Lattice φ4 Theory of Finite-Size Effects Above the Upper Critical Dimension
| Autor: | X. S. Chen, V. Dohm |
|---|---|
| Rok vydání: | 1998 |
| Předmět: |
Physics
General Physics and Astronomy Order (ring theory) Statistical and Nonlinear Physics Computer Science Applications Computational Theory and Mathematics Lattice (order) Periodic boundary conditions Field theory (psychology) Ising model Finite set Scaling Condensed Matter - Statistical Mechanics Mathematical Physics Lattice model (physics) Mathematical physics |
| Zdroj: | International Journal of Modern Physics C. :1073-1105 |
| ISSN: | 1793-6586 0129-1831 |
| DOI: | 10.1142/s012918319800100x |
| Popis: | We present a perturbative calculation of finite-size effects near $T_c$ of the $\phi^4$ lattice model in a $d$-dimensional cubic geometry of size $L$ with periodic boundary conditions for $d > 4$. The structural differences between the $\phi^4$ lattice theory and the $\phi^4$ field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters.One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finite $\xi/L$ where $\xi$ is the bulk correlation length. At $T_c$, the large-$L$ behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to $T_c$ of the lattice model, such as $T_{max}(L)$ of the maximum of the susceptibility $\chi$, are found to scale asymptotically as $T_c - T_{max}(L) \sim L^{-d/2}$, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict $\chi_{max} \sim L^{d/2}$ asymptotically. On a quantitative level, the asymptotic amplitudes of this large -$L$ behavior close to $T_c$ have not been observed in previous MC simulations at $d = 5$ because of nonnegligible finite-size terms $\sim L^{(4-d)/2}$ caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of the $L^{(4-d)/2}$ and $L^{4-d}$ terms predicted by our theory. Comment: Accepted in Int. J. Mod. Phys. C |
| Databáze: | OpenAIRE |
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