© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2301.03997 · Communications in Mathematical Physics

A Q-Operator for Open Spin Chains II: Boundary Factorization

Alec Cooper, Bart Vlaar, Robert Weston

Published Apr 25, 2024• 3 citations

Abstract

Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine $$\mathfrak {sl}_2$$ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

DOI: 10.1007/s00220-024-04973-0

Related papers

Factorization of \mathcal{R} -matrix and Baxter \mathcal{Q} -operators for genericsl(N) spin chains

Sergey É Derkachov, Alexander N Manashov

We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.

arXiv:0809.2050•Journal of Physics A Mathematical and Theoretical•Jan 21, 2009•25 citations

Q-operators for the open Heisenberg spin chain

Rouven Frassek, István M. Szécsényi

We construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.

arXiv:1509.04867•Nuclear Physics B•Oct 30, 2015•23 citations

The BaxterQ-operator for the gradedSL(2|1) spin chain

A.V. Belitsky, S É Derkachov, Gregory P. Korchemsky +1 more

We study an integrable non-compact superspin chain model that emerged in recent studies of the dilatation operator in the super-Yang–Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite dimensional representations of the SL (2|1) group. We extend the method of the Baxter Q -operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the -operators acting on the tensor product of two generic infinite dimensional SL (2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three ‘elementary’ transfer matrices which we identify as Baxter Q -operators. We establish functional relations between transfer matrices and use them to derive the T – Q relations for the Q -operators. The proposed construction can be generalized to integrable models based on supergroups of higher rank and, as distinct from the Bethe ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.

arXiv:hep-th/0610332•Journal of Statistical Mechanics Theory and Experiment•Jan 4, 2007•34 citations

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

S É Derkachov, Alexander N Manashov

It is shown that the transfer matrices of homogeneous s ℓ(2) invariant spin chains with generic spin, both closed and open, are factorized into the product of two operators. The latter satisfy the Baxter equation that follows from the structure of the reducible representations of the s ℓ(2) algebra.

arXiv:nlin/0512047•Journal of Physics A Mathematical and General•Mar 31, 2006•28 citations

A Q-operator for open spin chains I. Baxter’s TQ relation

Bart Vlaar, Robert Weston

We construct a Q -operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter’s TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of U q ( s l ^ 2 ) . The resulting Bethe equations are the same as those arising from Sklyanin’s algebraic Bethe ansatz.

arXiv:2001.10760•Journal of Physics A Mathematical and Theoretical•Apr 9, 2020•13 citations

Factorization identities and algebraic Bethe ansatz for $$ {D}_2^{(2)} $$ models

Rafael I. Nepomechie, Ana L. Retore

Abstract We express $$ {D}_2^{(2)} $$ D 2 2 transfer matrices as products of $$ {A}_1^{(1)} $$ A 1 1 transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain with an even number of sites that depends on a continuous parameter, which we interpret as the rapidity of the boundary.

arXiv:2012.08367•Journal of High Energy Physics•Mar 1, 2021•0 citations

Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine

\mathfrak {sl}_2

sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.

We construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.

We study an integrable non-compact superspin chain model that emerged in recent studies of the dilatation operator in the super-Yang–Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite dimensional representations of the SL (2|1) group. We extend the method of the Baxter Q -operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the -operators acting on the tensor product of two generic infinite dimensional SL (2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three ‘elementary’ transfer matrices which we identify as Baxter Q -operators. We establish functional relations between transfer matrices and use them to derive the T – Q relations for the Q -operators. The proposed construction can be generalized to integrable models based on supergroups of higher rank and, as distinct from the Bethe ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.

Factorization identities and algebraic Bethe ansatz for

{D}_2^{(2)}

models

Abstract We express

{D}_2^{(2)}

D 2 2 transfer matrices as products of

{A}_1^{(1)}

A 1 1 transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain with an even number of sites that depends on a continuous parameter, which we interpret as the rapidity of the boundary.