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Journal of Physics C Solid State Physics

Time-dependent correlations of the one-dimensional isotropic XY model

H B Cruz, L.L. Gonçalves

Published Jul 20, 1981• 44 citations
Publisher

Abstract

The authors study the time-dependent correlations of the free-ends isotropic XY model in a transverse field. The boundary effects on the longitudinal correlation, namely (S j z (t)S l z (0)), and bulk dynamic properties are considered at the temperature limits T= infinity and T=0- (for h>or= mod J mod ) where the results are known in closed form. The transverse correlation namely (S j x (t)S l x (0)) is calculated exactly at T=0 (for h>or= mod J mod ), and the boundary effects discussed. At T= infinity the known result for the bulk correlation is rederived by using a new combination of fermion operators.

DOI
10.1088/0022-3719/14/20/016

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Physical review. B, Solid state•Oct 1, 1974•9 citations
The authors study the time-dependent correlations of the free-ends isotropic XY model in a transverse field. The boundary effects on the longitudinal correlation, namely (S j z (t)S l z (0)), and bulk dynamic properties are considered at the temperature limits T= infinity and T=0- (for h>or= mod J mod ) where the results are known in closed form. The transverse correlation namely (S j x (t)S l x (0)) is calculated exactly at T=0 (for h>or= mod J mod ), and the boundary effects discussed. At T= infinity the known result for the bulk correlation is rederived by using a new combination of fermion operators.
Abstract The dynamics of the spin 1/2 one‐dimensional XY‐model in a transverse field is considered. The short‐time behaviour of the auto‐correlation function 〈S (t) S (0)〉 for the closed chain is obtained numerically by considering the exact results known for the open chain which are given in terms of Pfaffians. The bulk auto‐correlation is identified with the correlation 〈S (t) S (0)〉 for the open chain, and this identification is shown to be valid up to a characteristic time Jt ≅ 20. It is also shown, as expected, that the boundary effects are more pronounced for fields smaller than the exchange constant (h < |J|) and in the low temperature limit. The results are obtained for different temperatures and field strengths, and the numerical convergence is discussed in each case.
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.
Results are presented for the static and time-dependent behavior of the classical one-dimensional XY-model at finite temperatures. Equilibrium properties were determined using an importance sampling Monte Carlo method. A spin-dynamics computer simulation method was then used to evaluate spin-spin and energy-energy time-displaced correlation functions out to time t=10J−1 using initial states chosen from the equilibrium distribution. The transverse spin-spin auto correlation function decreases smoothly as a function of time. The longitudinal spin-spin correlation functions, however, show temperature dependent oscillatory behavior. At moderately high temperature the energy-energy correlations show diffusive t−1/2 behavior at long times with only a slow temperature dependence of the diffusion constant.
Computer calculations of the time-dependent spin and energy correlation functions of the classical one-dimensional XY\mathrm{XY}XY model at infinite temperature are reported. Plots of 〈SziSzi(t)〉〈{S}_{z}^{i}{S}_{z}^{i}(t)〉〈Szi​Szi​(t)〉, 〈SziSzi+1(t)〉〈{S}_{z}^{i}{S}_{z}^{i+1}(t)〉〈Szi​Szi+1​(t)〉, $〈{\ensuremath{\epsilon}}^{i}{\ensuremath{\epsilon}}^{i}(t)〉$, and (12)[〈SxiSxi(t)〉+〈SyiSyi(t)〉](\frac{1}{2})[〈{S}_{x}^{i}{S}_{x}^{i}(t)〉+〈{S}_{y}^{i}{S}_{y}^{i}(t)〉](21​)[〈Sxi​Sxi​(t)〉+〈Syi​Syi​(t)〉] out to times Jt=9Jt=9Jt=9 are given (${\ensuremath{\epsilon}}^{i}$ is the energy-density operator and $\ensuremath{\hbar}J$ is the exchange integral). Comparison is made with the exact results for the spin-1/2 XY\mathrm{XY}XY model. After an appropriate scaling of the exchange integral and normalization to the classical value at t=0t=0t=0 the values of the spin-1/2 functions are close to the values of their classical counterparts for times up to $\mathrm{Jt}\ensuremath{\approx}4$.