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arXiv:1810.11024 · Physical review. E

Emergent entropy production and hydrodynamics in quantum many-body systems

Tom Banks, Andrew Lucas

Published Feb 6, 2019• 20 citations
arXivHTMLPDF

Abstract

We study the dynamics of a locally conserved energy in ergodic, local many-body quantum systems on a lattice with no additional symmetry. The resulting dynamics is well approximated by a coarse grained, classical linear functional diffusion equation for the probability of all spatial configurations of energy. This is equivalent to nonlinear stochastic hydrodynamics, describing the diffusion of energy in physical spacetime. We find the absence of nonhydrodynamic slow degrees of freedom, a nonlinear fluctuation-dissipation theorem, and the emergence of a (weakly interacting) kinetic theory for hydrodynamic modes near thermal equilibrium. The observable part of the microscopic entropy obeys the local second law of thermodynamics, and quantitatively agrees with the phenomenological predictions of hydrodynamics. Our approach naturally generalizes to ergodic systems with additional symmetries, may lead to numerical algorithms to calculate diffusion constants for lattice models, and implies sufficiency conditions for a rigorous derivation of hydrodynamics in quantum systems.

DOI
10.1103/physreve.99.022105

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We study the dynamics of a locally conserved energy in ergodic, local many-body quantum systems on a lattice with no additional symmetry. The resulting dynamics is well approximated by a coarse grained, classical linear functional diffusion equation for the probability of all spatial configurations of energy. This is equivalent to nonlinear stochastic hydrodynamics, describing the diffusion of energy in physical spacetime. We find the absence of nonhydrodynamic slow degrees of freedom, a nonlinear fluctuation-dissipation theorem, and the emergence of a (weakly interacting) kinetic theory for hydrodynamic modes near thermal equilibrium. The observable part of the microscopic entropy obeys the local second law of thermodynamics, and quantitatively agrees with the phenomenological predictions of hydrodynamics. Our approach naturally generalizes to ergodic systems with additional symmetries, may lead to numerical algorithms to calculate diffusion constants for lattice models, and implies sufficiency conditions for a rigorous derivation of hydrodynamics in quantum systems.
The macroscopic hydrodynamic equations are derived for many-body systems in the local-equilibrium approach, using the Schrödinger picture of quantum mechanics. In this approach, statistical operators are defined in terms of microscopic densities associated with the fundamentally conserved quantities and other slow modes possibly emerging from continuous symmetry breaking, as well as macrofields conjugated to these densities. Functional identities can be deduced, allowing us to identify the reversible and dissipative parts of the mean current densities, to obtain general equations for the time evolution of the conjugate macrofields, and to establish the relationship to projection-operator methods. The entropy production is shown to be nonnegative by applying the Peierls-Bogoliubov inequality to a quantum integral fluctuation theorem. Using the expansion in the gradients of the conjugate macrofields, the transport coefficients are given by Green-Kubo formulas and the entropy production rate can be expressed in terms of quantum Einstein-Helfand formulas, implying its nonnegativity in agreement with the second law of thermodynamics. The results apply to multicomponent fluids and can be extended to condensed matter phases with broken continuous symmetries.
The second law of thermodynamics states that for a thermally isolated system entropy never decreases. Most physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and thus can be considered as in local equilibrium. For a many-body system in local equilibrium a stronger version of the second law applies which says that the entropy production at each spacetime point should be non-negative. In this paper we provide a proof of the second law for such systems and a first derivation of the local second law. For this purpose we develop a general non-equilibrium effective field theory of slow degrees of freedom from integrating out fast degrees of freedom in a quantum many-body system and consider its classical limit. The key elements of the proof are the presence of a Z2Z_2Z2​ symmetry, which can be considered as a proxy for local equilibrium and micro-time-reversibility, and a classical remnant of quantum unitarity. The Z2Z_2Z2​ symmetry leads to a local current from a procedure analogous to that used in the Noether theorem. Unitarity leads to a definite sign of the divergence of the current. We also discuss the origin of an arrow of time, as well as the coincidence of causal and thermodynamical arrows of time. Applied to hydrodynamics, the proof gives a first-principle derivation of the phenomenological entropy current condition and provides a constructive procedure for obtaining the entropy current.
