nontrivial fixed point, reflecting the suppressed pairing susceptibility.

Supermetal
----------

We coin a term, *supermetal*, to describe a state where the system resists ordering and remains a quantum critical state with power-law divergent susceptibilities. In this regard, the Gaussian fixed point is a noninteracting supermetal, and the nontrivial interacting fixed point is an interacting supermetal. Various susceptibilities in a noninteracting supermetal are determined by the DOS; they diverge with the DOS singularity exponent. In contrast, susceptibilities can have distinct exponents in an interacting supermetal. The one-loop analysis shows that the $s$-wave pairing susceptibility remains finite whereas the charge compressibility and spin susceptibility still diverge with the same exponent as that of the DOS. One may wonder whether an interacting supermetal is a Fermi liquid and whether the charge compressibility and spin susceptibility have the same exponents even with higher-order corrections. Those questions await an RG analysis to two-loop order in Sec. \[sec:field\_theory\].

The criticality of supermetals relies on scale invariance. Any relevant perturbations, such as $\mu$, $h$, and $\Delta$ considered above, introduce energy scales to the system, and they potentially drive the system toward ordering instabilities with broken symmetries. Here we outline possible consequences of relevant perturbations. Suppose that we add a finite chemical potential to force the system away from the critical point. At first, it drives the system to an electron or hole Fermi liquid state. However, a $2k_F$ singularity in the charge susceptibility arises and the Kohn–Luttinger mechanism for superconductivity [@Kohn-Luttinger] eventually takes effect. When a finite magnetic field is applied as a perturbation instead, it would trigger a phase transition to the ferromagnetic state.

Ward identity
-------------

In Sec. \[sec:scaling\_generic\], we derived the scaling relations for thermodynamic quantities and correlation functions. In the derivations, we used the fact that the chemical potential and frequency (or temperature) should have the same scaling exponent. This is a consequence of charge conservation. The Ward identity (more generally the Ward–Takahashi identity) describes the conservation law [@Ward; @Takahashi]. The identity is regarded as the quantum analog to Noether’s theorem. We show how it works in our present analysis. Also, the identity should hold even after an RG analysis, and thus it can be used to check the validity of an RG scheme, or specifically a choice of a cutoff.

Now we investigate the structure of the self-energy $\Sigma$. To be concrete, we look into the expansion of the self-energy Eq.  to find a relation between $\Sigma_\omega$ and $\Sigma_\mu$. The Ward identity concludes that $\Sigma_\omega$ and $\Sigma_\mu$ are equal at $T=0$: $$\label{eq:Ward_Sigma}
\Sigma_\omega = \Sigma_\mu.$$ The identity is based on charge conservation or the U(1) gauge invariance; the action and correlation functions are invariant under the transformations $\psi \mapsto e^{i\alpha(\bm{r},\tau)} \psi$ and $\bar{\psi} \mapsto \bar{\psi} e^{-i\alpha(\bm{r},\tau)}$ with a smooth scalar function $\alpha(\bm{r},\tau)$.

In the following, we consider the Ward identity from the diagrammatic point of view. We can relate the $\mu$-derivative of the self-energy to the vertex function corresponding to the coupling $\varphi\bar{\psi}\psi$. We note that the scalar field $\varphi$ is associated with the chemical potential $\mu$. We write the vertex function as $\Gamma_\mu (\omega+\omega',\omega)$, where we focus only on the frequency dependence in the following discussion. The vertex function modifies the coupling term $\bar{\psi}(\omega+\omega') \varphi(\omega') \psi(\omega)$ to be $\Gamma_\mu(\omega+\omega',\omega) \bar{\psi}(\omega+\omega') \varphi(\omega') \psi(\omega)$.

![ (a) Relation between the bare vertex and the noninteracting Green’s function. (b) Diagrammatic representation of the Ward–Takahashi identity. []{data-label="fig:WT"}](WT.pdf){width="0.9\hsize"}

To illustrate how Eq.  is derived, we make use of the equality $$G_{0}^{-1}(\bm{k},\omega+\omega') - G_{0}^{-1}(\bm{k},\omega) = i\omega',$$ and equivalently $$\label{eq:Ward_Green}
G_0(\bm{k},\omega+\omega') (i\omega') G_0(\bm{k},\omega)
= G_0(\bm{k},\omega) - G_0(\bm{k},\omega+\omega').$$ This equation is diagrammatically shown in Fig. \[fig:WT\](a). It relates the noninteracting vertex function $\Gamma_\mu = 1$ and the noninteracting Green’s function $G_0$. Now we add corrections to the self-energy, as depicted in Fig. \[fig:WT\](b) as shaded blobs. The dressed vertex function is obtained from the dressed self-energy by attaching the external scalar field $\varphi$ to every internal fermion line. Thus, considering Eq. , we find the Ward–Takahashi identity $$\begin{aligned}
&\quad G(\bm{k},\omega+\omega') (i\omega') \Gamma_\mu(\omega+\omega',\omega) G(\bm{k},\omega) \nonumber\\
&= G(\bm{k},\omega) - G(\bm{k},\omega+\omega').\end{aligned}$$ The full Green’s function $G(\bm{k},\omega)$ is given by $$G(\bm{k},\omega) = \frac{1}{i\omega - E_{\bm{k}} - \Sigma(\bm{k},\omega)},$$ with the full self-energy $\Sigma$. Taking the zero frequency limit $\omega' \to 0$, we obtain the Ward identity $$\Gamma_\mu(\omega,\omega) = 1-\frac{\partial\Sigma(\omega)}{\partial(i\omega)}.$$

The vertex function $\Gamma_\mu(\omega,\omega)$ is equivalent to the correction to the chemical potential; the chemical potential dressed by quantum corrections is $\mu\Gamma_\mu(\omega,\omega) = \mu (1-\Sigma_\mu)$. Therefore, the Ward identity confirms the relation Eq. . It immediately concludes $$\gamma_\mu = 1$$ from Eq. . The relation holds to all orders of perturbative calculations at $T=0$.

The result of the energy-shell RG analysis to one-loop order in Sec. \[sec:energy-shell\] satisfies the Ward identity. We notice that a frequency shell instead of the energy shell violates the Ward identity.

Field theory approach {#sec:field_theory}
=====================

This section focuses on the RG analysis from the field theory approach. To begin with, we briefly argue the two RG schemes: the energy-shell RG analysis and the field theory approach. We then confirm that it gives the same result as that from the energy-shell RG analysis at one-loop order. It is followed by the two-loop calculations to show the anomalous dimension and the correction to the energy dispersion.

RG schemes {#sec:schemes}
----------

An objective of RG analyses is to track the flow of parameters in a theory under a scale transformation. Here, we illustrate two different RG schemes: the Wilsonian approach, including the preceding energy-shell RG analysis, and the field theory approach. The common feature is to divide the integration manifold (frequency and momentum in the present case) into two parts and integrate out modes belonging to one of them. The two schemes differ in intervals of integrations. The first scheme involves an integration within a hard shell. In the energy-shell RG analysis, fluctuations inside the thin energy shell $E\in[-\Lambda,-\Lambda/b)$, $(\Lambda/b,\Lambda]$ are eliminated. This mode elimination followed by rescaling enables us to keep track of the change of parameters under a scale transformation. On the other hand, in the field theory approach, we integrate out all low-energy fluctuations below the cutoff $\Lambda$. Then, we deduce the RG flow of parameters by comparing results at different cutoffs $\Lambda$ and $\Lambda'$.

The two schemes have advantages in different aspects. In the Wilsonian approach, the frequency-momentum space is progressively integrated over, so the interpretation of the RG procedure is rather simple. The inclusion of low-energy modes results in a theory at low energies with different parameters. In spite of its simple interpretation, higher-loop calculations are not easy with the Wilsonian approach. In a one-loop calculation, we have only one shell to be concerned about. However, higher-loop diagrams consist of many internal lines (virtual states), so that we have to take care of shells for each of them. On the other hand, the field theory approach does not require such error-prone steps as it deals with all modes below the cutoff at once. This makes higher-loop calculations more tractable. Although not as intuitive as the Wilsonian approach, the field theory approach leads to the same results on critical phenomena. More descriptions about the comparison between the two schemes can be found in e.g. Ref. [@Shankar]. A brief review of the field theory approach is given in Appendix \[sec:field-theory\_review\].

Soft cutoff
-----------

In the field theory approach, we calculate the connected $N$-point correlation function $G^{(N)}$ or the one-particle irreducible $N$-point function $\Gamma^{(N)}$. If we face a UV divergence in calculating them, we need to cure the divergence to obtain physically meaningful results. There are several ways to do so; we here choose to employ the UV energy cutoff $\Lambda$ to make a comparison to the preceding energy-shell RG analysis.

The functions $G^{(N)}$ and $\Gamma^{(N)}$ can be obtained perturbatively with the noninteracting Green’s function $G_0$. We introduce the UV energy cutoff by suppressing the high-energy contributions in $G_0$. We define the noninteracting Green’s function with the energy cutoff $G_{0\Lambda}(\bm{k},\omega_n)$ as $$\begin{aligned}
G_{0\Lambda}(\bm{k},\omega_n) &= G_0(\bm{k},\omega_n) K_\Lambda(E_{\bm{k}}) \nonumber\\
&= \frac{K_\Lambda(E_{\bm{k}})}{i\omega_n - E_{\bm{k}}},\end{aligned}$$ with the UV energy cutoff factor $$K_\Lambda(E) = \frac{\Lambda^2}{\Lambda^2+E^2}.$$ Note that the cutoff factor smoothly varies from 0 to 1 and thus works as a soft energy cutoff. This is in contrast to the energy-shell RG analysis, where the interval of an energy integration is cut off abruptly at $\Lambda$ and $\Lambda/b$.

We can interpret the modified Green’s function as a Green’s function with an energy-dependent quasiparticle weight $K_\Lambda(E)$. The weight fades away in the high-energy limit $E\to\pm\infty$ to eliminate UV divergences, while $K_\Lambda(E) \to 1$ for energies much lower than the cutoff $\Lambda$. One may be tempted to see the modified Green’s function in a different way. For example, it can be rewritten as $$\begin{aligned}
G_{0\Lambda}(\bm{k},\omega_n)
&= \frac{1}{i\omega_n-E_{\bm{k}}} - \frac{1}{i\omega_n-E_{\bm{k}}} \frac{E_{\bm{k}}^2}{\Lambda^2+E_{\bm{k}}^2}. \end{aligned}$$ It may be viewed as a variation of the Pauli–Villars regularization, where the additional term cures a UV divergence but vanishes in the limit $\Lambda\to\infty$. However, we cannot think of it as a propagator with a large mass term since we cannot add a mass term for the electronic energy dispersion which is continuous and unbounded.

It should be noted that the cutoff factor $K_\Lambda(E)$ does not depend on frequency. It potentially causes a violation of the Ward identity, which would result in wrong conclusions. For example, if one chooses a cutoff factor of the form $\Lambda^2/(\Lambda^2+E^2+\omega_n^2)$, it invalidates the Ward identity. The absence of the frequency in the cutoff factor ensures the Ward identity.

