Coupled Ising Models Research Articles

The cubic fixed point at large N

By considering the renormalization group flow between N coupled Ising models in the UV and the cubic fixed point in the IR, we study the large N behavior of the cubic fixed points in three dimensions. We derive a diagrammatic expansion for the 1/N corrections to correlation functions. Leading large N corrections to conformal dimensions at the cubic fixed point are then evaluated using numeric conformal bootstrap data for the 3d Ising model.

Hilbert space fragmentation and Ashkin-Teller criticality in fluctuation coupled Ising models

We discuss the effects of exponential fragmentation of the Hilbert space on phase transitions in the context of coupled ferromagnetic Ising models in arbitrary dimension with special emphasis on the one dimensional case. We show that the dynamics generated by quantum fluctuations is bounded within spatial partitions of the system and weak mixing of these partitions caused by global transverse fields leads to a zero temperature phase with ordering in the local product of both Ising copies but no long range order in either species. This leads to a natural connection with the Ashkin-Teller universality class for general lattices. We confirm this for the periodic chain using quantum Monte Carlo simulations. We also point out that our treatment provides an explanation for pseudo-first order behavior seen in the Binder cumulants of the classical frustrated $J_1-J_2$ Ising model and the $q=4$ Potts model in 2D.

Nonequilibrium Fixed Points of Coupled Ising Models.

Driven-dissipative systems are expected to give rise to nonequilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their nonequilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely nonequilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase—reminiscent of a liquid-gas transition—and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) symmetry. However, they coalesce at a multicritical point, giving rise to a nonequilibrium model of coupled Ising-like order parameters described by a symmetry. Using a dynamical renormalization-group approach, we show that a pair of nonequilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the nonequilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes “hotter” and “hotter” at longer and longer wavelengths. Finally, we argue that this nonequilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.

Simulation of Structural Phase Transitions in Perovskite Methylhydrazinium Metal–Formate Frameworks: Coupled Ising and Potts Models

We present a theoretical model of methylhydrazinium CH3NH2NH2+ molecular cation ordering in perovskite [CH3NH2NH2][M(HCOO)3] (M = Mn, Mg, Fe, and Zn) formate frameworks, which exhibit two structural phase transitions. The proposed model is constructed by analyzing the available structural information and mapping the molecular cation states on a three-dimensional simple cubic lattice. The model includes the short-range Ising and Potts interactions between the dipolar CH3NH2NH2+ cations. We study the model using the Monte Carlo simulations as well as by the density functional theory calculations. The simulations indicate that our model accurately describes the methylhydrazinium cation arrangement in all three structural phases of the compounds. The calculated temperature dependences of the heat capacity and electric polarization are in good agreement with the experimental data. The simulations also allow us to obtain the characteristic energies of the molecular cation interactions for all members of the [CH3NH2NH2][M(HCOO)3] family.

Multicritical behavior of two coupled Ising models in the presence of a random field.

A system defined by two coupled Ising models, with a bimodal random field acting in one of them, is investigated. The interactions among variables of each Ising system are infinite ranged, a limit where mean field becomes exact. This model is studied at zero temperature, as well as for finite temperatures, representing physical situations which are appropriate for describing real systems, such as plastic crystals. A very rich critical behavior is found, depending directly on the particular choices of the temperature, couplings, and random-field strengths. Phase diagrams exhibiting ordered, partially ordered, and disordered phases are analyzed, showing the sequence of transitions through all these phases, similarly to what occurs in plastic crystals. Due to the wide variety of critical phenomena presented by the model, its usefulness for describing critical behavior in other substances is also expected.

One-point functions in integrable coupled minimal models

We propose exact vacuum expectation values of local fields for a quantum group restriction of the $C_2^{(1)}$ affine Toda theory which corresponds to two coupled minimal models. The central charge of the unperturbed models ranges from $c=1$ to $c=2$, where the perturbed models correspond to two magnetically coupled Ising models and Heisenberg spin ladders, respectively. As an application, in the massive phase we deduce the leading term of the asymptotics of the two-point correlation functions.

Coupled Ising models with disorder

In this paper we study the phase diagram of two Ising planes coupled by a standard spin-spin interaction with bond randomness in each plane. The whole phase diagram is analyzed by help of Monte Carlo simulations and field theory arguments.

Integrability of coupled conformal field theories

The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin ladders along the c < 1 discrete series.

Critical properties of the spin-1 Ashkin-Teller model from the differential operator method

In order to study the effect of the anisotropy parameter on the two coupled Ising models, we propose an Ashkin-Teller model (ATM) with spin variable S=1. Using an effective-field theory, we show that near the partially ordered phase ${\mathrm{PO}}_{1}$ that appears in spin-1/2 ATM, this model presents a new partially ordered phase ${\mathrm{PO}}_{2}$. The anisotropy parameter introduces a first-order phase transition between different phases. However, the model exhibits a rich variety of phase diagrams and higher-order critical points.

Randomly coupled Ising models.

We consider the phase diagram of two randomly coupled Ising models to mimic the successive phase transitions in plastic crystals. Detailed mean-field calculations are performed. Depending on the strength of the couplings, the phase diagrams display three ordered phases and some multicritical points. A tetracritical point is found to turn bicritical as the strength of the couplings increases. The nature of this multicritical point is then analyzed by means of a momentum-space renormalization-group calculation. Using the replica trick, we obtain an effective n-component spin Hamiltonian. The random coupling is found to be relevant and shown to have drastic effects on the multicritical behavior. The lower critical dimension is estimated to be ${\mathit{d}}_{\mathit{l}}$=2. In the n=0 limit, to first order in the parameter \ensuremath{\epsilon}=4-d, a system of seven recursion relations is obtained. Although there is a stable fixed point, it cannot be reached from physically acceptable initial conditions. We give arguments to support a runaway of the flow lines associated with a fluctuation-induced first-order transition.

Duality relations and the replica method for Ising models

Random Ising models are conveniently treated by the replica method. The authors study the duality relations for general symmetrically coupled Ising models on a square lattice in two dimensions. General conditions for self-duality are found and a number of new self-dual models are presented. Recently conjectured transition temperatures for the dilute Ising model, by Nishimori (1979), are shown not to obey these general conditions.