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arXiv:hep-th/9212135 · Physical Review Letters

Temperature correlations of quantum spins

Alexander Rudol'fovich Its, Anatoli G. Izergin, V. E. Korepin, N. A. Slavnov

Published Mar 15, 1993• 131 citations
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Abstract

We consider the isotropic XY model in a transverse magnetic field in one dimension. One can alternatively call this model the Heisenberg XXO antiferromagnet. We solve the problem of the evaluation of asymptotics of temperature correlations and explain the physical meaning of our result. To do this we represent the quantum correlation function as a tau function of a completely integrable differential equation. This is the well-known Ablowitz-Ladik lattice nonlinear Schr\"odinger differential equation.

DOI
10.1103/physrevlett.70.1704

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Journal of Applied Physics•Mar 15, 1984•14 citations
We consider the isotropic XY model in a transverse magnetic field in one dimension. One can alternatively call this model the Heisenberg XXO antiferromagnet. We solve the problem of the evaluation of asymptotics of temperature correlations and explain the physical meaning of our result. To do this we represent the quantum correlation function as a tau function of a completely integrable differential equation. This is the well-known Ablowitz-Ladik lattice nonlinear Schr"odinger differential equation.
We consider isotropic XY model in the transverse magnetic field on the one dimensional lattice. Another name of the model in Heisenberg XXO model of spin 1/2.We solved long standing problem of evaluation of temperature correlations. We first represent correlation function in the model, by means of completely integrable differential equation. This is famous Ablowitz-Ladik lattice Nonlinear Schrodinger equation.Correlation function is the τττ function of this differential equation. We solved this equation and evaluate asymptotics.
The quantum correlation functions for different exactly solvable (completely integrable) models are all constructed in a similar way. In Part IV our main example will be the nonlinear Schrödinger (NS) equation. We shall also comment on the other models: the Heisenberg antiferromagnet, the Hubbard model and so on. We shall discuss different approaches to quantum correlation functions. Our main approach leads to differential equations for quantum correlation functions. These are classical nonlinear differential equations; they are completely integrable and closely related to the original classical differential equation that was quantized (the NS equation in our case). In the first stage of calculation of correlation functions we represent them as a determinant of some Predholm integral operator (here we shall use the results of Part III). The correct language for the description of quantum correlation functions is the language of τ functions, which was developed in. This helps us to relate the differential equations for quantum correlation functions with the hierarchy of the original classical differential equation that was quantized (the NS equation in our example). In this way we solve the most difficult problem; that of the evaluation of time-and temperature-dependent correlation functions. This approach can be applied not only to correlation functions for the NS equation, but also to the sine-Gordon model, to non-relativistic fermions, to the Heisenberg antiferromagnetic with special anisotropy and to other models.
Astymptotics of temperature correlations is the most dificult problem of stat. mech. Recently it was solved for the impenetrable Bose Gas. The idea is to represent correlation function as τττ function of calssical completely integrable differential equation. Now we start to apply this program to isotropic XY model in the trasnsverse magnetic field. In this paper we are making first step. We representing correlation function as detrminanat of an integral operator.
An approach for studying phase transitions in systems with quantum degrees of freedom is discussed. Starting from the microscopic Hamiltonian of a quantum model, we first derive a set of exact differential equations for the free energy and the correlation functions describing the effects of fluctuations on the thermodynamics of the system. These equations reproduce the full renormalization group structure in the neighborhood of a critical point keeping, at the same time, full information on the nonuniversal properties of the model. As a concrete application we investigate the phase diagram of a Heisenberg antiferromagnet in a staggered external magnetic field. At long wavelengths the known relationship to the quantum nonlinear $\ensuremath{\sigma}$ model naturally emerges from our approach. By representing the two point function in an approximate analytical form, we obtain a closed partial differential equation which is then solved numerically. The results in three dimensions are in good agreement with available quantum Monte Carlo simulations and series expansions. More refined approximations to the general framework presented here and few applications to other models are briefly discussed.
We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.