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arXiv:1409.5001 · Physical Review B

Escort distribution function of work done and diagonal entropies in quenched Luttinger liquids

Balázs Dóra

Published Dec 22, 2014• 10 citations
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Abstract

We study the escort probability distribution function of work done during an interaction quantum quench of Luttinger liquids. It crosses over from the thermodynamic to the small system limit with increasing $a$, which is the order of the escort distribution, and depends on the universal combination $[|{K}_{i}\ensuremath{-}{K}_{f}{|/({K}_{i}+{K}_{F})]}^{a}$, with ${K}_{i}$, ${K}_{f}$ the initial and final Luttinger liquid parameters, respectively. From its characteristic function, the diagonal R\'enyi entropies and the many-body inverse participation ratio (IPR) are determined to evaluate the information content of the time-evolved wave function in terms of the eigenstates of the final Hamiltonian. The hierarchy of overlaps is dominated by that of the ground states. The IPR exhibits a crossover from Gaussian to power-law decay with increasing interaction quench parameter.

DOI
10.1103/physrevb.90.245132

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•Jan 1, 2024•0 citations
We study the escort probability distribution function of work done during an interaction quantum quench of Luttinger liquids. It crosses over from the thermodynamic to the small system limit with increasing aaa, which is the order of the escort distribution, and depends on the universal combination [∣Ki\ensuremath−Kf∣/(Ki+KF)]a[|{K}_{i}\ensuremath{-}{K}_{f}{|/({K}_{i}+{K}_{F})]}^{a}[∣Ki​\ensuremath−Kf​∣/(Ki​+KF​)]a, with Ki{K}_{i}Ki​, Kf{K}_{f}Kf​ the initial and final Luttinger liquid parameters, respectively. From its characteristic function, the diagonal R'enyi entropies and the many-body inverse participation ratio (IPR) are determined to evaluate the information content of the time-evolved wave function in terms of the eigenstates of the final Hamiltonian. The hierarchy of overlaps is dominated by that of the ground states. The IPR exhibits a crossover from Gaussian to power-law decay with increasing interaction quench parameter.
We study the relaxation dynamics of a nonequilibrium Tomonaga-Luttinger model after a sudden interaction switch-on (``quench''), focusing on a double-step initial momentum distribution function. In the framework of the nonequilibrium bosonization, the results are obtained in terms of singular Fredholm determinants that are evaluated numerically and whose asymptotics are found analytically. While the quasiparticle weights decay exponentially with time after the quench, this is not a relaxation into a thermal state, in view of the integrability of the model. The steady-state distribution emerging at infinite times retains two edges which support Luttinger-liquid-like power-law singularities smeared by dephasing. The obtained critical exponents and the dephasing length are found to depend on the initial nonequilibrium state.
We study the nonequilibrium dynamics of the Luttinger model after a quantum quench, when the initial state is a finite temperature thermal equilibrium state. The diagonal elements of the density matrix in the steady state show thermal features for high temperature initial states only, otherwise they retain highly nonthermal character. The time evolution of Uhlmann fidelity, which measures the distance between the time evolved and initial states, is evaluated for arbitrary initial temperatures and quench protocols. In the long time limit, the overlap between the time evolved and initial system decreases exponentially with the temperature with a universal prefactor. Within perturbation theory, the statistics of final total energy and work are numerically evaluated in the case of a sudden quench, which yield identical distributions at zero temperature. In both statistics, temperature effects are more significant in small systems. The Dirac-$\ensuremath{\delta}$ peak at the adiabatic ground state energy remains present in the probability distribution of the total energy, but disappears from the work distribution at nonzero initial temperatures.
We analyze the probability distribution function (PDF) of work done on a Luttinger liquid for an arbitrary finite duration interaction quench and show that it can be described in terms of a generalized Gibbs ensemble. We construct the corresponding density matrix with explicit intermode correlations, and determine the duration and interaction dependence of the probability of an adiabatic transition and the PDF of nonadiabatic processes. In the thermodynamic limit, the PDF of work exhibits a non-Gaussian maximum around the excess heat, carrying almost all the spectral weight. In contrast, in the small system limit most spectral weight is carried by a delta peak at the energy of the adiabatic process, and an oscillating PDF with dips at energies commensurate to the quench duration and with an exponential envelope develops. Relevance to cold atom experiments is also discussed.
Luttinger liquids (LLs) arise by coupling left- and right-moving particles through interactions in one dimension. This most natural partitioning of LLs is investigated by the momentum-space entanglement after a quantum quench using analytical and numerical methods. We show that the momentum-space entanglement spectrum of a LL possesses many universal features both in equilibrium and after a quantum quench. The largest entanglement eigenvalue is identical to the Loschmidt echo, i.e., the overlap of the disentangled and final wave functions of the system. The second largest eigenvalue is the overlap of the first excited state of the disentangled system with zero total momentum and the final wave function. The entanglement gap is universal both in equilibrium and after a quantum quench. The momentum-space entanglement entropy is always extensive and saturates fast to a time independent value after the quench, in sharp contrast to a spatial bipartitioning.
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.
The quantized conductance was produced due to dissipation within the Luttinger liquid from the parity inversion and time reversal symmetric interaction. For unitary time with Hermitian Luttinger liquid Hamiltonian and produced nonequilibrium conductivity and was renormalized due to interaction. We have studied the nonequilibrium dynamics and transport of parity inversion symmetric Luttinger liquid and the model was quenched from domain wall. The study was extended to initial ground state of a Hermitian interacting Luttinger liquid. We have used the model using the time evolving block decimation and have found the conductance inside the light core and depended on post-quench Hamiltonian and was quantized when the system was quenched to the non Hermitian Hamiltonian. The conductance was found when the system was quenched from an interacting initial state and showed that the initial state had no effect on conductance. The strength of the initial or final interactions in the case of Hermitian dynamics showed the quenched displayed a conductance that remained universal.