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Physical review. B, Condensed matter

Strong-coupling expansions for multidimensional Ising and O(n) spin systems

Gerald E. Sobelman

Published Aug 1, 1981• 3 citations
Publisher

Abstract

Recent lattice Hamiltonian strong-coupling calculations for the (1+1)-dimensional Ising, O(2), O(3), and O(4) models are extended to $d+1$ dimensions and to general $\mathrm{O}(n)$ symmetry. This allows a nontrivial numerical check on the rate of convergence of the strong-coupling expansions as the multidimensional models all have standard second-order phase transitions which have been well studied by other methods. Dlog-Pad\'e analysis of the fourth-order mass gap series for the (2+1)-dimensional Ising model show the critical exponent $\ensuremath{\nu}$ to be calculated to about the 10% level of accuracy. The behavior of $\ensuremath{\nu}$ as a function of $d$ and $n$ is exhibited and compared against known results. Mean-field behavior is recovered exactly in the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, while the Berlin-Kac spherical model is obtained in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit.

DOI
10.1103/physrevb.24.1493

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Journal of Mathematical Physics•Jul 1, 1969•128 citations
Recent lattice Hamiltonian strong-coupling calculations for the (1+1)-dimensional Ising, O(2), O(3), and O(4) models are extended to d+1d+1d+1 dimensions and to general O(n)\mathrm{O}(n)O(n) symmetry. This allows a nontrivial numerical check on the rate of convergence of the strong-coupling expansions as the multidimensional models all have standard second-order phase transitions which have been well studied by other methods. Dlog-Pad'e analysis of the fourth-order mass gap series for the (2+1)-dimensional Ising model show the critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν to be calculated to about the 10% level of accuracy. The behavior of \ensuremathν\ensuremath{\nu}\ensuremathν as a function of ddd and nnn is exhibited and compared against known results. Mean-field behavior is recovered exactly in the d\ensuremath→\ensuremath∞d\ensuremath{\rightarrow}\ensuremath{\infty}d\ensuremath→\ensuremath∞ limit, while the Berlin-Kac spherical model is obtained in the n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ limit.
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.
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.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1
We report results of the application of a recently proposed technique, which combines the cluster variation method (CVM) and Pad\'e approximants, to the analysis of the critical behavior of the face-centered-cubic Ising model. The highest-order CVM approximation ever considered for this lattice is used to estimate magnetization and susceptibility, which are then analyzed by means of Dlog Pad\'e approximants in order to extract information about the critical behavior, taking also into account the effects of confluent singularities. Our results are in good agreement with the most recent simulation and series expansion estimates and require a remarkably small numerical effort. \textcopyright{} 1996 The American Physical Society.
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.