---
abstract: |
    We calculate the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model within second order perturbation theory. Close to half-filling, interactions enhance anisotropies of the Fermi surface, but they never modify the topology of the Fermi surface in the weak coupling regime.\
author:
- |
    Christoph J. Halboth and Walter Metzner\
    [*Sektion Physik, Universität München*]{}\
    [*Theresienstra[ß]{}e 37, D-80333 München, Germany*]{}
title: |
    Fermi Surface of the 2D Hubbard Model\
    at Weak Coupling
---

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 {#section .unnumbered}

Since the discovery of high-temperature superconductivity, the structure of low-lying single-particle excitations in two-dimensional interacting Fermi systems has attracted much interest [@WYD; @DAG]. A key role is thereby played by the shape of the Fermi surface which determines the phase space for residual scattering processes and thus decay rates and Fermi liquid instabilities [@PKC]. In a recent Monte Carlo simulation of the two-dimensional Hubbard model Bulut, Scalapino and White [@BSW] have found that strong interactions may not only deform the Fermi surface of the non-interacting reference system, but may even lead to a different topology. While the Fermi surface of the non-interacting Hubbard model (with nearest-neighbor hopping) is always closed around the origin in $\bk$-space for densities $n<1$, the strongly interacting system exhibited a Fermi surface closed around $(\pi,\pi)$ for densities close to but below half-filling. This result raises the interesting question whether such a behavior occurs only in a strong coupling regime. Zlatić, Schotte and Schliecker [@ZSS] have recently argued that the Fermi surface topology can be changed by arbitrarily weak interactions in the limit $n \to 1$, since in that limit arbitrarily small deformations could lead to a topologically different shape.

In the following we will analyze the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model within second order perturbation theory. We show that interactions enhance the anisotropy of the Fermi surface for densities close to half-filling, but they do not change the topology by deforming a $(0,0)$-centered surface into a $(\pi,\pi)$-centered one in the weak coupling regime.

The deformation of the Fermi surface in two-dimensional Hubbard models has been studied already earlier by Schönhammer and Gunnarsson [@SG] and by Ossadnik [@OSS]. The aim of the former authors was to show that Kohn-Sham Fermi surfaces are in general not exact. The behavior of deformations in the limit $n \to 1$, however, has not been addressed in these earlier works.

 {#section-1 .unnumbered}

The Hubbard model Hamiltonian is given by $$\label{eq1}
 H = -t \sum_{\sg} \sum_{\bra j,j' \ket} 
   c^{\dag}_{j'\sg} \, c_{j\sg} \> + \>
   U \sum_j n_{j\up} \, n_{j\down}$$ where $t$ is the nearest-neighbor hopping amplitude and $U$ the on-site Coulomb repulsion. The operators $c^{\dag}_{j\sg}$ ($c_{j\sg}$) create (annihilate) fermions with spin projection $\sg$ on lattice site $j$ and $n_{j\sg} = c^{\dag}_{j\sg} c_{j\sg}$.

The non-interacting band-structure for nearest-neighbor hopping on a square lattice is $$\label{eq2}
 \eps^0_{\bk} = -2t \, (\cos k_x + \cos k_y)$$ This leads to a strictly convex Fermi surface centered around $\bk = (0,0)$ for densities $n<1$, to a diamond shaped surface at half-filling ($n\!=\!1$), and a $(\pi,\pi)$-centered Fermi surface for densities $n>1$ (see Fig. \[fig1\]).

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The Fermi surface of the interacting system can be obtained from the one-particle Green function $$\label{eq3}
 G(\bk,\om) = 
   {1 \over \om - (\eps^0_{\bk} - \mu) - \Sg(\bk,\om)}$$ where $\Sg(\bk,\om)$ is the self-energy. The chemical potential $\mu$ controls the average particle density $n$. The equation $$\label{eq4}
 \Re \, G^{-1}(\bk,\om) = \xik - (\eps^0_{\bk} - \mu) -
   \Re \, \Sg(\bk,\xik) = 0$$ determines the energy $\xi_{\bk} \equiv \eps_{\bk} - \mu$ of coherent single-particle excitations (quasi-particles) in the interacting system [@AGD]. The Fermi surface is the set of those points in $\bk$-space where the excitation energy vanishes, i.e. $\xi_{\bk_F} = 0$, and thus determined by $$\label{eq5}
 \eps^0_{\bk_F} - \mu \> + \> \Re\,\Sg(\bk_F,0) = 0$$ According to the Luttinger theorem [@LUT], the volume enclosed by the Fermi surface is related to the particle density by $$\label{eq6}
 n = 2 \int {d^2 k \over (2\pi)^d} \> 
   \Theta(\mu-\eps_{\bk})$$

