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We present a general formalism for deriving the thermodynamics of ferromagnets consisting of "atoms" carrying an arbitrary irreducible representation of and coupled through long-range two-body quadratic interactions. Using this formalism, we derive the thermodynamics and phase structure of ferromagnets with atoms in the doubly symmetric or doubly antisymmetric irreducible representations. The symmetric representation leads to a paramagnetic and a ferromagnetic phase with transitions similar to the ones for the fundamental representation studied before. The antisymmetric representation presents qualitatively new features, leading to a paramagnetic and two distinct ferromagnetic phases that can coexist over a range of temperatures, two of them becoming metastable. Our results are relevant to magnetic systems of atoms with reduced symmetry in their interactions compared to the fundamental case. 

The non-Abelian ferromagnet recently introduced by the authors, consisting of atoms in the fundamental representation of , is studied in the limit where becomes large and scales as the square root of the number of atoms . This model exhibits additional phases, as well as two different temperature scales related by a factor . The paramagnetic phase splits into a "dense" and a "dilute" phase, separated by a third-order transition and leading to a triple critical point in the scale parameter and the temperature, while the ferromagnetic phase exhibits additional structure, and a new paramagnetic-ferromagnetic metastable phase appears at the larger temperature scale. These phases can coexist, becoming stable or metastable as temperature varies. A generalized model in which the number of -equivalent states enters the partition function with a nontrivial weight, relevant, e.g., when there is gauge invariance in the system, is also studied and shown to manifest similar phases, with the dense-dilute phase transition becoming second-order in the fully gauge invariant case. 

We study the multiplicity of irreducible representations in the decomposition of 𝑛 fundamentals of 𝑆𝑈(𝑁) weighted by a power of their dimension in the large 𝑛 and large 𝑁 double scaling limit. A nontrivial scaling is obtained by keeping 𝑛∕𝑁2 fixed, which plays the role of an order parameter. We find that the system generically undergoes a fourth order phase transition in this parameter, from a dense phase to a dilute phase. The transition is enhanced to third order for the unweighted multiplicity, and disappears altogether when weighting with the first power of the dimension. This corresponds to the infinite temperature partition function of non-Abelian ferromagnets, and the results should be relevant to the thermodynamic limit of such ferromagnets at high temperatures. 

We study the thermodynamics of a non-abelian ferromagnet consisting of “atoms” each carrying a fun-damental representation of SU(N), coupled with long-range two-body quadratic interactions. We uncover a rich structure of phase transitions from non-magnetized, global SU(N)-invariant states to magnetized ones breaking global invariance to SU(N−1) ×U(1). Phases can coexist, one being stable and the other metastable, and the transition between states involves latent heat exchange, unlike in usual SU(2)ferro-magnets. Coupling the system to an external non-abelian magnetic field further enriches the phase structure, leading to additional phases. The system manifests hysteresis phenomena both in the magnetic field, as in usual ferromagnets, and in the temperature, in analogy to supercooled water. Potential applications are in fundamental situations or as a phenomenological model. 

We compute the multiplicity of the irreducible representations in the decomposition of the tensor product of an arbitrary number n of fundamental representations of SU(N), and we identify a duality in the representation content of this decomposition. Our method utilizes the mapping of the representations of SU(N) to the states of free fermions on the circle, and can be viewed as a random walk on a multidimensional lattice. We also derive the large-n limit and the response of the system to an external non-abelian magnetic field. These results can be used to study the phase properties of non-abelian ferromagnets and to take various scaling limits. 

We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

Abstract We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area is defined by closing the walks with a straight line across their endpoints and can assume half-integer values in lattice cell units. We also derive the length and area counting of walks with endpoints on specific straight lines and outline an approach for dealing with walks with fully fixed endpoints.