Formalities
-----------

### Structure of the RG analysis

To derive RG equations and see scaling properties, we calculate the one-particle irreducible $N$-point function $\Gamma^{(N)}_\Lambda$ with the cutoff $\Lambda$ and examine its cutoff dependence. The cutoff dependence is seen by comparing two $N$-point functions at different cutoffs; see Eq. . Specifically, we compare $\Gamma^{(N)}_\Lambda$ to one at a reference point $\Gamma^{(N)}_R$. The energy scale at the reference point is referred to as the renormalization scale. The procedure of fixing the model to the reference is equivalent to setting the initial parameters in the Wilsonian approach.

We first analyze the case with $T=h=\Delta=0$. We impose the renormalization conditions $$\begin{gathered}
\label{eq:condition_2}
\Gamma_R^{(2)}(k) = -i\omega_n + E_{\bm{k},0} -\mu_0, \\
\label{eq:condition_4}
\Gamma_R^{(4)}(k_1,k_2;k_3,k_4) = g_0,\end{gathered}$$ where the condition for $\Gamma^{(4)}$ should be considered at $k_1+k_2 = k_1+k_3 = k_1 +k_4 = 0$. The subscript $0$ denotes quantities at the renormalization scale. The interaction dresses the two-point and four-point functions and they acquire cutoff-dependent corrections. We here use the energy dispersion Eq.  for analysis. We assume that there is no additional term to the energy dispersion generated under the RG analysis, as we have discussed in Sec. \[sec:RG\_Wilson\_discussion\]. Then, the two-point and four-point functions at the cutoff $\Lambda$ can be expressed as $$\begin{gathered}
\label{eq:Gamma_2_Lambda}
\Gamma_\Lambda^{(2)} = -i\omega_n Z_\psi^{-1} + Z_{A_+}^{-1} A_+ k_+^{n_+} - Z_{A_-}^{-1} A_- k^{n_-} - Z_\mu^{-1} \mu, \\
\label{eq:Gamma_4_Lambda}
\Gamma_\Lambda^{(4)} = Z_g^{-1} g,\end{gathered}$$ where the corrections $Z_\psi$, $Z_{A_\pm}$, $Z_\mu$, and $Z_g$ are calculated perturbatively. The $N$-point functions at the renormalization scale and the cutoff $\Lambda$ are related by $$\begin{gathered}
\Gamma_R^{(N)} = Z_\psi^{N/2} \Gamma_\Lambda^{(N)}.\end{gathered}$$

The last equation leads to the RG equations. Since the left-hand side does not depend on the cutoff $\Lambda$, we obtain the differential equation $$\begin{aligned}
\Lambda \frac{d}{d\Lambda} \Gamma_R^{(N)} = 0.\end{aligned}$$ It leads to the Callan–Symanzik equation. We obtain for the one-particle irreducible $N$-point function

$$\begin{gathered}
\label{eq:CS}
\left[ \Lambda\frac{\partial}{\partial\Lambda} -\beta(\bar{g})\frac{\partial}{\partial\bar{g}} - \beta_\mu(\bar{g},\bar{\mu}) \frac{\partial}{\partial\bar{\mu}} - \beta_{A_+}(\bar{g},A_\pm) \frac{\partial}{\partial A_+} - \beta_{A_-}(\bar{g},A_\pm) \frac{\partial}{\partial A_-} - \frac{N}{2} \gamma_\psi(\bar{g}) \right] \Gamma_\Lambda^{(N)} = 0,\end{gathered}$$

where we use the dimensionless parameters defined in Eq. . The beta functions and $\gamma_\psi$ are defined by

$$\begin{gathered}
\label{eq:beta_g}
\beta(\bar{g}) = -\left( \Lambda\frac{\partial\bar{g}}{\partial\Lambda} \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}, \\
\label{eq:beta_mu}
\beta_\mu(\bar{g},\bar{\mu}) = -\left( \Lambda\frac{\partial\bar{\mu}}{\partial\Lambda} \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}, \\
\label{eq:beta_A}
\beta_{A_\pm}(\bar{g},A_\pm) = -\left( \Lambda\frac{\partial A_\pm}{\partial\Lambda} \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}, \\
\label{eq:gamma_def}
\gamma_\psi = -\left( \Lambda\frac{\partial}{\partial\Lambda} \ln Z_\psi \right)_{\bar{g}_0,\bar{\mu}_0,A_{\pm,0}}.\end{gathered}$$

We can rewrite the beta functions as

$$\begin{gathered}
\beta = \bar{g} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_g -2\gamma_\psi \right) , \\
\beta_\mu = \bar{\mu} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_\mu -\gamma_\psi \right) , \\
\beta_{A_\pm} = A_\pm \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_{A_\pm} -\gamma_\psi \right),\end{gathered}$$

since the renormalized values are given by

$$\begin{gathered}
\bar{g} = Z_g Z_\psi^{-2} \bar{g}_0, \\
\bar{\mu} = Z_\mu Z_\psi^{-1} \bar{\mu}_0, \\
A_\pm = Z_{A_\pm} Z_\psi^{-1} A_{\pm,0}.\end{gathered}$$

Those equations show that the field renormalization gives additional effects to the beta functions and hence the scaling properties.

### Solutions

The Callan–Symanzik equation can be solved by the method of characteristics; see Appendix \[sec:field-theory\_review\]. The beta functions describe the RG flows of the parameters:

$$\begin{gathered}
\label{eq:RG_g_beta}
\frac{d\bar{g}}{dl} = \beta(\bar{g}), \\
\label{eq:RG_mu_beta}
\frac{d\bar{\mu}}{dl} = \beta_{\mu} (\bar{g}, \bar{\mu}), \\
\label{eq:RG_A_beta}
\frac{dA_\pm}{dl} = \beta_{A_\pm} (\bar{g},A_\pm).\end{gathered}$$

$l=\ln{\Lambda_0/\Lambda}$ denotes the RG scale, measured relative to the renormalization scale $\Lambda_0$. Those RG equations are to be compared with those obtained by the energy-shell RG analysis in Sec. \[sec:energy-shell\]. In general, they are coupled differential equations and zeros of the beta functions determine fixed points.

When we expand the beta functions $\beta_\mu$ and $\beta_{A_\pm}$ around a fixed point with $\bar{g}^*$ as

$$\begin{gathered}
\beta_\mu(\bar{g}^*,\bar{\mu}) \approx \beta_\mu(\bar{g}^*,0) + \gamma_\mu(\bar{g}^*) \bar{\mu}, \\
\beta_{A_\pm}(\bar{g}^*,A_\pm) \approx \gamma_{A_\pm}(\bar{g}^*) A_{\pm},\end{gathered}$$

$\gamma_\mu(\bar{g}^*)$ and $\gamma_{A_\pm}(\bar{g}^*)$ give the exponents in the scaling region. Recall that $\gamma_\mu=1$ is required by the Ward identity, regardless of $\bar{g}$. We note that $\beta_\mu(\bar{g}^*,0)$ does not affect the scaling behavior apart from a shift of the chemical potential and that it can be neglected, as we have discusses in Sec. \[sec:energy-shell\_one-loop\]. With constant $\gamma_\mu(\bar{g}^*)$ and $\gamma_{A_\pm}(\bar{g}^*)$, we observe the scaling properties $$\begin{gathered}
\bar{\mu}(l) = \bar{\mu}_0 e^{l}, \quad
\label{eq:scaling_A_2}
A_\pm(l) = A_{\pm,0} e^{\gamma_{A_\pm}(\bar{g}^*)l}.\end{gathered}$$

The function $\gamma_\psi$ is ascribed to the anomalous dimension $\eta$ when it is computed at a fixed point. To see this, we solve the Callan–Symanzik equation ; see Appendix \[sec:field-theory\_review\] for details. The solution of the two-point function is given by $$\begin{aligned}
\label{eq:two-point_solution}
&\quad
\Gamma^{(2)}_\Lambda \left(e^{-l/n_\pm}k_{\pm,0}, e^{-l}\omega_0; \bar{g}(0), \bar{\mu}(0), A_\pm(0) \right) \nonumber\\
&= e^{-l} \Gamma_\Lambda^{(2)} \left( k_{\pm,0}, \omega_0; \bar{g}(l), \bar{\mu}(l), A_\pm(l) \right) \exp \left[ \int_0^l dl' \gamma_\psi(\bar{g}(l')) \right].\end{aligned}$$ We now examine the behavior in the critical region as a function of $\omega$, $k_+$, and $k_-$. Close to a fixed point, we use the values at the fixed point for $\bar{g}$, $\bar{\mu}$, and $\gamma_\psi$: $\bar{g}(0)=\bar{g}(l)=\bar{g}^*$, $\bar{\mu}(0)=\bar{\mu}(l)=0$, and $\gamma_\psi(\bar{g}^*)=\eta$. However, we have to keep the $l$ dependence of $A_\pm$ as they appear together with $k_\pm$. We assume the two-point function is a function of $A_+ k_+^{n_+}$, $A_- k_-^{n_-}$, $\omega$. Since those three quantities and $\Lambda$ have the dimension of energy, we can write the two-point function in the scaling region as $$\begin{aligned}
\Gamma^{(2)}_\Lambda \left( k_+, k_-, \omega; A_+, A_- \right) = \Lambda \hat{\Gamma}^{(2)}_\Lambda \left( \frac{A_+ k_+^{n_+}}{\Lambda}, \frac{A_- k_-^{n_-}}{\Lambda}, \frac{\omega}{\Lambda} \right),\end{aligned}$$ where $\hat{\Gamma}^{(2)}_\Lambda$ is a scaling function. Here, we do not assume homogeneity for $\hat{\Gamma}^{(2)}_\Lambda$ and determine the exponents for $A_+ k_+^{n_+}/\Lambda$, $A_- k_-^{n_-}/\Lambda$, and $\omega/\Lambda$, separately. In Eq. , $l$ is an arbitrary quantity; to inspect the scaling behavior in terms of $\omega$, we set $l=\ln (\omega_0/\omega)$ and $k_+=k_-=0$. The momentum dependence is considered in the same manner with $l=\ln (k_{\pm,0}/k_\pm)^{n_\pm}$ and Eq. . We then find $$\begin{aligned}
&\quad \Gamma^{(2)}_\Lambda (k_+,k_-,\omega) \nonumber\\
&\propto
\begin{cases}
\left(k_+^{n_+/[1+\gamma_{A_+}(\bar{g}^*)]}\right)^{1-\eta} & (k_-=\omega=0) \\
\left(k_-^{n_-/[1+\gamma_{A_-}(\bar{g}^*)]}\right)^{1-\eta} & (k_+=\omega=0) \\
\omega^{1-\eta} & (k_+=k_-=0).
\end{cases}\end{aligned}$$ It confirms the scaling relation of the two-point correlation function Eq.  along with the relation $G = [\Gamma^{(2)}]^{-1}$.

![image](fig_loop_7_1.pdf){width="\hsize"}

![image](fig_loop_7_2.pdf){width="0.95\hsize"}

One-loop calculations $\bm{(h=\Delta=0)}$
-----------------------------------------

We calculate the two-point and four-point functions to obtain the beta functions and $\gamma_\psi(\bar{g})$. This is accomplished by evaluating the perturbative corrections to the two-point and four-point functions (Figs. \[fig:loop\_two-point\] and \[fig:stu\]). As the corrections to the coupling constant $g$, there are three possible one-loop diagrams shown in Fig. \[fig:stu\]. To determine the perturbative correction $\delta g$, all diagrams should be evaluated with zero momentum transfer $q=0$, which is required by the renormalization condition Eq. . The three one-loop diagrams in Fig. \[fig:stu\](a) correspond to the BCS, density-density, and exchange channels (from left to right). Out of the three, the density-density channel does not contribute at one-loop order because of the Pauli exclusion principle for the contact interaction. This contribution is allowed when we assume the density-density interaction $\bar{\psi}_\sigma\bar{\psi}_{\sigma'} \psi_{\sigma'} \psi_\sigma$ with arbitrary spins $\sigma$, $\sigma'$. (For reference, we note that the three channels are referred to as the BCS, ZS (zero sound), and ZS$'$ in Ref. [@Shankar]; or $s$-, $t$-, and $u$-channels with the Mandelstam variables.)

