We study dynamical conservation laws and the propagation of thermodynamic structure in translation-invariant lattice quantum spin systems. Using Lieb-Robinson bounds [CMP 28 (1972), 251-257] in arbitrary dimension, and Araki's bounds [CMP 14 (1969), 120-157] in dimension one, we prove conservation of the specific entropy, specific energy, and specific relative entropy under suitable assumptions on the interactions. The transported local Hamiltonians admit surface energies that vanish per unit volume in the van Hove limit, and local observables remain a core for the transported generators at every finite time. Weak Gibbsianity is preserved for sufficiently small times in arbitrary dimension and for all times in dimension one. These results provide the analytical foundation for the structural theory of approach to equilibrium developed in a companion paper, and establish partial results toward the conjectures formulated there.
We study dynamical conservation laws and the propagation of thermodynamic structure in translation-invariant lattice quantum spin systems. Using Lieb-Robinson bounds [CMP 28 (1972), 251-257] in arbitrary dimension, and Araki's bounds [CMP 14 (1969), 120-157] in dimension one, we prove conservation of the specific entropy, specific energy, and specific relative entropy under suitable assumptions on the interactions. The transported local Hamiltonians admit surface energies that vanish per unit volume in the van Hove limit, and local observables remain a core for the transported generators at every finite time. Weak Gibbsianity is preserved for sufficiently small times in arbitrary dimension and for all times in dimension one. These results provide the analytical foundation for the structural theory of approach to equilibrium developed in a companion paper, and establish partial results toward the conjectures formulated there.
We continue the development, started in [AHP 25 (2023), 715-749], of a structural theory of approach to equilibrium in translation-invariant quantum spin systems. Dynamical Steady States (DSS) are defined as w*-limit points of the time-averaged evolution of an initial state. We show that if a DSS is an equilibrium state of a sufficiently regular interaction, then the specific energy of this interaction is necessarily a constant of the motion. Two distinct mechanisms lead to this conclusion, one based on dynamical surface energies and the other on conservation of specific relative entropy. We conjecture that these properties hold for a natural class of interactions. For a DSS admitting a physical equilibrium description, the associated equilibrium states are characterized by a dynamical Gibbs variational principle: they are precisely the minimizers of the specific relative entropy from the initial state over the physical equilibrium states. The achieved minimum equals the increase of specific entropy. The structural theory also yields characterizations of approach to thermal equilibrium, based on two model-dependent questions: restoration of a sufficiently local thermodynamic description at large times and classification of the corresponding local constants of motion. Once locality is restored, the constants of motion determine the possible asymptotic equilibrium states. In particular, if finite-range locality is restored and the energy is the only finite-range constant of the motion, then every DSS is a thermal equilibrium state at the temperature fixed by energy conservation. The finite-time dynamical input underlying this theory is established in a companion paper.
We prove that, for translationally invariant quantum spin chains with finite-range interactions, the existence of a specific conservation law implies the presence of infinitely many local conserved quantities, i.e., integrability. This shows that the standard hierarchy of local conserved quantities arising for nearest-neighbor Hamiltonians obtained via the canonical Yang-Baxter construction is already encoded in the lowest nontrivial conservation law, known as the Reshetikhin condition. Combined with recent rigorous results on nonintegrability, our theorem strongly restricts the possibility of partially integrable systems that admit only a finite but large number of local conserved quantities. Our work establishes a rigorous foundation for the systematic identification of new integrable models and deepens the algebraic understanding of conservation-law structures in quantum spin chains.
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.
