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arXiv:1701.06620 · Physical review. B./Physical review. B

Time evolution of the Luttinger model with nonuniform temperature profile

Edwin Langmann A, Joel L. Lebowitz, Vieri Mastropietro, Per Moosavi

Published Jun 23, 2017• 48 citations
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Abstract

We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile $T(x)$. As a specific example we consider the case when $T(x)$ is equal to ${T}_{L}$ $({T}_{R})$ far to the left (right). Using a series expansion in $\ensuremath{\epsilon}=2({T}_{R}\ensuremath{-}{T}_{L})/({T}_{L}+{T}_{R})$, we compute the energy density, the heat current density, and the fermion two-point correlation function for all times $t\ensuremath{\ge}0$. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of $T(x)$. For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in $\ensuremath{\epsilon}$. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on ${T}_{L,R}$ agrees with numerical results for the $XXZ$ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.

DOI
10.1103/physrevb.95.235142

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We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile T(x)T(x)T(x). As a specific example we consider the case when T(x)T(x)T(x) is equal to TL{T}_{L}TL​ (TR)({T}_{R})(TR​) far to the left (right). Using a series expansion in \ensuremathϵ=2(TR\ensuremath−TL)/(TL+TR)\ensuremath{\epsilon}=2({T}_{R}\ensuremath{-}{T}_{L})/({T}_{L}+{T}_{R})\ensuremathϵ=2(TR​\ensuremath−TL​)/(TL​+TR​), we compute the energy density, the heat current density, and the fermion two-point correlation function for all times t\ensuremath≥0t\ensuremath{\ge}0t\ensuremath≥0. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of T(x)T(x)T(x). For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on TL,R{T}_{L,R}TL,R​ agrees with numerical results for the XXZXXZXXZ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.
Recently, remarkably simple exact results were presented about the dynamics of heat transport in the local Luttinger model for nonequilibrium initial states defined by position-dependent temperature profiles. We present mathematical details on how these results were obtained. We also give an alternative derivation using only algebraic relations involving the energy-momentum tensor which hold true in any unitary conformal field theory (CFT). This establishes a simple universal correspondence between initial temperature profiles and the resulting heat-wave propagation in CFT. We extend these results to larger classes of nonequilibrium states. It is proposed that such universal CFT relations provide benchmarks to identify nonuniversal properties of nonequilibrium dynamics in other models.
Recent experimental advances on ultracold atomic gases and trapped ions have made it possible to simulate exactly solvable quantum systems of interacting particles. In particular, the feasibility of making rapid changes, so-called quantum quenches, to such set-ups has allowed experimentalists to probe non-equilibrium phenomena in closed interacting quantum systems. This, in turn, has spurred a considerable theoretical interest in quantum many-body systems out of equilibrium. In this thesis, we study non-equilibrium properties of quantum many-body systems in the framework of exactly solvable quantum field theory in one spatial dimension. Specific systems include interacting fermions described by the Luttinger model and effective descriptions of spin chains using conformal field theory (CFT). Special emphasis is placed on heat and charge transport, studied from the point of view of quench dynamics, and, in particular, the effects of breaking conformal symmetries on transport properties. Examples include the Luttinger model with non-local interactions, breaking Lorentz and scale invariance, and inhomogeneous CFT, which generalizes standard CFT in that the usual propagation velocity v is replaced by a function v(x) that depends smoothly on the position x, breaking translation invariance. The quench dynamics studied here is for quantum quenches between, in general, different smooth inhomogeneous systems. An example of this is the so-called smooth-profile protocol, in which the initial state is defined by, e.g., smooth inhomogeneous profiles of inverse temperature and chemical potential, and the time evolution is governed by a homogeneous Hamiltonian. Using this protocol, we compute exact analytical results for the full time evolution of the systems mentioned above. In particular, we derive finite-time results that are universal in the sense that the same relations between the non-equilibrium dynamics and the initial profiles hold for any unitary CFT. These results also make clear that heat and charge transport in standard CFT are purely ballistic. Finally, we propose and study an inhomogeneous CFT model with v(x) given by a random function. We argue that this model naturally emerges as an effective description of one-dimensional quantum many-body systems with certain static random impurities. Using tools from wave propagation in random media, we show that such impurities lead to normal and anomalous diffusive contributions to heat transport on top of the ballistic one known from standard CFT.
We establish a nonequilibrium scaling regime in the short-time evolution of one-dimensional interacting open quantum systems subject to a generic heating mechanism. This dynamical regime is characterized by uncompensated phonon production and a superdiffusive, universal scaling of quasiparticle lifetimes with momentum $\ensuremath{\sim}{q}^{\ensuremath{-}5/3}$, distinct from the finite- and zero-temperature cases. It is separated from a high-momentum regime by a time-dependent scale fading out as ${q}_{0}(t)\ensuremath{\sim}{t}^{\ensuremath{-}4/5}$. In the latter region we observe thermalization to an effective time-dependent equilibrium with linearly increasing temperature. By mapping out the dynamical phase diagram and computing the dynamical structure factor within an open-system Keldysh functional integral approach, we show how these predictions can be explored in cold-atom experiments by means of Bragg spectroscopy.
One of the most straightforward ways to study thermal properties beyond linear response is to monitor the relaxation of an arbitrarily large left-right temperature gradient ${T}_{L}\ensuremath{-}{T}_{R}$. In one-dimensional systems which support ballistic thermal transport, the local energy currents $\ensuremath{\langle}j(t)\ensuremath{\rangle}$ acquire a nonzero value at long times, and it was recently investigated whether or not this steady state fulfills a simple additive relation $\ensuremath{\langle}j(t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\rangle}=f({T}_{L})\ensuremath{-}f({T}_{R})$ in integrable models. In this paper, we probe the nonequilibrium dynamics of the Hubbard chain using density matrix renormalization group (DMRG) numerics. We show that the above form provides an effective description of thermal transport in this model; violations are below the finite-time accuracy of the DMRG. As a second setup, we study how an initially equilibrated system radiates into different nonthermal states (such as the vacuum).
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK{T}_{K}TK​ (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt{T}_{K}tTK​t. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.
We study the energy transport between two interacting spin chains which are initially separated, held at different temperatures, and subsequently put in contact. We consider the spin-1/2 XXZ model in the gapless regime and exploit its integrability properties to formulate an analytical ansatz for the nonequilibrium steady state even at temperatures where the low-energy Luttinger liquid description is not accurate. We apply our method to compute the steady energy current and benchmark it both with the known low-energy limit and at higher temperatures with numerical simulations. We find an excellent agreement even at high temperatures, where the Luttinger liquid prediction is shown to fail.