To one-loop order, the two-point and four-point functions acquire corrections to the chemical potential and the coupling constant, but not to field or the energy dispersion as we have seen in the energy-shell RG analysis. One-loop diagrams include $\Sigma_\text{H}$, $\Pi_\text{pp}$, and $\Pi_\text{ph}$ like Eq.  and , so the two-point and four-point functions become $$\begin{gathered}
\label{eq:Gamma_2_oneloop}
\Gamma^{(2)}_\Lambda = -i\omega_n + E_{\bm{k}} -\mu -g\Sigma_\text{H} +g\mu\Pi_\text{ph}, \\
\label{eq:Gamma_4_oneloop}
\Gamma^{(4)}_\Lambda = g - g^2 (\Pi_\text{pp}+\Pi_\text{ph}).\end{gathered}$$ These equations lead to

$$\begin{gathered}
Z_\psi^{-1} = 1, \quad Z_{A_\pm}^{-1} = 1, \\
Z_\mu^{-1} = 1 + \frac{g}{\mu}\Sigma_\text{H} - g\mu\Pi_\text{ph}, \\
Z_g^{-1} = 1 - g(\Pi_\text{pp}+\Pi_\text{ph}).\end{gathered}$$

Here we calculate the perturbative corrections with the soft cutoff $K_\Lambda$. The actual calculations for the beta functions require the $\Lambda$-derivatives instead of the corrections themselves. We thus obtain the one-loop corrections as follows:

$$\begin{aligned}
&\quad \Lambda\frac{\partial}{\partial\Lambda} \Sigma_\text{H} \nonumber\\
&= T\sum_n \int_{\bm{k}} G_0(\bm{k},\omega_n) \Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{k}}) \nonumber\\
&= T\sum_{n\geq0} \int dE D(E) \frac{-2E}{\omega_n^2 + E^2} \frac{2\Lambda^2 E^2}{(\Lambda^2 + E^2)^2} \nonumber\\
&= -\Lambda^2 \int dE D(E) \tanh\left(\frac{E}{2T}\right) \frac{E^2}{(\Lambda^2 + E^2)^2} \nonumber\\
&= -[D(\Lambda)-D(-\Lambda)] \Lambda \int_0^\infty dx \frac{x^{2-\epsilon}}{(1+x^2)^2} \tanh\left(\frac{\Lambda}{2T}x\right) \nonumber\\
&\equiv -\Lambda D(\Lambda) \tilde{c}_\text{H}(\bar{T}) ,\end{aligned}$$

$$\begin{aligned}
&\quad \Lambda\frac{\partial}{\partial\Lambda} \Pi_\text{pp} \nonumber\\
&= T\sum_n \int_{\bm{k}} G_0(\bm{k},\omega_n) G_0(-\bm{k},-\omega_n) \Lambda\frac{\partial}{\partial\Lambda} K^2_\Lambda(E_{\bm{k}}) \nonumber\\
&= T\sum_{n\geq0} \int dE D(E) \frac{2}{\omega_n^2 + E^2} \frac{4\Lambda^4 E^2}{(\Lambda^2+E^2)^3} \nonumber\\
&= 2\Lambda^4 \int dE D(E) \frac{E}{(\Lambda^2+E^2)^3} \tanh\left(\frac{E}{2T}\right) \nonumber\\
&= 2[D(\Lambda)+D(-\Lambda)] \int_0^\infty dx \frac{x^{1-\epsilon}}{(1+x^2)^3} \tanh\left(\frac{\Lambda}{2T}x\right) \nonumber\\
&\equiv D(\Lambda) \tilde{c}_\text{pp}(\bar{T}),\end{aligned}$$

$$\begin{aligned}
&\quad \Lambda\frac{\partial}{\partial\Lambda} \Pi_\text{ph} \nonumber\\
&= T\sum_{\omega_n} \int_{\bm{k}} G_0^2(\bm{k},\omega_n) \Lambda\frac{\partial}{\partial\Lambda} K^2_\Lambda(E_{\bm{k}}) \nonumber\\
&= T\sum_{n\geq0} \int dE D(E) \frac{-2(\omega_n^2-E^2)}{(\omega_n^2+E^2)^2} \frac{4\Lambda^4 E^2}{(\Lambda^2+E^2)^3} \nonumber\\
&= -\frac{\Lambda^4}{T} \int dE D(E) \frac{E^2}{(\Lambda^2+E^2)^3} \frac{1}{\cosh^2\left(\dfrac{E}{2T}\right)} \nonumber\\
&= -[D(\Lambda)+D(-\Lambda)] \frac{\Lambda}{T} \int_0^\infty dx \frac{x^{2-\epsilon}}{(1+x^2)^3} \frac{1}{\cosh^2\left(\dfrac{\Lambda}{2T}x\right)} \nonumber\\
&\equiv -D(\Lambda) \tilde{c}_\text{ph}(\bar{T}).\end{aligned}$$

As a result, we obtain the beta functions Eq.  and

$$\begin{gathered}
\beta(\bar{g}) = \epsilon\bar{g} - \bar{g}^2 \left[ \tilde{c}_\text{pp}(\bar{T})-\tilde{c}_\text{ph}(\bar{T}) \right], \\
\beta_\mu(\bar{g},\bar{\mu}) = \left[ 1-\tilde{c}_\text{ph}(\bar{T}) \bar{g} \right]\bar{\mu} + \tilde{c}_\text{H}(\bar{T}) \bar{g}.\end{gathered}$$

Note that the tree-level scaling terms appear from the definitions of the dimensionless parameters $\bar{g}_0=g_0 D(\Lambda)$ and $\bar{\mu}_0 = \mu/\Lambda$. The beta functions are to be compared with the result from the energy-shell RG analysis Eq. . To confirm, we first evaluate the coefficients $\tilde{c}_\text{H}$, $\tilde{c}_\text{pp}$, $\tilde{c}_\text{ph}$ at $T=0$:

$$\begin{gathered}
\tilde{c}_\text{H}(0) = \left( 1-\frac{D_-}{D_+} \right) \frac{\pi}{4} (1-\epsilon) \frac{1}{\cos\left(\dfrac{\pi\epsilon}{2}\right)}, \\
\tilde{c}_\text{pp}(0) = \left( 1+\frac{D_-}{D_+} \right) \frac{\pi}{4} \left( 1+\frac{\epsilon}{2} \right) \frac{\epsilon}{\sin\left(\dfrac{\pi\epsilon}{2}\right)}, \\
\tilde{c}_\text{ph}(0) = 0.\end{gathered}$$

The zeros of the beta function $\beta(\bar{g})$ give the two fixed points $$\label{eq:fixed_points_field}
\bar{g}_1^* = 0, \quad \bar{g}_2^* = \frac{\epsilon}{\tilde{c}_\text{pp}(0)} (>0).$$ We now find the Gaussian fixed point and the nontrivial interacting fixed point from the field theory approach. Although the value of $\bar{g}_2^*$ differs in the two schemes, resulting exponents for the thermodynamic quantities are not suffered from the difference since the exponents are not directly dependent on the coupling constant $\bar{g}$ at fixed point. We explicitly confirm this in the next subsection by calculating the beta functions for the magnetic field $\Delta$ and pairing field $\Delta$.

RG equations for $\bm{h}$ and $\bm{\Delta}$
-------------------------------------------

The beta functions for the magnetic field $h$ and pairing field $\Delta$ can be obtained from the corresponding vertex functions $\Gamma^{(2,h)}$ and $\Gamma^{(2,\Delta)}$, respectively. Perturbative corrections to them are depicted in Figs. \[fig:loop\_two-point\](b) and \[fig:loop\_two-point\](c). We impose the renormalization conditions

$$\begin{gathered}
\Gamma^{(2,h)}_R = Z_\psi \Gamma^{(2,h)}_\Lambda = h_0, \\
\Gamma^{(2,\Delta)}_R = Z_\psi \Gamma^{(2,\Delta)}_\Lambda = \Delta_0,\end{gathered}$$

where the vertex functions with the cutoff $\Lambda$ are expressed as $$\begin{gathered}
\Gamma^{(2,h)}_\Lambda = Z_h^{-1} h, \quad
\Gamma^{(2,\Delta)}_\Lambda = Z_\Delta^{-1} \Delta.\end{gathered}$$

To obtain the beta functions to one-loop order, it is sufficient to consider the Callan–Symnanzik equations without corrections to the energy dispersion and the chemical potential:

$$\begin{gathered}
\left[ \Lambda\frac{\partial}{\partial\Lambda} - \beta(\bar{g})\frac{\partial}{\partial \bar{g}} - \beta_h(\bar{g},\bar{h})\frac{\partial}{\partial\bar{h}} - \gamma_\psi(\bar{g}) \right] \Gamma^{(2;h)} = 0, \\
\left[ \Lambda\frac{\partial}{\partial\Lambda} - \beta(\bar{g})\frac{\partial}{\partial \bar{g}} - \beta_\Delta(\bar{g},\bar{\Delta})\frac{\partial}{\partial\bar{\Delta}} - \gamma_\psi(\bar{g}) \right] \Gamma^{(2;\Delta)} = 0,\end{gathered}$$

where the beta functions for the magnetic field and pairing field are defined by

$$\begin{gathered}
\label{eq:beta_h}
\beta_h(\bar{h}) = -\left( \Lambda\frac{\partial\bar{h}}{\partial\Lambda} \right)_{\bar{g}_0}, \\
\label{eq:beta_Delta}
\beta_{\bar{\Delta}}(\bar{g}) = -\left( \Lambda\frac{\partial\bar{\Delta}}{\partial\Lambda} \right)_{\bar{g}_0}.\end{gathered}$$

Using the relations $$\begin{gathered}
\bar{h} = Z_h Z_\psi^{-1} \bar{h}_0, \quad
\bar{\Delta} = Z_\Delta Z_\psi^{-1} \bar{\Delta}_0,\end{gathered}$$ the beta functions become

$$\begin{gathered}
\beta_h(\bar{g},\bar{h})
= \bar{h} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_h -\gamma_\psi \right), \\
\beta_\Delta(\bar{g},\bar{\Delta})
= \bar{\Delta} \left( -\Lambda\frac{\partial}{\partial\Lambda} \ln Z_\Delta -\gamma_\psi \right).\end{gathered}$$

They are related to the exponents $\gamma_h$ and $\gamma_\Delta$ when evaluated at a fixed point: $$\begin{gathered}
\beta_h(\bar{g}^*,\bar{h}) = \gamma_h(\bar{g}^*) \bar{h}, \quad
\beta_\Delta(\bar{g}^*,\bar{\Delta}) = \gamma_\Delta(\bar{g}^*) \bar{\Delta}.\end{gathered}$$

We calculate the vertex functions for $h$ and $\Delta$ to one-loop order and find $$\begin{gathered}
\Gamma^{(2,h)}_\Lambda = h, \quad
\Gamma^{(2,\Delta)}_\Lambda = \Delta - g\Delta\Pi_\text{pp}.\end{gathered}$$ We note that there is no perturbative correction to $\Gamma^{(2,h)}_\Lambda$ because of the spin-rotational symmetry of the model. The vertex functions lead to the beta functions

$$\begin{gathered}
\beta_h(\bar{g},\bar{h}) = \bar{h}, \\
\beta_\Delta(\bar{g},\bar{\Delta}) = \left[ 1-\tilde{c}_\text{pp}(\bar{T})\bar{g} \right] \bar{\Delta}.\end{gathered}$$

Now we confirm that the exponent for the pairing field $\Delta$ is the same independent of the RG schemes. Particularly at the nontrivial fixed point, we obtain $\beta_\Delta(\bar{g}_2^*) = (1-\epsilon)\bar{\Delta}$. This is consistent with the result from the energy-shell RG analysis. The coefficient $\tilde{c}_\text{pp}$, which determines the value of the coupling constant at the nontrivial fixed point, does not appear to the exponent of the pairing field.