We now calculate the interaction induced deformation of the Fermi surface to second order in the coupling strength $U$. The particle density is kept fixed by a suitable choice of an interaction-dependent chemical potential $\mu = \mu(n,U) = \mu_0 + 
\delta\mu$, where $\mu_0 = \mu_0(n)$ is the chemical potential corresponding to density $n$ at $U=0$. We denote the Fermi wave vectors of the non-interacting system by $\bk_{F0}$ and the deformation vectors by $\delta\bk_F = \bk_F - \bk_{F0}$.

To first order in $U$, the self-energy is a real constant: $\Sg_1(\bk,\om) = U n/2$. To keep the density fixed, the chemical potential has to be shifted accordingly by $\delta\mu_1 = U n/2$, which cancels $\Sg_1$ completely in $G$.

To second order in $U$, two Feynman diagrams contribute to the self-energy of the Hubbard model (see Fig. \[fig2\]).

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The second one is a real constant which is completely cancelled by a corresponding shift of the chemical potential.[^1] The first diagram, however, leads to a $\bk$-dependent contribution and generates a Fermi surface deformation. Expanding Eqs. (\[eq5\]) and (\[eq6\]) with $\bk_F = \bk_{F0} + 
\delta\bk_F$ and $\mu = \mu_0 + \delta\mu$ in powers of $U$ and comparing the second order terms, one obtains the relation $$\label{eq7}
 \nabla\eps^0_{\bk_{F0}} \cdot \delta\bk_{F2} = 
   \delta\mu_2 - \Sg_2(\bk_{F0},0)$$ for the deformation of the Fermi surface to second order in $U$, where the chemical potential shift is given by the Fermi surface average $$\label{eq8}
 \delta\mu_2 = {\int d^2k \: \delta(\eps^0_{\bk} \!-\! \mu) \,
   \Sg_2(\bk,0) \over
   \int d^2k \: \delta(\eps^0_{\bk} \!-\! \mu)}$$ Note that $\Sg_2$ is real at zero frequency. Of course there are many ways to define a map $\bk_{F0} \mapsto \bk_F$ between non-interacting and interacting Fermi surfaces. A natural choice is $\delta\bk_F =
\delta k_F \, \bn_{\bk_{F0}}$, where $\bn_{\bk_{F0}}$ is a unit vector normal to the non-interacting Fermi surface in $\bk_{F0}$. Eq. (\[eq7\]) then determines the modulus of the shift as $$\label{eq9}
  \delta k_{F2} = \delta k_{F2}(\bk_{F0}) = 
   {\delta\mu_2 - \Sg_2(\bk_{F0},0) \over v^0_{\bk_{F0}}}$$ where $v^0_{\bk_{F0}} = |\nabla\eps^0_{\bk_{F0}}|$ is the Fermi velocity of the non-interacting system. (See also the equivalent expression (4) in [@SG]) The second order self-energy has not yet been evaluated by purely analytical means. Using the spectral representation of the non-interacting propagator $G_0$, one can write the imaginary part of the self-energy contribution associated with the first Feynman diagram in Fig. \[fig2\] as $$\begin{aligned}
\label{eq10}
 \Im\Sg_{2a}(\bk,\om)& =&  - {\sgn(\om) \over \pi} \, U^2 \,
   \int {d^2 q \over (2\pi)^2} \, \int_0^{\om} d\nu \,\nonumber \\
 & &  \Im\Pi_0(\bq,\nu) \, \Im G_0(\bk\!-\!\bq,\om\!-\!\nu)\end{aligned}$$ where the imaginary part of the non-interacting polarisation bubble is given by $$\begin{aligned}
\label{eq11}
\Im\Pi_0(\bq,\nu) &=& {\sgn(\nu) \over \pi} \,
   \int {d^2 p \over (2\pi)^2} \, \int_{-\nu}^0 d\om' \,\qquad\qquad
 \nonumber \\
 & &\Im G_0(\bp,\om') \, \Im G_0(\bp+\bq,\om'+\nu)          \end{aligned}$$ and $\Im G_0(\bk,\om) = 
