Luttinger Liquid in non-Equilibrium Steady State

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 , 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 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.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 , 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 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.

arXiv:1210.5409Journal of Physics A Mathematical and TheoreticalFeb 15, 201338 citations

Interacting fermions and non-equilibrium properties of one-dimensional many-body systems

Per Moosavi

Recent experimental progress on ultracold atomic gases have opened up the possibility to simulate many-body systems out of equilibrium. We consider such a system described by the Luttinger model, which is a model of interacting fermions in one spatial dimension. It is well known that the Luttinger model is exactly solvable using bosonization. This also remains true for certain extensions of the model, e.g., where, in addition, the fermions are coupled to phonons. We give a self-contained account of bosonization, together with complete proofs, and show how this can be used to solve the Luttinger model and the above fermion-phonon model rigorously. The main focus is on non-equilibrium properties of the Luttinger model. We use the exact solution of the Luttinger model, with non-local interactions, to study the evolution starting from a non-uniform initial state with a position-dependent chemical potential. The system is shown to reach a current-carrying final steady state, in which the universal value of the electrical conductance, known from near-to-equilibrium settings, is recovered. We also study the effects of suddenly changing the interactions and show that the final state has memory of the initial state, which is, e.g., manifested by non- equilibrium exponents in its fermion two-point correlation functions.Recent experimental progress on ultracold atomic gases have opened up the possibility to simulate many-body systems out of equilibrium. We consider such a system described by the Luttinger model, which is a model of interacting fermions in one spatial dimension. It is well known that the Luttinger model is exactly solvable using bosonization. This also remains true for certain extensions of the model, e.g., where, in addition, the fermions are coupled to phonons. We give a self-contained account of bosonization, together with complete proofs, and show how this can be used to solve the Luttinger model and the above fermion-phonon model rigorously. The main focus is on non-equilibrium properties of the Luttinger model. We use the exact solution of the Luttinger model, with non-local interactions, to study the evolution starting from a non-uniform initial state with a position-dependent chemical potential. The system is shown to reach a current-carrying final steady state, in which the universal value of the electrical conductance, known from near-to-equilibrium settings, is recovered. We also study the effects of suddenly changing the interactions and show that the final state has memory of the initial state, which is, e.g., manifested by non- equilibrium exponents in its fermion two-point correlation functions.

Jan 1, 20160 citations

Transport properties of a two-lead Luttinger-liquid junction out of equilibrium: Fermionic representation

D. N. Aristov, P. Wölfle

The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage $V$ , if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage $V$, if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.

arXiv:1408.4914Physical Review BDec 8, 201411 citations

Many-particle correlations in a non-equilibrium Luttinger liquid

I V Protopopov, D B Gutman, A. D. Mirlin

We develop an operator-based approach to the problem of Luttinger liquid conductor in a non-equilibrium stationary state. We show that the coherent-state many-body fermionic density matrix as well as all fermionic correlation functions out of equilibrium are given by one-dimensional functional determinants of the Fredholm type. Thus, the model constitutes a remarkable example of a many-body problem where all the correlation functions can be evaluated exactly. On the basis of the general formalism we investigate four-point correlation functions of the fermions coming out of the Luttinger liquid wire. Obtained correlations in the fermionic distribution functions represent the combined effect of interaction and non-equilibrium conditions.We develop an operator-based approach to the problem of Luttinger liquid conductor in a non-equilibrium stationary state. We show that the coherent-state many-body fermionic density matrix as well as all fermionic correlation functions out of equilibrium are given by one-dimensional functional determinants of the Fredholm type. Thus, the model constitutes a remarkable example of a many-body problem where all the correlation functions can be evaluated exactly. On the basis of the general formalism we investigate four-point correlation functions of the fermions coming out of the Luttinger liquid wire. Obtained correlations in the fermionic distribution functions represent the combined effect of interaction and non-equilibrium conditions.

arXiv:1107.5561Journal of Statistical Mechanics Theory and ExperimentNov 2, 201116 citations

Quench dynamics of the Tomonaga–Luttinger model with momentum-dependent interaction

J Rentrop, Dirk Schuricht, V. Meden

We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.

arXiv:1203.0932New Journal of PhysicsJul 3, 201246 citations

Quench dynamics of the Tomonaga-Luttinger model with momentum-dependent interaction

Rentrop, J., Dirk Schuricht, V. Meden

Jan 1, 201251 citations

V

, if

V

is the largest energy scale compared to temperature

T

and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.