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arXiv:1505.03080 · Journal of Physics A Mathematical and Theoretical

Relaxation dynamics of local observables in integrable systems

Jacopo De Nardis, Lorenzo Piroli, Jean-Sébastien Caux

Published Oct 5, 2015• 61 citations
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Abstract

We show, using the quench action approach (Caux and Essler 2013 Phys. Rev. Lett. 110 257203 ), that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density–density correlation in the interacting Lieb–Liniger gas after a quench from a Bose–Einstein condensate. This shows an approach to equilibrium with power law t −3 which turns out to be independent of the post-quench interaction and of the considered observable.

DOI
10.1088/1751-8113/48/43/43ft01

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We show, using the quench action approach (Caux and Essler 2013 Phys. Rev. Lett. 110 257203 ), that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density–density correlation in the interacting Lieb–Liniger gas after a quench from a Bose–Einstein condensate. This shows an approach to equilibrium with power law t −3 which turns out to be independent of the post-quench interaction and of the considered observable.
We give a pedagogical introduction to the methodology of the Quench Action, which is an effective representation for the calculation of time-dependent expectation values of physical operators following a generic out-of-equilibrium state preparation protocol (for example a quantum quench). The representation, originally introduced in Caux and Essler (2013 Phys. Rev. Lett . 110 257203), is founded on a mixture of exact data for overlaps together with variational reasonings. It is argued to be quite generally valid and thermodynamically exact for arbitrary times after the quench (from short times all the way up to the steady state), and applicable to a wide class of physically relevant observables. Here, we introduce the method and its language, give an overview of some recent results, suggest a roadmap and offer some perspectives on possible future research directions.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.
Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.
We consider the time evolution of local observables after an interaction quench in the repulsive Lieb-Liniger model. The system is initialized in the ground state for vanishing interaction and then time-evolved with the Lieb-Liniger Hamiltonian for large, finite interacting strength c. We employ the Quench Action approach to express the full time evolution of local observables in terms of sums over energy eigenstates and then derive the leading terms of a 1/c expansion for several one and two-point functions as a function of time t>0 after the quantum quench. We observe delicate cancellations of contributions to the spectral sums that depend on the details of the choice of representative state in the Quench Action approach and our final results are independent of this choice. Our results provide a highly non-trivial confirmation of the typicality assumptions underlying the Quench Action approach.
The quench action formalism relies on the assumption that the overlap between a generic initial state $\left|Ψ_{0}\right\rangle $ and an eigenstate of an integrable model - defined through the rapidities $\left|k_{1},...k_{N}\right\rangle $ - can be written as: \begin{equation} \left\langle k_{1},...k_{N}\midΨ_{0}\right\rangle =\exp\left(-S_{Ψ_{0}}\left(ρ\left(k\right)\right)\right),\label{eq:Exponential} \end{equation} where ρ(k)ρ\left(k\right)ρ(k) is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and SΨ0S_{Ψ_{0}}SΨ0​​ is some smooth function of ρ(k)ρ\left(k\right)ρ(k) that depends on Ψ0Ψ_{0}Ψ0​. In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density ρ(k)ρ\left(k\right)ρ(k). In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the 1/c1/c1/c expansion for the overlap (with ccc being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.