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abstract: 'We propose and investigate an exactly solvable model of non-equilibrium Luttinger liquid on a star graph, modeling a multi-terminal quantum wire junction. The boundary condition at the junction is fixed by an orthogonal matrix $\S$, which describes the splitting of the electric current among the leads. The system is driven away from equilibrium by connecting the leads to heat baths at different temperatures and chemical potentials. The associated non-equilibrium steady state depends on $\S$ and is explicitly constructed. In this context we develop a non-equilibrium bosonization procedure and compute some basic correlation functions. Luttinger liquids with general anyon statistics are considered. The relative momentum distribution away from equilibrium turns out to be the convolution of equilibrium anyon distributions at different temperatures. Both the charge and heat transport are studied. The exact current-current correlation function is derived and the zero-frequency noise power is determined.'
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Ł[L]{} §[S]{}

1[\_1]{}

Ø[O]{} [**v**]{}

[**Luttinger Liquid in Non-equilibrium\
Steady State**]{}\

[Mihail Mintchev$^1$ and Paul Sorba$^2$]{}\
1.5 truecm

*${}^1$ Istituto Nazionale di Fisica Nucleare and Dipartimento di Fisica dell’Università di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy\
*

${}^2$ Laboratoire de Physique Théorique d’Annecy-le-Vieux, CNRS,\
9, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France

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

The universal features of a large class of one-dimensional quantum models, exhibiting gapless excitations with linear spectrum, are successfully described [@Hald; @Haldprl] by the Tomonaga-Luttinger (TL) liquid theory [@T50]-[@ML65]. This theory[^1] applies to various systems, including nanowire junctions and carbon nanotubes, which are available nowadays in experiment [@na1]-[@na3]. For this reason the study of non-equilibrium phenomena in the TL liquid phase attracts recently much attention [@GGM08]-[@Ines].

A typical non-equilibrium setup, considered in the literature, is the junction of two or more semi-infinite leads with electrons at different temperatures and/or chemical potentials. The junction is an interval of finite length $L$, where the electrons injected from the leads interact among themselves. This interaction drives the system away from equilibrium. Differently from the equilibrium TL liquid on the line, the non-equilibrium model defined in this way, is not exactly solvable. Nevertheless, it is extensively studied [@GGM08]-[@Ines] by various methods, including linear response theory, bosonization combined with the non-equilibrium Keldish formalism and perturbation theory.

One of the main goals of the present paper is to explore the possibility to construct and analyze an alternative [*exactly solvable*]{} model for a non-equilibrium TL junction. Since the universal features of such a system are expected to manifest themselves in the critical (scale invariant) limit, it is natural to shrink the domain of the non-equilibrium interaction to a point, taking $L\to 0$. For a complete description of the critical regime it is essential to take into account all point-like interactions, which ensure a unitary time evolution of the system. These interactions can be parametrized by a scattering matrix $\S$ localized in the junction point, as shown in the multi-terminal setup displayed in Fig.\[fig1\].

(600,120)(40,250) ![A junction with scattering matrix $\S$ and $n$ semi-infinite leads, connected at infinity to thermal reservoirs with temperatures $\beta_i$ and chemical potentials $\mu_i$.[]{data-label="fig1"}](fig1.pdf "fig:")

Each lead contains a TL liquid, which at infinity is in contact with a heat reservoir with (inverse) temperature $\beta_i$ and chemical potential $\mu_i$. Our first step below is to show that there exists a non-equilibrium steady state (NESS), which describes the TL configuration in Fig.\[fig1\]. This state is characterized by non-trivial time independent electric and heat currents, flowing in the leads. The scattering matrix $\S$ is implemented by imposing specific boundary conditions at the junction. It turns out that the boundary conditions, which describe the splitting of the electric steady current in the junction, lead to an exactly solvable problem. In fact, we establish the operator solution in this case and investigate the relative non-equilibrium correlation functions in the NESS representation.

The TL theory has been introduced originally [@T50]-[@ML65] for describing fermion systems. It has been understood later on [@Liguori:1999tw]-[@Calabrese:2007ty] that the fermion TL liquid is actually an element of a more general family of [*anyon*]{} TL liquids[^2], which obey Abelian braid statistics. In this paper we explore the general anyon TL liquid, obtaining the conventional fermionic and bosonic ones as a special cases.

From the two-point anyon correlation functions we extract the NESS distribution of the TL anyon excitations. In momentum space this non-equilibrium distribution is a [*nested convolution*]{} of equilibrium distributions at different temperatures and chemical potentials. As expected, the convolution depends on the scattering matrix $\S$, which drives the system away from equilibrium. We investigate also the NESS correlators of the electric and energy currents, describing in detail the charge and heat transport in the junction. The zero-frequency noise power is deduced from the two-point current-current correlation function, whose exact expression in terms of hypergeometric functions is established. We prove the breakdown of time reversal invariance as well.

The paper has the following structure. In the next section we construct non-equilibrium chiral fields in a NESS on a star graph modeling the junction. We derive here the non-equilibrium Casimir energy and the heat current and compare the latter with the conformal field theory result. In section 3 we develop a non-equilibrium finite temperature operator bosonization procedure. We also establish the operator solution, subject to the current splitting boundary condition at the junction. We show that this condition covers two different physical situations, corresponding to a junction with and without charge dissipation. The non-equilibrium correlation functions are investigated in section 4, where the anyon NESS distributions are derived. The charge and heat transport as well as the noise are also studied there. Section 5 provides a concise outlook of the paper and contains some general observations. The appendix collects some results about the asymptotic properties of the anyon NESS correlators.

Non-equilibrium chiral fields on a star graph
=============================================

The fundamental building blocks of bosonization away from equilibrium are the free massless scalar field ${\varphi}$ and its dual ${\widetilde{\varphi}}$. The fields ${\varphi}$ and ${\widetilde{\varphi}}$ propagate on a star graph $\Gamma$, which is shown in Fig. \[fig2\] and models the quantum wire junction.

(500,70)(-150,20) ![A star graph $\Gamma$ with $n$ edges modelling the junction of $n$ quantum wires.[]{data-label="fig2"}](fig2.pdf "fig:")

The edges $E_i$ are half-lines and each point $P$ in the bulk $\Gamma \setminus V$ of $\Gamma$ is uniquely determined by its coordinates $(x,i)$, where $x > 0$ is the distance to the vertex $V$ and $i=1,...,n$ labels the edge. Besides the massless Klein-Gordon equation, the fields ${\varphi}$ and ${\widetilde{\varphi}}$ satisfy the duality relations $$\der_t {\widetilde{\varphi}}(t,x,i) = - \der_x {\varphi}(t,x,i) \, ,\qquad 
\der_x {\widetilde{\varphi}}(t,x,i) = - \der_t {\varphi}(t,x,i) \, .
\label{dual}$$ The initial conditions are fixed by the equal-time canonical commutation relations $$[{\varphi}(t,x,i)\, ,\, {\varphi}(t,y,j)]_{{}_-} =
[{\widetilde{\varphi}}(t,x,i)\, ,\, {\widetilde{\varphi}}(t,y,j)]_{{}_-} = 0 \, ,
\label{ecc1}$$ $$[(\der_t{\varphi}) (t,x,i)\, ,\, {\varphi}(t,y,j)]_{{}_-}  =
[(\der_t{\widetilde{\varphi}})(t,x,i)\, ,\, {\widetilde{\varphi}}(t,y,j)]_{{}_-}  =
-\ri \delta_{ij}\delta (x-y) \, .
\label{ecc2}$$ In order to determine the dynamics completely, one must impose some boundary conditions at the vertex $x=0$. These conditions are conveniently formulated in terms of the combinations $${\varphi_{i,R}}(t-x) = {\varphi}(t,x,i) + {\widetilde{\varphi}}(t,x,i)\, , \qquad 
{\varphi_{i,L}}(t+x) = {\varphi}(t,x,i) - {\widetilde{\varphi}}(t,x,i)\, ,
\label{chi1}$$ which depend on $t-x$ and $t+x$ respectively and define right and left chiral fields ${\varphi_{i,Z}}$ on $\Gamma$. The most general [*scale invariant*]{} boundary conditions, generating a [*unitary time evolution*]{} of ${\varphi}$ and ${\widetilde{\varphi}}$, are parametrized by the orthogonal group $O(n)$ and read [@ks-00]-[@Bellazzini:2008mn] $${\varphi_{i,R}}(\xi ) = \sum_{j=1}^n \S_{ij}\, {\varphi}_{j,L}(\xi ) \, , \qquad \S \in O(n)\, .  
\label{bc1}$$ These simple conditions capture the universal features of the system and $\S$ has a straightforward physical interpretation: the vertex $V$ of $\Gamma$ represents a [*scale invariant point-like defect*]{}, $\S$ being the associated scattering matrix.

The non-equilibrium steady state $\Omega_{\beta,\mu_{{}_b}}$
------------------------------------------------------------

Our next step is to construct a steady state $\Omega_{\beta,\mu_{{}_b}}$, which captures the evolution of the chiral fields ${\varphi_{i,Z}}$ on $\Gamma$, whose edges are attached at infinity to thermal reservoirs at inverse temperatures $\beta_i$ as shown in Fig. \[fig1\]. In the boson case we take all chemical potentials to be equal[^3], setting $\mu_i = \mu_b$ in all reservoirs. The system is away from equilibrium if $\S$ contains at least one non-trivial transmission coefficient among reservoirs with different temperature. The construction of $\Omega_{\beta,\mu_{{}_b}}$, described below, follows the scheme developed in [@Mintchev:2011mx] and is based on scattering theory. It adapts to the case under consideration some modern ideas [@els-96]-[@st-06] about NESS. The framework is purely algebraic and generalizes the definition [@BR] of equilibrium Gibbs state over the algebra of canonical commutation relations (CCR).

We start by observing that the massless Klein-Gordon equation and the relations (\[dual\]) lead to the following representation $$\begin{aligned}
&&{\varphi_{i,R}}(\xi) = \int_0^\infty  \frac{{{\rm d}}k}{\pi \sqrt{2}}\, \sqrt {\Delta_\lambda (k)} 
\left [a_i^\ast (k) \e^{\ri k \xi} + a_i(k)\e^{-\ri k \xi}\right ] \, , 
\label{fil1} \\
&& {\varphi_{i,L}}(\xi) = \int_0^\infty  \frac{{{\rm d}}k}{\pi \sqrt{2}}\, \sqrt {\Delta_\lambda (k)}
\left [a_i^\ast (-k) \e^{\ri k \xi} + a_i(-k) \e^{-\ri k \xi}\right ] \, , 
\label{fil2}\end{aligned}$$ $\Delta_\lambda $ being some distribution to be fixed below. Using that $\S$ is a real matrix, the boundary condition (\[bc1\]) implies the constraints $$a_i(k) = \sum_{j=1}^n S_{ij}(k)\, a_j (-k) \, , \qquad 
a^\ast_i (k) = \sum_{j=1}^n  S_{ij}(k)\, a^\ast_ j(-k)\, ,    
\label{constr1}$$ where $$S(k) = \theta(-k) \S^t + \theta (k) \S\, ,  
\label{S}$$ $\theta$ is the Heaviside step function and $\S^t$ indicates the transpose of $\S$. From the equal-time commutation relations (\[ecc1\], \[ecc2\]) one infers that the elements $\{a_i(k),\, a^*_i(k)\, :\, k \in {\mbox{${\mathbb R}$}}, \, i=1,...,n\}$ generate the following deformation $\A$ of the algebra of CCR: $$[a_i(k)\, ,\, a_j(p)]_{{}_-} = [a^*_i (k)\, ,\, a^*_j (p)]_{{}_-} = 0 \, , 
\label{rta1}$$ $$[a_i(k)\, ,\, a^*_j (p)]_{{}_-} = 2\pi [\delta (k-p)\delta_{ij} + S_{ij}(k)\delta(k+p)] \, . 
\label{rta2}$$ Moreover, (\[ecc1\], \[ecc2\]) imply that $$|k| \Delta_\lambda (k) = 1\, . 
\label{distr1}$$ There exist a one-parameter family of tempered distributions, which solve this equation in ${\mbox{${\mathbb R}$}}$. A convenient representation of this family is given by [@Liguori:1997vd] $$\Delta_\lambda (k) = \frac{{{\rm d}}}{{{\rm d}}k} \left [\theta (k) \ln \frac {k}{\lambda}\right ]\, ,
\label{distr2}$$ where $\lambda >0$ is a free parameter with dimension of mass, having well-known infrared origin.

