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Physical review. B, Condensed matter

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

Gerhard H. Muller, Robert E. Shrock

Published Jan 1, 1984• 65 citations
Publisher

Abstract

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 $S=\frac{1}{2}$ isotropic $\mathrm{XY}$ 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 $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$, $m=0,1,2,3,\dots{}$, where ${\ensuremath{\omega}}_{0}=J$ for the $\mathrm{XY}$ 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 $\mathrm{XY}$ 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 $\mathrm{XY}$-like substances ${\mathrm{Cs}}_{2}$Co${\mathrm{Cl}}_{4}$ and Pr${\mathrm{Cl}}_{3}$ and the $S=\frac{1}{2}$ Ising-like substance CsCo${\mathrm{Cl}}_{3}$${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.

DOI
10.1103/physrevb.29.288

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

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Spin dynamics for the one-dimensional XY model at infinite temperature

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Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:

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Space-Time Correlations in Exchange-Coupled Paramagnets at Elevated Temperatures

Raza A. Tahir-Kheli, Daniel G. McFadden

The frequency-dependent correlations ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})\ensuremath{\equiv}{〈{{S}_{0}}^{z}(t){{S}_{n}}^{z}({t}^{\ensuremath{'}})〉}_{(\ensuremath{\omega})}$, $n=0,1,2,3, \mathrm{and} 4$, and their inverse-lattice Fourier transforms $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$, have recently been exactly calculated by Carboni and Richards for finite linear chains consisting of $N$ ($N=6,7,8,9, \mathrm{and} 10$) spins ($S=\frac{1}{2}$) in contact with a heat bath at temperature $T\ensuremath{\rightarrow}\ensuremath{\infty}$ and interacting via a nearest-neighbor isotropic Heisenberg exchange interaction. Carboni and Richards have used plausible extrapolation procedures to predict the corresponding correlations in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. As the extension of these exact calculations to two and three dimensions is of prohibitive difficulty, we have constructed an alternative theory, based upon a simple two-parameter Gaussian representation of the generalized diffusivity, for calculating these correlations for general spin and the dimensionality. To test the accuracy of this phenomenological theory (which is, however, free of any arbitrariness in the sense that the diffusivity is exactly specified by the second and the fourth frequency moments of the Fourier transform of the frequency-dependent correlation function, which are known at infinite temperature), we first compare our results for ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})$ with those given by Carboni and Richards for the one-dimensional spin-\textonehalf{} system and find the agreement to be excellent for $n=0$, good for $n=1,2$ and adequate for $n=3,4$. The comparison of the corresponding results for $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$ reveals the agreement to be quite satisfactory for $|\mathbf{K}|\ensuremath{\lesssim}\frac{1}{2}\ensuremath{\pi}$ but only adequate for the higher-K range, i.e., $\frac{1}{2}\ensuremath{\pi}\ensuremath{\lesssim}\mathbf{K}\ensuremath{\lesssim}\ensuremath{\pi}$. Further support in favor of our phenomenological theory is obtained from a comparison with another set of available exact "computer experiment" results, whereby ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$, ${\ensuremath{\Phi}}_{0}(t)$, and ${\ensuremath{\Phi}}_{1}(t)$ are accurately known for a three-dimensional (simple-cubic) lattice of infinite spins, i.e., $S\ensuremath{\rightarrow}\ensuremath{\infty}$. (This is the limit in which the classical spin system studied by Windsor corresponds to the quantum spin system of interest to us.) The agreement of our result with Windsor's is excellent. The salient features of our results are as follows: (i) In two dimensions, the divergence of ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$ becomes less sharp and disappears completely in three dimensions. (ii) The cutoff frequency, beyond which ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ is effectively zero, increases with the dimensionality, being in three dimensions about twice what it is in one dimension. (iii) There exists a system of reduced units, i.e., ${{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\rightarrow}{\ensuremath{\Phi}}_{n}(\ensuremath{\omega}){[S(S+1)]}^{\ensuremath{-}1}$, ${I}^{\ensuremath{'}}\ensuremath{\rightarrow}{[S(S+1)]}^{\frac{1}{2}}I$, and ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\rightarrow}(\frac{\ensuremath{\omega}}{{I}^{\ensuremath{'}}})$, in which the function ${I}^{\ensuremath{'}}{{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}({\ensuremath{\omega}}^{\ensuremath{'}})$ is approximately the same for all spins, and the accuracy of this law of corresponding states seems to increase with the increase in the dimensionality.

