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Physical Review

Spin Correlation Functions of the X−Y Model

Barry M. McCoy

Published Sep 10, 1968• 259 citations
Publisher

Abstract

We consider the one-dimensional $X\ensuremath{-}Y$ model of Lieb, Schulz, and Mattis and study the asymptotic behavior of each of the three correlation functions $〈{{\ensuremath{\sigma}}_{0}}^{i}{{\ensuremath{\sigma}}_{N}}^{i}〉={{\ensuremath{\rho}}_{N}}^{i}$, where $i=x$, $y$, or $z$. We study in detail the influence of $X\ensuremath{-}Y$ anisotropy by separately studying the correlation functions in both the isotropic and anisotropic cases at both nonzero and zero temperatures. For nonzero temperature we derive both low- and high-temperature expansions for all three correlations and show that these correlations go to zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The behavior near $T=0$ is studied in the isotropic case by considering the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit with $\mathrm{TN}$ fixed, while in the anisotropic case we must hold ${T}^{2}N$ fixed as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. In this manner we obtain the $T=0$ result that if the interaction is stronger in the $x$ direction, then ${{\ensuremath{\rho}}_{N}}^{x}$ approaches a constant exponentially while ${{\ensuremath{\rho}}_{N}}^{y}$ approaches zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We finally show that in the isotropic case at $T=0$ that ${{\ensuremath{\rho}}_{N}}^{x}={{\ensuremath{\rho}}_{N}}^{y}\ensuremath{\rightarrow}{N}^{\frac{\ensuremath{-}1}{2}}$. In all cases, at least the first two terms of the asymptotic series are explicitly given.

DOI
10.1103/physrev.173.531

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We consider the one-dimensional X\ensuremath−YX\ensuremath{-}YX\ensuremath−Y model of Lieb, Schulz, and Mattis and study the asymptotic behavior of each of the three correlation functions 〈\ensuremathσ0i\ensuremathσNi〉=\ensuremathρNi〈{{\ensuremath{\sigma}}_{0}}^{i}{{\ensuremath{\sigma}}_{N}}^{i}〉={{\ensuremath{\rho}}_{N}}^{i}〈\ensuremathσ0​i\ensuremathσN​i〉=\ensuremathρN​i, where i=xi=xi=x, yyy, or zzz. We study in detail the influence of X\ensuremath−YX\ensuremath{-}YX\ensuremath−Y anisotropy by separately studying the correlation functions in both the isotropic and anisotropic cases at both nonzero and zero temperatures. For nonzero temperature we derive both low- and high-temperature expansions for all three correlations and show that these correlations go to zero exponentially as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. The behavior near T=0T=0T=0 is studied in the isotropic case by considering the N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit with TN\mathrm{TN}TN fixed, while in the anisotropic case we must hold T2N{T}^{2}NT2N fixed as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. In this manner we obtain the T=0T=0T=0 result that if the interaction is stronger in the xxx direction, then \ensuremathρNx{{\ensuremath{\rho}}_{N}}^{x}\ensuremathρN​x approaches a constant exponentially while \ensuremathρNy{{\ensuremath{\rho}}_{N}}^{y}\ensuremathρN​y approaches zero exponentially as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. We finally show that in the isotropic case at T=0T=0T=0 that \ensuremathρNx=\ensuremathρNy\ensuremath→N\ensuremath−12{{\ensuremath{\rho}}_{N}}^{x}={{\ensuremath{\rho}}_{N}}^{y}\ensuremath{\rightarrow}{N}^{\frac{\ensuremath{-}1}{2}}\ensuremathρN​x=\ensuremathρN​y\ensuremath→N2\ensuremath−1​. In all cases, at least the first two terms of the asymptotic series are explicitly given.
We consider the anisotropic one-dimensional $X\ensuremath{-}Y$ model of Lieb, Schultz, and Mattis and study the three zero-temperature correlation functions ${\ensuremath{\rho}}_{\ensuremath{\mu}\ensuremath{\nu}}(R,t)=〈{S}_{1}^{\ensuremath{\mu}}(0){S}_{R+1}^{\ensuremath{\nu}}(t)〉$, where $\ensuremath{\mu}\ensuremath{\nu}=x,y,z$ but $\ensuremath{\mu}\ensuremath{\ne}\ensuremath{\nu}$. We obtain the first-order term of ${\ensuremath{\rho}}_{\mathrm{xy}}$ for $R\ensuremath{\rightarrow}\ensuremath{\infty}$ or $t\ensuremath{\rightarrow}\ensuremath{\infty}$ and exact expressions for ${\ensuremath{\rho}}_{\mathrm{xz}}$ and ${\ensuremath{\rho}}_{\mathrm{yz}}$.
We have calculated the spin-correlation functions (in the x, y, and z directions) in the linear spin-(1/2 XY system for both isotropic and anisotropic couplings. The study of zero-temperature correlation functions shows that a long-range order develops in the direction in which the coupling is stronger, but no such ordering exists for isotropic systems. We have studied the temperature variation of inverse correlation lengths in these directions and it has been found that at T=0 all the correlation lengths (${\ensuremath{\xi}}_{x}$, ${\ensuremath{\xi}}_{y}$, ${\ensuremath{\xi}}_{z}$) diverge in the isotropic case. On the other hand, in the anisotropic case, the correlation length diverges in the direction (either x or y) in which the coupling becomes stronger. The results are compared with the experimental data on Cs2{\mathrm{Cs}}_{2}Cs2​CoCl4{\mathrm{CoCl}}_{4}CoCl4​. The critical exponents (\ensuremath{\eta}',\ensuremath{\nu}') of the correlation function and correlation length are also calculated near the critical temperature (T=0) for different anisotropies, and it is found that the system behaves like an Ising model when a little anisotropy is introduced.
The long-time and long-distance asymptotic behavior of the xxx spin correlations at finite temperature in an anisotropic spin-1/2 XY chain is determined numerically. The decay of the correlations is exponential in both space and time. Similar exponential decay of correlations was already found earlier in the special case of the isotropic model, where analytical expressions for the decay rates could be derived via a mapping to a different model. While no such mapping is known for the anisotropic model, the asymptotic correlations can be very well approximated by a natural generalization of the known analytic results for the isotropic case.
The long-time and long-distance asymptotic behavior of the xxx spin correlations at finite temperature in an anisotropic spin-1/2 XY chain is determined numerically. The decay of the correlations is exponential in both space and time. Similar exponential decay of correlations was already found earlier in the special case of the isotropic model, where analytical expressions for the decay rates could be derived via a mapping to a different model. While no such mapping is known for the anisotropic model, the asymptotic correlations can be very well approximated by a natural generalization of the known analytic results for the isotropic case.
We present new results on the time-dependent correlation functions ${\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉$, $\ensuremath{\xi}=x,y$, at zero temperature of the one-dimensional S=12S=\frac{1}{2}S=21​ isotropic XY\mathrm{XY}XY model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$, m=0,1,2,3,…m=0,1,2,3,\dots{}m=0,1,2,3,…, where ${\ensuremath{\omega}}_{0}=J$ for the XY\mathrm{XY}XY model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the XY\mathrm{XY}XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the XY\mathrm{XY}XY-like substances Cs2{\mathrm{Cs}}_{2}Cs2​CoCl4{\mathrm{Cl}}_{4}Cl4​ and PrCl3{\mathrm{Cl}}_{3}Cl3​ and the S=12S=\frac{1}{2}S=21​ Ising-like substance CsCoCl3{\mathrm{Cl}}_{3}Cl3​${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.