Two-loop calculation and anomalous dimension {#sec:anomalous_dimension}
--------------------------------------------

An advantage of the field theory approach is considerable when we deal with higher-order corrections. In the following, we consider the two-loop corrections at $T=0$ for the anomalous dimension and the correction to the energy dispersion.

The field renormalization is seen from the frequency dependence of the self-energy. The linear term $\Sigma_\omega$ in Eq.  is given by $$\begin{aligned}
\Sigma_\omega &= \frac{\partial}{\partial(i\omega)} \Sigma(\bm{k}=\bm{0},\omega) \bigg|_{\omega=0}
\equiv \sum_{j\geq 2} g^j \Sigma_\omega^{(j)}.\end{aligned}$$ We expand $\Sigma$ with respect to the coupling constant $g$. Similarly, the renormalization of the energy dispersion arises from the momentum dependence of the self-energy $$\begin{aligned}
\Sigma_{\bm{k}} \equiv \Sigma(\bm{k},\omega=0) \equiv \sum_{j\geq 2} g^j \Sigma_{\bm{k}}^{(j)},\end{aligned}$$ and $\Sigma_\pm$ is obtained as $$\begin{gathered}
\Sigma_\pm = \frac{\partial \Sigma_{\bm{k}}}{\partial k_\pm^{n_\pm}} \bigg|_{k=0}.\end{gathered}$$ We have used that the fact that the one-loop correction, i.e., the Hartree contribution, does not yield the frequency or momentum dependence, so that the expansions with respect to the coupling constant begin at second order.

The renormalization condition Eq.  reads $$\begin{gathered}
Z_\psi^{-1} = 1- \Sigma_\omega, \quad
Z_{A_\pm}^{-1} = 1 + \Sigma_{k_\pm}.\end{gathered}$$ Then, the field renormalization $\gamma_\psi$ is expressed as $$\begin{aligned}
\label{eq:gamma_two-loop}
\gamma_\psi &= \Lambda\frac{\partial}{\partial\Lambda} \ln \left( 1 -\sum_{j\geq2} g^j \Sigma_\omega^{(j)} \right) \nonumber\\
&= -g^2 \Lambda\frac{\partial}{\partial\Lambda} \Sigma_\omega^{(2)} + O(g^3).\end{aligned}$$ Again, assuming that no additional term than the corrections to $A_\pm$ is generated from $\Sigma_{\bm{k}}$, we obtain the beta function for the coefficient of the energy dispersion $A_\pm$ as $$\begin{aligned}
\label{eq:beta_A_two-loop}
\beta_{A_\pm}
&= -\Lambda\frac{\partial A_\pm}{\partial\Lambda} \nonumber\\
&= A \left[ g^2 \Lambda\frac{\partial}{\partial\Lambda} (\Sigma_{\pm}^{(2)} + \Sigma_\omega^{(2)}) + O(g^3) \right] .\end{aligned}$$

We now calculate the two-loop correction to the self-energy $\Sigma^{(2)}$. The frequency and momentum dependent contribution can appear in the sunrise diagram, shown in Fig. \[fig:loop\](d) and \[fig:loop\_two-point\](a) as the rightmost term. It is calculated from $$\begin{aligned}
\label{eq:two-loop_function}
\Sigma^{(2)}(k) &= -\int_{pql} G_{0\Lambda}(p) G_{0\Lambda}(q) G_{0\Lambda}(l) \nonumber\\
&\qquad\times (2\pi)^{d+1} \delta(p+q-l-k).\end{aligned}$$ We use the shorthand notations $p=(\bm{p},\omega_p)$ and $\int_p = \int\frac{d\omega_p}{(2\pi)}\int\frac{d^dp}{(2\pi)^d}$. Then, we obtain the $\omega$-linear contribution

$$\begin{aligned}
\label{eq:two-loop_frequency}
&\quad -\Lambda\frac{\partial}{\partial\Lambda} \Sigma_\omega^{(2)} \nonumber\\
&= \frac{\partial}{\partial(i\omega_k)} \int_{pql} (2\pi)^{d+1} \delta(p+q-l-k) \frac{1}{i\omega_p-E_{\bm{p}}} \frac{1}{i\omega_q-E_{\bm{q}}} \frac{1}{i\omega_l-E_{\bm{l}}}
\Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \bigg|_{k=0} \nonumber\\
&= \left( \int_{\bm{p}\bm{q}}^+ \int_{\bm{l}}^- + \int_{\bm{p}\bm{q}}^- \int_{\bm{l}}^+ \right) (2\pi)^{d} \delta(\bm{p}+\bm{q}-\bm{l}) \frac{1}{(E_{\bm{p}}+E_{\bm{q}}-E_{\bm{l}})^2}
\Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \nonumber\\
&= 2\Lambda^{-2\epsilon} \left( \int_{\bar{\bm{p}}\bar{\bm{q}}}^+ \int_{\bar{\bm{l}}}^- + \int_{\bar{\bm{p}}\bar{\bm{q}}}^- \int_{\bar{\bm{l}}}^+ \right) \frac{(2\pi)^{d} \delta(\bar{\bm{p}}+\bar{\bm{q}}-\bar{\bm{l}})}{(E_{\bar{\bm{p}}}+E_{\bar{\bm{q}}}-E_{\bar{\bm{l}}})^2}
\frac{3E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2 + 2(E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2) + (E_{\bar{\bm{p}}}^2+E_{\bar{\bm{q}}}^2+E_{\bar{\bm{l}}}^2)}{(1+E_{\bar{\bm{p}}}^2)^2 (1+E_{\bar{\bm{q}}}^2)^2 (1+E_{\bar{\bm{l}}}^2)^2} \nonumber\\
&\equiv D^2(\Lambda) C^{(2)},\end{aligned}$$

and the momentum-dependent part $$\begin{aligned}
\label{eq:two-loop_momentum}
&\quad \Lambda\frac{\partial}{\partial\Lambda} \Sigma^{(2)}(\bm{k},\omega_k=0) \nonumber\\
&= -\int_{pql} (2\pi)^{d+1} \delta(p+q-l-k) \frac{1}{i\omega_p-E_{\bm{p}}} \frac{1}{i\omega_q-E_{\bm{q}}} \frac{1}{i\omega_l-E_{\bm{l}}}
\Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \bigg|_{\omega_k=0} \nonumber\\
&= -\left( \int_{\bm{p}\bm{q}}^+ \int_{\bm{l}}^- + \int_{\bm{p}\bm{q}}^- \int_{\bm{l}}^+ \right) (2\pi)^{d} \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \frac{1}{E_{\bm{p}}+E_{\bm{q}}-E_{\bm{l}}}
\Lambda\frac{\partial}{\partial\Lambda} K_\Lambda(E_{\bm{p}}) K_\Lambda(E_{\bm{q}}) K_\Lambda(E_{\bm{l}}) \nonumber\\
&= -2\Lambda^{-2\epsilon} \left( \int_{\bar{\bm{p}}\bar{\bm{q}}}^+ \int_{\bar{\bm{l}}}^- - \int_{\bar{\bm{p}}\bar{\bm{q}}}^- \int_{\bar{\bm{l}}}^+ \right) \frac{(2\pi)^{d} \delta(\bar{\bm{p}}+\bar{\bm{q}}-\bar{\bm{l}}-\bar{\bm{k}})}{|E_{\bar{\bm{p}}}+E_{\bar{\bm{q}}}-E_{\bar{\bm{l}}}|}
\frac{3E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2 + 2(E_{\bar{\bm{q}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{l}}}^2+E_{\bar{\bm{p}}}^2E_{\bar{\bm{q}}}^2) + (E_{\bar{\bm{p}}}^2+E_{\bar{\bm{q}}}^2+E_{\bar{\bm{l}}}^2)}{(1+E_{\bar{\bm{p}}}^2)^2 (1+E_{\bar{\bm{q}}}^2)^2 (1+E_{\bar{\bm{l}}}^2)^2} \nonumber\\
&\equiv D^2(\Lambda) C^{(2)}_{\bm{k}}.\end{aligned}$$

Here we denote the dimensionless quantities by adding bars; we define $\bar{\omega} = \omega/\Lambda$, $\bar{p}_+ = p_+/\Lambda^{1/n_1}$, and $\bar{p}_- = p_-/\Lambda^{1/n_2}$. The momentum is scaled by $\Lambda$ so that the energy becomes dimensionless: $E_{\bar{\bm{k}}} = E_{\bm{k}}/\Lambda$. $\int^{\pm}_{\bm{p}} = \int_{\bm{p}} \Theta (\pm E_{\bm{p}})$ stands for the momentum integral within the positive (negative) energy domain. The constraints on the momentum integrals emerge after the frequency integrals. They can be evaluated by identifying the position of poles on the complex plane, leading to the restricted regions of the momentum integrals.

We expect finite results for the two-loop results Eqs.  and at a saddle point of an energy dispersion because of the constraints on the momentum integrals $\int_{\bm{p}\bm{q}}^\pm \int_{\bm{l}}^\mp$. The two-loop contributions vanish at a band edge since there is no sign change in the energy dispersion.

Now we scrutinize the frequency-dependent part $\Sigma_\omega^{(2)}$, which is responsible to the field renormalization and hence the anomalous dimension. As we have discussed, the contribution vanishes at band edges and thus an anomalous dimension does not arise. It can be finite only at an energy saddle point. In addition, it is worth pointing out that the integrand of Eq.  is guaranteed to be positive. Therefore, if there exists a finite volume that satisfies the constraint of the momentum integrals, we find a finite result: $C^{(2)}>0$. The constraints on the momentum integrals can be rephrased as follows: There exists a momentum $\bm{l}=\bm{p}+\bm{q}$ such that $\operatorname{sgn}(E_{\bm{l}}) = -\operatorname{sgn}(E_{\bm{p}}) = -\operatorname{sgn}(E_{\bm{q}})$. Such a momentum $\bm{l}$ in general exists near a saddle point because the energy dispersion near a saddle point comprises two or more filled Fermi seas and the area is not convex. We do not further evaluate the expression of the two-loop correction as its value depends on the explicit form of the energy dispersion.

