On the number of ground states of the Edwards-Anderson spin glass model

Arguin, Louis-Pierre; Damron, Michael

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 28-62.

Les états fondamentaux du modèle de verre de spins de Edwards-Anderson (EA) sont étudiés sur des graphes infinis de degré fini. Les états fondamentaux sont les configurations de spins qui minimisent de manière locale l’Hamiltonien pour chaque ensemble fini de sommets. Un problème avec des implications importantes en physique et en mathématique est de déterminer le nombre d’états fondamentaux pour le modèle sur ${\mathbb{Z}}^d$ pour un $d>1$ donné. Ce problème est la version équivalente pour les modèles de verre de spins du problème du nombre de géodésiques infinies en percolation de premier passage et du nombre d’états fondamentaux du modèle d’Ising ferromagnétique désordonné. Il a été montré récemment par Newman, Stein et les deux auteurs que sur le demi-plan $\mathbb{Z}\times \mathbb{N}$, il existe un unique état fondamental (modulo un flip global des spins) produit par la limite faible des états fondamentaux des volumes finis pour un choix spécifique des conditions frontières. Dans cet article, nous étudions l’ensemble de tous les états fondamentaux sur le graphe infini $\mathbb{Z}\times \mathbb{N}$. Nous montrons que le nombre d’états fondamentaux est deux (correspondant à un flip global des spins) ou infini. Ceci est le premier résultat sur l’ensemble de tous les états fondamentaux pour une dimension non-triviale. Dans la première partie, nous développons des outils qui sont pertinents à la résolution du problème analogue sur ${\mathbb{Z}}^d$.

Ground states of the Edwards-Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on ${\mathbb{Z}}^d$ for any $d$. This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation or the number of ground states in the disordered ferromagnet. It was recently shown by Newman, Stein and the two authors that, on the half-plane $\mathbb{Z}\times \mathbb{N}$, there is a unique ground state (up to global flip) arising from the weak limit of finite-volume ground states for a particular choice of boundary conditions. In this paper, we study the entire set of ground states on the infinite graph, proving that the number of ground states on the half-plane must be two (related by a global flip) or infinity. This is the first result on the entire set of ground states in a non-trivial dimension. In the first part of the paper, we develop tools of interest to prove the analogous result on ${\mathbb{Z}}^d$.

Classification : 82D30, 82B44, 60K35
Mots clés : spin glasses, Edwards-Anderson model, ground states

[1] L.-P. Arguin, M. Damron, C. M. Newman and D. L. Stein. Uniqueness of ground states for short-range spin glasses in the half-plane. Comm. Math. Phys. 300 (2010) 641-657. | MR | Zbl

[2] I. Bieche, R. Maynard, R. Rammal and J. P. Uhry. On the ground states of the frustration model of a spin glass by a matching method of graph theory. J. Phys. A 13 (1980) 2553-2576. | MR

[3] K. Binder and A. P. Young. Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58 (1986) 801-976.

[4] R. M. Burton and M. Keane. Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989) 501-505. | MR | Zbl

[5] S. Edwards and P. W. Anderson. Theory of spin glasses. J. Phys. F 5 (1975) 965-974.

[6] D. S. Fisher and D. A. Huse. Ordered phase of short-range Ising spin-glasses. Phys. Rev. Lett. 56 (1986) 1601-1604.

[7] J. Fink. Towards a theory of ground state uniqueness. Excerpt from Ph.D. thesis, 2010.

[8] C. Hoffman. Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 (2005) 739-747. | MR | Zbl

[9] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, Berlin, 2002. | MR | Zbl

[10] G. Iacobelli and C. Külske. Metastates in finite-type mean-field models: Visibility, invisibility, and random restoration of symmetry. J. Stat. Phys. 140 (2010) 27-55. | MR | Zbl

[11] M. Loebl. Ground state incongruence in 2D spin glasses revisited. Electron. J. Combin. 11 (2004) R40. | MR | Zbl

[12] M. Mézard, G. Parisi and M. A. Virasoro. Spin Glass Theory and Beyond. World Scientific, Singapore, 1987. | MR | Zbl

[13] A. A. Middleton. Numerical investigation of the thermodynamic limit for ground states in models with quenched disorder. Phys. Rev. Lett. 83 (1999) 1672-1675.

[14] C. Newman. Topics in Disordered Systems. Birkhaüser, Basel, 1997. | MR | Zbl

[15] C. M. Newman and D. L. Stein. Are there incongruent ground states in 2D Edwards-Anderson spin glasses? Comm. Math. Phys. 224 (2001) 205-218. | MR | Zbl

[16] C. M. Newman and D. L. Stein. Topical review: Ordering and broken symmetry in short-ranged spin glasses. J. Phys. Cond. Mat. 15 (2003) R1319-R1364.

[17] M. Palassini and A. P. Young. Evidence for a trivial ground-state structure in the two-dimensional Ising spin glass. Phys. Rev. B 60 (1999) R9919-R9922.

[18] J. Wehr. On the number of infinite geodesics and ground states in disordered systems. J. Stat. Phys. 87 (1997) 439-447. | MR | Zbl

[19] J. Wehr and J. Woo. Absence of geodesics in first-passage percolation on a half-plane. Ann. Probab. 26 (1998) 358-367. | MR | Zbl