The above structure is very general and equations (\[fil1\],\[fil2\]) apply to any representation of the algebra $\A$, which is a simplified version of the so called reflection-transmission (RT) algebra [@Liguori:1996xr]-[@Mintchev:2003ue], describing factorized scattering in integrable models with point-like defects in one dimension. The Fock and the Gibbs state over $\A$ describe equilibrium physics and have been largely explored. We will investigate here the NESS $\Omega_{\beta,\mu_{{}_b}}$, which describes the physical situation shown in Fig. \[fig1\]. For this purpose we first observe that the sub-algebras $\A_\inc$ and $\A_\out$, generated by the elements $\{a_i(k),\, a^*_i(k)\, :\, k<0\}$ and $\{a_i(k),\, a^*_i(k)\, :\, k>0\}$ respectively, parametrize the [*asymptotic incoming*]{} and [*outgoing*]{} fields. Accordingly, both $\A_\inc$ and $\A_\out$ are conventional CCR algebras; in fact the $\delta (k+p)$ term in (\[rta2\]) vanishes if both momenta are negative or positive. It is worth stressing that (\[constr1\]) relate $\A_\inc$ with $\A_\out$ and that the whole RT algebra $\A$ can be generated via (\[constr1\]) either by $\A_\inc$, or by $\A_\out$. The main idea for constructing $\Omega_{\beta , \mu_b}$ is based on this kind of asymptotic completeness property. Starting with an equilibrium state on $\A_\inc$, we will extend it by means of (\[constr1\]) to a non-equilibrium state on the whole algebra $\A$. For this purpose we introduce the edge Hamiltonian and number operators $$h_i = \int_{-\infty}^0 \frac{{{\rm d}}k}{2\pi} |k| a^*_i(k) a_i(k)\, , \qquad 
n_i = \int_{-\infty}^0 \frac{{{\rm d}}k}{2\pi} a^*_i(k) a_i(k)\, ,   
\label{ac}$$ which describe the [*asymptotic*]{} dynamics at $t=-\infty$ (i.e. before the interaction) in terms of $\A_\inc$. Defining $$K = \sum_{i=1}^n \beta_i (h_i -\mu_b n_i) \, , \qquad \beta_i \geq 0\, , 
\label{kop}$$ we introduce the equilibrium Gibbs state over $\A_\inc$ in the standard way [@BR]. For any polynomial ${{\cal P}}$ over $\A_\inc$ we set $$\begin{aligned}
\left (\Omega_{\beta , \mu_b}\, ,\,  {{\cal P}}\bigl (a_i^*(k_i), a_j(p_j)\bigr ) \Omega_{\beta , \mu_b} \right ) \equiv 
\langle {{\cal P}}\bigl (a_i^*(k_i), a_j(p_j))\bigr ) \rangle_{\beta, \mu_b} = 
\nonumber \\
\frac{1}{Z} \tr \left [\e^{-K} {{\cal P}}\bigl (a_i^*(k_i), a_j(p_j)\bigr )\right ]\, ,  
\qquad \qquad \qquad 
\label{def1}\end{aligned}$$ where $k_i<0,\; p_j<0$ and $Z = \tr \left ( \e^{-K}\right )$. All the expectation values (\[def1\]) can be computed [@BR] by purely algebraic manipulations and can be expressed in terms of the two-point functions, which are written in terms of the familiar Bose distribution $$b_i (k) = \frac{\e^{-\beta_i [|k| - \mu_b]}}{1- \e^{-\beta_i [|k| - \mu_b]}} 
\label{fbd1}$$ in the following way $$\langle a_j^*(p)a_i(k) \rangle_{\beta, \mu_b} = 
b_i(k) \delta_{ij} 2\pi \delta (k-p)\, ,  
\label{2a}$$ $$\langle a_i(k)a_j^*(p)\rangle_{\beta, \mu_b} = [1+b_i(k)] 
\delta_{ij} 2\pi \delta (k-p)\, .  
\label{2b}$$ We stress that (\[2a\],\[2b\]) hold on $\A_\inc$, i.e. only for [*negative*]{} momenta. The common for all reservoirs chemical potential $\mu_b<0$ allows to avoid in (\[2a\],\[2b\]) the infrared singularity at $k=0$. We anticipate that $\mu_b$ has nothing to do with the [*fermion*]{} chemical potentials, appearing in the non-equilibrium bosonization procedure described in the next section, where the limit $\mu_b \to 0^-$ exist and will be performed.

The next step is to extend (\[def1\]-\[2b\]) to the whole RT algebra $\A$, namely to [*positive*]{} momenta. Employing (\[constr1\]) one finds $$\begin{aligned}
\langle a_j^*(p)a_i(k)\rangle_{\beta, \mu_b} = 
2\pi \Bigl\{\Bigl [\theta(-k)b_i(k) \delta_{ij}+ 
\theta(k)\sum_{l=1}^n \S_{il}\, b_l(k)\, \S^t_{lj}\Bigr ] \delta (k-p)  
\nonumber \\
+ \Bigl [\theta(-k)b_i(k)\, \S^t_{ij} + \theta(k)\S_{ij}\, b_j(k) \Bigr ] \delta (k+p) \Bigr\}\, .
\qquad \; \; \,  
\label{cor1}\end{aligned}$$ The expression for $\langle a_i(k)a_j^*(p)\rangle_{\beta, \mu}$ is obtained from (\[cor1\]) by the substitution $$b_i(k) \longmapsto 1+b_i(k) =  \frac{1}{1- \e^{-\beta_i [|k| - \mu_b]}} \, . 
\label{fbd2}$$

The final step is to compute a generic correlation function. By means of the commutation relations (\[rta1\],\[rta2\]), this problem is reduced to the evaluation of correlators of the form $\langle \prod_{m=1}^M a_{i_m}(k_{i_m}) \prod_{n=1}^N a^\ast_{j_n}(p_{j_n})\rangle_{\beta,\mu_b}$, which can be computed by iteration via $$\begin{aligned}
\langle \prod_{m=1}^M a_{i_m}(k_{i_m}) \prod_{n=1}^N a^{\ast j_n}(p_{j_n})\rangle_{\beta,\mu_b} = 
\qquad \qquad \qquad \nonumber \\
\delta_{MN}\, \sum_{m=1}^M \langle a_{i_1}(k_{i_1})a^{\ast j_m}(p_{j_m})\rangle_{\beta,\mu_b}  
\, \langle \prod_{m=2}^M a_{i_m}(k_{i_m}) 
\prod_{n=1\atop {n\not=m} }^N a^{\ast j_n}(p_{j_n}) \rangle_{\beta,\mu_b} \, .   
\label{gcf2}\end{aligned}$$

We would like to mention in conclusion that the use of the RT algebra $\A$ in the construction of $\Omega_{\beta, \mu_b}$ represents only a convenient choice of coordinates, which has a simple physical interpretation in terms of scattering data and applies to a variety of systems [@Mintchev:2011mx; @Caudrelier:2012xy] with point-like defects.

Energy density and energy transport in $\Omega_{\beta,\mu_b}$
-------------------------------------------------------------

In order to illustrate the physical properties of $\Omega_{\beta,\mu_b}$, it is instructive to investigate the non-equilibrium energy density and transport associated with the scalar field ${\varphi}$. The equations of motion imply the conservation $$\der_t \theta_{tt}(t,x,i)  - \der_x \theta_{xt}(t,x,i) = 0\, , 
\label{econs}$$ of the energy-momentum tensor $$\begin{aligned}
\theta_{tt}(t,x,i) &=& \frac{1}{2}: \left [ ({\partial}_t {\varphi})({\partial}_t {\varphi}) - 
{\varphi}({\partial}_x^2 {\varphi}) \right ]:(t,x,i)\, , 
\label{emtph1}\\
\theta_{xt}(t,x,i) &=& \frac{1}{2} :\left [ ({\partial}_x {\varphi})( {\partial}_t {\varphi}) - 
{\varphi}({\partial}_x {\partial}_t {\varphi})\right ] :(t,x,i)\, , 
\label{emtph2}\end{aligned}$$ where $:\cdots :$ denotes the normal product in the algebra $\A$. The boundary condition (\[bc1\]) implies the Kirchhoff rule $$\sum_{i=1}^n \theta_{xt} (t,0,i) = 0\, , 
\label{K1}$$ which, combined with (\[econs\]), ensures energy conservation.

The derivation of $\langle \theta_{tt}(t,x,i)\rangle_{\beta,\mu_b}$ and $\langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu_b}$ is based on the expectation value $$\begin{aligned}
\langle : {\varphi}(t_1,x_1,i_1) {\varphi}(t_2,x_2,i_2): \rangle_{\beta,\mu_b} = \qquad \qquad \qquad 
\nonumber \\
\int_0^\infty \frac{{{\rm d}}k}{2\pi}\Delta_\lambda (k) \Bigl \{\delta_{i_1i_2} b_{i_1}(k) \cos[k(t_{12}+x_{12}] + 
\S_{i_1i_2} b_{i_2}(k) \cos[k(t_{12}-{\widetilde{x}}_{12}] + 
\nonumber \\
b_{i_1}(k)\, \S^t_{i_1i_2} \cos[k(t_{12}+{\widetilde{x}}_{12}] 
+\sum_{j=1}^n \S_{i_1j}\, b_j(k)\, \S^t_{ji_2}  \cos[k(t_{12}-x_{12}]\Bigr \}\, , 
\label{ncf}\end{aligned}$$ where ${\widetilde{x}}_{12} = x_1+x_2$. Plugging (\[ncf\]) in the definitions (\[emtph1\],\[emtph2\]), one gets $$\begin{aligned}
\E_i(\beta, \mu_b) \equiv 
\langle \theta_{tt}(t,x,i)\rangle_{\beta,\mu_b} = \qquad \qquad \qquad \qquad 
\nonumber \\
\S_{ii} \int_0^\infty  \frac{{{\rm d}}k}{\pi}\, k\, \cos(2kx) b_i(k)+ \sum_{j=1}^n \left (\delta_{ij}+\S_{ij}^2 \right )
\frac{1}{2\pi \beta_j^2}\, {\rm Li}_2 \left (\e^{\beta_j \mu_b}\right )\, , \; \; \,   
\label{tr1}\\
\T_i(\beta, \mu_b) \equiv \langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu_b} = 
\sum_{j=1}^n \left (\delta_{ij}-\S_{ij}^2\right ) 
\frac{1}{2\pi \beta_j^2}\, {\rm Li}_2 \left (\e^{\beta_j \mu_b}\right )\, , \; \; \, 
\label{tr2}\end{aligned}$$ where ${\rm Li}_2$ is the dilogarithm function. Eq. (\[tr1\]) determines the Casimir energy, whereas (\[tr2\]) describes the energy (heat) transport. Both are time-independent, thus confirming that we are dealing with a steady state. Notice also that the energy density is $x$-dependent, which reflects the breaking of translation invariance by the junction and is consistent with the conservation law (\[econs\]). Since $\S$ is an orthogonal matrix, $\T_i(\beta,\mu_b)$ obviously satisfies the Kirchhoff’s rule (\[K1\]). The energy density $\E_i(\beta, \mu_b)$ can be written in the equivalent form $$\E_i(\beta, \mu_b) = 
\S_{ii} \int_0^\infty  \frac{{{\rm d}}k}{\pi}\, k\, \cos(2kx) b_i(k) + 
\frac{1}{\pi \beta_i^2}\, {\rm Li}_2 \left (\e^{\beta_i \mu_b}\right )
- \sum_{j=1}^n \left (\delta_{ij}-\S_{ij}^2 \right )
\frac{1}{2\pi \beta_j^2}\, {\rm Li}_2 \left (\e^{\beta_j \mu_b}\right )\, ,
\label{ntr}$$ where the last term vanishes at equilibrium and describes therefore the [*non-equilibrium contribution*]{} to the Casimir energy.

