Emergence of Hydrodynamic Spatial Long-Range Correlations in Nonequilibrium Many-Body Systems.

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances "develop long-range correlations as time passes." These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximized states. They require "the presence of interaction" and "at least two different fluid modes" and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasiparticle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.

Exact Large-Scale Fluctuations of the Phase Field in the Sine-Gordon Model.

We present the first exact theory and analytical formulas for the large-scale phase fluctuations in the sine-Gordon model, valid in all regimes of the field theory, for arbitrary temperatures and interaction strengths. Our result is based on the ballistic fluctuation theory combined with generalized hydrodynamics, and can be seen as an exact "dressing" of the phenomenological soliton-gas picture first introduced by Sachdev and Young [Phys. Rev. Lett. 78, 2220 (1997)PRLTAO0031-900710.1103/PhysRevLett.78.2220], to the modes of generalized hydrodynamics. The resulting physics of phase fluctuations in the sine-Gordon model is qualitatively different, as the stable quasiparticles of integrability give coherent ballistic propagation instead of diffusive spreading. We provide extensive numerical checks of our analytical predictions within the classical regime of the field theory by using Monte Carlo methods. We discuss how our results are of ready applicability to experiments on tunnel-coupled quasicondensates.

Nonequilibrium Dynamics and Weakly Broken Integrability.

Motivated by dynamical experiments on cold atomic gases, we develop a quantum kinetic approach to weakly perturbed integrable models out of equilibrium. Using the exact matrix elements of the underlying integrable model, we establish an analytical approach to real-time dynamics. The method addresses a broad range of timescales, from the intermediate regime of prethermalization to late-time thermalization. Predictions are given for the time evolution of physical quantities, including effective temperatures and thermalization rates. The approach provides conceptual links between perturbed quantum many-body dynamics and classical Kolmogorov-Arnold-Moser theory. In particular, we identify a family of perturbations which do not cause thermalization in the weakly perturbed regime.

Quantum Generalized Hydrodynamics.

Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed generalized hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to nonzero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multicomponent Luttinger liquid theory, with a small set of effective parameters that are fixed by the thermodynamic Bethe ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.

Thermalization of a Trapped One-Dimensional Bose Gas via Diffusion.

For a decade the fate of a one-dimensional gas of interacting bosons in an external trapping potential remained mysterious. We here show that whenever the underlying integrability of the gas is broken by the presence of the external potential, the inevitable diffusive rearrangements between the quasiparticles, quantified by the diffusion constants of the gas, eventually lead the system to thermalize at late times. We show that the full thermalizing dynamics can be described by the generalized hydrodynamics with diffusion and force terms, and we compare these predictions to numerical simulations. Finally, we provide an explanation for the slow thermalization rates observed in numerical and experimental settings: the hydrodynamics of integrable models is characterized by a continuity of modes, which can have arbitrarily small diffusion coefficients. As a consequence, the approach to thermalization can display prethermal plateau and relaxation dynamics with long polynomial finite-time corrections.

Entanglement Content of Quasiparticle Excitations.

We investigate the quantum entanglement content of quasiparticle excitations in extended many-body systems. We show that such excitations give an additive contribution to the bipartite von Neumann and Rényi entanglement entropies that takes a simple, universal form. It is largely independent of the momenta and masses of the excitations and of the geometry, dimension, and connectedness of the entanglement region. The result has a natural quantum information theoretic interpretation as the entanglement of a state where each quasiparticle is associated with two qubits representing their presence within and without the entanglement region, taking into account quantum (in)distinguishability. This applies to any excited state composed of finite numbers of quasiparticles with finite de Broglie wavelengths or finite intrinsic correlation length. This includes particle excitations in massive quantum field theory and gapped lattice systems, and certain highly excited states in conformal field theory and gapless models. We derive this result analytically in one-dimensional massive bosonic and fermionic free field theories and for simple setups in higher dimensions. We provide numerical evidence for the harmonic chain and the two-dimensional harmonic lattice in all regimes where the conditions above apply. Finally, we provide supporting calculations for integrable spin chain models and other interacting cases without particle production. Our results point to new possibilities for creating entangled states using many-body quantum systems.

Hydrodynamic Diffusion in Integrable Systems.

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.

Soliton Gases and Generalized Hydrodynamics.

We show that the equations of generalized hydrodynamics (GHD), a hydrodynamic theory for integrable quantum systems at the Euler scale, emerge in full generality in a family of classical gases, which generalize the gas of hard rods. In this family, the particles, upon colliding, jump forward or backward by a distance that depends on their velocities, reminiscent of classical soliton scattering. This provides a "molecular dynamics" for GHD: a numerical solver which is efficient, flexible, and which applies to the presence of external force fields. GHD also describes the hydrodynamics of classical soliton gases. We identify the GHD of any quantum model with that of the gas of its solitonlike wave packets, thus providing a remarkable quantum-classical equivalence. The theory is directly applicable, for instance, to integrable quantum chains and to the Lieb-Liniger model realized in cold-atom experiments.

