Geometric and topological aspects of quasi-free states on self-dual algebras (2022)
We focus on physical and mathematical results about equilibrium states of gapped and gapless Hamiltonians of lattice fermion and lattice spin systems. We use the algebraic formulation of quantum mechanics to describe and characterize such states as functionals on a fermionic observable algebra. The mathematical framework of the work is Araki's self-dual formalism [1]. We study equivalence classes of quadratic Hamiltonians defined as elements of self-dual CAR algebra ASD (K, Gamma). The self-dual algebra ASD (K, Gamma), defined in terms of a Hilbert space K (the one particle self-dual space) and a conjugation Gamma on K, can be used to describe fermion systems. A special class of states on ASD (K, Gamma) are the so-called quasi-free states, which can be completely determined by their 2-point functions. Among these states, we distinguish ground states and equilibrium states, so the theory is robust and convenient enough to describe a wide range of physical quantum systems. We explore two cases: 1. The zero temperature case, where the fluctuations are due only to quantum effects, and ground states display many of the relevant properties of the system. 2. The finite temperature case, where thermal fluctuations prevail and equilibrium states become more relevant. In the first case, the states are determined by special projections called basis projections. The topology of the space of all the basis projections is well known. There is a Z2-index (introduced by Araki, see [2]) that classifies these projections and this classification is lifted to the space of the states. In this work, we extend these results for lattices in any finite dimension and in the thermodynamic limit for systems that can even include disorder [3]. We also show that the index remains invariant along curves of Hamiltonians for which the gap remains open. Hence, any change in the value of the index is related to a gap closing. In the second case, the states are equilibrium states associated with systems with inverse finite temperature beta. These states are Gibbs states, which, as is well-known, satisfy the KMS condition. On the other hand, and following the self-dual formalism, given the uniqueness condition of the KMS states, there exist symbols S_beta in B(K), positive bounded and self-adjoint operators on the one-particle self-dual Hilbert space K, that determine the Gibbs states. The symbols S_beta do not always turn out to be basis projections, so unfortunately we do not have a topological characterization of the state space a priori (as in case 1). However, we have been able to study geometric aspects of the space of equilibrium states through the study of holonomies. In the literature there are several proposals to calculate geometric phases associated with states given by density operators, among them, we highlight the Uhlmann phase [4], [5]. There is a way to purify these equilibrium states and find basis projections in a larger space K^ = K + K (direct sum), in the same spirit as what is known as thermo-field dynamics. Using results from Kato [6], the calculation of holonomy and geometric phase in this larger space K^ is reduced to the study of these geometric aspects in the case of zero temperature. Applying the pertinent restrictions to the initial space, we recover the results of Uhlmann's proposal. In the concrete example of the Ising XY model, we also check that this geometric phase models a regime change given by the longitudinal magnetization of the system [7].