For a junction with $n=2$ wires there are two one-parameter families $$\S^+ = \left(\begin{array}{cc}\cos \theta & \sin \theta \\ 
-\sin \theta  & \cos \theta  \\ \end{array} \right)\, ,   \qquad 
\S^- = \left(\begin{array}{cc}\cos \theta  & \sin \theta \\ 
\sin \theta & -\cos \theta \\ \end{array} \right)\, , \quad \theta \in [0,2\pi)\, , 
\label{ex1}$$ of scattering matrices with ${\rm det} (\S^\pm) = \pm 1$. For both families one finds $$\T_1(\beta, \mu_b) = - \T_2(\beta, \mu_b) = 
\frac{\sin^2 \theta }{2\pi} \left [ \frac{1}{\beta_1^2} {\rm Li}_2 \left (\e^{\beta_1 \mu_b}\right ) - 
\frac{1}{\beta_2^2} {\rm Li}_2 \left (\e^{\beta_2 \mu_b}\right )\right ]\, , 
\label{ctr1}$$ which deserves a comment. The heath transport has been investigated recently in the framework of conformal field theory (CFT) for generic central charge $c$ in [@bd-12]. Eq. (\[ctr1\]) confirms the result of [@bd-12] for $c=1$ and extends this result in two directions: imperfect junction with transmission probability $\sin^2(\theta)$ and $\mu_b\not=0$. We observe in this respect that derivation of the dilogarithm terms in (\[ctr1\]) is problematic in a CFT context, because both $\beta_1 \mu_b$ and $\beta_2 \mu_b$ are non-trivial dimensionless parameters.

In the limit $\mu_b \to 0^-$ the $x$-dependent integral in (\[tr1\]) can be evaluated explicitly and one finds $$\begin{aligned}
\E_i(\beta, 0) &=&\frac{\pi}{12}\sum_{j=1}^n \left (\delta_{ij}+\S^2_{ij} \right )\frac{1}{\beta_j^2} + 
\S_{ii}\frac{1}{8\pi x^2} - \S_{ii}\frac{\pi}{2\beta_i^2 \left [\sinh \left (2\pi \frac{x}{\beta_i}\right )\right ]^2}\, , 
\label{tr3} \\
\T_i(\beta, 0) &=& \frac{\pi}{12}\sum_{j=1}^n \left (\delta_{ij} - \S_{ij}^2 \right )
\frac{1}{\beta_j^2}\, . \qquad 
\label{tr4}\end{aligned}$$

Chiral NESS correlators
-----------------------

In the non-equilibrium bosonization procedure, developed below, we will need the correlation functions of the chiral fields (\[fil1\],\[fil2\])) in the NESS $\Omega_{\beta, \mu_b}$. It is easily seen that all of them can be expressed in terms of the distribution $$w (\xi,\beta;\lambda, \mu_b)= 
\int_0^\infty \frac{{{\rm d}}k}{\pi} \Delta_\lambda (k) \left [\frac{\e^{-\beta [k - \mu_b]}}{1- \e^{-\beta [k - \mu_b]}} 
\e^{\ri k \xi} + \frac{1}{1- \e^{-\beta [k - \mu_b]}} 
\e^{-\ri k \xi}\right ]\, . 
\label{w}$$ The full $\lambda$-dependence and the singularity at $\mu_b=0$ of (\[w\]) are captured by [@Liguori:1999tw] $$w (\xi,\beta;\lambda, \mu_b) = \frac{1}{\pi} 
\left \{ \frac{2}{\beta |\mu_b|} \ln \frac{|\mu_b|}{\lambda} - 
\ln \left [2 \ri\, \sinh \left (\frac{\pi}{\beta} \xi - \ri \varepsilon \right ) \right ]\right \} + o(\mu_b)\, , 
\label{wa}$$ where $o(\mu_b)$ stands for $\lambda$-independent terms, which vanish in the limit $\mu_b \to 0^-$. It is convenient at this point to relate the (up to now free) infrared regularization parameter $\lambda $ to $\mu_b$ by means of $$\lambda = |\mu_b| \, .
\label{lambdamu}$$ The limit $\mu_b \to 0^-$ in (\[w\]) now exists and gives the distribution[^4] $$w (\xi,\beta )\equiv \lim_{\mu_b \to 0^-} w (\xi,\beta; \lambda = |\mu_b|, \mu_b) = 
-\frac{1}{\pi} \ln \left [2 \ri\, \sinh \left (\frac{\pi}{\beta} \xi - \ri \varepsilon \right ) \right ] \, ,  
\label{w1}$$ which is the fundamental block of the NESS chiral correlation functions. In fact, for the two-point correlators one gets $$\begin{aligned}
\langle {\varphi}_{i_1,L}(\xi_1) {\varphi}_{i_2,L}(\xi_2) \rangle_\beta &=& 
\delta_{i_1i_2}\, w (\xi_{12},\beta_{i_1})\, , 
\label{ll}\\
\langle {\varphi}_{i_1,L}(\xi_1) {\varphi}_{i_2,R}(\xi_2) \rangle_{\beta} &=& 
w (\xi_{12},\beta_{i_1})\, \S^t_{i_1i_2}\, , 
\label{lr} \\
\langle {\varphi}_{i_1,R}(\xi_1) {\varphi}_{i_2,L}(\xi_2) \rangle_{\beta} &=& 
\S_{i_1i_2}\, w (\xi_{12},\beta_{i_2})\, , 
\label{rl} \\ 
\langle {\varphi}_{i_1,R}(\xi_1) {\varphi}_{i_2,R}(\xi_2) \rangle_{\beta} &=& 
\sum_{j=1}^n \S_{i_1j}\, w (\xi_{12},\beta_j)\, \S^t_{ji_2}\, . 
\label{rr}\end{aligned}$$ where $\xi_{12}\equiv \xi_1-\xi_2$. As expected, the point-like interaction in the vertex of $\Gamma$ induce a non-trivial left-right mixing described by (\[lr\],\[rl\]).

Bosonization away from equilibrium
==================================

The possibility to express fermions in terms of bosons in 1+1 dimensional space-time has been discovered long ago by Jordan and Wigner [@JW28]. The bosonization technique in the Fock representation of the fields ${\varphi}$ and ${\widetilde{\varphi}}$ has been applied for solving the Tomonaga-Luttinger (TL) model in [@Hald]-[@ML65]. The framework has been extended later [@Liguori:1999tw; @Amaral:2005cqa] to the finite temperature Gibbs representation of ${\varphi}$ and ${\widetilde{\varphi}}$. Both the Fock and Gibbs representations describe equilibrium physics. Our goal in what follows we will to apply the NESS representation, constructed in the previous section, for investigating the non-equilibrium TL liquid in the multi-terminal configuration shown in Fig. \[fig1\].

The Tomonaga-Luttinger model on $\Gamma$
----------------------------------------

The bulk dynamics is governed by the TL Lagrangian density $${\cal L} = \ri \psi_1^*(\der_t - v_F\der_x)\psi_1 +  \ri \psi_2^*(\der_t + v_F\der_x)\psi_2\\ 
-g_+(\psi_1^* \psi_1+\psi_2^* \psi_2)^2 - g_-(\psi_1^* \psi_1-\psi_2^* \psi_2)^2 \, ,   
\label{lagr}$$ where $\{\psi_\alpha (t,x,i)\,:\, \alpha =1,2\}$ are complex fermion fields, $v_F>0$ is the Fermi velocity and $g_\pm \in {\mbox{${\mathbb R}$}}$ are the coupling constants.

The bulk theory has an obvious $U_L(1)\otimes U_R(1)$ symmetry. In fact, the Lagrangian density (\[lagr\]) is left invariant by the two independent phase transformations $$\psi_\a  \rightarrow \e^{\ri s_\alpha } \psi_\a \, , \qquad \qquad \; \, 
\psi^*_\a \rightarrow \e^{-\ri s_\alpha } \psi^*_\a \, ,  \qquad s_\a \in {\mbox{${\mathbb R}$}}\, ,\quad  \a =1,2\, . 
\label{symm1}$$ implying the current conservation laws $$\der_t \rho_{Z} (t,x,i) -v_F\der_x j_{Z}  (t,x,i)= 0\, , 
\label{conservation}$$ where the charge and current densities are given by $$\rho_{Z} (t,x,i) = 
\begin{cases}
[\psi^*_1\psi_1](t,x,i)\, , & Z=L\, , \\
[\psi^*_2\psi_2](t,x,i)\, , & Z=R\, ,
\end{cases} \qquad 
j_{Z} (t,x,i) = 
\begin{cases}
[\psi^*_1\psi_1](t,x,i)\, , & Z=L\, , \\
-[\psi^*_2\psi_2](t,x,i)\, , & Z=R\, . 
\end{cases} 
\label{lrcurrents}$$ The currents $j_{Z}(t,x,i)$ have simple physical meaning: $j_{L}(t,x,i)$ and $j_{R}(t,x,i)$ represent the particle excitations moving along the edge $E_i$ towards and away of the vertex $V$ respectively. Interpreting the vertex as a defect, which can be characterized by some scattering matrix, the currents $j_{L}$ and $j_{R}$ describe therefore the [*incoming*]{} and [*outgoing*]{} flows.