Equation  has a numerical factor $C^{(2)}$, which is independent of the cutoff $\Lambda$. From Eq. , we find the field renormalization $$\gamma_\psi = C^{(2)} \bar{g}^2 + O(\bar{g}^3).$$ This quantity gives the anomalous dimension when evaluated at a fixed point. It can be finite at the nontrivial fixed point to become $$\eta = C^{(2)} \bar{g}_2^{*2} + O(\bar{g}_2^{*3}) (>0).$$ A finite anomalous dimension concludes a non-Fermi liquid behavior at the nontrivial fixed point. This happens at a saddle point of an energy dispersion with a power-law DOS singularity.

The momentum-dependent part Eq.  can be considered in the same way. It becomes finite only at a saddle point of an energy dispersion, but not at a band edge. A finite $C_{\bm{k}}^{(2)}$ leads to the beta functions for $A_\pm$ $$\beta_{A_\pm} = A_\pm \left[ \bar{g}^2 \left( \left.\frac{\partial C_{\bm{k}}^{(2)}}{\partial k_\pm^{n_\pm}}\right|_{\bm{k}=0} - C^{(2)} \right) +O(\bar{g}^3) \right].$$ The quantity inside the square brackets is identified as $\gamma_{A_\pm}$ when evaluated at a fixed point.

Quasiparticle decay rate {#sec:lifetime}
========================

The preceding RG analyses focused on the real part of the self-energy or equivalently the two-point function. They give rise to the corrections to the action, which are captured through the RG equations. On the other hand, the imaginary part of the self-energy describes the damping of the quasiparticle, which is the focus of this section. It is generated by the interaction in the present model. Unlike the real part of the self-energy, the imaginary part can be calculated without a cutoff; we do not employ an RG method in this section, but integrate over the entire frequency and momentum space at once.

We calculate the quasiparticle decay rate $\Gamma(\bm{k},\omega)$, obtained from the retarded self-energy as $$\Gamma(\bm{k},\omega) = -\operatorname{Im}\Sigma^R(\bm{k},\omega).$$ The retarded self-energy $\Sigma^R(\bm{k},\omega)$ is calculated from the self-energy $\Sigma(\bm{k},\omega_n)$, with the analytic continuation of the Matsubara frequency to the real frequency: $i\omega_n = \omega + i\delta$ ($\delta$: infinitesimal positive quantity). In the presence of the contact interaction, a finite imaginary part of the self-energy $\Sigma^R$ emerges at two-loop order and higher. The one-loop correction, or the Hartree term $\Sigma_\text{H}$, does not yield a finite imaginary component. Here we consider the two-loop diagram (the sunrise diagram) \[Fig. \[fig:loop\](d)\] to calculate the quasiparticle decay rate $\Gamma$. Like Eq. , it is given by $$\begin{aligned}
\label{eq:two-loop_damping}
&\quad \Sigma^{(2)}(\bm{k},\omega_n) \nonumber\\
&= -T\sum_{\omega_p} T\sum_{\omega_q} T\sum_{\omega_l}
\int_{\bm{p}\bm{q}\bm{l}} G_0(\bm{p},\omega_p) G_0(\bm{q},\omega_q) G_0(\bm{l},\omega_l) \nonumber\\
&\quad\times \frac{(2\pi)^d}{T} \delta(\omega_p+\omega_q-\omega_l-\omega_n)  \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}),\end{aligned}$$ but we do not need a cutoff for the imaginary part.

The calculation of $\Sigma^{(2)}$ is standard and can be found in e.g. Ref. [@AGD]; we also show the derivation in Appendix \[sec:two-loop\_lifetime\] and just present the result here. The quasiparticle decay rate to two-loop order is given by $\Sigma^{(2)}$ after the analytic continuation: $$\begin{aligned}
&\quad \Gamma(\bm{k},\omega) \nonumber\\
&= -g^2 \operatorname{Im}\Sigma^{(2)R}(\bm{k},\omega) \nonumber\\
&= \frac{\pi}{4} g^2 \cosh\left(\frac{\omega}{2T}\right) \nonumber\\
&\quad \times \int_{\bm{p}\bm{q}} \frac{\delta(E_{\bm{p}}+E_{\bm{q}}-E_{\bm{p}+\bm{q}-\bm{k}})}{\cosh\left(\dfrac{E_{\bm{p}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{q}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{p}+\bm{q}-\bm{k}}}{2T}\right)} .\end{aligned}$$ This relation holds for an arbitrary energy dispersion $E_{\bm{k}}$.

We extract the temperature dependence by introducing dimensionless quantities in terms of temperature $T$: we define $\tilde{p}_\pm = p_\pm/T^{1/n_\pm}$, so that the energy dispersion satisfies $E_{\tilde{\bm{p}}}=E_{\bm{p}}/T$. Here we are interested in the low-frequency limit with $\omega \ll T$. By substituting $\bm{k}=\bm{0}$ and $\omega=0$, we obtain $$\begin{aligned}
\label{eq:decay_rate_0}
\Gamma &= \frac{\pi}{4} g^2 T^{1-2\epsilon} \int_{\tilde{\bm{p}}\tilde{\bm{q}}} \frac{\delta(E_{\tilde{\bm{p}}}+E_{\tilde{\bm{q}}}-E_{\tilde{\bm{p}}+\tilde{\bm{q}}})}{\cosh\left(\dfrac{E_{\tilde{\bm{p}}}}{2}\right) \cosh\left(\dfrac{E_{\tilde{\bm{q}}}}{2}\right) \cosh\left(\dfrac{E_{\tilde{\bm{p}}+\tilde{\bm{q}}}}{2}\right)} \nonumber\\
&\propto \bar{g}^2 T^{1-2\epsilon}.\end{aligned}$$ At the nontrivial fixed point $\bar{g} = \bar{g}_2^*$, the quasiparticle decay rate exhibits the temperature dependence $\Gamma\propto T^{1-2\epsilon}$.

In the Fermi liquid theory, when the temperature is much larger than the Fermi energy $\epsilon_F \ll T$, the decay rate is proportional to $T^2$. This result relies on the existence of the Fermi surface with finite DOS. On the other hand, the decay rate Eq.  is distinct from the Fermi liquid results, which reflects the divergent DOS at $\mu=0$. Also, we note that the result does not depend on whether the power-law divergent DOS is located at a saddle point or a band edge of the energy dispersion. This is in contrast to the anomalous dimension, which can only be found at a saddle point as we have discussed in Sec. \[sec:anomalous\_dimension\].

Summary and discussions {#sec:discussions}
=======================

We have analyzed electron interaction effects near a high-order VHS with a scale-invariant Fermi surface with a power-law divergent DOS. Scale invariance of the system allows an RG analysis to search for fixed points and a scaling analysis of thermodynamic quantities and correlation functions around the fixed points.

The one-loop RG analysis finds that electron interaction around high-order VHS offers a fermionic analog of the $\phi^4$ theory. We have identified the two RG fixed points: the Gaussian fixed point and the nontrivial interacting fixed point. The latter is an analog of the Wilson–Fisher fixed point in the $\phi^4$ theory. Like the $\phi^4$ theory, the Gaussian fixed point is unstable and the interacting fixed point is stable in terms of the RG flow of the coupling constant.

We performed a controlled RG analysis about the interacting fixed point owing to the smallness of the DOS singularity exponent $\epsilon$. The two-loop RG analysis reveals that the nontrivial interacting fixed point manifests a non-Fermi liquid behavior when the high-order VHS is realized at a saddle point of the energy dispersion. We diagnosed it with a finite anomalous dimension. Importantly, the DOS singularity exponent $\epsilon$ can be any rational number, associated with an analytic energy dispersion. This is in contrast to $\epsilon$ in the $\phi^4$ theory, where $\epsilon$ denotes the dimensionality of the system.

The system remains metallic at the interacting fixed point with power-law divergent susceptibilities, while the pairing susceptibility turns to be finite unlike the Gaussian fixed point. We term such a metallic state with power-law divergent susceptibilities but yet without a long-range order as a supermetal. In this sense, the Gaussian fixed point can be viewed as a noninteracting supermetal and the nontrivial fixed point as an interacting supermetal. The scaling exponents of thermodynamic quantities and correlation functions are calculated around the two fixed points.

It is worth drawing a comparison between a supermetal and a normal metal. Finite density in a normal metal invalidates scale invariance as it defines an innate length scale. The RG analysis with a closed Fermi surface is commonly referred to as Shankar’s RG [@Shankar]. It requires a judicious choice of RG transformations; Shankar’s RG takes the small cutoff limit to perform the RG analysis with the quartic interaction. As a result, only a very thin shell around the Fermi surface is considered. In addition, we should be aware of the non-analyticity of the charge susceptibility due to the innate length scale. It causes the BCS instability in a higher angular momentum channel at very low temperature through the Kohn–Luttinger mechanism [@Kohn-Luttinger]. The relation between the pairing susceptibility and Shankar’s RG is discussed in Ref. [@Parameswaran]; finite repulsion suppresses the $s$-wave pairing susceptibility.

Throughout the analysis, we assume that the DOS singularity exponent $\epsilon$ is a small positive quantity $(\epsilon>0)$. The formula for the DOS at a high-order VHS Eq.  can be used in the limit $\epsilon \to +0$. At a saddle point of an energy dispersion, the DOS has a logarithmic divergence instead of a power-law divergence. It then necessitates a UV cutoff in the momentum integral for the DOS, which is equivalent to subtract the DOS at a large energy: $$\begin{aligned}
D(E) - D(s\Lambda) &= D_{s} \left( |E|^{-\epsilon} -\Lambda^{-\epsilon} \right) \nonumber\\
&\to \epsilon D_s \ln \left(\frac{\Lambda}{|E|}\right) \quad (\epsilon\to+0)\end{aligned}$$ with $s=\operatorname{sgn}(E)$. Note that $\epsilon D_s$ is constant in the limit $\epsilon\to+0$, reflecting $1/\epsilon$ singularities in the coefficients $D_\pm$ for $d_\pm>0$. (At a band edge, we instead find a discontinuity without a divergence.) In two dimensions, we obtain a conventional VHS $E_{\bm{k}}=k_x^2-k_y^2$ as a saddle point of an energy dispersion. This is also a scale-invariant system; however, its logarithmically-divergent DOS $D(E)\propto \ln (\Lambda/E)$ raises technical issues in an RG analysis. The non-analyticity of the DOS hinders the RG analysis and the UV cutoff $\Lambda$ cannot be eliminated from RG equations [@Kallin; @Kapustin]. It occurs as a sequel that the low-energy physics is affected by the UV scale $\Lambda$.

One of the other theories for a non-Fermi liquid is the Hertz–Millis–Moriya theory for the quantum critical phenomenon in itinerant magnets [@Hertz; @Millis; @Moriya]. It describes the coupling between electrons with a Fermi surface and bosonic fluctuations near the magnetic transition. In this theory, low-energy modes of electrons are integrated out to yield a nonlocal effective action for the bosonic modes, which requires a justification. See reviews e.g. Refs. [@NFL1; @NFL2; @NFL3] for details and various other theories for non-Fermi liquids.

Our present analysis does not suffer from those three difficulties related to a closed Fermi surface, a logarithmic DOS, and the coexistence of gapless fermionic and bosonic modes. A scale-invariant Fermi surface with a power-law divergent DOS removes non-analyticity due to the logarithmic DOS and susceptibilities, and a small DOS singularity exponent guarantees that only the short-range interaction is relevant.