This Ph.D. thesis reports on progress in rigorously establishing hydrodynamic principles from the microscopic Hamiltonian dynamics of quantum many-body systems in a general, non-model-specific manner. Using the C*-algebra framework of statistical mechanics, we treat systems directly in the thermodynamic limit, primarily focusing on quantum lattice models where tools such as Lieb-Robinson bounds yield rigorous statements. We thus provide a proof-of-principle that large-scale behaviours can indeed be seen as emerging from microscopic dynamics, with mathematical proof. We first report on ergodicity results in short-range models with exponentially decaying or finite-range interactions. We show that time-averaged observables converge to their ensemble averages and decorrelate from all other observables almost everywhere within the light-cone defined by Lieb-Robinson bounds. This relaxation property indicates the loss of information at large scales, from which we prove a Boltzmann-Gibbs principle: at the Euler scaling limit of large time and distance, observables project onto hydrodynamic modes (extensive conserved quantities), within correlation functions. These results hold independently of microscopic details, capturing the physical idea that such details are lost at large space-time scales. Regarding finer scales of hydrodynamics, we discuss rigorous lower bounds on the strength of diffusion. We establish a general result on the clustering of n-th order connected correlations within C* dynamical systems. These results are applied to obtain a strictly positive lower bound on the diffusion constant of chaotic open quantum spin chains with nearest-neighbor interactions. This thesis underlines the universality of hydrodynamic principles, provides a framework for establishing them rigorously, and sets the stage for future progress toward the goal of proving the hydrodynamic equations.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
We provide a pure state formulation for hydrodynamics of isolated quantum many-body systems. A pure state describing quantum systems in local thermal equilibrium is constructed, which we call a local thermal pure quantum ($\ensuremath{\ell}\mathrm{TPQ}$) state. We show that the thermodynamic functional and the expectation values of local operators (including a real-time correlation function) calculated from the $\ensuremath{\ell}\mathrm{TPQ}$ state converge to those from a local Gibbs ensemble in the large fluid-cell limit. As a numerical demonstration, we investigate a one-dimensional spin chain and observe the hydrodynamic relaxation obeying Fourier's law. We further prove the second law of thermodynamics and the quantum fluctuation theorem, which are also validated numerically. The $\ensuremath{\ell}\mathrm{TPQ}$ formulation gives a useful theoretical basis to describe the emergent hydrodynamic behavior of quantum many-body systems furnished with a numerical efficiency applicable to both the nonrelativistic and relativistic regimes.
We develop a systematic effective field theory of hydrodynamics for many-body systems on the lattice with global continuous non-Abelian symmetries. Models with continuous non-Abelian symmetries are ubiquitous in physics, arising in diverse settings ranging from hot nuclear matter to cold atomic gases and quantum spin chains. In every dimension and for every flavor symmetry group, the low energy theory is a set of coupled noisy diffusion equations. Independence of the physics on the choice of canonical or microcanonical ensemble is manifest in our hydrodynamic expansion, even though the ensemble choice causes an apparent shift in quasinormal mode spectra. We use our formalism to explain why flavor symmetry is qualitatively different from hydrodynamics with other non-Abelian conservation laws, including angular momentum and charge multipoles. As a significant application of our framework, we study spin and energy diffusion in classical one-dimensional SU(2)-invariant spin chains, including the Heisenberg model along with multiple generalizations. We argue based on both numerical simulations and our effective field theory framework that non-integrable spin chains on a lattice exhibit conventional spin diffusion, in contrast to some recent predictions that diffusion constants grow logarithmically at late times. We show that the apparent enhancement of diffusion is due to slow equilibration caused by (non-Abelian) hydrodynamic fluctuations.