Physical review. B, Condensed matter

Spin-correlation functions of the anisotropicXYchain

Rita Basak, Ibha Chatterjee

Published Sep 1, 1989• 16 citations

Abstract

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 ${\mathrm{Cs}}_{2}$${\mathrm{CoCl}}_{4}$. 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.

DOI: 10.1103/physrevb.40.4627

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Spin Correlation Functions of the X−Y Model

Barry M. McCoy

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High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

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Journal of Applied PhysicsMar 1, 19692 citations

Partial Test of the Universality Hypothesis: The Case of Different Coupling Strengths in Different Lattice Directions

Gerald Paul, Eugene Stanley

Physical review. B, Solid stateApr 1, 197256 citations

High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

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Dynamical correlation functions for one-dimensional quantum spin systems: New results based on a rigorous approach

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Longitudinal Spin Correlation in Spin-1/2 DimerizedXYChain. II. Finite Temperature

Kiyomi Okamoto

We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation $R{\equiv}\langle S_{\imath}^{z}S_{\jmath}^{z}\rangle$ of the spin-1/2 dimerized XY chain at finite temperatures. The spin correlation R decays in the exponential law with the correlation length ξ( T ), which decreases with the increase of T . The present results are compared with those obtained by the use of the phase Hamiltonian method.

Journal of the Physical Society of JapanJun 15, 19893 citations

Physical review. B, Condensed matter

Spin-correlation functions of the anisotropicXYchain

Rita Basak, Ibha Chatterjee

Published Sep 1, 1989• 16 citations

Publisher

Abstract

{\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

{\mathrm{Cs}}_{2}$${\mathrm{CoCl}}_{4}

. 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.

DOI: 10.1103/physrevb.40.4627

Related papers

Spin Correlation Functions of the X−Y Model

Barry M. McCoy

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.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.

Physical ReviewSep 10, 1968259 citations

High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

David Jasnow, Michael Wortis

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.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.

Journal of Applied PhysicsMar 1, 19692 citations

Partial Test of the Universality Hypothesis: The Case of Different Coupling Strengths in Different Lattice Directions

Gerald Paul, Eugene Stanley

High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in ( $\frac{1}{{k}_{B}T}$ ) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no $z$ component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for $R=0$ , both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in ($\frac{1}{{k}_{B}T}$) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no $z$ component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for $R=0$, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).

Physical review. B, Solid stateApr 1, 197256 citations

High-Temperature Critical Indices for the Classical Anisotropic Heisenberg Model

David Jasnow, Michael Wortis

High-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heinsenberg model are calculated for various lattices and anisotropies through order ${T}^{\ensuremath{-}8}$ (close-packed lattices) and ${T}^{\ensuremath{-}9}$ (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices $\ensuremath{\gamma}$ (susceptibility), $\ensuremath{\nu}$ (correlation range), and $\ensuremath{\alpha}$ (specific heat) as functions of anisotropy. Our results are consistent with the hypothesis that the critical indices change only 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.High-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heinsenberg model are calculated for various lattices and anisotropies through order ${T}^{\ensuremath{-}8}$ (close-packed lattices) and ${T}^{\ensuremath{-}9}$ (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices $\ensuremath{\gamma}$ (susceptibility), $\ensuremath{\nu}$ (correlation range), and $\ensuremath{\alpha}$ (specific heat) as functions of anisotropy. Our results are consistent with the hypothesis that the critical indices change only 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.

Physical ReviewDec 10, 1968196 citations

Dynamical correlation functions for one-dimensional quantum spin systems: New results based on a rigorous approach

Gerhard H. Muller, Robert E. Shrock

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.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.

Journal of Applied PhysicsMar 15, 198414 citations

Longitudinal Spin Correlation in Spin-1/2 DimerizedXYChain. II. Finite Temperature

Kiyomi Okamoto

We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation $R{\equiv}\langle S_{\imath}^{z}S_{\jmath}^{z}\rangle$ of the spin-1/2 dimerized XY chain at finite temperatures. The spin correlation R decays in the exponential law with the correlation length ξ( T ), which decreases with the increase of T . The present results are compared with those obtained by the use of the phase Hamiltonian method.We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation $R{\equiv}\langle S_{\imath}^{z}S_{\jmath}^{z}\rangle$ of the spin-1/2 dimerized XY chain at finite temperatures. The spin correlation R decays in the exponential law with the correlation length ξ( T ), which decreases with the increase of T . The present results are compared with those obtained by the use of the phase Hamiltonian method.

Journal of the Physical Society of JapanJun 15, 19893 citations

{\ensuremath{\xi}}_{x}

{\ensuremath{\xi}}_{y}

{\ensuremath{\xi}}_{z}

{\mathrm{Cs}}_{2}$${\mathrm{CoCl}}_{4}

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.

\frac{1}{{k}_{B}T}

) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no

z

component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for

R=0

, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).

We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation

R{\equiv}\langle S_{\imath}^{z}S_{\jmath}^{z}\rangle

of the spin-1/2 dimerized XY chain at finite temperatures. The spin correlation R decays in the exponential law with the correlation length ξ( T ), which decreases with the increase of T . The present results are compared with those obtained by the use of the phase Hamiltonian method.