Linear quantum quench in the Heisenberg XXZ chain: Time-dependent Luttinger-model description of a lattice system

Abstract

We study variable-rate linear quenches in the anisotropic Heisenberg (XXZ) chain, starting at the XX point. This is equivalent to swithcing on a nearest neighbour interaction for hard-core bosons or an interaction quench for free fermions. The physical observables we investigate are: the energy pumped into the system during the quench, the spin-flip correlation function, and the bipartite fluctuations of the z component of the spin in a box. We find excellent agreement between exact numerics (infinite system time-evolving block decimation, iTEBD) and analytical results from bosonization, as a function of the quench time, spatial coordinate and interaction strength. This provides a stringent and much-needed test of Luttinger liquid theory in a non-equilibrium situation.PACS numbers: 71.10. Pm,75.10.Jm,05.70.Ln, While it is difficult to study genuine non-equilibrium dynamics in solid state systems due to the presence of many relaxation channels (phonons, impurities, interactions etc.), cold atoms in optical lattices provide an ideal laboratory for non-equilibrium investigations due to the high degree of control over various dissipation mechanisms. Cold-atom experiments in the past decade have explored a wide variety of non-equilibrium quantum dynamics in previously inaccessible regimes [1,2]. This has also led to an increasing amount of theoretical activity [2,3]. Key issues include thermalization as well as equilibration and their relation to integrability [2], pumping beyond the adiabatic limit or quantum fluctuation relations [4], and universal near-adiabatic dynamics in quantum critical systems [2,3]. Linear quenches occuring over a finite time can interpolate between the more familiar limits of an instantaneous quench and an adiabatic sweep. Very recently, a few experiments have examined the response of many-body experiments to such finitetime quenches [5,6]. It is thus of vital current interest to address the dynamics under linear sweeps of system parameters such as interaction strength.The response of a system to an external perturbation depends sensitively on its spatial dimension, as famously demonstrated in the experiment of Ref. [7]. There, one dimensional interacting bosons did not reach thermalization within the experimental timescale, while their higher dimensional realizations did. One dimensional systems are notoriously strongly correlated due to the limited phase space for scattering. The non-interacting ground state is immediately destroyed by interactions, forming a Luttinger liquid (LL) in many instances [8,9], and described by critical phenomena of collective modes with anomalous (non-integer) power-law dependence of correlation functions.Quantum quenches in Luttinger liquids have been addressed by several authors [10][11][12][13][14][15][16][17]. However, it is not clear to what extent the Luttinger liquid (LL) picture, which is a genuine low energy description, is applicable under non-equilibrium circumstances [17]. For an abrupt interaction change, certain observables revealed universal LL ...

Time evolution during and after finite-time quantum quenches in Luttinger liquids

“…A similar analysis has been performed for the dynamics of several observables in the XXZ chain after sudden quenches 13,14,17,29,33 as well as during linear ramps in the anisotropy. 44 …”

“…In the limit of hard-core bosons the Green function (71) corresponds to the spin-flip correlation function in the XXZ chain, which was studied by Pollmann et al 44 during linear quenches. More recently, Bernier et al 45 analysed the bosonic Green function during a linear ramp of the interaction strength and identified the front at which correlations form [similar to the second term in Eq.…”

“…Most previous works investigating such a setup have focused on linear quenches in which the interactions are ramped up at a constant speed. [43][44][45]51 We consider quench protocols starting from the non-interacting model and ranging over a finite quench time τ , ie, the coupling functions satisfy…”

“…They obtained perturbative results for the total energy and fermionic chiral Green function, which were later extended to spinspin correlation functions and compared with numerical simulations of the time evolution during the quench in the XXZ Heisenberg chain. 44 Bernier et al 45 went beyond the perturbative treatment by deriving exact results for the time evolution during linear quenches in terms of Bessel functions (see Sec. IV A).…”

Memory-preserving equilibration after a quantum quench in a one-dimensional critical model

“…In particular in the case of an instantaneous interaction quench in the Lieb-Liniger model, the presence of high-energy excitations [31,33] makes the LL approximation non-applicable and at least the special cases studied in the literature seem to suggest that the GGE expressed in terms of the exact Lieb-Liniger conserved charges is correct [33,34,38,44,45,71]. Nevertheless the LL approach turns out to describe correctly the quench dynamics in several other cases [72][73][74]. Quantum quenches in LL have been studied in [15-17, 75, 76] and a detailed comparison of our results with earlier works is presented in the Supplemental Material.…”