Solution of the random field XY magnet on a fully connected graph

We use large deviation theory to obtain the free energy of the XY model on a fully connected graph on each site of which there is a randomly oriented field of magnitude $h$. The phase diagram is obtained for two symmetric distributions of the random orientations: (a) a uniform distribution and (b) a distribution with cubic symmetry. In both cases, the ordered state reflects the symmetry of the underlying disorder distribution. The phase boundary has a multicritical point which separates a locus of continuous transitions (for small values of $h$) from a locus of first order transitions (for large $h$). The free energy is a function of a single variable in case (a) and a function of two variables in case (b), leading to different characters of the multicritical points in the two cases.

Generalized Gibbs Phase Rule and Multicriticality Applied to Magnetic Systems

A generalization of the original Gibbs phase rule is proposed in order to study the presence of single phases, multiphase coexistence, and multicritical phenomena in lattice spin magnetic models. The rule is based on counting the thermodynamic number of degrees of freedom, which strongly depends on the external fields needed to break the ground state degeneracy of the model. The phase diagrams of some spin Hamiltonians are analyzed according to this general phase rule, including general spin Ising and Blume–Capel models, as well as q-state Potts models. It is shown that by properly taking into account the intensive fields of the model in study, the generalized Gibbs phase rule furnishes a good description of the possible topology of the corresponding phase diagram. Although this scheme is unfortunately not able to locate the phase boundaries, it is quite useful to at least provide a good description regarding the possible presence of critical and multicritical surfaces, as well as isolated multicritical points.

Multicritical points of unitary matrix model with logarithmic potential identified with Argyres–Douglas points

In our recent publications, the partition function of the Gross–Witten–Wadia unitary matrix model with the logarithmic term has been identified with the [Formula: see text] function of a certain Painlevé system, and the double scaling limit of the associated discrete Painlevé equation to the critical point provides us with the Painlevé II equation. This limit captures the critical behavior of the [Formula: see text], [Formula: see text], [Formula: see text] supersymmetric gauge theory around its Argyres–Douglas 4D superconformal point. Here, we consider further extension of the model that contains the [Formula: see text]th multicritical point and that is to be identified with [Formula: see text] theory. In the [Formula: see text] case, we derive a system of two ODEs for the scaling functions to the free energy, the time variable being the scaled total mass and make a consistency check on the spectral curve on this matrix model.

Phase structure of the Euclidean three-dimensional O(1) ghost model

We have treated the Euclidean three-dimensional O(1) ghost model with a modified version of the effective average action (EAA) renormalization group (RG) method, developed by us. We call it Fourier–Wetterich RG approach and it is used to investigate the occurrence of a periodic condensate in terms of the functional RG. The modification involves additional terms in the ansatz of the EAA, corresponding to the Fourier-modes of the periodic condensate. The RG flow equations are derived keeping the terms up to the fourth order of the gradient expansion (GE), however the numerical calculations are conducted in the second order (or next-to-leading order, NLO) of the GE. The expansion of the flow equations around the nontrivial minimum of the local potential takes into account properly the vertices induced by the periodic condensate even if the wave function renormalization is set to be field-independent. The numerical analysis reveals several different phases with three multicritical points.