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Physical review. B, Condensed matter

Accurate results from Hamiltonian strong-coupling expansions of the Ising, planar, O(3) , and O(4) spin systems

Chris J. Hamer, John B. Kogut

Published Nov 1, 1979• 35 citations
Publisher

Abstract

Strong-coupling expansions of the quantum-Hamiltonian form of the Ising, $O(2)$, $O(3)$, and $O(4)$ 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 $O(2)$ 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.

DOI
10.1103/physrevb.20.3859

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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.
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 $\ensuremath{\beta}$ 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 $\ensuremath{\eta}$ 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​.
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We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.