Taku Matsui

FLUCTUATION THEOREM, NONEQUILIBRIUM STEADY STATES AND MACLENNAN- ZUBAREV ENSEMBLES OF A CLASS OF LARGE QUANTUM SYSTEMS

For an infinitely extended system consisting of a finite subsystem and several reservoirs, the time evolution of states is studied. Initially, the reservoirs are prepared to be in equilibrium with different temperatures and chemical potentials. If the time evolution is L1-asymptotic abelian, (i) steady states exist, (ii) they and their relative entropy production are independent of the way of division into a subsystem and reservoirs, and (iii) they are stable against local perturbations. The explicit expression of the relative entropy production and a KMS characterization of the steady states are given. And a rigorous definition of MacLennan-Zubarev ensembles is proposed. A noncommutative analog to the fluctuation theorem is derived provided that the evolution and an initial state are time reversal symmetric.

Entanglement, haag-duality and type properties of infinite quantum spin chains

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.

On the algebra of fluctuation in quantum spin chains

Abstract. We present a proof of the central limit theorem for a pair of mutually non-commuting 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.

MARKOV SEMIGROUPS ON UHF ALGEBRAS

We consider a class of Markov semigroups on UHF algebras. We establish the existence of dynamics for long range interactions. Our idea is a non-commutative extension of the argument for classical interacting particle systems. As a by-product we obtain sufficient conditions for unique ergodicity.

A characterization of pure finitely correlated states

We give a characterization of pure finitely correlated states (quantum Markov states) as zero energy states of UHF algebras.

BEC of free bosons on networks

We consider free bosons hopping on a network (infinite graph). The condition for Bose–Einstein condensation is given in terms of the random walk on a graph. In case of periodic lattices, we also consider boson moving in an external periodic potential and obatin the criterion for Bose–Einstein condensation.

Variational principle for non-equilibrium steady states of the XX model

We show that non-equilibrium steady states of the one-dimensional exactly-solved XX model can be characterized by the variational principle of free energy of a long range interaction and that they cannot be a KMS state for any C*-dynamical system of the UHF algebra.

Bosonic central limit theorem for the one-dimensional XY model

We prove the central limit theorem for Gibbs states and ground states of quasifree Fermions (bilinear Hamiltonians) and those of the off critical XY model on a one-dimensional integer lattice.

Spectral gap, and split property in quantum spin chains

In this article, we consider a class of ground states with spectral gap for quantum spin chains on an integer lattice and we prove that the factorization lemma of Hastings [“Topology and phases in fermionic systems,” J. Stat. Mech.: Theory Exp. 2008, L01001] implies split property (weak statistical independence) of left and right semi-infinite subsystems.

ON NON-COMMUTATIVE RUELLE TRANSFER OPERATOR

In this note, we study a non-commutative analogue of the Ruelle–Perron–Frobenius Transfer operators on UHF algebras.