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arXiv:hep-lat/9603002 · Physical review. B, Condensed matter

Strong-coupling analysis of two-dimensional O(N) σ models withN⩽2 on square, triangular, and honeycomb lattices

Massimo Campostrini B, Andrea Pelissetto C, Paolo Rossi, Ettore Vicari

Published Sep 1, 1996• 53 citations
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Abstract

The critical behavior of two-dimensional (2D) O(N) \ensuremath{\sigma} models with N\ensuremath{\le}2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N2 the critical behavior is of power-law type, and the exponents \ensuremath{\gamma} and \ensuremath{\nu} extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent \ensuremath{\sigma}=1/2, and with universality. The value \ensuremath{\sigma}=1/2 is confirmed within an uncertainty of few percent. The prediction \ensuremath{\eta}=1/4 is also roughly verified. For various values of N\ensuremath{\le}2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N\ensuremath{\le}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.7301

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The critical behavior of two-dimensional (2D) O(N) \ensuremath{\sigma} models with N\ensuremath{\le}2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N2 the critical behavior is of power-law type, and the exponents \ensuremath{\gamma} and \ensuremath{\nu} extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent \ensuremath{\sigma}=1/2, and with universality. The value \ensuremath{\sigma}=1/2 is confirmed within an uncertainty of few percent. The prediction \ensuremath{\eta}=1/4 is also roughly verified. For various values of N\ensuremath{\le}2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N\ensuremath{\le}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. \textcopyright{} 1996 The American Physical Society.
In three-dimensional O(N)O(N)O(N) models, we investigate the low-momentum behavior of the two-point Green's function G(x)G(x)G(x) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotationally invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice NNN-vector model, and 1/N1/N1/N expansion, field-theoretical methods within the ${\ensuremath{\varphi}}^{4}$ continuum formulation. Non-Gaussian corrections to the universal low-momentum behavior of G(x)G(x)G(x) are evaluated, and found to be very small. In nonrotationally invariant physical systems with O(N)O(N)O(N)-invariant interactions, the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent $\ensuremath{\rho}$, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At $N=\ensuremath{\infty}$ one finds $\ensuremath{\rho}=2$. We show that, for all values of N> 0N>~0N> 0, $\ensuremath{\rho}\ensuremath{\simeq}2$.
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.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann, and Jensen. We have computed various critical parameters and improved the estimates of others. Moreover, the properties of hyperscaling and of universality (spin-S independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin-S Ising model on the triangular, honeycomb, and kagom\'e lattices.
To allow analysis of square lattice O(2) model series, we have extended our q-fit method of analysis previously applied to triangular lattice models. We present analysis of our twelfth-order high-temperature series and of tenth-order strong-coupling series. The results from the high-temperature series are consistent with our earlier results \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, \ensuremath{\eta}=0.27\ifmmode\pm\else\textpm\fi{}0.03 for the triangular lattice with the reduced precision usually accompanying a reduction in coordination number. However, the analysis of the strong-coupling expansion, significantly smoothed by an Euler transformation, convincingly supports \ensuremath{\nu}=0.50\ifmmode\pm\else\textpm\fi{}0.05.
We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.