Abstract
We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [5]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Ising-like plus-minus symmetry, we also establish a local version of the Lee-Yang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves.
Similar content being viewed by others
References
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
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)
Biskup, M., Borgs, C., Chayes, J.T., Kotecký, R.: Phase diagrams of Potts models in external fields: I. Real fields. In preparation
Biskup, M., Borgs, C., Chayes, J.T., Kotecký, R.: Phase diagrams of Potts models in external fields: II. One complex field. In preparation
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)
Borgs, C., Imbrie, J.Z.: A unified approach to phase diagrams in field theory and statistical mechanics. Commun. Math. Phys. 123, 305–328 (1989)
Borgs, C., Kotecký, R.: A rigorous theory of finite-size scaling at first-order phase transitions. J. Statist. Phys. 61, 79–119 (1990)
Chang, S.-C., Shrock, R.: Ground state entropy of the Potts antiferromagnet on strips of the square lattice. Physica A 290, 402–430 (2001)
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)
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)
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
Doland, B.P., Johnston, D.A.: One dimensional Potts model, Lee-Yang edges, and chaos. Phys. Rev. E 65, 057103 (2002)
Federer, H.: Geometric Measure Theory. Berlin: Springer-Verlag, 1996
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
Friedli, S., Pfister, C.-E.: On the singularity of the free energy at first order phase transitions. Commun. Math. Phys. 245, 69–103 (2004)
Gamelin, T.W.: Complex Analysis. Undergraduate Texts in Mathematics, New York: Springer-Verlag, 2001
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
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)
Isakov, S.N.: Nonanalytic features of the first-order phase transition in the Ising model. Commun. Math. Phys. 95, 427–443 (1984)
Janke, W., Kenna, R.: Phase transition strengths from the density of partition function zeroes. Nucl. Phys. Proc. Suppl. 106, 905–907 (2002)
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)
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)
Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491–498 (1986)
Lee, K.-C.: Generalized circle theorem on zeros of partition functions at asymmetric first-order transitions. Phys. Rev. Lett. 73, 2801–2804 (1994)
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)
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)
Lu, W.T., Wu, F.Y.: Partition function zeroes of a self-dual Ising model. Physica A 258, 157–170 (1998)
Matveev, V., Shrock, R.: Complex-temperature properties of the Ising model on 2D heteropolygonal lattices. J. Phys.A: Math. Gen. 28, 5235–5256 (1995)
Matveev, V., Shrock, R.: Some new results on Yang-Lee zeros of the Ising model partition function. Phys. Lett. A 215, 271–279 (1996)
Milnor, J.W.: Topology from the Differentiable Viewpoint. Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press, 1997
Nashimori, H., Griffiths, R.B.: Structure and motion of the Lee-Yang zeros. J. Math. Phys. 24, 2637–2647 (1983)
Newman, C.M.: Zeros of the partition function for generalized Ising systems. Commun. Pure Appl. Math. 27, 143–159 (1974)
Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. (Russian), Theor. Math. Phys. 25(3), 358–369 (1975)
Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. Continuation. (Russian), Theor. Math. Phys. 26(1), 61–76 (1976)
Ruelle, D.: Extension of the Lee-Yang circle theorem. Phys. Rev. Lett. 26, 303–304 (1971)
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)
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)
Shrock, R.: Exact Potts model partition functions on ladder graphs. Physica A 283, 388–446 (2000)
Shrock, R., Tsai, S.-H.: Exact partition functions for Potts antiferromagnets on cyclic lattice strips. Physica A 275, 429–449 (2000)
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)
Sokal, A.D.: Chromatic roots are dense in the whole complex plane. Combin. Probab. Comput. 13, 221–261 (2004)
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)
Zahradník, M.: An alternate version of Pirogov-Sinai theory. Commun. Math. Phys. 93, 559–581 (1984)
Zahradník, M.: Analyticity of low-temperature phase diagrams of lattice spin models. J. Stat. Phys. 47, 725–455 (1987)
Author information
Authors and Affiliations
Additional information
Communicated by M. Aizenman
Reproduction of the entire article for non-commercial purposes is permitted without charge.
Rights and permissions
About this article
Cite this article
Biskup, M., Borgs, C., Chayes, J. et al. Partition Function Zeros at First-Order Phase Transitions: A General Analysis. Commun. Math. Phys. 251, 79–131 (2004). https://doi.org/10.1007/s00220-004-1169-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1169-5