Breakdown of the Generalized Gibbs Ensemble for Current-Generating Quenches

Abstract

We establish a relation between two hallmarks of integrable systems: the relaxation towards the generalized Gibbs ensemble (GGE) and the dissipationless charge transport. We show that the former one is possible only if the so called Mazur bound on the charge stiffness is saturated by local conserved quantities. As an example we show how a non-GGE steady state with a current can be generated in the one-dimensional model of interacting spinless fermions with a flux quench. Moreover an extended GGE involving the quasi-local conserved quantities can be formulated for this case.PACS numbers: 75.10.Pq, 05.60.Gg,05.70.Ln Recent advances in experiments on ultracold atoms together with new computational techniques have significantly broadened our understanding of relaxation processes in closed many-body quantum systems. It is commonly accepted that in generic macroscopic systems the long-time averages of local observables coincide with the results for the statistical Gibbs ensemble [1][2][3][4] and are uniquely determined by few parameters related to conserved quantities, in particular the system's energy and particle number. Due to the presence of macroscopic number of conserved quantities such a simple scenario is not applicable to integrable systems [5][6][7]. However, there is a large and still growing evidence that relaxation in the latter systems is consistent with the generalized Gibbs ensemble (GGE) [8][9][10][11][12], where the density matrix is determined not only by the Hamiltonian H and particle number N but also by other local conserved quantitiesIn this Letter we focus on the relaxation dynamics of one of the most studied integrable models: the model of interacting spinless fermions, being equivalent to the anisotropic Heisenberg (XXZ) model for which the set of Q i has been established [13,14]. We show that ρ GGE as generated only by local integrals of motion Q i doesn't exhaust all generic stationary states in the metallic (easy plane) regime. Instead, there are cases for which one should lift the requirement of locality of the conserved quantities and allow also for quasi-local integrals of motion [15,16]. In this Letter we call them non-GGE states, however we stress that these states can be viewed also as "extended GGE", where the extension concerns the locality of operators. Such operators have the parity opposite to local ones Q i . We identify one of such quasi-local quantities as the time-averaged particle current operator and we construct as well as verify it explicitly.It has been well recognized that integrable systems in spite of interaction reveal anomalous transport properties at finite inverse temperatures β = 1/T , e.g. the dissipationless particle current. This property is manifested by a nonvanishing charge stiffness D(β < ∞) [17][18][19], which in turn is bounded from below by the local conservation laws via the Mazur bound [18,20]. The dissipationless transport and the relaxation towards GGE are probably the most prominent hallmarks of integrability, still they have been studied indep...

Entanglement prethermalization in the Tomonaga-Luttinger model

“…, (34) because it is known that integrable systems relax to the GGE [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. Here, the parameters {λ k↑↓ } are determined by the initial values of the conserved quantities…”

Finite-temperature charge transport in the one-dimensional Hubbard model

“…If this initial condition leads to a finite value of jQ i , then the current will never completely decay back to zero. A simple example is the quench of a flux piercing a ring, which has been studied in this context [37].…”

Non-equilibrium dynamics of isolated quantum systems

“…This issue has been considered already in a few manuscripts appeared in the last months [69][70][71][72][73], but a conclusive answer has not yet been provided. The most probable scenario concerns the presence of additional charges which are not yet explicitly known and that do not appear in the standard GGE construction [55].…”

“…For example, it is well known that in the gapless regime (|Δ| < 1) and in the presence of a current, there are additional quasi-local (i.e. with a kernel decaying with the distance and not with support on a finite interval as the standard ones) which must be added to the GGE [71]. These charges however do not explain the GGE failure in the gapped regime.…”