Recent work on the finite temperature correlators of the Ising chain in a transverse field is reviewed. The crossover phase diagram in the coupling-constant and temperature plane is discussed, and various exact results and conjectures for static and dynamic correlators are presented.
We consider finite-temperature properties of the Ising chain in a transverse field in the vicinity of its zero-temperature, second-order quantum phase transition. New universal crossover functions for static and dynamic correlators of the “spin” operator are obtained. The static results follow from an early lattice computation of McCoy, and a method of analytic continuation in the space of coupling constants. The dynamic results are in the “renormalized classical” region and follow from a proposed mapping of the quantum dynamics to the Glauber dynamics of a classical Ising chain.
Two sets of nonlinear differential equations are derived and discussed for the time-dependent correlations between x-components of spins (S = 12) in an Ising chain in the presence of a transverse magnetic field. The equations are independent of temperature which enters only through the initial conditions for the correlations. The equations are valid for the (general) inhomogeneous case in which the exchange coupling as well as the magnetic field depend on the sites in the chain. In the derivation use is made of a general formulation of the thermodynamic Wick theorem. For the homogeneous case a nonlinear differential-difference equation is derived, generalizing the Painlevé III equation found previously at zero temperature in the scaling limit. The finite-temperature field theory limit is discussed also.
The interplay of spin and charge fluctuations in the random transverse-field Ising spin chain on the fermionic space is investigated. The finite chemical potential, which controls the charge fluctuations, leads to the appearance of the quantum critical region in the phase diagram where the magnetic correlations are quenched by nonmagnetic sites. Regions of nonmonotonous temperature dependence of spin-spin correlation length appear at nonzero $\ensuremath{\mu}.$ The results on the one-fermion density of states of the model are discussed.
We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime ∆> 1, extending our previous results to the full parameter plane of the antiferromagnetic chain ( ∆> −1 and arbitrary field h). The factorized formulae are numerically efficient and allow for taking the isotropic limit ( ∆ = 1) and the Ising limit ( ∆ = ∞). At the critical field separating the fully polarized phase from the Néel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite ∆ inducing unexpected crossover phenomena in the correlations. 1