Exact thermal properties of free-fermionic spin chains

Abstract

An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.

Uhlmann Fidelity and Fidelity Susceptibility for Integrable Spin Chains at Finite Temperature: Exact Results

“…In this section, we briefly recall the methods and results from [22]. To begin, we note that from the transformation to fermionic quasiparticles, the total Hilbert space H can be written as a tensor product of 4-dimensional Hilbert subspaces corresponding to each pair of momenta…”

“…It has been argued that in the thermodynamic limit only Z + F is relevant, i.e., that the partition function is governed by the Positive Fermionic part corresponding to the first part of Equation (34) [20]. While this is true in many cases, one has to be careful; see [22] for details. Explicitly, the Positive Fermionic part of the partition function has the form…”

“…Later, Kapitonov and Il'inskii [21] provided a derivation of full partition function using integrals over Grassmann variables. In our recent work [22], we provide an elementary method to derive exact expressions for partition function and characteristic function of a wide class of observables using only the structure of Hilbert space. The big discrepancy, even in the thermodynamic limit, between the fidelity susceptibility of lowest energy states in positive and negative parity subspaces, was first reported by Damski and Rams [23].…”

“…However, to our best knowledge, an exact treatment of the fidelity between thermal states of integrable spin chains in the complete Hilbert spaces seems to be lacking at the time of writing. We aim at filling this gap by using methods developed in [22]. We provide exact expressions for fidelity between two arbitrary thermal states in the quantum XY models and analyze the temperature dependence of fidelity susceptibility in a paradigmatic test-bed for quantum critical phenomena, the quantum Ising model in a transverse field.…”

“…In the remaining part of the paper, we first review the diagonalization of the XY model with particular stress on the exact treatment of the parity subspaces in Section 2. In Section 3 we recall the methods developed in [22] and use them to derive the full expression for Uhlmann fidelity between arbitrary thermal states in Section 4. We then focus on fidelity susceptibility computed at the critical point, analyze its temperature dependence and compare it with the simplified expression obtained considering only the positive parity contribution; see Section 5.…”