scholarly journals MODIFIED TETRAHEDRON EQUATION AND RELATED 3D INTEGRABLE MODELS, II

1996 ◽  
Vol 11 (02) ◽  
pp. 313-327 ◽  
Author(s):  
H.E. BOOS
Keyword(s):  
General Model ◽  
Spin Model ◽  
Parametric Family ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
Tetrahedron Equation ◽  
Cubic Lattice ◽  
Analytical Formulae ◽  
Integrable Spin ◽  
Free Parameters

This work is a continuation of Ref. 13, where the Boltzmann weights for the N-state integrable spin model on the cubic lattice has been obtained only numerically. In this paper we present the analytical formulae for this model in a particular case. Here the Boltzmann weights depend on six free parameters including the elliptic modulus. The obtained solution allows us to construct a two-parametric family of the commuting two-layer transfer matrices. The presented model is expected to be simpler for a further investigation in comparison with a more general model mentioned above.

scholarly journals A (2 + 1)-dimensional integrable spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures

1997 ◽  
Vol 233 (4-6) ◽  
pp. 391-396 ◽  
Author(s):  
R. Myrzakulov ◽  
S. Vijayalakshmi ◽  
G.N. Nugmanova ◽  
M. Lakshmanan
Keyword(s):  
Coherent Structures ◽  
Spin Model ◽  
Integrable Spin

scholarly journals GENERALIZED YANG-BAXTER EQUATION

1993 ◽  
Vol 08 (24) ◽  
pp. 2299-2309 ◽  
Author(s):  
R. M. KASHAEV ◽  
YU. G. STROGANOV
Keyword(s):  
Single Layer ◽  
Lattice Models ◽  
Baxter Equation ◽  
Explicit Solutions ◽  
Generalized Equations ◽  
Two Dimensional ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
Dimensional Lattice ◽  
Potts Models

A generalization of the Yang-Baxter equation is proposed. It enables us to construct integrable two-dimensional lattice models with commuting two-layer transfer matrices, while single-layer ones are not necessarily commutative. Explicit solutions to the generalized equations are found. They are related with Boltzmann weights of the sl (3) chiral Potts models.

scholarly journals PHASE TRANSITION IN THE HEISENBERG FULLY-FRUSTRATED SIMPLE CUBIC LATTICE

2011 ◽  
Vol 25 (12n13) ◽  
pp. 929-936 ◽  
Author(s):  
V. THANH NGO ◽  
D. TIEN HOANG ◽  
H. T. DIEP
Keyword(s):  
Phase Transition ◽  
Statistical Physics ◽  
Spin Model ◽  
Order Transition ◽  
Heisenberg Spin ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Second Order Transition ◽  
Frustrated Spin

The phase transition in frustrated spin systems is a fascinating subject in statistical physics. We show the result obtained by the Wang–Landau flat histogram Monte Carlo simulation on the phase transition in the fully frustrated simple cubic lattice with the Heisenberg spin model. The degeneracy of the ground state of this system is infinite with two continuous parameters. We find a clear first-order transition in contradiction with previous studies which have shown a second-order transition with unusual critical properties. The robustness of our calculations allows us to conclude this issue putting an end to the 20-year long uncertainty.

scholarly journals Exact and Approximate Compression of Transfer Matrices for Graph Homomorphisms

2008 ◽  
Vol 11 ◽  
pp. 1-14 ◽  
Author(s):  
Per Håkan Lundow ◽  
Klas Markström
Keyword(s):  
Transfer Matrix ◽  
Three Dimensional ◽  
Transfer Matrices ◽  
Square Grid ◽  
Cubic Lattice ◽  
Matrix Compression ◽  
Graph Homomorphisms

AbstractThe aim of this paper is to extend the previous work on transfer matrix compression in the case of graph homomorphisms. For H-homomorphisms of lattice-like graphs we demonstrate how the automorphisms of H, as well as those of the underlying lattice, can be used to reduce the size of the relevant transfer matrices. As applications of this method we give currently best known bounds for the number of 4- and 5-colourings of the square grid, and the number of 3- and 4-colourings of the three-dimensional cubic lattice. Finally, we also discuss approximate compression of transfer matrices.

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.

scholarly journals QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Gwenäel Ferrando ◽  
Rouven Frassek ◽  
Vladimir Kazakov
Keyword(s):  
Lie Algebra ◽  
Finite Length ◽  
Spin Chains ◽  
Transfer Matrices ◽  
Full System ◽  
Functional Relations ◽  
Integrable Spin ◽  
Integrable Spin Chains

Abstract We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the Dr Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through r + 1 basic Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length.

scholarly journals MODIFIED TETRAHEDRON EQUATIONS AND RELATED 3D INTEGRABLE MODELS, I

1995 ◽  
Vol 10 (28) ◽  
pp. 4041-4063 ◽  
Author(s):  
H.E. BOOS ◽  
V.V. MANGAZEEV ◽  
S.M. SERGEEV
Keyword(s):  
Free Parameter ◽  
Three Dimensional ◽  
Elliptic Functions ◽  
Integrable Models ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
New Family

Using a modified version of the tetrahedron equations we construct a new family of N- state three-dimensional integrable models with commuting two-layer transfer matrices. We investigate a particular class of solutions to these equations and parametrize them in terms of elliptic functions. The corresponding models contain one free parameter, k—an elliptic modulus.

scholarly journals Multiparameter Statistical Models from Braid Matrices: Explicit Eigenvalues of Transfer Matrices , Spin Chains, Factorizable Scatterings for All

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
B. Abdesselam ◽  
A. Chakrabarti
Keyword(s):  
Statistical Models ◽  
Linear Equations ◽  
Spin Chains ◽  
Transfer Matrices ◽  
Circular Permutations ◽  
Constant Coefficients ◽  
Free Parameters ◽  
Class Of Matrices ◽  
Zero Sum

For a class of multiparameter statistical models based on braid matrices, the eigenvalues of the transfer matrix are obtained explicitly for all . Our formalism yields them as solutions of sets of linear equations with simple constant coefficients. The role of zero-sum multiplets constituted in terms of roots of unity is pointed out, and their origin is traced to circular permutations of the indices in the tensor products of basis states induced by our class of matrices. The role of free parameters, increasing as withN, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for allN. Inverse Cayley transforms of the Yang-Baxter matrices corresponding to our braid matrices are obtained for allN. They provide potentials for factorizableS-matrices. Main results are summarized, and perspectives are indicated in the concluding remarks.

Properties of the spin model above Tc on the sample cubic lattice

1975 ◽  
Vol 53 (4) ◽  
pp. 313-314 ◽  
Author(s):  
D.J. Austen ◽  
D.D. Betts
Keyword(s):  
Spin Model ◽  
Cubic Lattice
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