Our analysis relies on scale invariance of the Fermi surface at a high-order VHS. To realize it in materials, there must be a single high-order VHS in the Brillouin zone at the energy range in focus. Otherwise, an energy or a length scale appears, which violates scale invariance. In reality, materials could have multiple VHS points at the same energy in the Brillouin zone, related by symmetry. Even though each VHS point is separately scale invariant, the distance among the VHS points set the length scale, potentially leading to instabilities. However, for a certain parameter range before an ordering instability takes place, there could exist a scaling region where thermodynamic or transport quantities follow scaling properties. For example, when temperature $T$ and the carrier density $n$ are control parameters, a physical quantity $Q$ follows the scaling relation $$Q(T,n) = T^a \hat{\mathcal{F}}\left( n T^{-(1-\epsilon)} \right)$$ around the Gaussian fixed point, where the exponent $a$ is determined by a dimensional analysis of $Q$ and $\hat{\mathcal{F}}$ is a scaling function.

The connection between VHS and superconductivity was proposed in relation to an electronic mechanism for superconductivity [@Hirsch; @Newns], and electronic systems with multiple conventional VHS (i.e., logarithmically divergent DOS) in the Brillouin zone are first studied in the context of copper oxide superconductors [@Dzyaloshinskii; @Schulz; @Lederer; @Furukawa; @LeHur; @Raghu]. There are two VHS points at the high symmetry points $(\pi,0)$ and $(0,\pi)$. As the spectral weight is peaked at those points, the VHS points are also called hot spots. The large DOS and Fermi surface nesting at the wave vector that connects the hot spots invoke $d$-wave superconductivity or a density-wave state. Similar analyses are applied for doped monolayer graphene [@Nandkishore; @Gonzalez] and twisted bilayer graphene [@Isobe]. The existence of multiple VHS requires a set of coupling constants to describe interactions as scattering processes among the hot spots. Also, the overlap of the Fermi surfaces around the hot spots is quantified by so-called nesting parameters in the particle-hole and particle-particle channels. The nesting parameters control the RG flow of the coupling constants, and hence determine the resultant instability if exists.

When there are multiple high-order VHS points (or high-order and conventional VHS points) in the Brillouin zone, the formal structure of the coupled RG equations for the coupling constants resembles that for the conventional VHS case. However, we could find the differences between the two cases. First, the coupling constants have finite scaling dimensions for high-order VHS due to the power-law DOS. Recall that for the contact interaction, the product of the coupling constant and the DOS becomes dimensionless. It results in the tree-level scaling of the coupling constants, and the RG equations acquire linear terms with respect to the coupling constants. As we have seen, the coexistence of the linear and quadratic terms give rise to fixed points in the weak coupling region. An RG flow around those fixed points would describe the weak-coupling behavior. Nevertheless, quadratic terms would be dominant in identifying instabilities, where the coupling constants becomes large; see also Ref. [@multicritical1].

Another difference appears in nesting parameters. With conventional VHS, Fermi surface nesting yields an additional logarithm to that of the DOS, including the double logarithm in the BCS susceptibility. However, a high-order VHS does not alter the analytic properties but only the prefactors of susceptibilities are affected; cf. Appendix \[sec:susceptibilities\] for the comparison between the particle-hole and particle-particle susceptibilities at a high-order VHS. It relaxes nesting conditions for instabilities with high-order VHS. In addition, from a theoretical point of view, the absence of double logarithms retrieves the analytic tractability of RG equations.

We acknowledge helpful discussions with A. V. Chubukov, E. Fradkin, S.-S. Lee, and especially E. Berg. This work was supported by DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0018945. LF was supported in part by a Simons Investigator Award from the Simons Foundation.

One-loop RG analysis at finite temperature {#sec:finite-temperature}
==========================================

We work on the RG equations at finite temperature $T \neq 0$. Temperature has a dimension of energy, so that it is one of relevant perturbations. Here we consider the one-loop RG equations from the energy-shell RG analysis. At finite temperature, the one-loop corrections $\Sigma_\text{H}$, $\Pi_\text{pp}$, and $\Pi_\text{ph}$ should be evaluated with $T\neq 0$. To order $l$, we obtain

$$\begin{gathered}
\Sigma_\text{H} = T\sum_{\omega_n} \int_{\bm{k}}^> G_0(\bm{k},\omega_n) \simeq l c_\text{H}(\bar{T}) \Lambda D(\Lambda) , \\
\Pi_{\text{pp}} = T\sum_{\omega_n} \int_{\bm{k}}^> G_0(\bm{k},\omega_n) G_0(-\bm{k},-\omega_n) \simeq l c_\text{pp}(\bar{T}) D(\Lambda) , \\
\Pi_\text{ph} = T\sum_{\omega_n} \int_{\bm{k}}^> G_0(\bm{k},\omega_n) G_0(\bm{k},\omega_n) \simeq l c_\text{ph}(\bar{T}) D(\Lambda) ,\end{gathered}$$

where we introduce the dimensionless temperature $\bar{T}=T/\Lambda$. Again, all quantities are evaluated at zero external frequency and momentum, so that the results depend only on the DOS. The temperature-dependent dimensionless coefficients $c_{\mu}$, $c_\text{pp}$, and $c_\text{ph}$ are

$$\begin{gathered}
\label{eq:c_mu}
c_\text{H}(\bar{T}) =  \frac{1}{2} \left( 1-\frac{D_-}{D_+} \right) \tanh\left(\frac{1}{2\bar{T}}\right), \\
\label{eq:c_pp}
c_\text{pp}(\bar{T}) = \frac{1}{2} \left( 1+\frac{D_-}{D_+} \right) \tanh\left(\frac{1}{2\bar{T}}\right), \\
\label{eq:c_ph}
c_\text{ph}(\bar{T}) = \frac{1}{2\bar{T}} \left( 1+\frac{D_-}{D_+} \right) \frac{1}{2\cosh^2\left(\dfrac{1}{2\bar{T}}\right)}.\end{gathered}$$

Unlike the calculation at $T=0$, $\Pi_\text{ph}$ becomes finite for $T\neq 0$, while the correction to the field or the energy dispersion remains absent to one-loop order.

In the analysis at zero temperature, we rescale the frequency in Eq. . At finite temperature, rescaling of the Matsubara frequency leads to rescaling of temperature [@Millis]. Temperature obeys the same scaling relation as that for the frequency: $T' = bT$.

Including the temperature-dependent factors $c_{\mu}(\bar{T})$, $c_\text{pp}(\bar{T})$, and $c_\text{ph}(\bar{T})$, we obtain the changes of the parameters at an RG step as

$$\begin{gathered}
T' = b T, \\
\mu' \simeq b [\mu + lc_\text{H}(\bar{T})gD(\Lambda) - lc_\text{ph}(\bar{T})\mu gD(\Lambda) ], \\
h' = b h, \\
\Delta' \simeq b [\Delta - lc_\text{pp}(\bar{T})gD(\Lambda)], \\
g' \simeq b^{\epsilon} \{ g - l [c_\text{pp}(\bar{T}) + c_\text{ph}(\bar{T})] g^2 D(\Lambda) \}.\end{gathered}$$

Then, we reach the RG equations

$$\begin{gathered}
\frac{d\bar{T}}{dl} = \bar{T}, \\
\frac{d\bar{\mu}}{dl} = [ 1- c_\text{ph} (\bar{T})\bar{g} ] \bar{\mu} + c_\mu(\bar{T}) \bar{g}, \\
\frac{d\bar{h}}{dl} = \bar{h}, \\
\frac{d\bar{\Delta}}{dl} = [ 1- c_\text{pp}(\bar{T})\bar{g} ] \bar{\Delta}, \\
\frac{d\bar{g}}{dl} = \epsilon \bar{g} - \left[ c_\text{pp}(\bar{T})+c_\text{ph}(\bar{T}) \right] \bar{g}^2.\end{gathered}$$

Since temperature $T$ is relevant and its fixed point is located at $T=0$, the fixed points of the parameters are found at $T=0$, as we discussed in the main part.

Brief review of the field theory approach to RG analyses {#sec:field-theory_review}
========================================================

Here we describe a field theory approach to RG equations, in light of the Wilsonian approach. We derive the Callan–Symanzik equation and the beta functions, which show how scale-dependent parameters affect physical quantities. The following descriptions are partly motivated by the references [@Zinn-Justin; @Peskin; @Amit; @Nair].

For the sake of clarity, we consider a theory with a scalar field $\phi$ and a set of dimensionless parameters $\{\bar{g}_\nu\}$, where we write the action as $S[\phi;\bar{g}]$. The partition function is given by $$\mathcal{Z} = \int D\phi e^{-S[\phi;\bar{g}]}.$$ When the model suffers from UV divergences, i.e., perturbative loop corrections have UV divergences, we need to cure them to extract meaningful information. Those UV divergences can be regularized by removing UV modes from the model. To this end, we decompose the field $\phi$ depending on the energy range to which they contribute: $\phi_{\Lambda'}^{\Lambda}$ accounts for the energy between $\Lambda'$ and $\Lambda$. Then, we redefine the partition function as $$\mathcal{Z}_{\Lambda_0}[\bar{g}_{0}] = \int D\phi_0^{\Lambda_0} e^{-S_{\Lambda_0}[\phi_0^{\Lambda_0};\bar{g}_{0}]}.$$ It does not obviously have a UV divergence because no UV modes are included. Here the energy scale $\Lambda_0$ works as a UV energy cutoff. (This is equivalent to impose the effective action to be finite at $\Lambda_0$ without a cutoff. From this viewpoint, counterterms are introduced to cure UV divergences. See below for the effective action.)

The scale $\Lambda_0$ is an arbitrary energy scale to regularize UV divergences. The next thing we should check is how a change of the characteristic energy scale affects the theory. To see it, we define the effective action at an energy scale $\Lambda(<\Lambda_0)$ as $$\begin{aligned}
S_\Lambda^\text{eff}[Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)] = -\ln \left[ \int D\phi_\Lambda^{\Lambda_0} e^{-S_{\Lambda_0}[\phi_0^\Lambda + \phi_\Lambda^{\Lambda_0};\bar{g}_{0}]} \right].\end{aligned}$$ We require the effective action $S^\text{eff}$ to have the same form as the action $S$. Then, the partition function can be written as $$\begin{aligned}
\mathcal{Z}_{\Lambda_0} [\bar{g}_{0}]
&= \int D\phi_0^\Lambda D\phi_\Lambda^{\Lambda_0} e^{-S_{\Lambda_0}[\phi_0^\Lambda + \phi_\Lambda^{\Lambda_0};\bar{g}_{0}]} \nonumber\\
&= \int D\phi_0^{\Lambda} e^{-S^\text{eff}_\Lambda [Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)]} \nonumber\\
&\equiv \mathcal{Z}_\Lambda [Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)].\end{aligned}$$ This is simply rewriting of the partition function with the effective action at the scale $\Lambda$. We relate the partition functions at different scales to find $$\begin{aligned}
\mathcal{Z}_\Lambda [Z_\Lambda^{1/2} \phi_0^\Lambda;\bar{g}(\Lambda)]
= \mathcal{Z}_{\Lambda'} [Z_{\Lambda'}^{1/2} \phi_0^{\Lambda'};\bar{g}(\Lambda')].\end{aligned}$$ This equality tells us that we have the same partition function defined at different energy scales $\Lambda$ and $\Lambda'$, together with the changes of the weight $Z$ and the parameters $\bar{g}_a$.

