scholarly journals Noncompact sl(N) Spin Chains: BGG-Resolution, Q-Operators and Alternating Sum Representation for Finite-Dimensional Transfer Matrices

2011 ◽  
Vol 97 (2) ◽  
pp. 185-202 ◽  
Author(s):  
Sergey E. Derkachov ◽  
Alexander N. Manashov
Keyword(s):  
Spin Chains ◽  
Transfer Matrices ◽  
Finite Dimensional

ON COMMUTANTS, DUAL PAIRS AND NON-SEMISIMPLE ALGEBRAS FROM STATISTICAL MECHANICS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 675-705 ◽  
Author(s):  
PAUL P. MARTIN ◽  
DAVID S. MCANALLY
Keyword(s):  
Dual Pair ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Complex Vector ◽  
Dual Pairs ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Special Case

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that Hn(q), Uq(sl2) act as a dual pair on the tensored vector representation for all q.

Quantum spin chains and classical integrable systems

2018 ◽  
Author(s):  
Anton Zabrodin
Keyword(s):  
Integrable Systems ◽  
Quantum Spin ◽  
Spin Chains ◽  
Algebraic Equations ◽  
Quantum Spin Chains ◽  
Many Body ◽  
Finite Dimensional ◽  
The Family ◽  
Classical Correspondence ◽  
Remarkable Relation

This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.

scholarly journals Integrable quenches in nested spin chains II: fusion of boundary transfer matrices

2019 ◽  
Vol 2019 (6) ◽  
pp. 063104 ◽  
Author(s):  
Lorenzo Piroli ◽  
Eric Vernier ◽  
Pasquale Calabrese ◽  
Balázs Pozsgay
Keyword(s):  
Spin Chains ◽  
Transfer Matrices

scholarly journals ANALYTICAL BETHE ANSATZ FOR OPEN SPIN CHAINS WITH SOLITON NONPRESERVING BOUNDARY CONDITIONS

2006 ◽  
Vol 21 (07) ◽  
pp. 1537-1554 ◽  
Author(s):  
D. ARNAUDON ◽  
A. DOIKOU ◽  
L. FRAPPAT ◽  
É. RAGOUCY ◽  
N. CRAMPÉ
Keyword(s):  
Boundary Conditions ◽  
Bethe Ansatz ◽  
Spin Chains ◽  
Full Generality ◽  
Finite Dimensional ◽  
Algebraic Treatment ◽  
Algebraic Framework ◽  
Matrix Eigenvalues ◽  
Bethe Equations

We present an "algebraic treatment" of the analytical Bethe ansatz for open spin chains with soliton nonpreserving (SNP) boundary conditions. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic [Formula: see text] open SNP spin chain possessing on each site an arbitrary representation. As a result, we obtain the Bethe equations in their full generality. The classification of finite dimensional irreducible representations for the twisted Yangians are directly linked to the calculation of the transfer matrix eigenvalues.

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 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.

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