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arXiv:2011.01159 · SciPost Physics

The folded spin-1/2 XXZ model: II. Thermodynamics and hydrodynamics with a minimal set of charges

Lenart Zadnik, Kemal Bidzhiev, Maurizio Fagotti

Published May 3, 2021• 70 citations
arXivHTMLPDF

Abstract

We study the (dual) folded spin-1/2 XXZ model in the thermodynamic limit. We focus, in particular, on a class of ``local'' macrostates that includes Gibbs ensembles. We develop a thermodynamic Bethe Ansatz description and work out generalised hydrodynamics at the leading order. Remarkably, in the ballistic scaling limit the junction of two local macrostates results in a discontinuity in the profile of essentially any local observable.

DOI
10.21468/scipostphys.10.5.099

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We study the (dual) folded spin-1/2 XXZ model in the thermodynamic limit. We focus, in particular, on a class of ``local'' macrostates that includes Gibbs ensembles. We develop a thermodynamic Bethe Ansatz description and work out generalised hydrodynamics at the leading order. Remarkably, in the ballistic scaling limit the junction of two local macrostates results in a discontinuity in the profile of essentially any local observable.
The anomalous nature of spin transport in the XXZ quantum spin chain has been a topic of theoretical interest for some time. Here, the integrability of the underlying dynamics leads to a ballistic component of the spin current, characterized by a spin Drude weight, which measures the degree of divergence of the zero-frequency spin conductivity. However, this quantity had previously proven to be beyond the reach of standard Bethe ansatz techniques. Here, the authors show that a recently developed hydrodynamic formalism for quantum integrable models may be used to compute the spin Drude weight. They also propose a numerical scheme to obtain hydrodynamic predictions for finite-time energy transport. This suggests that the hydrodynamic approach captures completely the ballistic component that dominates transport at long times and distances in the gapless regime of the XXZ model.
We study an effective Hamiltonian generating time evolution of states on intermediate time scales in the strong-coupling limit of the spin-1/2 XXZ model. To leading order, it describes an integrable model with local interactions. We solve it completely by means of a coordinate Bethe Ansatz that manifestly breaks the translational symmetry. We demonstrate the existence of exponentially many jammed states and estimate their stability under the leading correction to the effective Hamiltonian. Some ground state properties of the model are discussed.
Abstract The thermodynamic Bethe ansatz approach to the study of integrable quantum field theories was introduced in the early 90s. Since then it has been known that the thermodynamic Bethe ansatz equations can be recast in the form of Y -systems. These Y -systems have a number of interesting properties, notably in the high-temperature limit their solutions are constants from which the central charge of the ultraviolet fixed point can be obtained and they are typically periodic functions, with period proportional to the dimension of the perturbing field. In this letter we discuss the derivation of Y -systems when the standard thermodynamic Bethe ansatz equations are replaced by generalised versions, describing generalised Gibbs ensembles. We shown that for many integrable quantum field theories, there is a large class of distinct generalised Gibbs ensembles which share the same Y -system.
We set up a hydrodynamic description of the non-equilibrium dynamics of sine-Gordon quantum field theory for generic coupling. It is built upon an explicit form of the Bethe Ansatz description of general thermodynamic states, with the structure of the resulting coupled integral equations encoded in terms of graphical diagrams. The resulting framework is applied to derive results for the Drude weights of charge and energy. Quantities associated with the charge universally have fractal dependence on the coupling, which is notably absent from those associated with energy, a feature explained by the different roles played by reflective scattering in transporting these quantities. We then present far-from-equilibrium results, including explicit time evolution starting from bipartite initial states and dynamical correlation functions. Our framework can be applied to explore numerous other aspects of non-equilibrium dynamics, which opens the way to a wide array of theoretical studies and potential novel experimental predictions.
We investigate the modular properties of Generalised Gibbs Ensembles (GGEs) in two dimensional conformal field theories. These are obtained by inserting higher spin charges in the expressions for the partition function of the theory. We investigate the particular case where KdV charges are inserted in the GGE. We first determine an asymptotic expression for the transformed GGE. This expression is an expansion in terms of the zero modes of all the quasi-primary fields in the theory, not just the KdV charges. While these charges are non-commuting they can be re-exponentiated to give an asymptotic expression for the transformed GGE in terms of another GGE. As an explicit example we focus on the Lee-Yang model. We use the Thermodynamic Bethe Ansatz in the Lee-Yang model to first replicate the asymptotic results, and then find additional energies that need to be included in the transformed GGE in order to find the exact modular transformation.
Abstract We study the low-temperature transport properties of out-of-equilibrium XXZ spin-1/2 chains. We consider the protocol where two semi-infinite chains are prepared in two thermal states at small but different temperatures and suddenly joined together. We focus on the qualitative and quantitative features of the profiles of local observables, which at large times t and distances x from the junction become functions of the ratio ζ = x / t . By means of the generalized hydrodynamic equations, we analyse the rich phenomenology arising by considering different regimes of the phase diagram. In the gapped phases, variations of the profiles are found to be exponentially small in the temperatures, but described by non-trivial functions of ζ . We provide analytical formulae for the latter, which give accurate results also for small but finite temperatures. In the gapless regime, we show how the three-step conformal predictions for the profiles of energy density and energy current are naturally recovered from the hydrodynamic equations. Moreover, we also recover the recent non-linear Luttinger liquid predictions for low-temperature transport: universal peaks of width Δ ζ ∝ T emerge at the edges of the light cone in the profiles of generic observables. Such peaks are described by the same function of ζ for all local observables.