Theory of Critical-Point Scattering and Correlations. II. Heisenberg Models

Ritchie, Douglas; Fisher, Michael E.

doi:10.1103/physrevb.5.2668

Cited by 195 publications

(41 citation statements)

References 47 publications

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“…(3.4), h(q) = [(q/2)/ sin(q/2)] 2 captures lattice effects [37], and the finite-size scaling function g l (x) is chosen as a simple generalization of a Lorentzian:…”

Section: B Static Structure Factor At Criticalitymentioning

confidence: 99%

Spin-dynamics simulations of the three-dimensionalXYmodel: Structure factor and transport properties

Krech

Landau

1999

Phys. Rev. B

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We present extensive Monte-Carlo spin dynamics simulations of the classical XY model in three dimensions on a simple cubic lattice with periodic boundary conditions. A recently developed efficient integration algorithm for the equations of motion is used, which allows a substantial improvement of statistics and large integration times. We find spin wave peaks in a wide range around the critical point and spin diffusion for all temperatures. At the critical point we find evidence for a violation of dynamic scaling in the sense that independent components of the dynamic structure factor S(q, ω) require different dynamic exponents in order to obtain scaling. Below the critical point we investigate the dispersion relation of the spin waves and the linewidths of S(q, ω) and find agreement with mode coupling theory. Apart from strong spin wave peaks we observe additional peaks in S(q, ω) which can be attributed to two-spin wave interactions. The overall lineshapes are also discussed and compared to mode coupling predictions. Finally, we present first results for the transport coefficient D(q, ω) of the out-of-plane magnetization component at the critical point, which is related to the thermal conductivity of 4 He near the superfluid-normal transition.

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“…(3.4), h(q) = [(q/2)/ sin(q/2)] 2 captures lattice effects [37], and the finite-size scaling function g l (x) is chosen as a simple generalization of a Lorentzian:…”

Section: B Static Structure Factor At Criticalitymentioning

confidence: 99%

Spin-dynamics simulations of the three-dimensionalXYmodel: Structure factor and transport properties

Krech

Landau

1999

Phys. Rev. B

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“…Only recently have consistent, but significantly more precise, experimental results been reported in low gravity superfluid experiments [41]. Also the precision of results coming from high temperature expansions [18][19][20][21][22][23][24][25][26][27][28][29] and various numerical simulations [30][31][32][33][34][35][36][37][38][39][40] on the lattice has kept steadily improving.…”

Section: Exponentsmentioning

confidence: 99%

Precise determination of critical exponents and equation of state by field theory methods

Zinn-Justin¹

2001

Physics Reports

139

166

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Renormalization group, and in particular its Quantum Field Theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized φ 4 3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N -vector model [1,2] and of the equation of state of the 3D Ising model [3]. These results are among the most precise available probing field theory in a nonperturbative regime.Precise calculations first require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large order behaviour of perturbation series has then allowed to develop efficient "summation" techniques, based on Borel transformation and conformal mapping [4].We first discuss exponents and describe our recent results [2]. Compared to exponents, the determination of the scaling equation of state of the 3D Ising model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived.Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer Talk given at the Conference "Renormalization Group 2000, Taxco (Mexico), 11-15 Jan. 1999 * Laboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique 2 resources, many new calculations will become feasible. In addition to the infinite volume quantities, finite size universal quantities would also be of interest, to provide a more direct contact with numerical simulations. Let us also mention dynamical observables, a largely unexplored territory.

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“…In particular, the theory has already been formulated also for the D-vector model [3], i.e., a system of classical D-dimensional spins with nearest-neighbor interaction. In the present paper the D-vector model SCOZA is further investigated and the simplest version of the SCOZA that we have developed is solved numerically in the case of three-dimensional spins (classical Heisenberg model) on a threedimensional lattice, and the results are compared with Monte Carlo simulations [4,5,6] and Padè approximants based on high-temperature expansions [7,8,9,10,11]. An analysis of what is lacking in this simple application to the D-vector model then guides us in formulating a more sophisticated version of the theory that takes into account the influence of transverse correlations on the longitudinal susceptibility and as a result, yields a more complete and accurate description of the spin-wave properties of the model.…”

Section: Introductionmentioning

confidence: 99%

“…b: from high-temperature expansions [10]. c: from high-temperature expansions [9]. For the SC lattice, the inverse critical temperature predicted by the quantum hierarchical reference theory (QHRT) [22] in the classical limit is also reported.…”

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confidence: 99%

Self-consistent Ornstein–Zernike approximation for three-dimensional spins

Pini

Høye²,

Stell

2002

Physica A: Statistical Mechanics and its Applications

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An Ornstein-Zernike approximation for the two-body correlation function embodying thermodynamic consistency is applied to a system of classical Heisenberg spins on a three-dimensional lattice. The consistency condition determined in a previous work is supplemented by introducing a simplified expression for the meansquare fluctuations of the spin on each lattice site. The thermodynamics and the correlations obtained by this closure are then compared with approximants based on extrapolation of series expansions and with Monte Carlo simulations. The comparison reveals that many properties of the model, including the critical temperature, are very well reproduced by this simple version of the theory, but that it shows substantial quantitative error in the critical region, both above the critical temperature and with respect to its rendering of the spontaneous magnetization curve. A less simple but conceptually more satisfactory version of the SCOZA is then developed, but not solved, in which the effects of transverse correlations on the longitudinal susceptibility is included, yielding a more complete and accurate description of the spin-wave properties of the model.

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Theory of Critical-Point Scattering and Correlations. II. Heisenberg Models

Cited by 195 publications

References 47 publications

Spin-dynamics simulations of the three-dimensionalXYmodel: Structure factor and transport properties

Spin-dynamics simulations of the three-dimensionalXYmodel: Structure factor and transport properties

Precise determination of critical exponents and equation of state by field theory methods

Self-consistent Ornstein–Zernike approximation for three-dimensional spins

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