Abstract
We develop a method for precise asymptotic analysis of partition functions near first-order phase transitions. Working in a (ν+1)-dimensional cylinder of volumeL×...×L×t, we show that leading exponentials int can be determined from a simple matrix calculation providedt≧v logL. Through a careful surface analysis we relate the off-diagonal matrix elements of this matrix to the surface tension andL, while the diagonal matrix elements of this matrix are related to the metastable free energies of the model. For the off-diagonal matrix elements, which are related to the crossover length from hypercubic (L=t) to cylindrical (t=∞) scaling, this includes a determination of the pre-exponential power ofL as a function of dimension. The results are applied to supersymmetric field theory and, in a forthcoming paper, to the finite-size scaling of the magnetization and inner energy at field and temperature driven first-order transitions in the crossover region from hypercubic to cylindrical scaling.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BF] Bricmont, J., Fröhlich, J.: Statistical mechanical methods in particle structure analysis of lattice field theories II: Scalar and surface models. Commun. Math. Phys.98, 553–578 (1985)
[BI1] Borgs, C., Imbrie, J.Z.: A unified approach to phase diagrams in field theory and statistical mechanics. Commun. Math. Phys.123, 305–328 (1989)
[BI2] Borgs, C., Imbrie, J.Z.: Crossover finite-size scaling at first-order transitions, preprint (1992)
[BK] Borgs, C., Kotecký, R.: Finite-size effects at asymmetric first-order phase transitions, preprint. A rigorous theory of finite-size scaling at first order phase transitions. J. Stat. Phys.61, 79 (1990)
[BFW] Borgs, C., Fröhlich, J., Waxler, R.: The phase structure of the largeN lattice Higgs model. Nucl. Phys.B328, 611–638 (1989)
[BKM] Borgs, C., Kotechý, R., Miracle-Solé, S.: Finite-size scaling for Potts models. J. Stat. Phys.62, 529 (1991)
[BZ] Brézin, E., Zinn-Justin, J.: Finite size effects in phase transitions. Nucl. Phys.B257, 867–893 (1985)
[CJBZ] Cabrera, G.G., Jullien, R., Brézin, E., Zinn-Justin, J.: Test of finite-size scaling in first order phase transitions. J. Phys.47, 1305–1313 (1986)
[D] Dobrushin, R.L.: Gibbs states describing the coexistence of phases for a three-dimensional Ising model. Theor. Prob. Appl.17, 582–600 (1972). Investigation of Gibbsian states for three-dimensional lattice systems. Theor. Prob. Appl.18, 253–271 (1973)
[FB] Fisher, M.E., Berker, A.N.: Scaling for first-order phase transitions in thermodynamic and finite systems. Phys. Rev.B26, 2507–2513 (1982)
[G] Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys.27, 103–136 (1972)
[HKZ] Holichy, P., Kotecký, R., Zahradnik, M.: Rigid interfaces for lattice models at low temperatures. J. Stat. Phys.50, 755–812 (1988)
[I] Isakov, S.N.: Nonanalytic features of the first order phase transition in the Ising model. Commun. Math. Phys.95, 427–433 (1984)
[IJW] Imbrie, J.Z., Janowsky, S.A., Weitsman, J.: Space dependent Dirac operators and effective quantum field theory for fermions. Commun. Math. Phys.135, 421–442 (1991)
[J] Janowsky, S.A.: The phase structure of the two-dimensionalN=2 Wess-Zumino model. Harvard University thesis (1990)
[JLW] Jaffe, A., Lesniewski, A., Weitsman, J.: Index of a family of Dirac operators on loop space. Commun. Math. Phys.112, 75–88 (1987); The loop spaceS 1 ↔ ℝ and supersymmetric quantum fields. Ann. Phys.183 337–351 (1988); The two-dimensional,N=2 Wess-Zumino models on a cylinder. Commun. Math. Phys.114, 147–165 (1988)
[JW1] Janowsky, S.A., Weitsman, J.: The phase structure of the two-dimensionalN=2 Wess-Zumino model. Commun. Math. Phys.142, 25–66 (1991)
[JW2] Janowsky, S.A., Weitsman, J.: A vanishing theorem for supersymmetric quantum field theory and finite-size effects in multiphase cluster expansions. Commun. Math. Phys.143, 85–97 (1991)
[LMMRS] Lanait, L., Messager, A., Miracle-Solé, S., Ruiz, J., Shlossman, S.: Interfaces in the Potts Model I: Pirogov-Sinai theory of the Fortuin-Kastelyn representation. Commun. Math. Phys140, 81–92 (1991)
[P] Privman, V. (ed.): Finite-size scaling and numerical simulation of statistical systems. Singapore: World Scientific 1990
[PF] Privman, V., Fisher, M.E.: Finite-size effects at first-order transitions J. Stat. Phys.33, 385–417 (1983)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Research partially supported by the A. P. Sloan Foundation and by the NSF under DMS-8858073
Research partially supported by the NSF under DMS-8858073 and DMS-9008827
Rights and permissions
About this article
Cite this article
Borgs, C., Imbrie, J.Z. Finite-size scaling and surface tension from effective one dimensional systems. Commun.Math. Phys. 145, 235–280 (1992). https://doi.org/10.1007/BF02099138
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02099138