The Ising model on random lattices in arbitrary dimensions

Abstract

We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.

Introduction

Random matrix models [1], [2], [3] provide a statistical theory of random discretized Riemann surfaces. The amplitudes of the ribbon Feynman graphs of their perturbative expansion support a $1/N$ expansion [4] (where N is the size of the matrices) indexed by the genus of the surfaces. In the large N limit the planar graphs corresponding to surfaces of spherical topology dominate. The genus is related (by the Gauss–Bonnet theorem) to the Einstein–Hilbert action on the two-dimensional surfaces and the large N parameter appears as the inverse Newton constant.

The planar graphs which lead this expansion proliferate exponentially, like $K^n$, with the number of vertices n. The sum over random lattices of spherical topology is thus convergent for small enough coupling constant and the planar free energy is finite. Planar graphs can be counted precisely through algebraic equations [5], as they are related to trees [6], [7], [8], the universal structures behind such equations. When the coupling constant grows, the free energy becomes dominated by graphs with a large number of vertices and exhibits a critical behavior. It is in this regime that the system reaches its continuum limit and the critical exponents can be evaluated.

In the seminal paper [9], followed by [10], [11], Kazakov et al. solved the two-dimensional Ising model on random geometries using a two-matrix model. Surprisingly this solution turned out to be simpler than the one on a fixed lattice. The importance of this work was further enhanced by the discovery of the KPZ correspondence [12], [13], [14] which relates two-dimensional conformal field theories coupled or not to Liouville gravity, and by the introduction of the double scaling limit of random matrices which combines all genera [15], [16], [17]. The back and forth navigation between two-dimensional statistical systems on fixed and random geometries led to early computations of novel critical exponents [18]. Their values have been later confirmed by rigorous probabilists, through many developments in particular rewarded by the Fields medals of W. Werner and S. Smirnov. The transformation of the two-dimensional statistical mechanics landscape was so deep that it could be considered a change of paradigm. Statistical models on random geometry appear somehow more fundamental and ordinary statistical physics on a fixed lattice as a quenched version in which the fluctuating geometry has been frozen.

It would be highly desirable to generalize such ideas and results to more than two dimensions. Random matrices generalize in higher dimensions to random tensors [19], [20], [21], whose perturbative expansion performs a sum over random higher dimensional geometries. But until recently the key to analytical rather than numerical results, namely the $1/N$ expansion, was missing.

That situation has changed with the discovery of such a $1/N$ expansion [22], [23], [24] for colored [25], [26], [27] random tensors. The amplitude of their graphs supports a $1/N$ expansion indexed by the degree, a positive integer, which plays in higher dimensions the role the genus played in two dimensions. The leading order graphs, baptized melonic [28], triangulate the d-dimensional sphere. They form a summable series and map to colored d-ary trees [28], [29]. When the coupling constant approaches its critical value, the free energy exhibits a critical behavior and, like in matrix models, the colored tensor models reach their continuum limit dominated by triangulations with an infinite number of simplices. The entropy exponent of the melonic series, analogous to the string susceptibility ${\gamma }_{\text{string}}=-1/2$ of the 1-matrix model for the pure gravity universal class, is ${\gamma }_{\text{melons}}=1/2$ [28].

Colored random tensors are a promising tractable discretization of quantum gravity in three and more dimensions and the subject is developing fast [30], [31], [32], [33]. The understanding of the leading (melonic) order of colored tensor models allows the study of the coupling of statistical systems to random geometries in arbitrary dimension. Concretely one needs to develop an algorithm for solving tensor models and study the critical behavior of various particular examples. As some of the more powerful tools of matrix models, like the reduction to eigenvalues, are absent in tensor models, the algorithm we present here relies on combining Schwinger–Dyson (SD) equations with a factorization property characteristic of melonic graphs [28].

We illustrate our method on the Ising model coupled to random melonic triangulations. It is generated by a (colored) two-tensor model, the straightforward generalization of the matrix model studied by Kazakov [9] in arbitrary dimensions. It is presented in Section 2. Unlike in two dimensions, we find in Section 3 that the model does not exhibit a phase transition in the continuum limit. The large N limit corresponds to the limit of zero bare Newtonʼs constant. As such it is only a limiting case of numerical simulations. However, our results are in agreement with numerical simulations for small Newtonʼs constant [34] associated to the (cold) branched-polymer phase.¹ An open question is therefore to understand why and how the progressive freezing of the random geometry would let the ordinary Ising phase transition appear.