Physical Review•Jun 10, 1969•63 citations
We present new results on the time-dependent correlation functions \ensuremathΞn(t)=4〈S0\ensuremathξ(t)Sn\ensuremathξ〉{\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉\ensuremathΞn​(t)=4〈S0\ensuremathξ​(t)Sn\ensuremathξ​〉, \ensuremathξ=x,y\ensuremath{\xi}=x,y\ensuremathξ=x,y, at zero temperature of the one-dimensional S=12S=\frac{1}{2}S=21​ isotropic XY\mathrm{XY}XY model (h=\ensuremathγ=0h=\ensuremath{\gamma}=0h=\ensuremathγ=0) and of the transverse Ising (TI) model at the critical magnetic field (h=\ensuremathγ=1h=\ensuremath{\gamma}=1h=\ensuremathγ=1). Both models are characterized by special cases of the Hamiltonian H=\ensuremath−J\ensuremathΣl[(1+\ensuremathγ)SlxSl+1x+(1\ensuremath−\ensuremathγ)SlySl+1y+hS1z]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}]H=\ensuremath−J\ensuremathΣl[(1+\ensuremathγ)Slx​Sl+1x​+(1\ensuremath−\ensuremathγ)Sly​Sl+1y​+hS1z​]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions \ensuremathΞ0(0){\ensuremath{\Xi}}_{0}(0)\ensuremathΞ0​(0) and on the singularities of their frequency-dependent Fourier transforms \ensuremathΦ0\ensuremathξ\ensuremathξ(\ensuremathω){\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})\ensuremathΦ0\ensuremathξ\ensuremathξ​(\ensuremathω). We have also determined the latter functions by high-precision numerical calculations. The functions \ensuremathΦ0\ensuremathξ\ensuremathξ(\ensuremathω){\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})\ensuremathΦ0\ensuremathξ\ensuremathξ​(\ensuremathω), \ensuremathξ=x,y\ensuremath{\xi}=x,y\ensuremathξ=x,y, have singularities at the infinite sequence of frequencies \ensuremathω=m\ensuremathω0\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}\ensuremathω=m\ensuremathω0​, m=0,1,2,3,…m=0,1,2,3,\dots{}m=0,1,2,3,…, where \ensuremathω0=J{\ensuremath{\omega}}_{0}=J\ensuremathω0​=J for the XY\mathrm{XY}XY model and \ensuremathω0=2J{\ensuremath{\omega}}_{0}=2J\ensuremathω0​=2J for the TI model. In the TI case, the leading singularities in \ensuremathφ0xx(\ensuremathω){\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})\ensuremathφ0xx​(\ensuremathω) are alternately one-sided and two-sided power-law singularities, the first two of which (at \ensuremathω=0,2J\ensuremath{\omega}=0,2J\ensuremathω=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 \ensuremathω=0,J\ensuremath{\omega}=0,J\ensuremathω=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 \ensuremathω\ensuremath≠0\ensuremath{\omega}\ensuremath{\ne}0\ensuremathω\ensuremath=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 CsCo{\mathrm{Cl}}_{3}$${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}O.
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.
New exact results are presented on the long-time asymptotic expansions of the T=0T=0T=0 autocorrelation functions, and on the leading singularities of their frequency-dependent Fourier transforms, for the one-dimensional S=12S=\frac{1}{2}S=21​ isotropic XY\mathrm{XY}XY model and the S=12S=\frac{1}{2}S=21​ transverse Ising model at the critical field. High-precision numerical calculations of the latter functions are also given, and experiments are proposed to observe the functional behaviors that are found.
Dynamic spin correlation functions 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​〉 for the one-dimensional S=1/2 XX model H=-Jtsumi{\mathit{Jtsum}}_{\mathit{i}}Jtsumi​{Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Si+1x{\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}Si+1x​+Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Si{\mathit{S}}_{\mathit{i}}Si​ $_{+1}^{\mathit{y}}\mathrm{}}$ are calculated exactly for finite open chains with up to N=10 000 spins. Over a certain time range the results are free of finite-size effects and thus represent correlation functions of an infinite chain (bulk regime) or a semi-infinite chain (boundary regime). In the bulk regime, the long-time asymptotic decay as inferred by extrapolation is Gaussian at T=\ensuremath{\infty}, exponential at 0T\ensuremath{\infty}, and power law (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$) at T=0, in agreement with exact results. In the boundary regime, a power-law decay is obtained at all temperatures; the characteristic exponent is universal at T=0 (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$) and at 0T\ensuremath{\infty}(\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$), but is site dependent at T=\ensuremath{\infty}. In the high-temperature regime (T/J\ensuremath{\gg}1) and in the low-temperature regime (T/J\ensuremath{\ll}1), crossovers between different decay laws can be observed in 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​〉. Additional crossovers are found between bulk-tpe and boundary-type decay for i=j near the boundary, and between spacelike and timelike behavior for i\ensuremath{\ne}j.