We then aim to calculate the $N$-point correlation function with the cutoff $\Lambda$ and the parameters $\bar{g}_a(\Lambda)$: $$\begin{aligned}
&\quad \langle \phi_0^{\Lambda}(k_1) \cdots \phi_0^{\Lambda}(k_N) \rangle_{\Lambda;\bar{g}(\Lambda)} \nonumber\\
&= \frac{1}{\mathcal{Z}_{\Lambda}[\phi_0^\Lambda;\bar{g}(\Lambda)]} \int D\phi_0^{\Lambda} \phi_0^{\Lambda}(k_1) \cdots \phi_0^{\Lambda}(k_N) e^{-S^\text{eff}_{\Lambda}[\phi_0^{\Lambda};\bar{g}(\Lambda)]}.\end{aligned}$$ When all momenta $k_a$ correspond to energies below $\Lambda$ and $\Lambda'$, we find $\phi_0^{\Lambda}(k_a) = \phi_0^{\Lambda'}(k_a)$, which enables us to relate the $N$-point correlation functions at different scales as $$\begin{aligned}
&\quad Z_{\Lambda}^{-N/2} \langle \phi_0^{\Lambda}(k_1) \cdots \phi_0^{\Lambda}(k_N) \rangle_{\Lambda;\bar{g}(\Lambda)} \nonumber\\
&= Z_{\Lambda'}^{-N/2} \langle \phi_0^{\Lambda'}(k_1) \cdots \phi_0^{\Lambda'}(k_N) \rangle_{\Lambda';\bar{g}(\Lambda')}.\end{aligned}$$ We now write this relation with the connected $N$-point correlation function $G^{(N)}$: $$\begin{aligned}
Z_{\Lambda}^{-N/2} G_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{k_a\})
= Z_{\Lambda'}^{-N/2} G_{\Lambda';\bar{g}(\Lambda')}^{(N)}(\{k_a\}).\end{aligned}$$ The scale dependence of this equality can be written in the form of a differential equation: $$\begin{aligned}
\left[ \Lambda\frac{\partial}{\partial\Lambda} -\beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} + \frac{N}{2}\gamma(\bar{g}) \right] G_{\Lambda;\bar{g}_\nu(\Lambda)}^{(N)}(\{k_a\}) = 0.\end{aligned}$$ Note that the repeated index is summed over. This equation is called the Callan–Symanzik equation [@Callan; @Symanzik1; @Symanzik2] for the connected $N$-point correlation function $G^{(N)}$ with the beta functions $\beta_\nu$ and the field renormalization $\gamma$ defined by

$$\begin{gathered}
\beta_\nu(\bar{g}) = -\left(\Lambda\frac{\partial\bar{g}_\nu}{\partial\Lambda}\right)_{\bar{g}_{0}}, \\
\gamma(\bar{g}) = -\left(\Lambda\frac{\partial}{\partial\Lambda} \ln Z_{\Lambda}\right)_{\bar{g}_{0}}.\end{gathered}$$

The correlation functions are obtained from perturbative calculations. Actually, it is more straightforward to obtain the one-particle irreducible $N$-point function $\Gamma^{(N)}$ instead of the $N$-point correlation function $G^{(N)}$. When a model involves a quartic interaction $\phi^4$ without a cubic term $\phi^3$, $\Gamma^{(2)}$ and $\Gamma^{(4)}$ are given by $$\begin{gathered}
\Gamma^{(2)}(k) = [G^{(2)}(k)]^{-1}, \\
\Gamma^{(4)}(k_1,k_2,k_3,k_4) = \frac{G^{(4)}(k_1,k_2,k_3,k_4)}{G^{(2)}(k_1) G^{(2)}(k_2) G^{(2)}(k_3) G^{(2)}(k_4)}.\end{gathered}$$ For the definition of $\Gamma^{(N)}$ from the effective action, see the references [@Zinn-Justin; @Peskin; @Amit; @Nair]. Roughly speaking, $\Gamma^{(N)}$ corresponds to the coefficient of the $\phi^N$ term in the effective action. Again, using the fact that $\phi_0^{\Lambda}(k_a) = \phi_0^{\Lambda'}(k_a)$ holds when the energy corresponding to the momentum $k_a$ is smaller than $\Lambda$ and $\Lambda'$, we find the relation $$\begin{aligned}
\label{eq:Gamma_scale}
Z_{\Lambda}^{N/2} \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{k_a\})
= Z_{\Lambda'}^{N/2} \Gamma_{\Lambda';\bar{g}(\Lambda')}^{(N)}(\{k_a\}).\end{aligned}$$ It results in the Callan–Symanzik equation for $\Gamma^{(N)}$: $$\begin{aligned}
\left[ \Lambda\frac{\partial}{\partial\Lambda} -\beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} - \frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{k_a\}) = 0.\end{aligned}$$

So far, we have compared $G^{(N)}$ or $\Gamma^{(N)}$ at different cutoffs $\Lambda$ and $\Lambda'$, so that the dependence on $\Lambda$ is explicit. However, this comparison is still theoretical; our aim is to compare the two theories with the same cutoff $\Lambda$. For this sake, we rescale the coordinate to change the cutoff. Suppose we have the scaling relations

$$\begin{gathered}
\Lambda' = b \Lambda, \\
k'_j = b^{d_{k_j}} k_j, \\
\phi_0^{\Lambda'}(k') = b^{d_\phi} \phi_0^{\Lambda}(k),\end{gathered}$$

where $k_j$ is a component of $k=(\bm{k},\omega)$ and $d_{\mathcal{O}}$ denotes the scaling (energy) dimension of $\mathcal{O}$. Those relations lead to $$\begin{aligned}
\label{eq:Gamma_scale_2}
\Gamma_{\Lambda';\bar{g}(\Lambda')}^{(N)}(\{k_a'\})
= b^{d_{\Gamma^{(N)}}} \Gamma_{\Lambda;\bar{g}(\Lambda')}^{(N)}(\{k_a\}),\end{aligned}$$ Rescaling the momentum forces the cutoff $\Lambda'$ back to $\Lambda$ with the overall factor $b^{d_{\Gamma^{(N)}}}$, but this process does not alter the dimensionless parameters $\bar{g}_\nu$. From Eqs.  and , we find the relation $$\begin{aligned}
\label{eq:scale_Wilsonian}
\Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{ b^{-d_{k_j}} k_{a,j} \})
= \frac{Z_{\Lambda/b;\bar{g}(\Lambda/b)}^{N/2}}{Z_{\Lambda;\bar{g}(\Lambda)}^{N/2}} b^{-d_{\Gamma^{(N)}}} \Gamma_{\Lambda;\bar{g}(\Lambda/b)}^{(N)}(\{ k_{a,j} \}).\end{aligned}$$ Importantly, this equation compares the $N$-point function $\Gamma^{(N)}$ with the same cutoff $\Lambda$ but at different momenta and parameters. The interpretation of the scale-dependent parameters can be found from this equation: The parameters $\bar{g}_\nu(\Lambda/b)$ describe the physics at scale $k_j/b^{d_{k_j}}$.

This procedure actually illustrates rescaling in the Wilsonian RG scheme. We rewrite Eq.  as $$\begin{aligned}
\Gamma_{\Lambda;\bar{g}(\Lambda/b)}^{(N)}(\{ k_{a,j} \})
= \frac{Z_{\Lambda;\bar{g}(\Lambda)}^{N/2}}{Z_{\Lambda/b;\bar{g}(\Lambda/b)}^{N/2}} b^{d_{\Gamma^{(N)}}} \Gamma_{\Lambda;\bar{g}(\Lambda)}^{(N)}(\{ b^{-d_{k_j}} k_{a,j} \}),\end{aligned}$$ which can be interpreted in the following way. We integrate out fluctuations between the range of $(\Lambda/b,\Lambda]$ (corresponding to $Z_{\Lambda;\bar{g}(\Lambda)}^{N/2}/Z_{\Lambda/b;\bar{g}(\Lambda/b)}^{N/2}$) and rescale the field and parameters (multiplying the factor $b^{d_{\Gamma^{(N)}}}$) to obtain the new action with a different coupling constant but with the same cutoff $\Lambda$.

We can also write down the Callan-Symanzik equation to describe the momentum dependence, instead of the cutoff $\Lambda$. We differentiate Eq.  with respect to $b$ and then set $b=1$ to obtain $$\begin{aligned}
\left[ d_{k_j} k_{a,j} \frac{\partial}{\partial k_{a,j}} + \beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} - d_{\Gamma^{(N)}} + \frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}}^{(N)}(\{k_{a,j}\}) = 0.\end{aligned}$$ Equivalently, we can introduce a factor $b$ to scale all momenta $b^{d_{k_j}} k_{a,j}$ at once, so the Callan–Symanzik equation becomes $$\begin{aligned}
\left[ b\frac{\partial}{\partial b} + \beta_\nu(\bar{g})\frac{\partial}{\partial\bar{g}_\nu} - d_{\Gamma^{(N)}} + \frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}}^{(N)}(\{ b^{d_{k_j}} k_{a,j} \}) = 0.\end{aligned}$$

The Callan–Symanzik equation can be solved by the method of characteristics. With a parameter $l$, we obtain the differential equations

$$\begin{gathered}
\frac{db}{dl} = b, \\
\frac{d\bar{g}_\nu}{dl} = \beta_\nu(\bar{g}), \\
\frac{d\Gamma_{\Lambda;\bar{g}}^{(N)}(\{ b^{d_{k_j}}k_{a,j} \})}{dl} = \left[ d_{\Gamma^{(N)}} -\frac{N}{2}\gamma(\bar{g}) \right] \Gamma_{\Lambda;\bar{g}}^{(N)}(\{ b^{d_{k_j}}k_{a,j} \}).\end{gathered}$$

The solution to the first equation is straightforward:

$$b(l) = e^l,$$

with the initial condition $b(0)=1$. We write the solution to the second formally as $$\bar{g}_\nu(l) = \int_0^l dl' \beta_\nu(\bar{g}(l')).$$ Also, the formal solution to the third equation is $$\begin{aligned}
&\quad \Gamma_{\Lambda;\bar{g}(l)}^{(N)}(\{ e^{d_{k_j} l} k_{a,j} \}) \nonumber\\
&= e^{d_{\Gamma^{(N)}}l} \Gamma_{\Lambda;\bar{g}(0)}^{(N)}(\{ k_{a,j} \}) \exp \left[ -\frac{N}{2}\int_{0}^{l} dl' \gamma(\bar{g}(l')) \right].\end{aligned}$$

Changing $k_{a,j}$ to $e^{-d_{k_j} l} k_{a,j}$, we can express the $N$-point function as $$\begin{aligned}
&\quad \Gamma_{\Lambda;\bar{g}(0)}^{(N)}(\{ e^{-d_{k_j} l} k_{a,j} \}) \nonumber\\
&= e^{-d_{\Gamma^{(N)}}l} \Gamma_{\Lambda;\bar{g}(l)}^{(N)}(\{ k_{a,j} \}) \exp \left[ \frac{N}{2}\int_{0}^{l} dl' \gamma(\bar{g}(l')) \right].\end{aligned}$$ This equation describes the parameters $\bar{g}_\nu$ effectively behaves as if they are $\bar{g}_\nu(l)$ with small momenta $e^{-d_{k_j} l} k_{a,j}$ $(l>0)$.

