scholarly journals The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance

2020 ◽  
Author(s):  
Paul Melotti
Keyword(s):  
Gibbs Measure ◽  
Partition Functions ◽  
Free Fermion ◽  
Vertex Model ◽  
Invariant Model

AbstractWe study the eight-vertex model at its free-fermion point. We express a new “switching” symmetry of the model in several forms: partition functions, order-disorder variables, couplings, Kasteleyn matrices. This symmetry can be used to relate free-fermion 8V-models to free-fermion 6V-models, or bipartite dimers. We also define new solution of the Yang–Baxter equations in a “checkerboard” setting, and a corresponding Z-invariant model. Using the bipartite dimers of Boutillier et al. (Probab Theory Relat Fields 174:235–305, 2019), we give exact local formulas for edge correlations in the Z-invariant free-fermion 8V-model on lozenge graphs, and we deduce the construction of an ergodic Gibbs measure.

scholarly journals Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

2018 ◽  
Vol 174 (1) ◽  
pp. 1-27 ◽  
Author(s):  
F. Colomo ◽  
A. G. Pronko ◽  
A. Sportiello
Keyword(s):  
Free Fermion ◽  
Vertex Model ◽  
Arctic Curve ◽  
Shaped Domain

scholarly journals QUANTUM GROUP APPROACH TO A SOLUBLE VERTEX MODEL WITH GENERALIZED ICE RULE

1996 ◽  
Vol 11 (10) ◽  
pp. 1747-1761
Author(s):  
C.L. SOW ◽  
T.T. TRUONG
Keyword(s):  
Free Energy ◽  
Quantum Mechanics ◽  
Positive Integer ◽  
Quantum Group ◽  
Algebraic Structure ◽  
Square Lattice ◽  
Free Fermion ◽  
Vertex Model ◽  
Group Approach ◽  
Operator Structure

Using the representation of the quantum group SL q(2) by the Weyl operators of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertex is subjected to a generalized form of the so-called “ice rule,” its property is studied in detail and its free energy calculated with the method of quantum inverse scattering. Remarkably, in analogy with the usual six-vertex model, there exists a “free-fermion” limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the fermion-boson correspondence.

scholarly journals Characterizing partition functions of the vertex model

2012 ◽  
Vol 350 (1) ◽  
pp. 197-206 ◽  
Author(s):  
Jan Draisma ◽  
Dion C. Gijswijt ◽  
László Lovász ◽  
Guus Regts ◽  
Alexander Schrijver
Keyword(s):  
Partition Functions ◽  
Vertex Model

scholarly journals Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Kohei Motegi
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Free Fermion ◽  
Vertex Model ◽  
Fermion Model ◽  
Free Fermion Model

We apply the Izergin-Korepin analysis to the study of the projected wavefunctions of the generalized free-fermion model. We introduce a generalization of the L-operator of the six-vertex model by Bump-Brubaker-Friedberg and Bump-McNamara-Nakasuji. We make the Izergin-Korepin analysis to characterize the projected wavefunctions and show that they can be expressed as a product of factors and certain symmetric functions which generalizes the factorial Schur functions. This result can be seen as a generalization of the Tokuyama formula for the factorial Schur functions.

scholarly journals Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

2019 ◽  
Vol 2019 (757) ◽  
pp. 159-195 ◽  
Author(s):  
Michael Wheeler ◽  
Paul Zinn-Justin
Keyword(s):  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Structure Constants ◽  
Product Rules ◽  
Grothendieck Polynomials

AbstractWe study the Littlewood–Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang–Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood–Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson–Tao and Vakil.

scholarly journals Integrable probability: stochastic vertex models and symmetric functions

2018 ◽  
Author(s):  
Alexei Borodin ◽  
Leonid Petrov
Keyword(s):  
Symmetric Functions ◽  
Height Function ◽  
Rational Functions ◽  
Universality Class ◽  
Integral Representations ◽  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Schur Polynomials ◽  
Higher Spin

This chapter presents the study of a homogeneous stochastic higher spin six-vertex model in a quadrant. For this model concise integral representations for multipoint q-moments of the height function and for the q-correlation functions are derived. At least in the case of the step initial condition, these formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six-vertex model, ASEP, various q-TASEPs, and associated zero-range processes. The arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the higher spin six-vertex model for suitable domains; they generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six-vertex model.

scholarly journals Domino tilings and the six-vertex model at its free-fermion point

2006 ◽  
Vol 39 (33) ◽  
pp. 10297-10306 ◽  
Author(s):  
Patrik L Ferrari ◽  
Herbert Spohn
Keyword(s):  
Free Fermion ◽  
Vertex Model ◽  
Domino Tilings
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