Large-Scale Description of Interacting One-Dimensional Bose Gases: Generalized Hydrodynamics Supersedes Conventional Hydrodynamics.

The theory of generalized hydrodynamics (GHD) was recently developed as a new tool for the study of inhomogeneous time evolution in many-body interacting systems with infinitely many conserved charges. In this Letter, we show that it supersedes the widely used conventional hydrodynamics (CHD) of one-dimensional Bose gases. We illustrate this by studying "nonlinear sound waves" emanating from initial density accumulations in the Lieb-Liniger model. We show that, at zero temperature and in the absence of shocks, GHD reduces to CHD, thus for the first time justifying its use from purely hydrodynamic principles. We show that sharp profiles, which appear in finite times in CHD, immediately dissolve into a higher hierarchy of reductions of GHD, with no sustained shock. CHD thereon fails to capture the correct hydrodynamics. We establish the correct hydrodynamic equations, which are finite-dimensional reductions of GHD characterized by multiple, disjoint Fermi seas. We further verify that at nonzero temperature, CHD fails at all nonzero times. Finally, we numerically confirm the emergence of hydrodynamics at zero temperature by comparing its predictions with a full quantum simulation performed using the NRG-TSA-abacus algorithm. The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.

Diffusion and Signatures of Localization in Stochastic Conformal Field Theory.

We define a simple model of conformal field theory in random space-time environments, which we refer to as stochastic conformal field theory. This model accounts for the effects of dilute random impurities in strongly interacting critical many-body systems. On one hand, surprisingly, although impurities are separated by macroscopic distances, we find that the infinite-time steady state is factorized on microscopic lengths, a signature of the emergence of localization. The stationary state also displays vanishing energy current and strong uncorrelated spatial fluctuations of local observables. On the other hand, at finite times, the transient shows a crossover from ballistic to diffusive energy propagation. In this regime and a Markovian limit, concentrating on current-generating initial states with a temperature imbalance, we show that the energy current and density satisfy simple dissipative hydrodynamic equations. We describe the space-time scales at which nonequilibrium currents exist. We show that a light-cone effect subsists in the presence of impurities although a momentum burst propagates transiently on a diffusive scale only.

Monte Carlo method for critical systems in infinite volume: The planar Ising model.

In this paper we propose a Monte Carlo method for generating finite-domain marginals of critical distributions of statistical models in infinite volume. The algorithm corrects the problem of the long-range effects of boundaries associated to generating critical distributions on finite lattices. It uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of "holographic" boundary condition that encodes the presence of an infinite volume beyond it. We check the quality of the distribution obtained in the case of the planar Ising model by comparing various observables with their infinite-plane prediction. We accurately reproduce planar two-, three-, and four-point of spin and energy operators. We also define a lattice stress-energy tensor, and numerically obtain the associated conformal Ward identities and the Ising central charge.

Entanglement entropy of highly degenerate States and fractal dimensions.

We consider the bipartite entanglement entropy of ground states of extended quantum systems with a large degeneracy. Often, as when there is a spontaneously broken global Lie group symmetry, basis elements of the lowest-energy space form a natural geometrical structure. For instance, the spins of a spin-1/2 representation, pointing in various directions, form a sphere. We show that for subsystems with a large number m of local degrees of freedom, the entanglement entropy diverges as d/2 logm, where d is the fractal dimension of the subset of basis elements with nonzero coefficients. We interpret this result by seeing d as the (not necessarily integer) number of zero-energy Goldstone bosons describing the ground state. We suggest that this result holds quite generally for largely degenerate ground states, with potential applications to spin glasses and quenched disorder.

Bipartite entanglement entropy in massive two-dimensional quantum field theory.

Recently, Cardy, Castro Alvaredo, and the author obtained the first exponential correction to saturation of the bipartite entanglement entropy at large region lengths in massive two-dimensional integrable quantum field theory. It depends only on the particle content of the model, and not on the way particles scatter. Based on general analyticity arguments for form factors, we propose that this result is universal, and holds for any massive two-dimensional model (also out of integrability). We suggest a link of this result with counting pair creations far in the past.

New method for studying steady states in quantum impurity problems: the interacting resonant level model.

We develop a new perturbative method for studying any steady states of quantum impurities, in or out of equilibrium. We show that steady-state averages are completely fixed by basic properties of the steady-state (Hershfield's) density matrix along with dynamical "impurity conditions." This gives the full perturbative expansion without Feynman diagrams (matrix products instead are used), and "resums" into an equilibrium average that may lend itself to numerical procedures. We calculate the universal current in the interacting resonant level model (IRLM) at finite bias V to first order in Coulomb repulsion U for all V and temperatures. We find that the bias, like the temperature, cuts off low-energy processes. In the IRLM, this implies a power-law decay of the current at large V (also recently observed by Boulat and Saleur at some finite value of U).