We study the time evolution of a 1D interacting fermion system described by the Luttinger model starting from a non-equilibrium 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 = 2(T R − 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 ≥ 0. For local (delta-function) interaction, the first two are computed to all orders giving simple exact expressions involving the Schwarzian derivative of the integral of T (x). For non-local interaction, breaking scale invariance, we compute the non-equilibrium steady state (NESS) to all orders and the evolution to first order in . The heat current in the NESS is universal even when conformal invariance is broken by the interaction, 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 non-universal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.
We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian H λ with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t > 0 via a Hamiltonian H λ ′ which differs from H λ by the strength of the interaction. Asymptotically in time, as t → ∞, after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference µ + − µ − between right-(+) and left-(−) moving fermions obtained from the two-point correlation function. Both I and µ + − µ − depend on λ and λ ′ . Only for the case λ = λ ′ = 0 does µ + − µ − equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, G = I (µ + − µ − ), has a universal value equal to the conductance quantum e 2 h for the spinless case.
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.
We discuss an extension of the (massless) Thirring model describing interacting fermions in one dimension which are coupled to phonons and where all interactions are local. This fermion-phonon model can be solved exactly by bosonization. We present a construction and solution of this model which is mathematically rigorous by treating it as a limit of a Luttinger-phonon model. A self-contained account of the mathematical results underlying bosonization is included, together with complete proofs.Comment: 59 pages, LaTeX, 1 figure; minor updates to original submission, final published version; minor fix to agree with published versio
We propose and study a conformal field theory (CFT) model with random position-dependent velocity that, as we argue, naturally emerges as an effective description of heat transport in onedimensional quantum many-body systems with certain static random impurities. We present exact analytical results that elucidate how purely ballistic heat waves in standard CFT can acquire normal and anomalous diffusive contributions due to our impurities. Our results include impurity-averaged Green's functions describing the time evolution of the energy density and the heat current, and an explicit formula for the thermal conductivity that, in addition to a universal Drude peak, has a nontrivial real regular contribution that depends on details of the impurities.
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