On the Algebra of Fluctuation in Quantum Spin Chains

Isao Harada, Keiji Fukui, Tokihisa Takao +7 more

Yoshiko Ogata

Taku Matsui

We present a proof of the central limit theorem for a pair of mutually noncommuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem. KEY WORDS: central limit theorem; KMS states; quantum spin chain

Fabio Benatti, Federico Carollo, Roberto Floreanini B +1 more

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.

Bruno Nachtergaele

Berislav Buča, Tomaž Prosen

Abstract. We show how to exploit algebraic relations among the operators (or matrices) which constitute the non-equilibrium matrix product steady state of a boundary driven quantum spin chain to find partial differential equations determining all the m -point correlation functions in the continuum (or thermodynamic) limit. These partial differential equations, the order of which is determined by scaling of the non-equilibrium partition function, are readily solved if we also know the boundary conditions. In this way we can avoid resorting to representation theory of the matrix product algebra. We apply our methods to study the distributions, or moments, of the magnetization and the spin current observables in boundary driven open XXZ spin chains, as well as some connected correlation functions. Interestingly, we find that the transverse connected correlation functions are thermodynamically non-decaying and long range at the isotropic point .