The current splitting boundary condition
----------------------------------------

It is well known [@Hald]-[@ML65] that the TL model (\[lagr\]) is exactly solvable on the line ${\mbox{${\mathbb R}$}}$. On the graph $\Gamma$ the situation is more involved, because one should take into account the boundary conditions in the vertex $V$. The conditions $$\psi_1(t,0,i) = \sum_{j=1}^n \UU_{ij} \psi_2(t,0,j) \, , \qquad \UU\in U(n)\, , 
\label{bc2}$$ which work in the free case $g_-=g_+=0$, do not lead [@nfll-99] to exactly solvable problem after switching on the TL interactions. The fact that on ${\mbox{${\mathbb R}$}}$ the quartic bulk interactions in (\[lagr\]) are solved exactly via bosonization suggest to try boundary conditions which, differently from (\[bc2\]), are formulated in terms of real boson fields. In this spirit and according our previous comments on the chiral currents (\[lrcurrents\]), it is quite natural to consider $$j_R(t,0,i) = \sum_{k=1}^n {\mbox{${\mathbb J}$}}_{ik}\, j_L(t,0,k) \, ,  \qquad {\mbox{${\mathbb J}$}}\in O(n)\, , 
\label{bc3}$$ which has been proposed and explored first in the two-terminal case in [@SS]. An advantage of (\[bc3\]) is the direct interpretation in terms of [*gauge invariant physical observables*]{}, which represent the basic building blocks of algebraic quantum field theory (see e.g. [@H]). In fact, (\[bc3\]) describes the splitting in the vertex $V$ of the outgoing current $j_R(t,0,i)$ along the edge $E_i$ in incoming currents $j_L(t,0,k)$ along the edges $E_k$. For this reason we refer to ${\mbox{${\mathbb J}$}}$ the as the [*current splitting matrix*]{} and show in the next subsection that ${\mbox{${\mathbb J}$}}$ actually coincides with the boson scattering matrix $\S$.

Operator solution of the TL model on $\Gamma$
---------------------------------------------

Referring for the details to [@Bellazzini:2008fu], we recall here the anyon operator solution of the TL model on a star graph $\Gamma$. The solution provides a [*unified description*]{} of all anyon Luttinger liquids and is expressed in terms of the chiral fields (\[fil1\],\[fil2\]) and the parameters $\sigma,\, \tau \in {\mbox{${\mathbb R}$}}$ and the sound velocity $v\in {\mbox{${\mathbb R}$}}$ as follows: $$\begin{aligned}
\psi_1(t,x,i) &=&
\eta_i :\e^{\ri \sqrt {\pi} \left [\sigma {\varphi_{i,R}}(vt-x) + \tau {\varphi_{i,L}}(vt+x)\right ]}:\, , 
\label{psi1}\\
\psi_2(t,x,i) &=&
\eta_i:\e^{\ri \sqrt {\pi} \left [\tau {\varphi_{i,R}}(vt-x) + \sigma {\varphi_{i,L}}(vt+x)\right ]}:\, . 
\label{psi2}\end{aligned}$$ Here $: \cdots :$ denotes the normal product in the RT algebra $\A$ and $\eta_i$ are some Klein factors, controlling the statistics of $\psi_\alpha $. In this respect we impose the general anyon exchange relation $$\psi_\alpha^*(t,x_1,i_1) \psi_\alpha (t,x_2,i_2) = 
\e^{(-1)^{\alpha} \ri \pi \k \eps(x_{12})} \psi_\alpha (t,x_2,i_2)\psi_\alpha^*(t,x_1,i_1)\, , 
\qquad x_1\not=x_2\, , 
\label{an1}$$ where $\eps(x)$ is the sign function and $\k > 0$ is the so called [*statistical parameter*]{} which interpolates between bosons ($\k$ - even integer) and fermions ($\k$ - odd integer). A simple realization of the Klein factors is $$\eta_i = \frac{1}{\sqrt {2\pi}}:\e^{\pi\ri (\gamma_i + \gamma^*_i)}: \, , 
\label{KL1}$$ where $\{\gamma_i,\, \gamma^*_i \, :\, i=1,...,n\}$ generate the auxiliary algebra $$[\gamma_i\, ,\, \gamma_j] = [\gamma^*_i\, ,\, \gamma^*_j] = 0\, , \qquad 
[\gamma_i\, ,\, \gamma^*_j] = \ri \frac{\k}{2} \epsilon_{ij} \, , 
\label{KL2}$$ with $\epsilon_{ij}=-1$ for $i<j$, $\epsilon_{ii}=0$ and $\epsilon_{ij}=1$ for $i>j$.

In order to fix the solution (\[psi1\],\[psi2\]) completely, one should determine the parameters $\sigma$, $\tau$ and $v$ in terms of coupling constants $g_\pm$ and the statistical parameter $\k$. Using a standard short distance expansion and (\[psi1\],\[psi2\]) one gets the charge and current densities[^5] $$\rho_\pm (t,x,i) \equiv (:\psi^*_1\psi_1: \pm  :\psi^*_2\psi_2:)(t,x,i) = 
\frac{-1}{2\sqrt {\pi }\zeta_\pm } 
\left [(\der {\varphi_{i,R}})(vt-x) \pm (\der {\varphi_{i,L}})(vt+x)\right ] , 
\label{rhopm}$$ $$j_\pm (t,x,i) = 
\frac{v}{2\sqrt {\pi }v_F \zeta_\pm} 
\left [(\der {\varphi_{i,R}})(vt-x) \mp (\der {\varphi_{i,L}})(vt+x)\right ]\, , 
\label{jpm}$$ where for convenience the variables $$\zeta_\pm =\tau\pm\sigma \, . 
\label{zpm}$$ have been introduced. The normalization of (\[rhopm\]) is fixed [@Bellazzini:2008fu] by the Ward identities associated with the electric charge $Q_+$ and the helicity $Q_-$ defined by $$Q_\pm = \sum_{i=1}^n \int_0^\infty {{\rm d}}x\, \rho_\pm(t,x,i)\, .  
\label{charges}$$ The normalization of (\[jpm\]) in turn is determined by the conservation law $$\der_t \rho_\pm (t,x,i) - v_F\der_x j_\pm (t,x,i) = 0\, .  
\label{vcons}$$ Plugging (\[psi1\],\[psi2\],\[rhopm\]) in the quantum equations of motion $$\begin{aligned}
\ri [\der_t +(-1)^\alpha v_F\der_x] \psi_\alpha (t,x,i)= \nonumber 
\qquad \qquad \qquad \qquad \\
2 [g_+:\rho_+(t,x,i)\psi_\alpha : (t,x,i) -(-1)^\alpha g_- :\rho_-(t,x,i)\psi_\alpha :(t,x,i)]\, , 
\label{TLeqm}\end{aligned}$$ one finds $$\begin{aligned}
v \zeta_+^2 &=& v_F \k +\frac{2}{\pi}g_+ \, , 
\label{sys2}\\
v \zeta_-^2 &=& v_F \k +\frac{2}{\pi}g_- \, .  
\label{sys3}\end{aligned}$$ Moreover, the exchange relation (\[an1\]) implies $$\zeta_+\, \zeta_- = \kappa\, . 
\label{sys1}$$ Eqs. (\[sys1\],\[sys2\],\[sys3\]) provide a system for determining $v$ and $\zeta_\pm$ (or equivalently $\sigma$ and $\tau$) in terms of $\k$ and $g_\pm$. The solution is $$\begin{aligned}
\zeta_\pm^2 &=& \kappa
\left(\frac{\pi \kappa v_F+2g_+}{\pi \kappa v_F+2g_-}\right)^{\pm 1/2}\, , 
\label{z}\\ 
v&=&\frac{\sqrt{(\pi \kappa v_F+2g_-)(\pi \kappa v_F+2g_+)}}{\pi \kappa}\, , 
\label{v}\end{aligned}$$ where the positive roots are taken in the right hand side. The relations (\[z\]) and (\[v\]) represent the anyon generalization [@Bellazzini:2008fu] of the well known result for canonical fermions $\k=1$, where an alternative and frequently used notation [@voit-95] is $$g_2=2(g_+-g_-)\, , \qquad g_4=2(g_+ + g_-)\, , \qquad K=\zeta_-^2=\zeta_+^{-2}\, . 
\label{altnot}$$

Considering the general anyon solution (\[z\], \[v\]), we assume in what follows that the parameters $\{\k,\, g_\pm\}$ belong to the domain $${\cal D} =\{\k>0,\; 2g_\pm> -\pi \kappa v_F\} \, ,  
\label{physcond}$$ which ensures that $\sigma$, $\tau$ and $v$ are real and finite.

Let us discuss finally the current splitting boundary condition (\[bc3\]) and establish the relation between ${\mbox{${\mathbb J}$}}$ and $\S$. Expressing the chiral currents $j_Z$ in terms of the chiral fields ${\varphi_{i,Z}}$, one finds $$\begin{aligned}
j_R(t,x,i) &=& \frac{1}{2} (\zeta_-j_- + \zeta_+j_+)(t,x,i) = \frac{v}{2\sqrt {\pi }v_F} \der {\varphi_{i,R}}(vt-x) \, , 
\label{qcc1} \\
j_L(t,x,i) &=& \frac{1}{2} (\zeta_-j_- - \zeta_+j_+)(t,x,i)= \frac{v}{2\sqrt {\pi }v_F} \der {\varphi_{i,L}}(vt+x) \, ,
\label{qcc2}\end{aligned}$$ which, according to (\[bc1\]) satisfy the current splitting boundary condition (\[bc3\]), provided that $${\mbox{${\mathbb J}$}}= \S \in O(n) \, . 
\label{bc4}$$

The symmetry content of the TL junction is strongly influenced by (\[bc4\]). The point is that in the presence of a defect the continuity equation (\[vcons\]) alone is not enough to ensure the electric charge conservation. A direct computation shows indeed that $$\der_t Q_+ =  \frac{v}{2\sqrt {\pi} \zeta_+} \sum_{k=1}^n 
\left (1-\sum_{i=1}^n \S_{ik} \right )(\der \varphi_{k,L})(vt)\, . 
\label{sym1}$$ The independence of ${\varphi_{i,L}}$ implies that the electric charge $Q_+$ is conserved if and only if $$\sum_{i=1}^n \S_{ik} = 1\, , \qquad \forall\; k=1,...,n\, ,   
\label{K3}$$ which, as expected, is equivalent to the Kirchhoff rule $$\sum_{i=1}^n j_+ (t,0,i) = 0\, . 
\label{K2}$$ Since $\S \in O(n)$, one infers from (\[K3\]) that $\S^t$ satisfies (\[K3\]) as well. Therefore, the electric charge $Q_+$ is conserved for those $\S$, whose entries along each column (line) sum up to 1. In geometric terms, these scattering matrices belong to the stability subgroup $O_{\bf v}\subset O(n)$ of the $n$-vector ${\bf v} = (1,1,...,1)$. An explicit parametrization of $O_{\bf v}$ in terms of angular variables is given in [@Bellazzini:2009nk].

Summarizing, the condition $\S\in O(n)$ guaranties the energy conservation in the TL-junction. Concerning the electric charge $Q_+$, one must distinguish two different regimes. $Q_+$ is conserved for $\S\in O_{\bf v}$. If instead $\S$ belongs to the complement ${\widetilde{O}}_{\bf v} \equiv  
O(n)\setminus O_{\bf v}$, there is an external incoming or outgoing charge flow in the junction and $Q_+$ is not conserved. The possibility to describe such imperfect junctions is a remarkable feature of the current splitting boundary condition (\[bc3\]). The physical details about the charge transport in the junction are discussed in section 4.2 below.

NESS representation and chemical potentials
-------------------------------------------

The crucial property of the operator solution (\[psi1\],\[psi2\],\[z\],\[v\]) is that it is universal, meaning that it applies for any representation of the chiral field algebra generated by ${\varphi_{i,Z}}$. The Fock and Gibbs representations have been largely studied and describe the equilibrium properties of the TL model on $\Gamma$. In order to explore the behavior of the Luttinger liquid away from equilibrium, we investigate below the operator solution in the NESS representation of the RT algebra $\A$, constructed in section 2.2.

