Finite-temperature correlation function for the one-dimensional quantum Ising model: The virial expansion

Abstract

We rewrite the exact expression for the finite temperature two-point correlation function for the magnetization as a partition function of some field theory. This removes singularities and provides a convenient form to develop a virial expansion (the expansion in powers of soliton density).To calculate correlation functions in strongly correlated systems is not an easy task, even if the corresponding models happen to be integrable. For models with dynamically generated spectral gaps the most powerful technique is the formfactor approach pioneered by Karowski et. al.[1] and perfected by Smirnov [2]. This approach works wonderfully for zero temperature, but encounters difficulties at T = 0. These difficulties are related to singularities in the operator matrix elements (formfactors). These singularities exist for operators nonlocal with respect to solitons, they originate from forward scattering processes and their treatment requires careful infrared regularization. Despite long efforts a correct regularization has not been yet found. However, for models of free fermions (such as the XY model or the Quantum Ising model), there are alternative means to calculate the correlation functions which allow to bypass the above problems. These alternative approaches include the determinant representation of the correlation functions [3], [4] and the semiclassical method [5] (which may have much wider application, see [6], [7]). For these results to have a greater use one has to establish their relationship with the formfactor approach. A step in this direction was made in [8] where the semiclassical results [5], [6], [7] were reproduced by summing up the leading singularities in the formfactor expansion. Such summation was restricted to the leading order in the soliton density n ∼ exp(−M/T ) (M is the spectral gap). In this paper we describe a formfactor-based representation for correlation functions which, though in its present form is valid only for models of free fermions, is rather suggestive and may give rise to useful generalizations in the future. For the Quantum Ising (QI) model, which is the main object of this paper, this procedure naturally gives rise to a virial expansion of the dynamical spin susceptibility. The QI model Hamiltonian is

Wigner crystal physics in quantum wires

“…The resulting Hamiltonian H t + H NN is the Hamiltonian of the transverse field Ising model [67]. Here the parameters t and v are related to the parameters of the original model.…”

“…It turns out that the Hamiltonian takes a much simpler form if one rotates σ x → −σ z first. The representation of the spin operators in terms of (spinless) fermions, then yields the noninteracting Hamiltonian [67] …”

Form factors in equilibrium and non-equilibrium mixed states of the Ising model

“…There is still relatively little known about the structure of two-point functions in such general situations, and how this structure depends on the mixed state. Correlation functions in thermal states (or on the cylinder) have been widely studied in general QFT (see for instance the books [19]), and with more precision in massive integrable QFT [6,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. The structure is relatively well understood, although still much work needs to be done in integrable QFT in order to get as powerful large-distance or large-time expansions as for vacuum two-point functions.…”

Integral equations and long-time asymptotics for finite-temperature Ising chain correlation functions

“…(recall the functions g ± (θ) and h ± (θ), (22) and (40) respectively). The form of the initial scattering data a(θ), b(θ) above constitute the main results of this section.…”

“…A lot is known about these correlation functions, but the analytic continuation to real time is plagued by singularities, so more work is needed in order to investigate the finite-temperature correlation functions in time-like regions. Recent approaches to the problem include a semiclassical method applicable to the regime of small temperature [24,23], an approach, working in a similar regime, based on identifying the leading singularities of operator matrix elements [1], a virial expansion (in powers of soliton density) [22], and a low-temperature expansion using a regularisation of infinite-volume form factors and an appropriate re-summation scheme [11]. Here, we take a different approach based on solving a famous set of nonlinear partial differential equations (PDEs) satisfied by the thermal correlation functions [21,19,13], a generalisation of the nonlinear ordinary differential equations (Painlevé III) occurring at zero temperature [26,25,3,13].…”