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Physical review. B, Condensed matter

Weak-coupling series and the critical indices of Zp spin systems in two dimensions

Chris J. Hamer, John B. Kogut

Published Oct 1, 1980• 16 citations
Publisher

Abstract

The quantum Hamiltonian version of ${Z}_{p}$ spin systems in two dimensions is used to study these model's weak-coupling phases. Twelve-term series are presented for the specific heats, magnetizations, and susceptibilities for $p=2, 3, 5, 6, \mathrm{and} 12$. Series analyses yield useful estimates of the critical indices for the three-state model. We suggest that the magnetization index $\ensuremath{\beta}$ lies within the interval 0.106-0.110. For $p>4$ the series indicate a phase transition below the self-dual point as predicted by Elitzur, Pearson, and Shigemitsu. The specific-heat series, however, show no anomaly at these points. Estimates of $\ensuremath{\eta}$ at the lower critical points are presented for $p=5, 6, \mathrm{and} 12$, and they agree with the expected result $\frac{4}{{p}^{2}}$.

DOI
10.1103/physrevb.22.3378

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The quantum Hamiltonian version of Zp{Z}_{p}Zp​ spin systems in two dimensions is used to study these model's weak-coupling phases. Twelve-term series are presented for the specific heats, magnetizations, and susceptibilities for p=2,3,5,6,and12p=2, 3, 5, 6, \mathrm{and} 12p=2,3,5,6,and12. Series analyses yield useful estimates of the critical indices for the three-state model. We suggest that the magnetization index \ensuremathβ\ensuremath{\beta}\ensuremathβ lies within the interval 0.106-0.110. For p>4p>4p>4 the series indicate a phase transition below the self-dual point as predicted by Elitzur, Pearson, and Shigemitsu. The specific-heat series, however, show no anomaly at these points. Estimates of \ensuremathη\ensuremath{\eta}\ensuremathη at the lower critical points are presented for p=5,6,and12p=5, 6, \mathrm{and} 12p=5,6,and12, and they agree with the expected result 4p2\frac{4}{{p}^{2}}p24​.
Strong-coupling expansions of the quantum-Hamiltonian form of the Ising, O(2)O(2)O(2), O(3)O(3)O(3), and O(4)O(4)O(4) models in two dimensions are analyzed. The critical point, correlation-length exponent $\ensuremath{\nu}$, and magnetization index $\ensuremath{\beta}$ are found exactly for the Ising model. The susceptibility index $\ensuremath{\gamma}$ and the specific-heat index $\ensuremath{\alpha}$ are computed to within 1% of their known values. The phase transition of the O(2)O(2)O(2) model is found and the correlation length is predicted to diverge with an essential singularity $\ensuremath{\xi}(T)\ensuremath{\sim}\mathrm{exp}[b{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\sigma}}]$ where $\ensuremath{\sigma}=0.7\ifmmode\pm\else\textpm\fi{}0.1$. The index of the spin-spin correlation function $\ensuremath{\eta}$ is predicted to be 0.26\ifmmode\pm\else\textpm\fi{}0.03 at the critical point in agreement with experiment. The non-Abelian spin systems are argued to exist only in a disordered phase. The asymptotic freedom of these models plays an important role in these arguments by ruling out the possibility of weak-coupling phase transitions.
We review the estimates for the critical indices of the Ising model below TC{T}_{C}TC​ and conclude (for 3 dimensions): ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.066_{\ensuremath{-}0.04}^{}{}_{}{}^{+0.16}$, $\ensuremath{\beta}=0.312_{\ensuremath{-}0.005}^{}{}_{}{}^{+0.002}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.310_{\ensuremath{-}0.05}^{}{}_{}{}^{+0.04}$ for the specific heat, magnetization, and magnetic susceptibility, respectively. For 2 dimensions, we estimate ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75_{\ensuremath{-}0.00}^{}{}_{}{}^{+0.01}$. In order to explain previous estimates of ${\ensuremath{\alpha}}^{\ensuremath{'}}=0$, we point out that a low power can, in practice, look deceptively like a logarithm. Finally, we discuss the behavior of the specific heat at constant magnetization.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.
The Hamiltonian formulation of lattice spin systems is used to study the critical properties of a truncated quantum O(n)O(n)O(n)-spin model in one and ddd spatial dimensions for arbitrary nnn. Strong-coupling expansions for the mass gap, ground-state energy density, and susceptibility have been computed up to 14th order and used to search for phase transitions. For n=0n=0n=0 we obtain the exponents for the self-avoiding random-walk problem. In the O(2)O(2)O(2) case we find that the correlation length diverges with an essential singularity at the critical point in 1 + 1 dimensions. No phase transition is found for the O(3)O(3)O(3) and O(4)O(4)O(4) models in the same dimension. Exponents for other values of nnn are presented and higher dimensions are considered.
In this paper we report analyses of high temperature series expansions for the spin [Formula: see text] XY model on the triangular and square lattices. Quantities for which series are analyzed include the fluctuation in the transverse magnetization, fourth order fluctuations in the same quantity, second and fourth moments of the transverse spin–spin correlations, specific heat, and entropy. The evidence favours a phase transition at a finite temperature with conventional power law critical singularities. Scaling seems to hold but hyperscaling seems to be violated. Estimates for critical exponents include γ = 2.50 ± 0.3. Δ = 2.38 ± 0.2, and ν = 143 ± 0.10. The specific heat exhibits no singular behaviour at Tc.