The first step in this direction is the introduction of the [*fermion*]{} chemical potentials $$\mu_i = k_F -V_i \, , 
\label{fcp1}$$ where $k_F$ defines the Fermi energy for $\k=1$ and $V_i$ is the external voltage applied to the thermal reservoir in the edge $E_i$ of Fig. \[fig2\]. In what follows we keep $k_F$ fixed and vary eventually the gate voltages $V_i$. As already mentioned, the [*boson*]{} chemical potential $\mu_b<0$ has been introduced for avoiding some infrared singularities at the boson level and has nothing to do with $\mu_i$. In fact, in the chiral correlators (\[ll\]-\[rr\]) we already performed the limit $\mu_b \to 0^-$. In order to recover $\mu_i$, following [@Liguori:1999tw] we introduce the shift $\alpha_\mu$, defined by $${\varphi_{i,L}}(\xi) \longmapsto (\alpha_\mu {\varphi_{i,L}})(\xi ) = {\varphi_{i,L}}(\xi) -\frac{\xi}{\sqrt \pi\, \zeta_+}\, \mu_i 
\label{fcp2}$$ and, consistently with the boundary condition (\[bc1\]), $${\varphi_{i,R}}(\xi) \longmapsto (\alpha_\mu {\varphi_{i,R}})(\xi ) = {\varphi_{i,R}}(\xi) -\frac{\xi}{\sqrt \pi\, \zeta_+}\sum_{j=1}^n \S_{ij} \mu_j \, . 
\label{fcp3}$$ The transformations (\[fcp1\],\[fcp2\]) extend to an automorphism $\alpha_{\mu}$ on the whole algebra generated by the chiral fields ${\varphi_{i,Z}}$, which is directly implemented in the operator solution (\[psi1\],\[psi2\],\[rhopm\],\[jpm\]). At this stage the TL correlation functions in the NESS are defined by $$\langle \O_1[{\varphi}_{i_1,Z}] \cdots \O_k[{\varphi}_{i_k,Z}] \rangle_{\beta,\mu} = 
\langle \O_1[\alpha_\mu {\varphi}_{i_1,Z}] \cdots 
\O_k[\alpha_\mu {\varphi}_{i_k,Z}]\rangle_{\beta}\, . 
\label{gcf}$$

In the rest of the paper we focus on the correlation functions (\[gcf\]), which capture the physical properties of the Luttinger liquid with the current splitting boundary condition (\[bc3\]) away from equilibrium. We will show in particular that (\[gcf\]) satisfy the Kubo-Martin-Schwinger (KMS) condition [@BR; @H] at equilibrium, which justifies the introduction of the chemical potentials $\mu_i$ by means of (\[fcp2\],\[fcp3\]).

Non-equilibrium TL correlation functions
========================================

Anyon correlators
-----------------

We derive here the two-point correlators of $\psi_\alpha (t,x,i)$ defined by (\[psi1\], \[psi2\]) in the NESS and discuss their properties. For this purpose we extend away from equilibrium the finite temperature results of [@Liguori:1999tw]. Using (\[gcf2\]), for $\psi_1$ one finds $$\begin{aligned}
\langle \psi_1^*(t_1,x_1,i)\psi_1(t_2,x_2,j)\rangle_{\beta, \mu} = 
A_{ij}\, B_{ij}(t_{1,2},x_{1,2};\mu) \times
\qquad \qquad \qquad \qquad 
\nonumber \\
\left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ {\widetilde{x}}_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}^t} 
\left \{\frac{1}{\frac{\beta_j}{\pi} \sinh \left [\frac{\pi}{\beta_j}(vt_{12} - {\widetilde{x}}_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}} \quad \; \; 
\nonumber \\
\left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) -
\ri \varepsilon \right ]}\right \}^{\tau^2 \delta_{ij}} 
\prod_{k=1}^n
\left \{\frac{1}{\frac{\beta_k}{\pi} \sinh \left [\frac{\pi}{\beta_k}(vt_{12} - x_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma^2 \S_{ik}\S_{kj}^t} \, ,
\label{anc1}\end{aligned}$$ where $$A_{ij} = \frac{\e^{\ri \pi^2 \kappa \epsilon_{ij}/2}}{2\pi} 
\left (\frac{1}{2\ri}\right )^{(\sigma^2 +\tau^2)\delta_{ij} +\sigma \tau (\S_{ij}+\S^t_{ij})} \, , 
\label{anc2}$$ $$B_{ij}(t_{1,2},x_{1,2};\mu) = \e^{\ri \{\tau [(vt_1+x_1)\mu_i-(vt_2+x_2)\mu_j]+
\sigma[(vt_1-x_1)\sum_{k=1}^n\S_{ik}\mu_k -(vt_2-x_2)\sum_{k=1}^n\S_{jk}\mu_k]\}/(\sigma+\tau)} \, .
\label{anc4}$$ The $\psi_2$-correlator has the analogous form, $$\langle \psi_2^*(t_1,x_1,i)\psi_2(t_2,x_2,j)\rangle_{\beta, \mu} = (\ref{anc1})\quad {\rm with}\quad \sigma 
\leftrightarrow \tau \, . 
\label{anc22}$$

The TL junction involves two types of $\psi_1$-$\psi_2$ interactions. First, the Lagrangian (\[lagr\]) contains a $\psi_1$-$\psi_2$ [*bulk coupling*]{} proportional to $(g_+-g_-)$. Second, the current splitting boundary condition (\[bc3\]) provides an additional [*boundary interaction*]{} described by the mixed left-right correlators (\[lr\], \[rl\])). Consequently, the mixed $\psi_1$-$\psi_2$ correlators are non-trivial and have the form $$\begin{aligned}
\langle \psi_1^*(t_1,x_1,i)\psi_2(t_2,x_2,j)\rangle_{\beta, \mu} = 
{\widetilde{A}}_{ij}\, {\widetilde{B}}_{ij}(t_{1,2},x_{1,2};\mu) \times
\qquad \qquad \qquad \qquad 
\nonumber \\
\left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ {\widetilde{x}}_{12}) -
\ri \varepsilon \right ]}\right \}^{\tau^2 \S_{ij}^t} 
\left \{\frac{1}{\frac{\beta_j}{\pi} \sinh \left [\frac{\pi}{\beta_j}(vt_{12} - {\widetilde{x}}_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma^2 \S_{ij}} \quad \; \; 
\nonumber \\
\left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma \tau \delta_{ij}} 
\prod_{k=1}^n
\left \{\frac{1}{\frac{\beta_k}{\pi} \sinh \left [\frac{\pi}{\beta_k}(vt_{12} - x_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ik}\S_{kj}^t} \, ,
\label{manc1}\end{aligned}$$ where $${\widetilde{A}}_{ij} = \frac{\e^{\ri \pi^2 \kappa \epsilon_{ij}/2}}{2\pi} 
\left (\frac{1}{2\ri}\right )^{\sigma^2 \S_{ij} +\tau^2 \S^t_{ij} +2\sigma \tau \delta_{ij}} \, , 
\label{manc2}$$ $${\widetilde{B}}_{ij}(t_{1,2},x_{1,2};\mu) = \e^{\ri \{\tau [(vt_1+x_1)\mu_i-(vt_2-x_2)\sum_{k=1}^n\S_{jk}\mu_k]+
\sigma[(vt_1-x_1)\sum_{k=1}^n\S_{ik}\mu_k-(vt_2+x_2)\mu_j ]\}/(\sigma+\tau)} \, .
\label{manc4}$$ Finally, $$\langle \psi_2^*(t_1,x_1,i)\psi_1(t_2,x_2,j)\rangle_{\beta, \mu} = (\ref{manc1})\quad {\rm with}\quad \sigma 
\leftrightarrow \tau \, . 
\label{manc22}$$

As expected, in the equilibrium limit $\beta_i = \beta$ and $\mu_i=\mu$ for all $i$, the correlators (\[anc1\])–(\[manc22\]) simplify and satisfy the KMS condition, which represents a non-trivial check both on the computation and on the shift (\[fcp2\], \[fcp3\]) introducing the chemical potentials. Let us consider for instance (\[anc1\]), which in this limit takes the form $$\begin{aligned}
\langle \psi_1^*(t_1,x_1,i)\psi_1(t_2,x_2,j)\rangle_{\beta, \mu} = A_{ij}\, 
\e^{\ri [\tau (vt_{12}+x_{12}) + \sigma (vt_{12}-x_{12})]\mu /(\sigma+\tau)}\times
\qquad \qquad 
\nonumber \\
\left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12}+ {\widetilde{x}}_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}^t} 
\left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12} - {\widetilde{x}}_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}} \quad \; \; 
\nonumber \\
\left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12}+ x_{12}) -
\ri \varepsilon \right ]}\right \}^{\tau^2 \delta_{ij}} 
\left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12} - x_{12}) -
\ri \varepsilon \right ]}\right \}^{\sigma^2 \delta_{ij}} \, . \quad 
\label{anc5}\end{aligned}$$ Recalling that the KMS automorphism $\varrho_s$ acts on $\psi_\alpha$ as follows, $$\left [\varrho_s \psi_\alpha\right ] (t,x,i) = \e^{\ri s \mu}\, \psi_\alpha (t + s/v,x,i)\, , 
\label{anc6}$$ one can check that the equilibrium correlator (\[anc5\]) satisfies the KMS condition $$\langle \psi_1^*(t_1,x_1,i)\left [\varrho_{s+\ri \beta}\psi_1\right ](t_2,x_2,j)\rangle_{\beta, \mu} 
= \langle \left [\varrho_{s}\psi_1\right ](t_2,x_2,j)\psi_1^*(t_1,x_1,i)\rangle_{\beta, \mu}  
\label{anc7}$$ for all values of the statistical parameter $\k$.

The critical scaling dimensions $d_i$ can be extracted from (\[anc1\])–(\[manc22\]) in the limit $\beta_i \to \infty$ and $\mu_i \to 0$. Because of the operator mixing, this is a subtle issue, which has been discussed in full detail in [@Bellazzini:2009nk]. One gets, $$d_i = \frac{1}{2}(\sigma^2 + \tau^2) + \sigma \tau s_i \, , \qquad i=1,...,n \, ,  
\label{dimensions0}$$ where $s_i=\pm 1$ are the eigenvalues of $\S$. As already observed in [@Bellazzini:2006kh], the impact of the vertex interaction is captured by the term $\sigma \tau s_i$, which preserves unitarity in the sense of conformal field theory because $d_i \geq 0$.