In this Thesis I present the results of four research projects I carried out during my Ph.D. training at SISSA. The broad topic of my research was the mechanism of relaxation in isolated, integrable quantum systems. Roughly speaking, integrable quantum systems are one-dimensional lattice models or field theories that feature an extensively large number of conservation laws beyond energy and momentum conservation, which greatly influence their dynamical behaviour. Integrable systems have enjoyed a considerable interest in the last decade, mainly due to the development of ultracold atomic techniques. These techniques made it possible to implement experimentally a wealth of one-dimensional quantum models whose significance had been purely theoretical before. The theoretical interest in integrable systems stems in part from the fact that in many aspects they are exactly solvable. Exact analytical formulas are available for their spectrum and their equilibrium thermodynamic properties, creating an incentive to study analytically their non-equilibrium behaviour as well. Exact results for the dynamics of integrable systems may serve as a benchmark for confirming theories in statistical mechanics or testing approximative methods. To this day, computing analytically the entire time evolution of observables in a genuinely interacting integrable system is prohibitively difficult in general. In this Thesis, I concentrate on the description of the system at long times after it is brought out of equilibrium. Homogeneous, translationally invariant integrable systems are expected to relax to a stationary state, in which local observables are described by a generalization of the Gibbs ensemble that takes into consideration all the conservation laws beyond the energy. Two of the projects presented here concern this relaxed state. In Chapter 2, a method is presented for computing the Rényi entropies of the Generalized Gibbs Ensemble describing the relaxed state in the integrable Heisen- berg XXZ spin chain. Rényi entropies provide information about the relaxed state and its entanglement properties, going beyond the ordinary von Neumann thermo- dynamic entropy. The method presented there is very general; in principle it works in any integrable model and any translationally invariant thermodynamic macrostate. An alternative to the GGE in describing relaxed states after a quantum quench is the Quench Action approach. This approach is based on the explicit analytical knowledge of the overlaps between the initial state and the Hamiltonian eigenstates. In Chapter 3, this approach is generalized to a spin-1 integrable spin chain, the SU(3) invariant Lai–Sutherland model. The evolution of entanglement is then studied using the quasiparticle picture, which features ballistically propagating classical particle pairs carrying entanglement. If the requirement of translational invariance of the initial state is relaxed, various transport phenomena arise in integrable systems. The evolution of slowly varying, smoothly distributed states is described by a generalization of hydrodynamics, which takes into consideration the infinitely many local conservation laws. At the scale of non-diffusive, ballistic transport, the main feature of this generalized hydrodynamics (GHD) is the existence of continuity equations for the occupation number of each mode. Further two projects are presented in this Thesis which are concerned with GHD.I n Chapter 4, a numerical method is presented for computing the evolution of entanglement entropies within GHD. This method combines the quasiparticle picture of entanglement with a molecular dynamics simulation of GHD, the so-called fleagas. Numerical evidence is shown that this method reproduces previously known analytical results in GHD. Its versatility is demonstrated by computing another entanglement-related quantity, the mutual information, for which no analytical results are available. Finally, in Chapter 5, the GHD framework is generalized to a system of spin-1/2 fermions with repulsive contact interaction, the Yang–Gaudin model. It is shown that at low temperatures, the dynamics of the system is described by the superposition of two uncoupled, inhomogeneous conformal field theories. This fact can be interpreted as a separation of the spin and the charge degrees of freedom of the system.
Abstract Equilibrium statistical mechanics rests on the assumption of chaotic dynamics of a system modulo the conservation laws of local observables: extremization of entropy immediately gives Gibbs’ ensemble (GE) for energy conserving systems and a generalized version of it (GGE) when the number of local conserved quantities is more than one. Through the last decade, statistical mechanics has been extended to describe the late-time behaviour of periodically driven (Floquet) quantum matter starting from a generic state. The structure built on the fundamental assumptions of ergodicity and identification of the relevant conservation laws in this inherently non-equilibrium setting. More recently, it has been shown that the statistical mechanics of Floquet systems has a much richer structure due to the existence of emergent conservation laws: these are approximate but stable conservation laws arising due to the drive, and are not present in the undriven system. Extensive numerical and analytical results support perpetual stability of these emergent (though approximate) conservation laws, probably even in the thermodynamic limit. This banks on the recent finding of a sharp threshold for Floquet thermalization in clean, interacting non-integrable Floquet systems. This indicates to the possibility of stable Floquet phases of matter in disorder-free systems. This review intends to give a self-contained theoretical overview of these developments for a broad physics audience. We conclude by briefly surveying the current experimental scenario.
We discuss the properties of two open quantum systems with a general class of irreversible quantum dynamics. First we study Lieb-Robinson bounds in a quantum lattice systems. The time-dependent generator of the dynamics of the system is of the Lindblad-Kossakowski type. This generator satisfies some suitable decay condition in space. We show that the dynamics with a such generator on a finite system is a well-defined quantum dynamics in a sense of a norm-continuous cocycle of unit preserving completely positive maps. After proving Lieb-Robinson bounds for a finite lattice systems we also show the existence of the thermodynamic limit of the dynamics. We show that in a strong limit there exits a strongly continuous cocycle of unit preserving completely positive maps. Which means that the dynamics exists in an infinite system, where Lieb-Robinson bounds also holds. In the second part of the paper we consider a system that consists of a beam of two-level atoms that pass one by one through the microwave cavity. The atoms are randomly excited and there is exactly one atom present in the cavity at any given moment. We consider both the ideal and leaky cavity and study the time asymptotic behavior of the state of the cavity. We show that the number of photons increases indefinitely in the case of the ideal cavity. In case of leaky cavity we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the energy flux in both the ideal and leaky and the entropy production for the ideal cavity.