-\pi \, \sgn(\om) \, \delta\big[\om - (\eps_{\bk}^0 - \mu_0) \big]$. Performing the integrals over $\om'$, $p_x$ and $q_x$ in (\[eq10\]) and (\[eq11\]) analytically (eliminating thus three $\delta$-functions), one obtains the expression $$\begin{aligned}
\label{eq12}
\Im\Sg_{2a}(\bk,\om)= - \sgn(\om) \, {U^2 \over 64\pi^3 t^2} \, 
   \int\! dq_y \int\! dp_y \int_0^{\om}\! d\nu&&\nonumber \\ 
\qquad \qquad \sum_{q_x^0,p_x^0} 
   \left[ {\Theta(\mu_0\!-\!\eps_{\bp}^0) - 
    \Theta(\mu_0\!-\!\eps_{\bp+\bp}^0)
   \over \sqrt{(1 \!-\! f^2)(4\sin^2(q_x/2) - g^2)}} 
   \right]_{p_x = p_x^0 \atop q_x = q_x^0}&&\end{aligned}$$ with the functions $$\label{eq13}
 \left. \begin{array}{rcl}
   f & = & (\nu - \om - \mu_0)/2t - \cos(k_y-q_y) \\
   g & = & \nu/2t + \cos(p_y+q_y) - \cos(p_y)
   \end{array} \right.$$ The summation variables $q_x^0$ and $p_x^0$ in (\[eq12\]) are the roots of the set of equations $$\label{eq14}
 \left. \begin{array}{rcl}
   \cos(k_x \!-\! q_x)                & = & f \\ 
   2\sin(p_x \!+\! q_x/2) \sin(q_x/2) & = & g
   \end{array} \right.$$ The remaining three-fold integral is easily computed numerically (e.g. via a standard Monte-Carlo routine on a work-station). Note that the representation in (\[eq12\]) is particularly suitable for a high resolution of the low-energy limit $\om \to 0$, since the integration region shrinks with $\om$. The full self-energy function can be reconstructed from its imaginary part $\Im\Sg_{2a}$ by a simple Hilbert transform $$\label{eq15}
 \Sg_{2a}(\bk,\om) = - \,\pi^{-1} \int_{-\infty}^{\infty} d\om'
   {|\Im\Sg_{2a}(\bk,\om')| \over 
    \om - \om' + i0^+ \sgn(\om)}$$ The constant contribution $\Sg_{2b}$ from the second diagram in Fig. \[fig2\] need not be calculated since it is completely cancelled by a corresponding shift $\delta\mu_{2b}$ of the chemical potential. Our numerical results for the second order self-energy agree with those published recently by Zlatić et al. [@ZSS], who computed $\Sg_{2a}$ via a sequence of fast Fourier transforms. They also agree with earlier results by Schweitzer and Czycholl [@SC] and by Ossadnik [@OSS].

 {#section-2 .unnumbered}

To discuss explicit results for the Fermi surface deformation we introduce polar coordinates in $\bk$-space. Points on the Fermi surface of the non-interacting system are thus specified by an angle $\phi$. Due to the discrete symmetries of the square lattice it is sufficient to consider angles between $0$ and $45$ degrees. In Fig. \[fig3\] we show results for $\delta k_{F2}/U^2$ as a function of $\phi$ for various densities $n$.

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Here and in the following we set the hopping amplitude $t=1$. At low densities $n < 0.6$ weak interactions tend to compensate anisotropies of the non-interacting Fermi surface while at densities $0.7 < n < 1$ anisotropies are further enhanced by interactions, as observed already earlier in unpublished work by Ossadnik [@OSS]. The diamond shaped Fermi surface at half-filling is of course not affected at all by interactions, as a consequence of the particle-hole symmetry of the Hubbard model (\[eq1\]). More generally, particle-hole symmetry maps the Fermi surface at density $n$ onto the surface for density $2-n$ by a simple $(\pi,\pi)$-shift in $\bk$-space (the diamond at half-filling is thereby mapped onto itself).