The infinite-temperature space- and time-dependent spin-correlation functions grx(t){g}_{r}^{x}(t)grx​(t) are studied for the one-dimensional XY\mathrm{XY}XY model. Numerical calculations are performed to obtain the exact autocorrelation function g0x(t){g}_{0}^{x}(t)g0x​(t) for chains containing 5, 7, and 9 spins (S=12S=\frac{1}{2}S=21​). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that g0x(t){g}_{0}^{x}(t)g0x​(t) for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that grx(t){g}_{r}^{x}(t)grx​(t) for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for g0x(t){g}_{0}^{x}(t)g0x​(t) is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
The frequency-dependent correlations ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})\ensuremath{\equiv}{〈{{S}_{0}}^{z}(t){{S}_{n}}^{z}({t}^{\ensuremath{'}})〉}_{(\ensuremath{\omega})}$, n=0,1,2,3,and4n=0,1,2,3, \mathrm{and} 4n=0,1,2,3,and4, and their inverse-lattice Fourier transforms $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$, have recently been exactly calculated by Carboni and Richards for finite linear chains consisting of NNN (N=6,7,8,9,and10N=6,7,8,9, \mathrm{and} 10N=6,7,8,9,and10) spins (S=12S=\frac{1}{2}S=21​) in contact with a heat bath at temperature $T\ensuremath{\rightarrow}\ensuremath{\infty}$ and interacting via a nearest-neighbor isotropic Heisenberg exchange interaction. Carboni and Richards have used plausible extrapolation procedures to predict the corresponding correlations in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. As the extension of these exact calculations to two and three dimensions is of prohibitive difficulty, we have constructed an alternative theory, based upon a simple two-parameter Gaussian representation of the generalized diffusivity, for calculating these correlations for general spin and the dimensionality. To test the accuracy of this phenomenological theory (which is, however, free of any arbitrariness in the sense that the diffusivity is exactly specified by the second and the fourth frequency moments of the Fourier transform of the frequency-dependent correlation function, which are known at infinite temperature), we first compare our results for ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})$ with those given by Carboni and Richards for the one-dimensional spin-\textonehalf{} system and find the agreement to be excellent for n=0n=0n=0, good for n=1,2n=1,2n=1,2 and adequate for n=3,4n=3,4n=3,4. The comparison of the corresponding results for $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$ reveals the agreement to be quite satisfactory for $|\mathbf{K}|\ensuremath{\lesssim}\frac{1}{2}\ensuremath{\pi}$ but only adequate for the higher-K range, i.e., $\frac{1}{2}\ensuremath{\pi}\ensuremath{\lesssim}\mathbf{K}\ensuremath{\lesssim}\ensuremath{\pi}$. Further support in favor of our phenomenological theory is obtained from a comparison with another set of available exact "computer experiment" results, whereby ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$, ${\ensuremath{\Phi}}_{0}(t)$, and ${\ensuremath{\Phi}}_{1}(t)$ are accurately known for a three-dimensional (simple-cubic) lattice of infinite spins, i.e., $S\ensuremath{\rightarrow}\ensuremath{\infty}$. (This is the limit in which the classical spin system studied by Windsor corresponds to the quantum spin system of interest to us.) The agreement of our result with Windsor's is excellent. The salient features of our results are as follows: (i) In two dimensions, the divergence of ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$ becomes less sharp and disappears completely in three dimensions. (ii) The cutoff frequency, beyond which ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ is effectively zero, increases with the dimensionality, being in three dimensions about twice what it is in one dimension. (iii) There exists a system of reduced units, i.e., ${{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\rightarrow}{\ensuremath{\Phi}}_{n}(\ensuremath{\omega}){[S(S+1)]}^{\ensuremath{-}1}$, ${I}^{\ensuremath{'}}\ensuremath{\rightarrow}{[S(S+1)]}^{\frac{1}{2}}I$, and ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\rightarrow}(\frac{\ensuremath{\omega}}{{I}^{\ensuremath{'}}})$, in which the function ${I}^{\ensuremath{'}}{{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}({\ensuremath{\omega}}^{\ensuremath{'}})$ is approximately the same for all spins, and the accuracy of this law of corresponding states seems to increase with the increase in the dimensionality.