A remarkable special case is obtained by setting $g_+=g_-\equiv g$. In this case the bulk $\psi_1$-$\psi_2$ coupling vanishes and one is left only with the boundary interaction induced by the current splitting boundary condition (\[bc3\]). From (\[v\]) one gets $$v=v_F + \frac{2g}{\pi \kappa }\, , 
\label{nv}$$ and (using (\[sys1\]) with $\k > 0$ and $\tau \geq 0$) $$\sigma =0\, , \quad \tau = \sqrt {\kappa }\, .
\label{nst}$$ Inserting (\[nst\]) in (\[anc1\])–(\[anc22\]) and localizing the fields in the same edge (i.e. setting $i=j$), one finds that the correlation functions simplify to $$\begin{aligned}
C_{11}(vt_{12}+x_{12},i;\beta,\mu) \equiv 
\langle \psi_1^*(t_1,x_1,i)\psi_1(t_2,x_2,i)\rangle_{\beta, \mu} = 
\nonumber \\
\frac{1}{2\pi} 
\left (\frac{1}{2\ri}\right )^\k 
\e^{\ri (vt_{12}+ x_{12})\mu_i}
\left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) -
\ri \varepsilon \right ]}\right \}^{\k}\, ,
\label{panc1}\end{aligned}$$ $$\begin{aligned}
C_{22}(vt_{12}-x_{12},i;\beta,\mu) \equiv 
\langle \psi_2^*(t_1,x_1,i)\psi_2(t_2,x_2,i)\rangle_{\beta, \mu} = 
\qquad \quad 
\nonumber \\
\frac{1}{2\pi} 
\left (\frac{1}{2\ri}\right )^\k 
\prod_{k=1}^n \e^{\ri (vt_{12}- x_{12})\S_{ik}\mu_k} 
\left \{\frac{1}{\frac{\beta_k}{\pi} \sinh \left [\frac{\pi}{\beta_k}(vt_{12}- x_{12}) -
\ri \varepsilon \right ]}\right \}^{\k \S_{ik}^2}. 
\label{panc2}\end{aligned}$$ The condition $g_+=g_-$ and the left-right asymmetry of the NESS construction in section 2 imply that only left moving (incoming) excitations contribute to $C_{11}$, which therefore coincides with the equilibrium correlator [@Liguori:1999tw]. All the non-equilibrium features are captured by $C_{22}$, which involves only right moving (outgoing) excitations. In fact, in spite of being localized in the edge $E_i$ of the graph, (\[panc2\]) depends on the temperatures and chemical potentials of the all $n$ edges.

It is instructive for this reason to derive and compare the Fourier transforms of (\[panc1\], \[panc2\]). We will show first that they can be expressed in terms of the finite temperature TL [*anyon distribution*]{} discovered in [@Liguori:1999tw]. Consider in fact $${\widehat{C}}_{11}(E,p,i;\beta,\mu) \equiv \int_{-\infty}^\infty {{\rm d}}t 
\int_{-\infty}^\infty {{\rm d}}x\, \e^{-\ri (Evt +px)}\, C_{11}(vt +x,i;\beta,\mu)\, . 
\label{ad1}$$ Plugging (\[panc1\]) in (\[ad1\]) one gets $${\widehat{C}}_{11}(E,p,i;\beta,\mu) = \frac{\pi^\kappa}{v} \delta(E-p)\, d(p-\mu_i,\beta_i;\k) \, , 
\label{ad2}$$ where the $\delta$-function fixes the dispersion relation and $d$ is the equilibrium anyon momentum distribution [@Liguori:1999tw] $$\begin{aligned}
d(p,\beta;\k) = \frac{\beta^{1-\k} \e^{-\beta p/2}}{2\pi}\, B \left (\frac{\k}{2} - \frac{\ri}{2\pi} \beta p \, , 
\frac{\k}{2} + \frac{\ri}{2\pi} \beta p\right ) = \qquad \qquad \qquad 
\nonumber \\
\frac{\beta^{1-\k} \e^{-\beta p/2}}{2\pi \Gamma (\k)}\,   
\Gamma \left (\frac{\k}{2} - \frac{\ri}{2\pi} \beta p\right ) 
\Gamma \left (\frac{\k}{2} + \frac{\ri}{2\pi} \beta p\right ) \, , \qquad \k>0\, , 
\label{ad3}\end{aligned}$$ $B$ and $\Gamma$ being the beta and gamma functions (Euler’s integrals of first and second kind respectively). Notice that for $\k\not=1$ the distribution (\[ad3\]) depends on both $\beta$ and $p$ and not only on the dimensionless combination $\beta p$. Eq. (\[ad3\]) defines a smooth function of $p\in {\mbox{${\mathbb R}$}}$, which satisfies $$\lim_{p\to 0}d(p,\beta;\k) = 
\frac{\beta^{1-\k} \Gamma^2(\k/2)}{2\pi \Gamma (\k)}\, , 
\label{as0}$$ and has the following asymptotic behavior: $$\begin{aligned}
\lim_{p\to \infty}d(p,\beta;\k) &=& 0\, , \quad \forall\;  \k>0\, , 
\label{as1}\\
\lim_{p\to -\infty}d(p,\beta;\k) &=& 
\begin{cases}
0\, , \quad &0<\k<1\, ,\\
1\, , \quad &\k=1\, ,\\
\infty \, , \quad &\k >1\, . 
\end{cases}
\label{as2}\end{aligned}$$ For positive integer $\k$ (i.e. for fermions and bosons) the distribution (\[ad3\]) simplifies to $$d(p,\beta;\k) = 
\begin{cases}
\frac{1}{(1+\e^{\beta p})} \frac{\beta^{-2(n-1)}}{[2(n-1)]!} \prod_{j=1}^{n-1}
\bigl |j - \frac{1}{2} - \frac{\ri}{2\pi} p\beta \bigr |^2\, , & \k = 2n-1 \, , \\
\\
\frac{1}{(\e^{\beta p}-1)} \frac{2\pi \beta^{-2n} }{(2n-1)! p} \prod_{j=0}^{n-1}
\bigl |j- \frac{\ri}{2\pi} p\beta \bigr |^2\, , & \k = 2n\, , 
\end{cases} 
\label{intad}$$ where $n=1,2,...$ and the familiar Fermi and Bose distributions appear as prefactors. The first two fermion and boson distributions are $$\begin{aligned}
d(p, \beta;1) = \frac{1}{(1+\e^{\beta p})}\, , \qquad \qquad 
d(p, \beta; 3) = \frac{1}{(1+\e^{\beta p})} \frac{(\pi^2+p^2\beta^2)}{8\pi^2\beta^2}\, , 
\label{ad4f} \\
d(p, \beta; 2) = \frac{1}{(\e^{\beta p}-1)} \frac{p}{2\pi}\, , \qquad 
d(p, \beta; 4) = \frac{1}{(\e^{\beta p}-1)} \frac{p(4\pi^2+p^2\beta^2)}{48\pi^3\beta^2}\, .
\label{ad4b}\end{aligned}$$ As expected, in the fermion point $\k=1$ of the TL liquid one gets the familiar Fermi distribution. In spite of the fact that the remaining boson and fermion points ($\k=2,3,...$) have been established in [@Liguori:1999tw; @Ilieva:2000cj] more then a decade ago, to our knowledge their physical meaning and potential applications of (\[intad\]) have not been fully explored.

(500,110)(-120,20) ![The distribution $d$ at fixed temperature $\beta=1$ for $\k=1/2$ (dotted blue line) and $\k=3/2$ (dashed black line) compared to the Fermi distribution $\k=1$ (continuous red curve).[]{data-label="fig3"}](fig3.pdf "fig:")

(500,110)(-120,20) ![The distribution $d$ at fixed $\k=1/4$ for different temperatures $\beta =0.2$ (continuos red line), $\beta=0.4$ (dashed black line) and $\beta=0.8$ (dotted blue line).[]{data-label="fig4"}](fig4.pdf "fig:")

In order to give an idea about the anyon distributions in the interval $0<\k< 1$, we show some of them in Fig.\[fig3\], where the standard Fermi distribution (continuous red curve) is given for comparison. Fig.\[fig4\] displays the behavior of the anyon distribution (\[ad3\]) for fixed $\k=1/4$ and different temperatures. For $0<\k<1$ and with decreasing of the temperature $T\sim 1/\beta$ one observes the formation of a sharp peak at $p=0$ (in agreement with eq. (\[as0\])), which signals a condensation-like phenomenon [@Liguori:1999tw].

Concerning the Fourier transform of (\[panc2\]), it is useful to consider first the case when all the temperatures are equal ($\beta_i = \beta$), the system being driven away from equilibrium only by the voltages $V_i$. In this case $$\begin{aligned}
C_{22}(vt_{12}-x_{12},i;\beta,\mu) \equiv 
\langle \psi_2^*(t_1,x_1,i)\psi_2(t_2,x_2,i)\rangle_{\beta, \mu} = 
\qquad \quad 
\nonumber \\
\frac{1}{2\pi} 
\left (\frac{1}{2\ri}\right )^\k 
\e^{\ri (vt_{12}- x_{12})\sum_{k=1}^n\S_{ik}\mu_k} 
\left \{\frac{1}{\frac{\beta}{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12}- x_{12}) -
\ri \varepsilon \right ]}\right \}^{\k}. 
\label{panc22}\end{aligned}$$ and therefore $${\widehat{C}}_{22}(E,p,i:\beta,\mu) = \frac{\pi^\kappa}{v} \delta(E-p)\, d(p-\sum_{k=1}^n\S_{ik}\mu_k,\beta;\k)\, . 
\label{ad22}$$ One has still the equilibrium distribution, with the energy shifted by a linear combination of the chemical potentials $\mu_k$, whose coefficients are the $\S$-matrix elements.

Finally, in the coordinate space the general expression (\[panc2\]) is a product of $C_{11}$-factors with different temperatures and chemical potentials. One gets therefore in momentum space the nested convolution formula $$\begin{aligned}
{\widehat{C}}_{22}(E,p,i:\beta,\mu) = \frac{\pi^\kappa}{v} \delta(E-p)\, 
\int_{-\infty}^\infty \frac{{{\rm d}}k_1}{2\pi} \int_{-\infty}^\infty \frac{{{\rm d}}k_2}{2\pi} 
\cdots \int_{-\infty}^\infty \frac{{{\rm d}}k_{n-1}}{2\pi} \times \qquad 
\nonumber \\
d(k_1-\S_{i1}\mu_1,\beta_1;\k\S^2_{i1})d(k_2-\S_{i2}\mu_2-k_1,\beta_2;\k\S^2_{i2})
\cdots d(p-\S_{in}\mu_n-k_{n-1},\beta_n;\k\S^2_{in})\, .  
\nonumber \\ 
\label{ad223}\end{aligned}$$ Being a convolution of distributions, (\[ad223\]) is also a well defined distribution. The NESS $\Omega_{\beta, \mu}$ has therefore a remarkable property: the associated [*non-equilibrium distribution*]{} is simply a [*convolution of equilibrium distributions*]{} with different temperatures and chemical potentials.

Since the general form of (\[ad223\]) is quite complicated, it is instructive to consider below the case $n=2$ and $\mu_1=\mu_2=0$, focusing on $$D_2(p;\beta_1,\beta_2;\k,\theta) = 
\int_{-\infty}^\infty \frac{{{\rm d}}k}{2\pi}\, d(k,\beta_1;\k \cos^2 \theta )d(p-k,\beta_2;\k \sin^2 \theta)\, , 
\qquad \theta \in [0,\pi)\, .  
\label{b1}$$ Using the $x$-space representation, at equal temperatures one finds the relation $$D_2(p;\beta,\beta;\k,\theta) = d(k,\beta ;\k)\, , \qquad \forall\; \theta \in [0,\pi)\, .  
\label{b2}$$ For $\beta_1 \not=\beta_2$ the convolution $D_2$ defines a new distribution. Since we were not able to determine its explicit analytic form, we give some plots which are obtained numerically. The plots in Figs. \[fig5\] and \[fig6\] illustrate the behavior of $D_2$ for different values of $\beta_{1,2}$, $\k$ and $\theta$. We see that even for $\beta_1 \not=\beta_2$ the distribution $D_2$ is similar to $d$ with a kind of “effective" temperature and statistical parameter depending on $\beta_{1,2}$, $\k$ and $\theta$.