Let us now clarify whether weak interactions can modify the Fermi surface topology at densities close to half-filling. The first Fermi point that may reach the Brillouin zone boundary (and thus introduce a different topology) upon increasing $U$ is obviously the one at $\phi = 0$, since it is closer to the zone boundary than any other Fermi point already in the non-interacting system and in addition $\delta k_F(\phi)$ is maximal for $\phi = 0$ (if $n > 0.7$). Quantitative information on the Fermi surface deformation in the “critical” regime $n \to 1$ and $\phi \to 0$ is provided in Fig.  \[fig4\], where we have plotted $\delta k_{F2}/U^2$ as a function of density for various small angles $\phi$.

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Within numerical accuracy, $\delta k_{F2}(\phi)$ tends to $0$ for $n \to 1$ for all $\phi$, as expected from particle-hole symmetry and continuity. Finally, in Fig. \[fig5\] we show the critical coupling strength $U_c(n)$ that is required to make $\bk_F = \bk_{F0} + \delta\bk_{F2}$ reach the Brillouin zone boundary (at the point $(\pi,0)$). Close to half-filling, $U_c(n)$ behaves linearly as a function of density and extrapolates to a rather big [*finite*]{} value in the limit $n \to 1$. Hence, interactions do not modify the Fermi surface topology of the two-dimensional Hubbard model within the perturbatively controlled weak coupling regime at any density.

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Zlatić et al. have calculated the critical coupling strength from second order perturbation theory for $n = 0.97$ (only), where they obtained a smaller $U_c$ (by a factor of about $2$) than we did, although our results for the self-energy agree. The discrepancy arises because these authors have determined the new Fermi surface by directly solving Eq. (\[eq5\]) with the second order self-energy, while we have expanded $\delta k_F$ to second order in $U$, which is the order we really control. In the small $U$ limit both procedures yield the same shift to order $U^2$, but quantitative differences arise for finite $U$. The qualitative result that $U_c(n)$ remains finite in the limit $n \to 1$ is thereby not affected.

In summary, we have calculated the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model within second order perturbation theory. Close to half-filling, interactions enhance anisotropies of the Fermi surface, but they never modify the topology of the Fermi surface in the weak coupling regime.

[*Acknowledgement*]{} We are very grateful to Dieter for numerous valuable discussions.

[99]{} See, for example, Proceedings of the International Conference on [*Materials and Mechanisms of Superconductivity: High Temperature Superconductors IV*]{}, ed. P. Wyder (North Holland 1994). For a review of numerical results, see E. Dagotto, Rev. Mod. Phys. [**66**]{}, 763 (1994). For a review on the Fermi surface structure and other properties of high-$T_c$ superconductors, see W.E. Pickett, H. Krakauer, R.E. Cohen and D.J. Singh, Science [**255**]{}, 46 (1992). N. Bulut, D.J. Scalapino and S.R. White, Phys. Rev. B [**50**]{}, 7215 (1994). V. Zlatić, K.D. Schotte and G. Schliecker, Phys. Rev. B [**52**]{}, 3639 (1995). K. Schönhammer and O. Gunnarsson, Phys. Rev. B [**37**]{}, 3128 (1988). P. Ossadnik, Diploma Thesis (Universität Köln 1990). A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, [*Methods of Quantum Field Theory in Statistical Physics*]{} (Dover, New York 1975). J.M. Luttinger, Phys. Rev. [**119**]{}, 1153 (1960). W. Kohn and J.M. Luttinger, Phys. Rev. [**118**]{}, 41 (1960). H. Schweitzer and G. Czycholl, Z. Phys. B [**83**]{}, 93 (1991).

[^1]: We note that this diagram is “anomalous” in the sense of Kohn and Luttinger [@KL], i.e. it vanishes if the thermodynamic limit is taken strictly at $T=0$, while it yields a finite contribution if the zero temperature limit is taken after the thermodynamic limit.
