scholarly journals From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices

Axioms ◽  
2012 ◽  
Vol 1 (2) ◽  
pp. 186-200 ◽  
Author(s):  
Jeffrey R. Schmidt
Keyword(s):  
Necessary Condition ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Commuting Transfer Matrices

scholarly journals TRACE MAPS, INVARIANTS, AND SOME OF THEIR APPLICATIONS

1993 ◽  
Vol 07 (06n07) ◽  
pp. 1527-1550 ◽  
Author(s):  
M. BAAKE ◽  
U. GRIMM ◽  
D. JOSEPH
Keyword(s):  
Ising Model ◽  
Algebraic Structure ◽  
Electronic Spectra ◽  
Transfer Matrices ◽  
Two Level Systems ◽  
Commuting Transfer Matrices ◽  
Trace Maps

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x, y, z)=x2+y2+z2−2xyz−1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.

A NEW INTEGRABLE MODEL OF STRONGLY CORRELATED ELECTRONS WITH HUBBARD-LIKE INTERACTION

2001 ◽  
Vol 15 (06n07) ◽  
pp. 213-218
Author(s):  
XIANG-YU GE
Keyword(s):  
Bethe Ansatz ◽  
Integrable Model ◽  
Strongly Correlated Electrons ◽  
The Other ◽  
Competitive Interactions ◽  
Correlated Electrons ◽  
Strongly Correlated ◽  
Transfer Matrices ◽  
Ansatz Method ◽  
Commuting Transfer Matrices

A new completely integrable model of strongly correlated electrons is proposed which describes two competitive interactions: one is the correlated one-particle hopping, the other is the Hubbard-like interaction. The integrability follows from the fact that the Hamiltonian is derivable from a one-parameter family of commuting transfer matrices. The Bethe ansatz equations are derived by algebraic Bethe ansatz method.

scholarly journals Correlations and commuting transfer matrices in integrable unitary circuits

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Pieter W. Claeys ◽  
Jonah Herzog-Arbeitman ◽  
Austen Lamacraft
Keyword(s):  
Correlation Functions ◽  
Exceptional Point ◽  
Baxter Equation ◽  
Matrix Formalism ◽  
Transfer Matrices ◽  
Local Conservation ◽  
Vacuum States ◽  
Integrable Spin ◽  
Commuting Transfer Matrices ◽  
Bethe Equations

We consider a unitary circuit where the underlying gates are chosen to be \check{R}Ř-matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.

General Square Lattice Vertex Models

2019 ◽  
pp. 430-453
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Square Lattice ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Spectral Parameters ◽  
Vertex Models ◽  
Integrable Quantum ◽  
Special Cases ◽  
Mechanical Models ◽  
The One ◽  
Ice Model

Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.

scholarly journals Density matrices in integrable face models

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Holger Frahm ◽  
Daniel Westerfeld
Keyword(s):  
Functional Equations ◽  
Model Parameters ◽  
Nearest Neighbour ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Density Matrices ◽  
Reduced Density Matrices ◽  
Local Operators ◽  
Reduced Density ◽  
Solid Type

Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian \mathbold{su(2)_3}𝐬𝐮(2)3 or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be ‘factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.

Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus>1

1987 ◽  
Vol 123 (5) ◽  
pp. 219-223 ◽  
Author(s):  
Helen Au-Yang ◽  
Barry M. McCoy ◽  
Jacques H.H. Perk ◽  
Shuang Tang ◽  
Mu-Lin Yan
Keyword(s):  
Transfer Matrices ◽  
Potts Models ◽  
Commuting Transfer Matrices

scholarly journals Liouville reflection operator, affine Yangian and Bethe ansatz

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Litvinov ◽  
Ilya Vilkoviskiy
Keyword(s):  
Field Theory ◽  
Bethe Ansatz ◽  
Transfer Matrices ◽  
Reflection Operator ◽  
Integrable Structure ◽  
Conformal Field ◽  
Long Wave ◽  
The Family ◽  
Commuting Transfer Matrices ◽  
The One

Abstract In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik Okounkov R-matrix. We discuss relation between RLL and current realization of the affine Yangian of $$ \mathfrak{gl}(1) $$ gl 1 . We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.

scholarly journals Cylindric Hecke Characters and Gromov–Witten Invariants via the Asymmetric Six-Vertex Model

2020 ◽  
Author(s):  
Christian Korff
Keyword(s):  
Hecke Algebras ◽  
Square Lattice ◽  
Vertex Operators ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Genus Zero ◽  
Planar Lattice ◽  
Infinite Dimensional ◽  
Structure Constants ◽  
Restriction Functor

AbstractWe construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov–Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual characters for whose computation we give several explicit combinatorial formulae. One of these expressions is a generalisation of Ram’s formula for irreducible Hecke characters and uses cylindric broken rim hook tableaux. We show that the latter are in bijection with so-called ‘ice configurations’ on a cylindrical square lattice, which define the asymmetric six-vertex model in statistical mechanics. A key ingredient of our construction is an extension of the boson-fermion correspondence to Hecke algebras and employing the latter we find new expressions for Jing’s vertex operators of Hall–Littlewood functions in terms of the six-vertex transfer matrices on the infinite planar lattice.

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