(500,110)(-120,20) ![The distribution $D_2$ at fixed $\k=1/2$ and $\theta =\pi/4$ for different temperatures $(\beta_1,\beta_2) =(1/2,1)$ (continuos red line), $(\beta_1,\beta_2) =(1,2)$ (dashed black line) and $(\beta_1,\beta_2) =(2,4)$ (dotted blue line).[]{data-label="fig5"}](fig5.pdf "fig:")

(500,110)(-120,20) ![The distribution $D_2$ at fixed $\beta_1=1$, $\beta_2=6$ and $\theta=\pi/6$ for statistical parameters $\k =1/3$ (continuos red line), $\k=2/3$ (dashed black line) and $\k=4/3$ (dotted blue line).[]{data-label="fig6"}](fig6.pdf "fig:")

Summarizing, we derived above the two-point TL anyon correlation functions away from equilibrium. The results (\[anc1\],\[manc1\]) are expressed as products of $\S$-dependent powers of equilibrium correlators at different temperatures. In agreement with this fact the momentum space anyon NESS distribution is the convolution of equilibrium anyon distributions (\[ad223\]). The equilibrium limit satisfies the KMS conditions. At criticality one is dealing with a $c=1$ conformal field theory, whose anomalous dimensions (\[dimensions0\]) depend not only on the coupling constants $g_\pm$, but also on the scattering matrix $\S$. The above technique allows to compute higher anyon correlation functions as well, but in order to investigate the transport properties of the system, we concentrate below on the electric and energy current correlators away from equilibrium.

Charge and heat transport
-------------------------

The charge transport in the NESS is described by $$\begin{aligned}
\langle \der_t Q_+ \rangle_{\beta, \mu} &=& \frac{v}{2\pi \zeta^2_+} \sum_{i,j=1}^n 
(\S_{ij} - \delta_{ij}) \mu_j\, , 
\label{ct00}\\
\langle j_+(t,x,i)\rangle_{\beta, \mu} &=& \frac{v}{2\pi v_F \zeta^2_+} 
\sum_{j=1}^n (\delta_{ij}-\S_{ij})\mu_j \, , 
\label{ct01}\end{aligned}$$ which follow by substituting (\[jpm\],\[sym1\]) in (\[gcf\]). Eq. (\[ct00\]) describes the external charge flow in the junction: it is constant in time and is incoming for $\langle \der_t Q_+ \rangle_{\beta, \mu} >0$ and outgoing for $\langle \der_t Q_+ \rangle_{\beta, \mu} <0$. Eq. (\[ct01\]) determines instead the value of the currents along the leads. The charge balance $$\langle \der_t Q_+ \rangle_{\beta, \mu} +
v_F \sum_{i=1}^n\langle j_+(t,x,i)\rangle_{\beta, \mu} = 0  
\label{ct02}$$ is satisfied and represents an useful check. If $\S\in O_{\bf v}$, the electric charge is conserved $\langle \der_t Q_+ \rangle_{\beta, \mu} =0$ and the $k_F$-dependence in (\[ct01\]) drops out, leading to $$\langle j_+(t,x,i)\rangle_{\beta, \mu} = \frac{v}{2\pi v_F \zeta^2_+} 
\sum_{j=1}^n (\S_{ij}-\delta_{ij})V_j \, . 
\label{ct1}$$ The current (\[ct1\]) satisfies the Kirchhoff rule (\[K2\]) and vanishes at equilibrium ($V_i=V$ for all $i$) as it should be. The dependence on the statistical parameter $\k$ is explicit and deserves a comment. In the physical domain $\cal D$, defined by (\[physcond\]), the overall coefficient in front of the sum in (\[ct1\]) is positive, $$G(g_-,\k)\equiv \frac{v}{2\pi v_F\zeta_+^2} = \frac{\pi \kappa v_F + 2g_-}{2 \pi^2 \kappa^2 v_F} >0\, . 
\label{ct3}$$ For $g_->0$ the coefficient $G$ decreases monotonically with $\k>0$. For $g_-<0$ one has that $\k>-2g_-/\pi v_F$ in the physical domain $\cal D$. In this case the coefficient $G$ increases in the interval $-2g_-/\pi v_F<\k < -4g_-/\pi v_F$, reaching the maximal value $-v_F/16g_-$ and decreases for $\k>-4g_-/\pi v_f$. This behavior is illustrated in Fig. \[fig7\].

(500,110)(-120,20) ![Behavior of $G$ at $g_-=1$ (dashed black line) and $g_-=-1$ (continuous red line).[]{data-label="fig7"}](fig7.pdf "fig:")

The current (\[ct1\]) is proportional to the applied external voltages. Non-linear effects are absent in the critical regime under consideration, which implies the conductance tensor $$\G_{ij} =G(g_-,\k)\sum_{j=1}^n (\S_{ij}-\delta_{ij})\, . 
\label{ct2}$$ We see that the NESS approach, adopted in this paper, confirms the result for $\G$, obtained previously for $\k=1$ by different methods, including renormalization group techniques [@lrs-02; @emabms-05], linear response theory [@Bellazzini:2006kh; @Bellazzini:2008mn] and conformal field theory [@coa-03; @hc-08]. The novelty in (\[ct2\]) is the explicit dependence on the statistical parameter $\k$, shown in Fig. \[fig7\].

A similar computation gives the energy (heat) flow $$\langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu} = 
\frac{v^2}{8\pi \zeta_+^2}\left [ \mu_i^2 - \left (\sum_{j=1}^n \S_{ij}\mu_j \right )^2\right ] + 
\frac{\pi v^2}{12}\sum_{j=1}^n \left (\delta_{ij} - \S_{ij}^2 \right )\frac{1}{\beta_j^2}\, , 
\label{ct04}$$ which satisfies the Kirchhoff rule (\[K1\]) for all $\S \in O(n)$. For $\S\in O_{\bf v}$ the expression (\[ct04\]) takes the form $$\begin{aligned}
\langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu} = 
\qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\
\frac{v^2}{8\pi \zeta_+^2}\left [ V_i^2 - 2k_F \sum_{j=1}^n (\delta_{ij}-\S_{ij})V_j - 
\left (\sum_{j=1}^n \S_{ij}V_j \right )^2\right ] + 
\frac{\pi v^2}{12}\sum_{j=1}^n \left (\delta_{ij} - \S_{ij}^2 \right )\frac{1}{\beta_j^2}\, . 
\label{ct4}\end{aligned}$$ The heat flow depends therefore not only on $k_F$ and the voltages $V_i$, but also on the temperatures $\beta_i$.

As before, we consider for illustration the case of $n=2$ wires. Inserting the scattering matrices (\[ex1\]) in (\[ct01\]) one has $$\langle \der_t Q_+ \rangle_{\beta, \mu} =
\begin{cases} 
Gv_F[(\mu_1 +\mu_2)(\cos \theta-1)-(\mu_1-\mu_2)\sin \theta ]\, ,   & \; \;   {\rm det}\, \S =1\, , \\
Gv_F[(\mu_1 +\mu_2)(\sin \theta -1)+(\mu_1-\mu_2)\cos \theta ]\, ,   & \; \;   {\rm det}\, \S =-1\, . \\
\end{cases}
\label{ex2}$$ Therefore, $$\langle \der_t Q_+ \rangle_{\beta, \mu} = 0 \Longrightarrow 
\begin{cases} 
\theta =0 \quad \Longrightarrow j_+(t,x,1) = j_+(t,x,2) =0 \, ,   & \; \;   {\rm det}\, \S =1\, , \\
\theta =\pi/2 \Longrightarrow  j_+(t,x,1) = -j_+(t,x,2)=G(\mu_1-\mu_2)\, ,   & \; \;   {\rm det}\, \S =-1\, , \\
\end{cases}
\label{ex3}$$ corresponding respectively to full reflection (disconnected edges) and complete transmission in the junction. Finally, $$\begin{aligned}
\langle \theta_{xt}(t,x,1)\rangle_{\beta,\mu} = -\langle \theta_{xt}(t,x,2)\rangle_{\beta,\mu}=
\qquad \qquad \qquad \nonumber \\ 
\frac{v^2}{8\pi \zeta_+^2}[(\mu_1^2-\mu_2^2)\sin^2 \theta  -\mu_1\mu_2 \sin 2\theta ] + 
\frac{\pi v^2\sin^2 \theta}{12}\left (\frac{1}{\beta_1^2}-\frac{1}{\beta_2^2}\right ) \, ,  
\label{ex4}\end{aligned}$$ for both families in (\[ex1\]).

Quantum noise
-------------

In this section we derive the [*noise power*]{} in the TL junction in Fig. \[fig1\]. For this purpose we need [@bb-00] the two-point [*connected*]{} current-current correlator $$\begin{aligned}
\langle j_+(t_1,x_1,i) j_+(t_2,x_2,j) \rangle_{\beta, \mu}^{\rm conn} \equiv 
\qquad \qquad \qquad \qquad 
\nonumber \\
\langle j_+(t_1,x_1,i) j_+(t_2,x_2,j) \rangle_{\beta, \mu} - 
\langle j_+(t_1,x_1,i)\rangle_{\beta, \mu}\langle j_+(t_2,x_2,j) \rangle_{\beta, \mu}\, . 
\label{N1}\end{aligned}$$ After some algebra one finds $$\begin{aligned}
\langle j_+(t_1,x_1,i) j_+(t_2,x_2,j) \rangle_{\beta, \mu}^{\rm conn} \equiv 
\qquad \qquad \qquad \qquad 
\nonumber \\
\left (\frac{v}{2 v_f \zeta_+}\right )^2 
\Biggl \{\frac{1}{\beta_i^2 \sinh^2\left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) -\ri \varepsilon \right ]} \delta_{ij} +
\sum_{l=1}^n \S_{il} \frac{1}{\beta_l^2 \sinh^2\left [\frac{\pi}{\beta_l}(vt_{12}- x_{12}) -
\ri \varepsilon \right ]} \S_{lj}^t
\quad \nonumber \\
 -\S_{ij} \frac{1}{\beta_j^2 \sinh^2\left [\frac{\pi}{\beta_j}(vt_{12}-{\widetilde{x}}_{12}) -\ri \varepsilon \right ]}
- \frac{1}{\beta_i^2 \sinh^2\left [\frac{\pi}{\beta_i} (vt_{12}+{\widetilde{x}}_{12}) -\ri \varepsilon \right ]} \S^t_{ij} \Biggr \} \, . \quad 
\label{N2}\end{aligned}$$ One easily verifies that the equilibrium limit ($\beta_i \to \beta$ for all $i$) of (\[N2\]) satisfies the KMS condition $$\langle j_+(t_1,x_1,i) [\varrho_{s+\ri \beta} j_+](t_2,x_2,j) \rangle_{\beta, \mu}^{\rm conn} = 
\langle [\varrho_{s} j_+](t_2,x_2,j) j_+(t_1,x_1,i)  \rangle_{\beta, \mu}^{\rm conn}\, ,
\label{currKMS1}$$ where the KMS automorphism $\varrho$ acts on $j_+$ as follows, $$[\varrho_s j_+](t,x,i) = j_+(t+s/v,x,i) \, . 
\label{currKMS2}$$

