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arXiv:1810.08227 · Physical review. B./Physical review. B

Subdiffusive front scaling in interacting integrable models

Vir B. Bulchandani, Christoph Karrasch

Published Mar 29, 2019• 45 citations
arXivHTMLPDF

Abstract

We show that any interacting integrable model possesses a class of initial states for which the leading corrections to ballistic transport are subdiffusive rather than diffusive. These initial states are natural to realize experimentally and include the domain-wall initial condition that has been the object of much recent scrutiny. Upon performing numerical matrix product state simulations in the spin-$1/2$ XXZ chain, we find that such states can exhibit subdiffusive ${t}^{1/3}$ scaling of fronts of spin, energy, and entanglement entropy across the entire range of anisotropies. This demonstrates that Tracy-Widom scaling is not incompatible with model interactions, as was previously believed.

DOI
10.1103/physrevb.99.121410

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We show that any interacting integrable model possesses a class of initial states for which the leading corrections to ballistic transport are subdiffusive rather than diffusive. These initial states are natural to realize experimentally and include the domain-wall initial condition that has been the object of much recent scrutiny. Upon performing numerical matrix product state simulations in the spin-1/21/21/2 XXZ chain, we find that such states can exhibit subdiffusive t1/3{t}^{1/3}t1/3 scaling of fronts of spin, energy, and entanglement entropy across the entire range of anisotropies. This demonstrates that Tracy-Widom scaling is not incompatible with model interactions, as was previously believed.
We study the far-from-equilibrium dynamics of domain-wall initial states in the integrable lattice Landau--Lifshitz spin chain. Building on the linear spin-wave description of domain-wall spreading, we derive the long-wavelength magnon dispersion directly from the microscopic integrable Hamiltonian and use it to characterize the propagation and broadening of the resulting wavefronts. The dispersion interpolates between a quadratic form at the isotropic point and a linear form in the easy-plane regime. In the isotropic limit, the quadratic dispersion leads to diffusive broadening, whereas in the easy-plane regime the domain-wall front propagates ballistically. Keeping the leading nonlinear correction to the linear dispersion gives a cubic dispersive term, which produces a characteristic t1/3t^{1/3}t1/3 broadening of the ballistic front. We further show that the sign of this cubic correction changes at a critical anisotropy, leading to a reversal of the dispersive wake from trailing to advancing. The resulting critical point occurs at Δc=1/7Δ_c=1/7Δc​=1/7, corresponding to γc≃1.047γ_c\simeq 1.047γc​≃1.047. Numerical simulations of the full nonlinear spin dynamics confirm the analytical scaling and demonstrate the coexistence of the amplitude-independent spin-wave front with amplitude-dependent nonlinear soliton propagation.
We theoretically study quantum spin transport in a one-dimensional folded XXZ model with an alternating domain-wall initial state via the Bethe ansatz technique, exactly demonstrating that a probability distribution of finding a leftmost up spin with an appropriate scaling variable converges to the Tracy-Widom distribution for the Gaussian unitary ensemble (GUE), which is a universal distribution for the largest eigenvalue of GUE under a soft-edge scaling limit. Our finding presented here offers a first exact derivation of the GUE Tracy-Widom distribution in the dynamics of the interacting quantum model not being mapped to noninteracting fermions via the Jordan-Wigner transformation. On the basis of the exact solution of the folded XXZ model and our numerical analysis of the XXZ model, we discuss a universal behavior for the probability of finding the leftmost up spin in the XXZ model.
We show that hydrodynamic diffusion is generically present in many-body, one-dimensional interacting quantum and classical integrable models. We extend the recently developed generalized hydrodynamic (GHD) to include terms of Navier-Stokes type, which leads to positive entropy production and diffusive relaxation mechanisms. These terms provide the subleading diffusive corrections to Euler-scale GHD for the large-scale nonequilibrium dynamics of integrable systems, and arise due to two-body scatterings among quasiparticles. We give exact expressions for the diffusion coefficients. Our results apply to a large class of integrable models, including quantum and classical, Galilean and relativistic field theories, chains, and gases in one dimension, such as the Lieb-Liniger model describing cold atom gases and the Heisenberg quantum spin chain. We provide numerical evaluations in the Heisenberg XXZ spin chain, both for the spin diffusion constant, and for the diffusive effects during the melting of a small domain wall of spins, finding excellent agreement with time-dependent density matrix renormalization group numerical simulations.
We study the ballistic-to-diffusive transition induced by the weak breaking of integrability in a boundary-driven XXZ spin chain. Studying the evolution of the spin current density Js{\mathcal{J}}^{s}Js as a function of the system size LLL, we show that, accounting for boundary effects, the transition has a nontrivial universal behavior close to the XX limit. It is controlled by the scattering length ${L}^{*}\ensuremath{\propto}{V}^{\ensuremath{-}2}$, where VVV is the strength of the integrability-breaking term. In the XXZ model, the interplay of interactions controls the emergence of a transient ``quasiballistic'' regime at length scales much shorter than L∗{L}^{*}L∗. This parametrically large regime is characterized by a strong renormalization of the current which forbids a universal scaling, unlike the XX model. Our results are based on matrix product operator numerical simulations and agree with perturbative analytical calculations.
The nonequilibrium dynamics of domain wall initial states in a classical anisotropic Heisenberg chain exhibits a striking coexistence of apparently linear and nonlinear behaviors: the propagation and spreading of the domain wall can be captured quantitatively by linear, i.e., noninteracting, spin wave theory absent its usual justifications; while, simultaneously, for a wide range of easy-plane anisotropies, emission can take the place of stable solitons-a process and objects intrinsically associated with interactions and nonlinearities. The easy-axis domain wall only has transient dynamics, the isotropic one broadens diffusively, while the easy-plane one yields a pair of ballistically counterpropagating domain walls which, unusually, broaden subdiffusively, their width scaling as t^{1/3}.
We study the dynamics of an isotropic spin-12\frac{1}{2}21​ Heisenberg chain starting in a domain-wall initial condition where the spins are initially up on the left half-line and down on the right half-line. We focus on the long-time behavior of the magnetization profile. We perform extensive time-dependent density-matrix renormalization-group simulations (up to t=350t=350t=350) and find that the data are compatible with a diffusive behavior. Subleading corrections decay slowly blurring the emergence of the diffusive behavior. We also compare our results with two alternative scenarios: superdiffusive behavior and enhanced diffusion with a logarithmic correction. We finally discuss the evolution of the entanglement entropy.