×

zbMATH — the first resource for mathematics

Partition function zeros at first-order phase transitions: a general analysis. (English) Zbl 1088.82010
The paper is dedicated to the study of the complex phase diagram of a number of lattice spin models with one complex interaction parameter \(z\). The basic idea is to represent the partition function \(Z^{ per}_L\) in a box of side \(L\) with periodic boundary conditions in the form \[ Z^{ per}_L (z) \simeq \sum_{m=1}^r q_m \exp(- f_m (z) L^d), \] where \(m\) numbers phases, \(f_m\) are not necessarily analytic functions – metastable free energies of the phases, \(q_m \) are positive integers describing the degeneracy of the phases. A number of assumptions under which the above representation holds are formulated. Under these assumptions equations whose solutions give the location of zeros of \(Z^{ per}_L \) are derived. It is shown that the error term is of order \(e^{-L}\). In the limit \(L \rightarrow + \infty\), the zeros are shown to concentrate on phase boundaries which are simple curves ending in the so called multiple points, at which more than two phases have the same value of \(f_m\).

MSC:
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Beraha, S., Kahane, J., Weiss, N.J.: Limits of zeroes of recursively defined families of polynomials. In: G.-C. Rota (ed.), Studies in Foundations and Combinatorics (Advances in Mathematics Supplementary Studies, Vol. 1), New York: Academic Press, 1978, pp. 213-232 · Zbl 0477.05034
[2] Biskup, M., Borgs, C., Chayes, J.T., Kleinwaks, L.J., Kotecký, R.: General theory of Lee-Yang zeros in models with first-order phase transitions. Phys. Rev. Lett. 84:21, 4794-4797 (2000)
[3] Biskup, M., Borgs, C., Chayes, J.T., Kotecký, R.: Phase diagrams of Potts models in external fields: I. Real fields. In preparation · Zbl 0977.82012
[4] Biskup, M., Borgs, C., Chayes, J.T., Kotecký, R.: Phase diagrams of Potts models in external fields: II. One complex field. In preparation · Zbl 0977.82012
[5] Biskup, M., Borgs, C., Chayes, J.T., Kotecký, R.: Partition function zeros at first-order phase transitions: Pirogov-Sinai theory. J. Stat. Phys. 116, 97-155 (2004) · Zbl 1142.82328
[6] Borgs, C., Imbrie, J.Z.: A unified approach to phase diagrams in field theory and statistical mechanics. Commun. Math. Phys. 123, 305-328 (1989) · Zbl 0668.60092
[7] Borgs, C., Kotecký, R.: A rigorous theory of finite-size scaling at first-order phase transitions. J. Statist. Phys. 61, 79-119 (1990)
[8] Chang, S.-C., Shrock, R.: Ground state entropy of the Potts antiferromagnet on strips of the square lattice. Physica A 290, 402-430 (2001) · Zbl 0972.82023
[9] Chang, S.-C., Shrock, R.: T=0 partition functions for Potts antiferromagnets on lattice strips with fully periodic boundary conditions. Physica A 292, 307-345 (2001) · Zbl 0972.82032
[10] Chen, C.-N., Hu, C.-K., Wu, F.Y.: Partition function zeros of the square lattice Potts model. Phys. Rev. Lett.76, 169-172 (1996) · Zbl 1042.82541
[11] Dobrushin, R.L.: Estimates of semiinvariants for the Ising model at low temperatures. In: R.L. Dobrushin et al. (ed.), Topics in statistical and theoretical physics. F. A. Berezin memorial volume, Transl. Ser. 2, Vol. 177(32), Providence: Am. Math. Soc. (1996) pp. 59-81 · Zbl 0873.60074
[12] Doland, B.P., Johnston, D.A.: One dimensional Potts model, Lee-Yang edges, and chaos. Phys. Rev. E 65, 057103 (2002)
[13] Federer, H.: Geometric Measure Theory. Berlin: Springer-Verlag, 1996 · Zbl 0874.49001
[14] Fisher, M.E.: The nature of critical points. In: W.E. Brittin (ed.), Lectures in Theoretical Physics, Vol 7c (Statistical physics, weak interactions, field theory), Boulder: University of Colorado Press, 1965, pp. 1-159
[15] Friedli, S., Pfister, C.-E.: On the singularity of the free energy at first order phase transitions. Commun. Math. Phys. 245, 69-103 (2004) · Zbl 1075.82009
[16] Gamelin, T.W.: Complex Analysis. Undergraduate Texts in Mathematics, New York: Springer-Verlag, 2001 · Zbl 0978.30001