The explicit expression (\[N2\]) contains fundamental physical information about the NESS. First of all, since[^6] $$\langle j_+(t_1,x_1,i_1) j_+(t_2,x_2,i_2)\rangle_{\beta,\mu}^{\rm conn} \not = \\
\overline {\langle j_+(-t_1,x_1,i_1) j_+(-t_2,x_2,i_2)\rangle}{}_{\beta,\mu}^{\rm conn} 
\label{cc2}$$ the NESS breaks down [*time reversal*]{} invariance, even if the junction interaction preserves it, i.e. if $\S=\S^t$ [@Bellazzini:2009nk]. Nevertheless, [*time translation*]{} invariance is preserved, which allows one to use the conventional definition [@bb-00] of [*noise power*]{} $$P_{ij}(\beta; x_1, x_2 ; \omega) \equiv \int_{-\infty}^\infty {{\rm d}}t\,  \e^{i \omega t} \, 
\langle j_x(t,x_1,i) j_x(0,x_2,j) \rangle_{\beta, \mu}^{\rm conn}\, . 
\label{N3}$$ Eq. (\[N3\]) defines a complex matrix whose entries can be expressed [@PBM] in terms of the hypergeometric function ${}_2F_1$, namely $$\begin{aligned}
P_{ij}(\beta; x_1, x_2 ; \omega) = \left (\frac{v}{2 v_F \zeta_+}\right )^2 
\bigl \{
\bigl [F_-(\omega, \beta_i, x_{12}) - F_+(\omega, \beta_i,x_{12})\bigr ] \beta_i^{-1} \delta_{ij} + 
\qquad \qquad \nonumber \\ 
\sum_{l=1}^n\S_{il}\beta_l^{-1}\bigl [F_-(\omega, \beta_l, -x_{12}) - F_+(\omega, \beta_l, -x_{12})\bigr ]\S^t_{lj} -
\qquad \qquad \qquad \qquad \nonumber \\
\S_{ij}\beta_j^{-1}\bigl [F_-(\omega, \beta_j,-{\widetilde{x}}_{12}) - F_+(\omega, \beta_j,-{\widetilde{x}}_{12})\bigr ] - 
\bigl [F_-(\omega, \beta_i, {\widetilde{x}}_{12}) - F_+(\omega, \beta_i,{\widetilde{x}}_{12})\bigr ] \beta_i^{-1} \S^t_{ij} 
\bigl \}\, , \nonumber \\
\label{A1}\end{aligned}$$ with $$F_\pm (\omega, \beta, x) = \frac{\e^{\pm 2\pi x/\beta }}{\ri \omega \beta \pm 2\pi v}\;  
{}_2F_1 \left (2, 1\pm \frac{\ri \omega \beta}{2 \pi v}, 2 \pm  
\frac{\ri \omega \beta}{2 \pi v}, \e^{\pm 2\pi x/\beta }\right )\, . 
\label{A2}$$ From (\[A1\], \[A2\]) one can deduce the zero-frequency limit ([*zero-frequency noise power*]{}) $$P_{ij}(\beta) \equiv \lim_{\omega \to 0^+} P_{ij}(\beta; x_1, x_2 ; \omega) \, . 
\label{N4}$$ Using $$\lim_{\omega \to 0^+} \left [F_- (\omega; \beta, x) - 
F_+ (\omega; \beta, x)\right ] = \frac{1}{2\pi v}\, , 
\label{N5}$$ one gets $$P_{ij}(\beta) = 
\frac{G(g_-,\k)}{v_F}
\left (\beta_i^{-1}\delta_{ij} - \S_{ij}\beta_j^{-1} - \beta_i^{-1} \S^t_{ij} + 
\sum_{l=1}^n \S_{il}\beta_l^{-1} \S^t_{lj} \right )\, , 
\label{N6}$$ where $G$, defined by (\[ct3\]), captures the dependence (see Fig.\[fig7\]) of the noise on the statistical parameter $\k$. As expected, $P_{ij}(\beta)$ turns out to be a $x_{1,2}$-independent real symmetric matrix. If the electric charge is conserved ($\S \in O_{\bf v}$), the noise power (\[N6\]) satisfies in addition the Kirchhoff rule $$\sum_{i=1}^n P_{ij}(\beta ) = \sum_{j=1}^n P_{ij}(\beta ) = 0\, .  
\label{N7}$$ The expression (\[N6\]) admits the typical Johnson-Nyquist $\beta^{-1}$ behavior and shows the non-trivial interplay between the different temperatures and the scattering matrix. For example, in the two-terminal case with $\S=\S^+$ one finds $$P^+ = \frac{G}{v_Fk_B}\left(\begin{array}{cc}T_1(1-\cos \theta)^2 +T_2 \sin^2 \theta  & 
(T_1-T_2)(1-\cos \theta)\sin \theta \\ 
(T_1-T_2)(1-\cos \theta)\sin \theta & 
T_2(1-\cos \theta)^2 + T_1\sin^2 \theta \\ \end{array} \right)\, , 
\label{N8}$$ where $T=(k_B\beta)^{-1}$ is the absolute temperature and $k_B$ is the Boltzmann constant. The eigenvalues of $P^+$ $$p^+_i= \frac{2G}{v_Fk_B}(1-\cos \theta)T_i\, \geq 0\, , \qquad 
i=1,2 
\label{N9}$$ are nonnegative in agreement with the positivity of the two-point function (\[N1\]). Analogous result holds for $P^-$ corresponding to $\S^-$.

Outlook and conclusions
=======================

In this paper we constructed and investigated an exactly solvable model of a non-equilibrium Luttinger junction. The basic points of our approach are:

(i)

:   a scale invariant point-like interaction, which is described by a scattering matrix $\S$ and drives the system away from equilibrium;

(ii)

:   a representation generated by a NESS $\Omega_{\beta, \mu}$, which encodes the point-like interaction in the chiral fields ${\varphi_{i,Z}}$;

(iii)

:   an exact operator solution of the TL model (in terms of ${\varphi_{i,Z}}$) on a star graph with the current splitting boundary condition in the vertex;

(iv)

:   an extension of the conventional fermion Luttinger liquid to anyon statistics.

Combining these ingredients, we derived the basic correlation functions in the state $\Omega_{\beta, \mu}$. The essential characteristic features of these functions are:

(a)

:   the non-equilibrium two-point anyon correlations are products of $\S$-dependent powers of equilibrium correlations at the temperatures and chemical potentials of the heat baths, connected to the leads;

(b)

:   accordingly, the corresponding momentum space distribution is the convolution of equilibrium anyon distributions at different temperatures and chemical potentials;

(c)

:   the Fourier transform of the leading terms in the large distance expansion of the anyon correlations gives Cauchy-Lorentz distributions, which after convolution reproduce themselves with appropriate width and median;

(d)

:   in the critical limit one has a $c=1$ conformal field theory with $\S$-dependent anomalous dimensions, which are explicitly derived;

(e)

:   the expected breakdown of time reversal invariance is manifest in the current-current correlator.

We investigated in detail the energy and charge transport in the junction for all values of the statistical parameter. The energy is conserved for $\S \in O(n)$, which covers both possibilities of a junction without and with electric charge dissipation. In the latter case we determined the exact expression for the charge flow leaving or entering the junction. The connected current-current correlation is a linear combination of hypergeometric functions. The associated zero-frequency noise power depends linearly on the temperatures.

Our investigation above has been focused essentially on the critical properties of anyon Luttinger liquids away from equilibrium. It will be interesting to study the noncritical aspects as well. The generalization of the results of this paper beyond the Luttinger liquid paradigm, when the nonlinearity of the dispersion relation becomes essential, is also a challenging open problem.

We thank B. Douçot and I. Safi for an inspiring discussion, which stimulated our interest in non-equilibrium Luttinger liquids. M.M. would like also to thank the Laboratoire de Physique Théorique d’Annecy-le-Vieux for the kind hospitality during the preparation of the manuscript.

Large space separation asymptotics
==================================

The behavior of the correlators (\[panc1\],\[panc2\]) at large large space separation $|x_{12}| \gg \beta_i$ is encoded in $$\begin{aligned}
C_{11}(x,i;\beta,\mu) &=& \left (\frac{1}{2\ri}\right )^\k \left (\frac{2\pi}{\beta_i}\right )^\k 
L\left (x;\frac{\pi \k}{\beta_i},-\mu_i\right ) + \cdots \, ,  
\label{ld1} \\ 
C_{22}(x,i;\beta,\mu) &=&  \left (\frac{1}{2\ri}\right )^\k \prod_{k=1}^n \left (\frac{2\pi}{\beta_k}\right )^\k 
L\left (x;\frac{\pi \k\S^2_{ik}}{\beta_k},-\S_{ik}\mu_k\right ) +\cdots \, ,  
\label{ld2}\end{aligned}$$ where $x\gg \beta_i >0$, the dots stand for sub-leading contributions and $$L(x;\gamma,\mu) \equiv \frac{1}{2\pi} \e^{-\ri \mu x - \gamma x}\, , \quad x>0\, . 
\label{ld3}$$ The Fourier transform $${\widehat{L}}(p;\gamma,\mu) \equiv \int_{-\infty}^\infty {{\rm d}}x\, \e^{\ri px} L(x;\gamma,\mu) = 
\frac{1}{\pi} \frac{\gamma}{\gamma^2 + (p-\mu)^2} 
\label{ld4}$$ is the familiar Cauchy-Lorentz distribution[^7], where $\gamma$ is the half width at half maximum and $\mu$ is the statistical median. Using that the class of Cauchy-Lorentz distributions is closed under convolution, one finds $$\begin{aligned}
{\widehat{C}}_{11}(p,i;\beta,\mu) &\equiv&  \int_{-\infty}^\infty {{\rm d}}x\, \e^{\ri px} C_{11}(x,i;\beta,\mu) 
\sim {\widehat{L}}\left (p;\frac{\pi \k}{\beta_i},-\mu_i\right ) + \cdots \, ,  
\label{ld5} \\ 
{\widehat{C}}_{22}(p,i;\beta,\mu) &\equiv&  \int_{-\infty}^\infty {{\rm d}}x\, \e^{\ri px} C_{22}(x,i;\beta,\mu) 
\sim {\widehat{L}}\left (p;\pi \k\sum_{j=1}^n\frac{\S^2_{ij}}{\beta_j},-\sum_{j=1}^n\S_{ij}\mu_j\right ) +\cdots \, . 
\nonumber \\
\label{ld6}\end{aligned}$$

Summarizing, the Fourier transform of the leading term in the long distance expansion of both (\[panc1\]) and (\[panc2\]) is a Cauchy-Lorentz distribution. Notice that the width and the median of (\[ld6\]) depend on the temperatures and chemical potentials of all heat baths, as well as on $\S$.

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[^1]: For some more recent reviews we refer to [@voit-95]-[@egg-08].

[^2]: Similar results hold [@PKA09; @Calabrese:2008fv] for the Lieb-Lineger and Calogero-Sutherland models.

[^3]: This choice will not prevent us to deal in the fermion case below with arbitrary $\mu_i$.

[^4]: The $\ri \varepsilon$ prescription, adopted throughout the paper, indicates as usual the weak limit $\varepsilon \to 0^+$.

[^5]: Without loss of generality we assume in what follows $\tau \geq 0$ and $\tau \not= \pm \sigma$.

[^6]: The bar indicates complex conjugation.

[^7]: Known also as non-relativistic Breit-Wigner distribution.
