. In this paper we extend the theory of mean-field-dynamical semigroups given in [DW1,Du1] to treat the irreversible mean-field dynamics of quasi-local mean-field observables. These are observables which are site averaged except within a region of tagged sites. In the thermodynamic limit the tagged sites absorb the whole lattice, but also become negligible in proportion to the bulk. We develop the theory in detail for a class of interactions which contains the mean-field versions of quantum lattice interactions with infinite range. For this class we obtain an explicit form of the dynamics in the thermodynamic limit. We show that the evolution of the bulk is governed by a flow on the one-particle state space, whereas the evolution of local perturbations in the tagged region factorizes over sites, and is governed by a cocycle of completely positive maps. We obtain an H-theorem which suggests that local disturbances typically become completely delocalized for large times, and we show this...
. In this paper we extend the theory of mean-field-dynamical semigroups given in [DW1,Du1] to treat the irreversible mean-field dynamics of quasi-local mean-field observables. These are observables which are site averaged except within a region of tagged sites. In the thermodynamic limit the tagged sites absorb the whole lattice, but also become negligible in proportion to the bulk. We develop the theory in detail for a class of interactions which contains the mean-field versions of quantum lattice interactions with infinite range. For this class we obtain an explicit form of the dynamics in the thermodynamic limit. We show that the evolution of the bulk is governed by a flow on the one-particle state space, whereas the evolution of local perturbations in the tagged region factorizes over sites, and is governed by a cocycle of completely positive maps. We obtain an H-theorem which suggests that local disturbances typically become completely delocalized for large times, and we show this...
We study the dynamics of local correlation functions in dissipative mean-field systems. This is done by extending the abstract notion of a mean-field dynamical semigroups on C*-algebra given in [1], from an evolution on site-averaged observables, to one on a class of local observables. Conditions are established under which this generalized mean-field dynamics factorises, in the thermodynamic limit, into contributions from disjoints for single site observables in this limit. We demonstrate that these conditions are satisfied for a large class of model systems
Following the work of P. Bona [J. Math. Phys. 29, 2223 (1988)], a class of infinite mean-field quantum lattice systems is considered within the framework of algebraic quantum mechanics, and the following question is discussed: Which are the possible dynamical descriptions of the system, i.e., in which representations π of the system’s quasilocal C*-algebra 𝒜 do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged C*-algebra 𝒞π containing π(𝒜) and (some of) the classical observables of the system in π. To this end we associate a ‘‘classical state space’’ Eπ to any representation π (which is a subset of the phase-space E⊇RL defined by Bona for the largest possible representation), and show how it is characterized by the π-normal states, thereby obtaining a way to determine Eπ explicitly in many cases. The answer to the question posed then reads: The limiting (Heisenberg) dynamics exists in π on 𝒞π if and only if the set Eπ is invariant under the flow φQt on E which describes the time evolution of classical observables. Conversely, to any invariant, closed subset B of E, there exists a dynamical description of the quantum system with classical state space B.
Abstract We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.
We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ``state-dependent" rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-)local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.
Think globally, act locally. Summary Dynamical mean-field theory is designed to treat systems with local effective interactions that are strong compared with the independent-particle terms that lead to delocalized band-like states. Interactions are taken into account by a many-body calculation for an auxiliary system, a site embedded in a dynamical mean field, that is chosen to best represent the coupling to the rest of the crystal. The methods are constructed to be exact in three limits: interacting electrons on isolated sites, a lattice with no interactions, and the limit of infinite dimensions d → ∞ where mean-field theory is exact. This chapter is devoted to the general formulation, the single-site approximation where the calculation of the self-energy is mapped onto a self-consistent quantum impurity problem, and instructive examples for the Hubbard model on a d → ∞ Bethe lattice. Further developments and applications are the topics of Chs. 17–21. One of the most rewarding features of condensed matter theory is the ability to address difficult problems from different points of view. The preceding Chs. 9–15 present an approach based on perturbation expansions in the Coulomb interaction. In particular, the GW approximation for the self-energy has proven to be extremely successful in describing electronic spectra of many materials, as described in Ch. 13. The methods can be applied to the ordered states of materials with d and f states, for example, ferromagnetic Ni and anti-ferromagnetic NiO, as described in Secs. 13.4 and 20.7. However, the GW and related approximations have difficulties in treating cases with degenerate or nearly degenerate states and low-energy excitations. Present-day methods do not describe phenomena like the fluctuations of local moments in a ferromagnetic material above the Curie temperature or the insulating character of NiO in the paramagnetic phase; more generally, they have difficulty describing strong correlation. The topic of this chapter and Chs. 17–21 is dynamical mean-field theory, which is also a Green's function method in which the key quantity is the self-energy. However, it is designed to treat strong interactions for electrons in localized atomic-like states, such as the d and f states in transition metals, lanthanide and actinide elements and compounds.
Abstract We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ‘state-dependent’ rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-) local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.