COMMUTING TRANSFER MATRICES AND LINK POLYNOMIALS

1992 ◽  
Vol 03 (02) ◽  
pp. 205-212 ◽  
Author(s):  
V.F.R. JONES
Keyword(s):  
Statistical Mechanics ◽  
Transfer Matrices ◽  
Crossing Numbers ◽  
Commuting Transfer Matrices

Borrowing from an argument in statistical mechanics we give a machine for constructing pairs of links with the same skein polynomials. Examples are generally not mutants (Kauffman polynomials differ) and can have small crossing numbers, e.g. the coincidence V41 #41 = V89, is explained.

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.

Star-Triangle and Star-Star Relations in Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 27-37 ◽  
Author(s):  
R. J. Baxter
Keyword(s):  
Statistical Mechanics ◽  
Potts Model ◽  
Original Model ◽  
State Model ◽  
Nearest Neighbour ◽  
Transfer Matrices ◽  
Zamolodchikov Model ◽  
Adequate Condition

The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here we show that it satisfies a "star-to-reverse-star" (or simply star-star) relation, even though we know of no star-triangle relation for this model. For any nearest-neighbour checkerboard model, we show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a "twisted" Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for "solvability".

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
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