Finite-temperature transport in one-dimensional quantum lattice models

Abstract

The last decade has witnessed an impressive progress in the theoretical understanding of transport properties of clean, one-dimensional quantum lattice systems. Many physically relevant models in one dimension are Bethe-ansatz integrable, including the anisotropic spin-1/2 Heisenberg (also called spin-1/2 XXZ chain) and the Fermi-Hubbard model. Nevertheless, practical computations of, for instance, correlation functions and transport coefficients pose hard problems from both the conceptual and technical point of view. Only due to recent progress in the theory of integrable systems on the one hand and due to the development of numerical methods on the other hand has it become possible to compute their finite temperature and nonequilibrium transport properties quantitatively. Most importantly, due to the discovery of a novel class of quasilocal conserved quantities, there is now a qualitative understanding of the origin of ballistic finite-temperature transport, and even diffusive or super-diffusive subleading corrections, in integrable lattice models. We shall review the current understanding of transport in one-dimensional lattice models, in particular, in the paradigmatic example of the spin-1/2 XXZ and Fermi-Hubbard models, and we elaborate on state-of-the-art theoretical methods, including both analytical and computational approaches. Among other novel techniques, we discuss matrix-product-states based simulation methods, dynamical typicality, and, in particular, generalized hydrodynamics. We will discuss the close and fruitful connection between theoretical models and recent experiments, with examples from both the realm of quantum magnets and ultracold quantum gases in optical lattices.

Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54

“…Over the last two decades intense efforts by both experimentalists and theorists have greatly advanced our understanding of isolated quantum matter out of equilibrium [1][2][3][4][5][6][7][8][9]. It is now established that at asymptotically large times expectation values of local observables relax to time-independent numbers in translationally invariant systems [6,9], while they follow slow dynamics governed by emergent classical hydrodynamic equations [10][11][12] when the translational invariance is broken. This surprising onset of relaxation in isolated systems is induced by the effective bath created by the system on its own parts, and the final (quasi) stationary state of the system is determined by the conservation laws with local densities [13].…”

Quantum versus Classical Dynamics in Spin Models: Chains, Ladders, and Square Lattices

“…Compared to ∆ = 1, the main difference is a change of the exponent α from 2/3 to 1/2. Hence, this value indicates a diffusive decay, which is by now well known to occur the regime ∆ > 1, even in the case of the integrable quantum system 18 .…”

“…Moreover, plotted in a double-logarithmic representation [Fig. 1 (b)], we find that the hydrodynamic power-law tail C (M) (t) ∝ t −α at intermediate times is well described by α ≈ 2/3, which suggests superdiffusive transport within the Kardar-Parisi-Zhang (KPZ) universality class 18,19,57,59,62,63 (for more details see Sec. IV A 1 below).…”

“…In the context of the autocorrelation function C(t), the emergence of hydrodynamics reflects itself in terms of a a power-law tail 18 ,…”

“…From a theoretical point of view, quantum spin systems routinely serve as test beds to study concepts such as the eigenstate thermalization hypothesis [8][9][10][11][12] or the phenomenon of many-body localization 13,14 . Moreover, in the case of one-dimensional chain geometries, the integrability of certain spin models, accompanied by the existence of an extensive set of (quasi-)local conserved charges [15][16][17] , paves the way to obtain analytical insights, e.g., regarding their transport and relaxation behavior in the thermodynamic limit [18][19][20][21] . At the same time, the development of sophisticated numerical techniques [22][23][24] has significantly advanced our understanding of out-ofequilibrium processes in quantum spin models.…”