Before concluding the section, we note that $\gamma$ corresponds to the anomalous dimension. Let us consider the two-point function $\Gamma^{(2)}$, which corresponds to the inverse of the two-point correlation function. At a fixed point, the beta function vanishes, and hence both $\bar{g}$ and $\gamma$ are constant; we set $\bar{g}^*$ and $\eta$, respectively. Here we consider the frequency dependence, so we choose $k_j = \omega_0$. Since $d_\omega=1$, we obtain $$\begin{aligned}
\Gamma_{\Lambda;\bar{g}^*}^{(2)}(e^{-l} \omega_0) = e^{-d_{\Gamma^{(2)}} l} \Gamma_{\Lambda;\bar{g}^*}^{(2)}(\omega_0) e^{\eta l}.\end{aligned}$$ As $l$ is arbitrary, we set $l=\ln(\omega_0/\omega)$ to find $$\begin{aligned}
\Gamma_{\Lambda;\bar{g}^*}^{(2)}(\omega) = \omega^{d_{\Gamma^{(2)}}-\eta} \omega_0^\eta \Gamma_{\Lambda;\bar{g}^*}^{(2)}(\omega_0)
\propto \omega^{d_{\Gamma^{(2)}}-\eta}.\end{aligned}$$ A naive power counting predicts $\Gamma^{(2)}\propto \omega^{d_{\Gamma^{(2)}}}$, but actually it behaves differently with the exponent $d_{\Gamma^{(2)}}-\eta$. The deviation $\eta$ corresponds to the anomalous dimension.

Two-loop self-energy for the quasiparticle lifetime {#sec:two-loop_lifetime}
===================================================

The quasiparticle damping is captured by a finite imaginary part of the self-energy $\Sigma$. In a series of perturbative expansions, the lowest order correction appears at second order, which is diagrammatically shown in Fig. \[fig:loop\](d). Its contribution is given in Eq.  and here we calculate it explicitly.

We first calculate the Matsubara summations. With the standard procedure, we can convert the summations into the contour integrals on the complex plane, to obtain $$\begin{aligned}
&\quad \Sigma^{(2)}(\bm{k},\omega_n) \nonumber\\
&= -\frac{1}{(2\pi)^2} \int_{\bm{p}\bm{q}\bm{l}} (2\pi)^d \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \int_{-\infty}^{\infty} d\omega_p d\omega_l \nonumber\\
&\quad \times \bigg\{ G_0(\bm{p},\omega_n-i\omega_p) [ \operatorname{Im}G_0^R(\bm{q},\omega_l) \operatorname{Im}G_0^R(\bm{l},\omega_p+\omega_l) -\operatorname{Im}G_0^R(\bm{q},\omega_l-\omega_p) \operatorname{Im}G_0^R(\bm{l},\omega_l) ]  \coth\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_l}{2T}\right) \nonumber\\
&\qquad + \operatorname{Im}G_0^R(\bm{p},\omega_p) [ G_0(\bm{q},\omega_n+i\omega_p-i\omega_l) \operatorname{Im}G_0^R(\bm{l},\omega_l) + \operatorname{Im}G_0^R(\bm{q},\omega_l) \cdot G_0(\bm{l},-\omega_n-i\omega_p-i\omega_l) ] \tanh\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_l}{2T}\right) \bigg\},\end{aligned}$$ We define the noninteracting retarded (advanced) Green’s function $G_0^R$ $(G_0^A)$ as $$G_0^{R/A}(\bm{k},\omega) = \frac{1}{\omega - E_{\bm{k}} \pm i\delta}.$$

The retarded function is obtained by the analytic continuation $i\omega_n = \omega + i\delta$. We insert $\int d\omega_q \delta(\omega_p+\omega_q-\omega_l-\omega)$ to write the self-energy $\Sigma^{(2)R}$ in a symmetric form: $$\begin{aligned}
&\quad \Sigma^{(2)R}(\bm{k},\omega) \nonumber\\
&= -\frac{1}{(2\pi)^2} \int_{\bm{p}\bm{q}\bm{l}} (2\pi)^d \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \int_{-\infty}^{\infty} d\omega_p d\omega_q d\omega_l \delta(\omega_p+\omega_q-\omega_l-\omega) \nonumber\\
&\quad\times \bigg\{ G_0^R(\bm{p},\omega_p) \operatorname{Im}G_0^R(\bm{q},\omega_q) \operatorname{Im}G_0^R(\bm{l},\omega_l)
\coth\left(\frac{\omega_l-\omega_q}{2T}\right) \left[ \tanh\left(\frac{\omega_q}{2T}\right) - \tanh\left(\frac{\omega_l}{2T}\right) \right] \nonumber\\
&\qquad + \operatorname{Im}G_0^R(\bm{p},\omega_p) \cdot G_0^R(\bm{q},\omega_q) \operatorname{Im}G_0^R(\bm{l},\omega_l) \tanh\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_l}{2T}\right) \nonumber\\
&\qquad + \operatorname{Im}G_0^R(\bm{p},\omega_p) \operatorname{Im}G_0^R(\bm{q},\omega_q) \cdot G_0^A(\bm{l},\omega_l) \tanh\left(\frac{\omega_p}{2T}\right) \tanh\left(\frac{\omega_q}{2T}\right) \bigg\}.\end{aligned}$$ Now we take the imaginary part to obtain $$\begin{aligned}
&\quad \operatorname{Im}\Sigma^{(2)R}(\bm{k},\omega) \nonumber\\
&= \frac{1}{(2\pi)^2} \cosh\left(\frac{\omega}{2T}\right) \int_{\bm{p}\bm{q}\bm{l}} (2\pi)^d \delta(\bm{p}+\bm{q}-\bm{l}-\bm{k}) \int_{-\infty}^{\infty} d\omega_p d\omega_q d\omega_l \delta(\omega_p+\omega_q-\omega_l-\omega) \nonumber\\
&\quad\times \operatorname{Im}G_0^R(\bm{p},\omega_p) \operatorname{Im}G_0^R(\bm{q},\omega_q) \operatorname{Im}G_0^R(\bm{l},\omega_l)
\frac{1}{\cosh\left(\dfrac{\omega_p}{2T}\right) \cosh\left(\dfrac{\omega_q}{2T}\right) \cosh\left(\dfrac{\omega_l}{2T}\right)} \nonumber\\
&= -\frac{\pi}{4} \cosh\left(\frac{\omega}{2T}\right) \int_{\bm{p}\bm{q}} \frac{\delta(\omega-E_{\bm{p}}-E_{\bm{q}}+E_{\bm{p}+\bm{q}-\bm{k}})}{\cosh\left(\dfrac{E_{\bm{p}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{q}}}{2T}\right) \cosh\left(\dfrac{E_{\bm{p}+\bm{q}-\bm{k}}}{2T}\right)} ,\end{aligned}$$ where we use the relation $\operatorname{Im}G_0^R(\bm{k},\omega) = -\operatorname{Im}G_0^A(\bm{k},\omega) = -\pi \delta(\omega-E_{\bm{k}})$.

Susceptibilities at a high-order VHS {#sec:susceptibilities}
====================================

A divergent DOS $D(E)$ accompanies divergent susceptibilities. Here we consider the noninteracting susceptibilities in the particle-hole and particle-particle channels, $\chi_\text{ph}$ and $\chi_\text{pp}$, respectively: $$\begin{gathered}
\chi_\text{ph} (\bm{q},\omega;T) = \int_{\bm{p}} \frac{f(\xi_{\bm{p}+\bm{q}})-f(\xi_{\bm{p}})}{\omega+i\delta-\xi_{\bm{p}+\bm{q}}+\xi_{\bm{p}}}, \\
\chi_\text{pp} (\bm{q},\omega;T) = \int_{\bm{p}} \frac{f(\xi_{\bm{p}+\bm{q}})-f(-\xi_{-\bm{p}})}{\omega+i\delta-\xi_{\bm{p}+\bm{q}}-\xi_{\bm{p}}},\end{gathered}$$ where $f(\xi)=(e^{\xi/T}+1)^{-1}$ is the Fermi–Dirac distribution and $\xi_{\bm{p}} = E_{\bm{p}}-\mu$. In the following, we focus on the static susceptibilities $(\omega=0)$. At $\mu = 0$, the noninteracting susceptibilities follow the scaling relations for momentum $\bm{q}$ and temperature $T$, described by $$\begin{gathered}
\chi_\text{ph}(\bm{q},\omega=0;T) = \kappa^{-\epsilon} \hat{\chi}_\text{ph} \left( \frac{q_+^{n_+}}{\kappa}, \frac{q_-^{n_-}}{\kappa}, \frac{T}{\kappa} \right), \\
\chi_\text{pp}(\bm{q},\omega=0;T) = \kappa^{-\epsilon} \hat{\chi}_\text{pp} \left( \frac{q_+^{n_+}}{\kappa}, \frac{q_-^{n_-}}{\kappa}, \frac{T}{\kappa} \right),\end{gathered}$$ where $\hat{\chi}_\text{ph}$ and $\hat{\chi}_\text{pp}$ are the scaling functions. Those scaling behaviors of the noninteracting susceptibilities lead to the approximate relations $$\begin{gathered}
\chi_\text{ph}(\bm{q},\omega=0;T) \sim \max(T^{-\epsilon}, q_+^{-\epsilon n_+}, q_-^{-\epsilon n_-}), \\
\chi_\text{pp}(\bm{q},\omega=0;T) \sim \max(T^{-\epsilon}, q_+^{-\epsilon n_+}, q_-^{-\epsilon n_-}).\end{gathered}$$

Although the explicit forms depend on the specific form of the energy dispersion, the temperature dependence reflects only the DOS Eq. : $$\begin{gathered}
\label{eq:diff_ph}
\chi_\text{ph}(T) = T^{-\epsilon} (D_++D_-) (2^{1+\epsilon}-1) \Gamma(1-\epsilon) [-\zeta(-\epsilon)],  \\
\label{eq:diff_pp}
\chi_\text{pp}(T) = \frac{1}{\epsilon} T^{-\epsilon} (D_++D_-) (2^{1+\epsilon}-1) \Gamma(1-\epsilon) [-\zeta(-\epsilon)],\end{gathered}$$ where $\zeta(s)$ is the Riemann zeta function. We see that $\chi_\text{pp}$ is larger than $\chi_\text{ph}$ by the numerical factor $1/\epsilon$.

For the case of a high-order VHS with a power-law divergent DOS, differences between $\chi_\text{ph}$ and $\chi_\text{pp}$ appear as the prefactors of the terms $T^{-\epsilon}$, $q_+^{-\epsilon n_+}$, $q_-^{-\epsilon n_-}$ as we have shown in Eqs.  and . This is in contrast to the conventional VHS with a logarithmically divergent DOS; while $\chi_\text{ph}$ has a logarithmic divergence reflecting the DOS, $\chi_\text{pp}$ exhibits a double logarithmic divergence. The additional logarithm in the BCS channel appears in the presence of time-reversal symmetry, which can be regarded as the Fermi surface nesting with itself.

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