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

Abstract

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.

Critical region ofD-dimensional spins: extension and analysis of the hierarchical reference theory

“…However for D > 1 this will be modified somewhat since then horizontal isotherms are expected at phase coexistence as is the case for the MSM and GMSM (D = ∞). As already mentioned above, this horizontal slope was found by numerical investigation of SCOZA [30] for D = 3. For this case the critical exponent for the curve of coexistence became the one of the MSM, β = 1/2, within numerical accuracy.…”

“…The SCOZA problem for D-dimensional spins has been considered earlier by Høye and Stell [25]. This problem was also solved numerically [30] for D = 3. A special feature of the numerical results for this case is that the isotherms are horizontal at phase coexistence with a mean field type curve of coexistence with critical index [30], β = 1/2.…”

“…This problem was also solved numerically [30] for D = 3. A special feature of the numerical results for this case is that the isotherms are horizontal at phase coexistence with a mean field type curve of coexistence with critical index [30], β = 1/2. This clearly differs from best estimates β ≈ 0.33 and the SCOZA value β = 0.35 for D = 1; but horizontal isotherms at coexistence are correct.…”

The uniform quantized electron gas revisited

“…With SCOZA the strength of the effective interaction is tuned by changing the effective temperature, while with HRT the effective interaction is built (and subsequently modified) by gradually including its wave vectors in Fourier space (renormalization). Both SCOZA and HRT turned out to give very accurate results, also in the challenging critical region [4][5][6].…”

Increase in Collector Current in Hot-Electron Transistors Controlled by Gate Bias

“…Its real breakthrough came in 1996 [8], when a reformulation of the SCOZA partial differential equation (PDE) made access to subcritical temperatures possible. Since then, the SCOZA has been applied to a few discrete [8][9][10][11][12][13][14][15] and continuum systems [5,6,16]. Here we focus on the continuum case, where applications have been restricted up to now to the one-component case and to hard-core (HC) interactions with an adjacent attractive potential built up by linear combination of up to two Yukawa tails (offering the possibility to approximate a Lennard-Jones (LJ) interaction rather accurately [6]).…”