[17] Gibbs, J.W.: Elementary Principles of Statistical Mechanics. In: J.W. Gibbs (ed.), The Collected Works, Vol. II, New Haven, CT: Yale University Press, 1948 · Zbl 0031.13504
[18] Glumac, Z., Uzelac, K.: The partition function zeros in the one-dimensional q-state Potts model. J. Phys. A: Math. Gen. 27, 7709-7717 (1994) · Zbl 0844.60096
[19] Isakov, S.N.: Nonanalytic features of the first-order phase transition in the Ising model. Commun. Math. Phys. 95, 427-443 (1984)
[20] Janke, W., Kenna, R.: Phase transition strengths from the density of partition function zeroes. Nucl. Phys. Proc. Suppl. 106, 905-907 (2002) · Zbl 01739493
[21] Kenna, R., Lang, C.B.: Scaling and density of Lee-Yang zeros in the four-dimensional Ising model. Phys. Rev. E 49, 5012-5017 (1994)
[22] Kim, S.-Y., Creswick, R.J.: Yang-Lee zeros of the q-state Potts model in the complex magnetic field plane. Phys. Rev. Lett. 81, 2000-2003 (1998) · Zbl 0944.82007
[23] Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491-498 (1986) · Zbl 0593.05006
[24] Lee, K.-C.: Generalized circle theorem on zeros of partition functions at asymmetric first-order transitions. Phys. Rev. Lett. 73, 2801-2804 (1994)
[25] Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions: II. Lattice gas and Ising model. Phys. Rev. 87, 410-419 (1952) · Zbl 0048.43401
[26] Lieb, E.H., Sokal, A.D.: A general Lee-Yang theorem for one-component and multi-component ferromagnets. Commun. Math. Phys. 80, 153-179 (1981)
[27] Lu, W.T., Wu, F.Y.: Partition function zeroes of a self-dual Ising model. Physica A 258, 157-170 (1998)
[28] Matveev, V., Shrock, R.: Complex-temperature properties of the Ising model on 2D heteropolygonal lattices. J. Phys.A: Math. Gen. 28, 5235-5256 (1995) · Zbl 0868.60103
[29] Matveev, V., Shrock, R.: Some new results on Yang-Lee zeros of the Ising model partition function. Phys. Lett. A 215, 271-279 (1996) · Zbl 0972.82513
[30] Milnor, J.W.: Topology from the Differentiable Viewpoint. Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press, 1997 · Zbl 1025.57002
[31] Nashimori, H., Griffiths, R.B.: Structure and motion of the Lee-Yang zeros. J. Math. Phys. 24, 2637-2647 (1983) · Zbl 0544.76145
[32] Newman, C.M.: Zeros of the partition function for generalized Ising systems. Commun. Pure Appl. Math. 27, 143-159 (1974)
[33] Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. (Russian), Theor. Math. Phys. 25(3), 358-369 (1975)
[34] Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. Continuation. (Russian), Theor. Math. Phys. 26(1), 61-76 (1976)
[35] Ruelle, D.: Extension of the Lee-Yang circle theorem. Phys. Rev. Lett. 26, 303-304 (1971)
[36] van Saarloos, W., Kurtze, D.A.: Location of zeros in the complex temperature plane: Absence of Lee-Yang theorem. J. Phys. A: Math. Gen. 18, 1301-1311 (1984)
[37] Salas, J., Sokal, A.D.: Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I. General theory and square-lattice chromatic polynomial, J. Stat. Phys. 104(3-4), 609-699 (2001) · Zbl 1100.82509
[38] Shrock, R.: Exact Potts model partition functions on ladder graphs. Physica A 283, 388-446 (2000)
[39] Shrock, R., Tsai, S.-H.: Exact partition functions for Potts antiferromagnets on cyclic lattice strips. Physica A 275, 429-449 (2000)
[40] Sokal, A.D.: Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combin. Probab. Comput. 10(1), 41-77 (2001) · Zbl 0999.82022
[41] Sokal, A.D.: Chromatic roots are dense in the whole complex plane. Combin. Probab. Comput. 13, 221-261 (2004) · Zbl 1100.05040
[42] Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions: I. Theory of condensation. Phys. Rev. 87, 404-409 (1952) · Zbl 0048.43305
[43] Zahradník, M.: An alternate version of Pirogov-Sinai theory. Commun. Math. Phys. 93, 559-581 (1984)
[44] Zahradník, M.: Analyticity of low-temperature phase diagrams of lattice spin models. J. Stat. Phys. 47, 725-455 (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.