Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

Abstract

We study the return probability and its imaginary (τ) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy |∆| < 1. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to τ 2 , which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region |∆| < 1 the decay for large times t is nowhere continuous as a function of anisotropy: it is either gaussian at root of unity or exponential otherwise. As an aside, we also determine that the front moves as x f (t) = t √ 1 − ∆ 2 , by analytic continuation of known arctic curves in the six vertex model. Exactly at |∆| = 1, we find the return probability decays as e −ζ(3/2) √ t/π t 1/2 O(1). It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench. arXiv:1707.06625v3 [cond-mat.stat-mech]

Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

“…When presenting our results in the next section, we compare them to the prior work in detail. We remark that inhomogeneous quenches in one-dimensional systems are being addressed in a wider context of interacting integrable models such as the XXZ chain [42,43] (the result of the latter reference for the Loschmidt echo is consistent with our result in the appropriate limit) and nonintegrable models such as the interacting resonant level model [44][45][46][47], nonintegrable Ising spin chain [48] and various models of the molecular and superconducting junctions [49][50][51][52].…”

Dynamics and level statistics of interacting fermions in the lowest Landau level

“…In the 'tight-binding chain' picture, this would mean unit filling for the first third or last third of the chain, and zero filling for the rest of the chain. This would then be analogous to the dynamics considered in, e.g. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], for spin or tight-binding systems, when the initial state has all fermions or all -spins on one side of the chain. In such cases, the single-particle hopping leads to transport of particles from the filled side to the empty side, with a current-carrying non-equilibrium steady state arising dynamically within the light cone region.…”

“…This has an obvious analogy in fermionic tight-binding chains, and is also analogous to a spin chain where some contiguous block start in a spin-up state and the rest start in a spin-down state, e.g.the 'domain wall' state        ñ | . For tight-binding fermionic chains and spin chains, this type of dynamics by now has been studied in some detail [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Related situations, e.g.when one/both of the regions are not completely full/empty, are also of interest and sometimes involve similar physics [23,[32][33][34][35][36][37][38][39][40][41].…”

“…For local Hamiltonians, an intermediate current-carrying region grows linearly as a 'light cone', within which a nonequilibrium steady state emerges. The form of the non-equilibrium steady state [14,21,23], the nature of the boundary of the light-cone [22,23], spreading of entanglement [23], and the effects of interactions [15,16,21,28] and integrability [25,27] are some of the many aspects that have been investigated. In this literature, the spreading dynamics is driven by single-particle hopping, as opposed to the correlated pair hopping driving our LLL dynamics.…”

Quantum return probability of a system of N non-interacting lattice fermions

“…For instance, for free lattice fermions with domain-wall initial condition, the return probability was found to obey a pure Gaussian decay in time [37,38]. More recently, the XXZ quantum spin chain with domain-wall initial condition has been investigated in the whole massless phase, where the anisotropy parameter obeys |∆| < 1 [39]. There, the decay of the return probability is found to be either Gaussian or exponential, depending on whether the angle parameter γ (defined by ∆ = cos γ) is commensurate to π or not.…”