Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Hamiltonians for six-vertex models
59
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we explain a connection between a family of free-fermionic six-vertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials.
rate research
Read More
In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $Delta<1$. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $a=b=1$ and $cge1$, and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.
Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a biproduct, we construct Demazure atoms on Kashiwaras $mathcal{B}_infty$ crystal and give new algorithms for computing Lascoux-Schutzenberger keys.
We discuss transpose (sometimes called universal exchange or all-to-all) on vertex symmetric networks. We provide a method to compare the efficiency of transpose schemes on two different networks with a cost function based on the number processors and wires needed to complete a given algorithm in a given time.
For a graph $G$, we associate a family of real symmetric matrices, $mathcal{S}(G)$, where for any $M in mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $min(m,n)ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.
A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $ell$ and a non-empty planar graph $H$ with at least $ell-1$ connected components, there exists a function $f_{H, ell}:mathbb{N}rightarrow mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, ell}(k)$ that meets all such minor-models of $H$. This function $f_{H, ell}$ is independent with the number $m$ of given sets, and thus, our result generalizes Maders $mathcal S$-path Theorem, by applying $ell=2$ and $H$ to be the one-vertex graph. We prove that such a function $f_{H, ell}$ does not exist if $H$ consists of at most $ell-2$ connected components.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
