Related papers: Cluster integrable systems and spin chains
We present a general strategy to extend quantum cluster algorithms for S=1/2 spin systems, such as the loop algorithm, to systems with arbitrary size of spins. In general, the partition function of a high-S spin system is represented in…
In this work the notion of Hamiltonian chain is presented as applied to anisotropic oscillator potentials especially defined on three and four dimensional Euclidean spaces. A Hamiltonian chain is a sequence of superintegrable Hamiltonians…
We derive the coherent state representation of the integrable spin chain Hamiltonian with non-compact supersymmetry group G=SU(1,1|1). By passing to the continuous limit, we find a spin chain sigma model describing a string moving on the…
We introduce a new set of boundary conditions for three-dimensional higher spin gravity with gauge group $SL(3,\mathbb{R})\times SL(3,\mathbb{R})$, where its dynamics at the boundary is described by the members of the modified Boussinesq…
We examine classical and quantum aspects of the planar non-compact spin system coupled with Chern-Simons gauge field in the presence of background charge. We first define our classical spin system as non- relativistic non-linear sigma model…
Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the…
We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${\Sigma_g \times S^1}$ with a topological twist along $\Sigma_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation…
We introduce a formalism for describing holomorphic blocks of 3d quiver gauge theories using networks of Ding-Iohara-Miki algebra intertwiners. Our approach is very direct and gives an explicit identification of the blocks with…
By using a class of `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry (HS) spin…
We describe the Poisson ideals and attached symplectic geometry of a cluster algebra with compatible Poisson structure. We apply these results to determine the spectrum of a quantum cluster algebra. As an application, we describe the…
The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a…
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…
We study supersymmetric $SU(N-4)$ gauge theories with a symmetric tensor and $N$ antifundamental representations. The theory with $W=0$ has a dual description in terms of a non-chiral $Spin(8)$ theory with one spinor and $N$ vectors. This…
We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by…
The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized"…
Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schr\"odinger family of…
Recent theoretical studies raised the possibility of a realization of spin nematic states in the S=1 triangular lattice compound NiGa_2S_4. We study the bilinear-biquadratic spin 1 chain in a zig-zag geometry by means of the density matrix…
We show that Seiberg-like duality of $\mathcal{N}=1$ gauge theory coupled with tensor chiral fields and fundamental chiral fields works if the meson spectrum built from the tensor fields takes particular form: a) It should be truncated; b)…
We present a construction of integrable quantum spin chains where local spin-spin interactions are weighted by ``position''-dependent potential containing abelian non-local spin dependance. This construction applies to the previously…
The form of the spectral curve for $4d$ $N=2$ supersymmetric Yang-Mills theory with matter fields in the fundamental representation of the gauge group suggests that its $1d$ integrable counterpart should be looked for among (inhomogeneous)…