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Tensor networks provide a natural language for non-invertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a non-invertible operator implementing…
The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants - one with an integral operator having an Appell function kernel and…
We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants,…
A dynamic model of the asymmetric Ising glass is presented: an Ising model with antiferromagnet bonds with probabilities q arranged at random in a ferromagnetic matrix. The dynamics is introduced by changing the arrangement of the…
The correlation function of the two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor expansion are…
Spin-orbit entangled pseudospins hold promise for a wide array of exotic magnetism ranging from a Heisenberg antiferromagnet to a Kitaev spin liquid depending on the lattice and bonding geometry, but many of the host materials suffer from…
The spin-orbital model for triply degenerate t_2g electrons on a triangular lattice has been shown to be dominated by dimers: the phase diagram contains both strongly resonating, compound spin-orbital dimer states and quasi-static,…
In this training course report, I briefly present the one- and two-matrix models as tools for the study of conformal field theories with boundaries. In a first part, after a short historical presentation of random matrices, I present the…
In this work we deal with doubly decorated Ising-Heisenberg models on planar lattices. Applying the generalized decoration-iteration transformation we obtain exact results for the antiferromagnetic version of the model. The existence of a…
Hadwiger's transversal theorem gives necessary and sufficient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful…
K\H{o}nig's edge-coloring theorem for bipartite graphs and Vizing's edge-coloring theorem for general graphs are celebrated results in graph theory and combinatorial optimization. Schrijver generalized K\H{o}nig's theorem to a framework…
We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromodynamics with two colours and dynamical fermions at nonzero quark chemical potential mu. In this…
We study the perturbative approach to the Wilsonian integration of noncommutative gauge theories in the matrix representation. We begin by motivating the study of noncommutative gauge theories and reviewing the matrix formulation. We then…
We show that a finite coloring of an amenable group contains `many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables. This gives the first combinatorial proof and extensions of Bergelson and…
A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…
A quantum anti-ferromagnetic spin-1 model is characterised on a 2D lattice with the following requirements: i) The Hamiltonian is made out of nearest neighbour interactions. ii) It is homogeneous, translational and rotational invariant.…
We formulate a new geometrical string on the euclidean lattice. It is possible to find such spin systems with local interaction which reproduce the same surface dynamics.In the three-dimensional case this spin system is a usual Ising…
Quantum nonrelativistic systems with $2\times2$ matrix potentials are investigated. Physically, they simulate charged or neutral fermions with non-trivial dipole momenta, interacting with an external electric field. Assuming rotationally…
We introduce the 3-colour noncommutative quantum field theory model in two dimensions. For this model we prove a generalised Ward-Takahashi identity, which is special to coloured noncommutative QFT models and has no underlying continuous…
We discuss basic features and new developments in recently proposed induced gauge theory solvable in any number of dimensions in the limit of infinite number of colours. Its geometrical (string) picture is clarified, using planar graph…