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.
Highlights
Quantum many-body spin systems that are exactly solvable and exhibit a quantum phase transition have been key to advance our understanding of critical phenomena in the quantum domain
Quasi-free fermion models have been discussed in the context of quantum thermodynamics, as a test-bed to explore work statistics and fluctuation theorems [43,44,45,46] and as a working substance in a quantum thermodynamic cycle [47]
We have provided an exact treatment of the thermal equilibrium properties for a class of integrable spin chains that admit a description in terms of free fermions
Summary
Quantum many-body spin systems that are exactly solvable and exhibit a quantum phase transition have been key to advance our understanding of critical phenomena in the quantum domain. The one-dimensional XY model and the closely-related transverse-field quantum Ising model (TFQIM) occupy a unique status, and are paradigmatic test-beds of quantum critical behavior [1,2,3,4] They belongs to a family of models that admit an exact diagonalization by a combination of Jordan-Wigner and Fourier transformations, yielding a formulation of the system in terms of free fermions [5, 6]. ˆcn andcn† are ladder Fermionic operators at site n, which satisfy anti-commutation relationsci, ˆc†j = δi,j andci, ˆcj = ˆci†, ˆc†j = 0 This is in contrast to the Pauli matrices, which satisfy commutation relations σ †n, σ −m = δn,mσzn and σzn, σ ±m = ±2δn,mσ ±n with σ ±n = σnx ± iσny.
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