scholarly journals Factorization of the transfer matrices for the quantumsell(2) spin chains and Baxter equation

2006 ◽  
Vol 39 (16) ◽  
pp. 4147-4159 ◽  
Author(s):  
S É Derkachov ◽  
A N Manashov
Keyword(s):  
Spin Chains ◽  
Baxter Equation ◽  
Transfer Matrices

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 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 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 Quantum Spectral Curve of γ-Twisted 𝒩 = 4 SYM Theory and Fishnet CFT

2018 ◽  
Vol 30 (07) ◽  
pp. 1840010 ◽  
Author(s):  
Vladimir Kazakov
Keyword(s):  
Algebraic Structure ◽  
Spectral Curve ◽  
Scaling Limit ◽  
Integrable Model ◽  
Quantization Condition ◽  
Spin Chains ◽  
Hasse Diagram ◽  
Baxter Equation ◽  
Heisenberg Spin Chain ◽  
Wheel Graphs

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of [Formula: see text] SYM, including its [Formula: see text]-deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the [Formula: see text]-system — the full system of Baxter [Formula: see text]-functions of the underlying integrable model. The algebraic structure of the [Formula: see text]-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for [Formula: see text]-functions organized into the Hasse diagram. When supplemented with analyticity conditions on [Formula: see text]-functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First, we demonstrate the spectral equations on the example of [Formula: see text] and [Formula: see text] Heisenberg (super)spin chains. Supersymmetry [Formula: see text] occurs as a simple “rotation” of the Hasse diagram for a [Formula: see text] system. Then we apply this method to the spectral problem of [Formula: see text]/CFT4-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on [Formula: see text]-functions which involve an infinitely branching Riemann surface and a set of Riemann–Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of [Formula: see text]-twisted [Formula: see text] SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models — the bi-scalar theory — the QSC degenerates into the [Formula: see text]-system for integrable non-compact Heisenberg spin chain with conformal, [Formula: see text] symmetry. We describe the QSC derivation of Baxter equation and the quantization condition for particular fishnet graphs — wheel graphs, and review numerical and analytic results for them.

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 Yang-Baxter and the Boost: splitting the difference

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Marius de Leeuw ◽  
Chiara Paletta ◽  
Anton Pribytok ◽  
Ana L. Retore ◽  
Paul Ryan
Keyword(s):  
Hilbert Space ◽  
Sigma Model ◽  
Spin Chains ◽  
Baxter Equation ◽  
Regular Solutions ◽  
One Dimensional ◽  
Difference Form ◽  
The Difference ◽  
The One

In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times1616×16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2)𝔰𝔲(2)⊕𝔰𝔲(2) symmetry, which include the one-dimensional Hubbard model and the SS-matrix of the {AdS}_5 \times {S}^5AdS5×S5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

KASHIWARA-MIWA MODEL AND OPERATORS ACTING ON SPIN CHAINS

1994 ◽  
Vol 09 (27) ◽  
pp. 4701-4716
Author(s):  
ZHAN-NING HU
Keyword(s):  
Hopf Algebra ◽  
Quantum Algebra ◽  
Spin Chains ◽  
Baxter Equation ◽  
Elliptic Case ◽  
Special Case

In this paper, the Kashiwara-Miwa broken ZN model is discussed and a proof of the Yang-Baxter equation (YBE) is given, which shows that the YBE obtained by Hasegawa and Yamada is a special case. The Q operators acting on spin chains have a structure which is similar to that of the ordinary Hopf algebra, and the quantum algebra sl q(2) with qN=1 can be obtained from these operators as a limit of the elliptic case.

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