Boundary Yang-Baxter Equation Research Articles

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

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 sl2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}_2$$\\end{document} and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

The ghost algebra and the dilute ghost algebra

We introduce the ghost algebra, a two-boundary generalisation of the Temperley–Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang–Baxter equations, given existing solutions to the Yang–Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang–Baxter integrable.

Fusion hierarchies, T-systems and Y-systems for the dilute loop models on a strip

We study the dilute loop models on the geometry of a strip of width N. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T- and Y-systems, for these four boundary conditions and with a generic crossing parameter λ. For rational and thus a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-N expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small N. The present work complements a previous study (Morin-Duchesne and Pearce 2019 J. Stat. Mech. 1909) that investigated the dilute loop models with periodic boundary conditions.

Shor–Movassagh chain leads to unusual integrable model

The ground state of the Shor–Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states. The model is not solvable. We prove the integrability of the model without interacting part in this paper (free Shor–Movassagh). The Lax pair for the free Shor–Movassagh open chain is explicitly constructed. We further obtain the boundary K-matrices compatible with the integrability of the model on the open interval. Our construction provides a direct demonstration for the quantum integrability of the model, described by the Yang–Baxter algebra. Because the partial transpose of the R matrix is not invertible, the model does not have crossing unitarity and the integrable open chain cannot be constructed by the reflection equation (boundary Yang–Baxter equation).

Structure constants in mathcal{N} = 4 SYM at finite coupling as worldsheet g-function

We develop a novel nonperturbative approach to a class of three-point functions in planar mathcal{N} = 4 SYM based on Thermodynamic Bethe Ansatz (TBA). More specifically, we study three-point functions of a non-BPS single-trace operator and two determinant operators dual to maximal Giant Graviton D-branes in AdS5×S5. They correspond to disk one-point functions on the worldsheet and admit a simpler and more powerful integrability description than the standard single-trace three-point functions. We first introduce two new methods to efficiently compute such correlators at weak coupling; one based on large N collective fields, which provides an example of open-closed-open duality discussed by Gopakumar, and the other based on combinatorics. The results so obtained exhibit a simple determinant structure and indicate that the correlator can be interpreted as a generalization of g-functions in 2d QFT; an overlap between an integrable boundary state and a state corresponding to the single-trace operator. We then determine the boundary state at finite coupling using the symmetry, the crossing equation and the boundary Yang-Baxter equation. With the resulting boundary state, we derive the ground-state g-function based on TBA and conjecture its generalization to other states. This is the first fully nonperturbative proposal for the structure constants of operators of finite length. The results are tested extensively at weak and strong couplings. Finally, we point out that determinant operators can provide better probes of sub-AdS locality than single-trace operators and discuss possible applications.

Gauge theory and boundary integrability. Part II. Elliptic and trigonometric cases

We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki & Witten on a ℤ2 orbifold. We use this to construct semi- classical solutions of the boundary Yang-Baxter equation in the elliptic and trigonometric cases. A novel feature of the trigonometric case is that the ℤ2 action lifts to the gauge bundle in a z-dependent way. We construct several examples of K -matrices, and check that they agree with cases appearing in the literature.

On the classification of rational K-matrices

This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the ‘classical limit’ (the spectral parameter goes to infinity). This derivation uses only the boundary Yang–Baxter equation and the asymptotic expansions of the R-matrices. The result proves the previous assumption of the literature: if the original and the residual symmetry algebras are and then there exists a Lie-algebra involution of for which the invariant sub-algebra is . In addition, we study a K-matrix which is not invertible in the ‘classical limit’. It is shown that its symmetry algebra is not reductive but a semi-direct sum of reductive and solvable Lie-algebras.

Integrable Matrix Product States from boundary integrability

We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to “operator valued” solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation. We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the “square root relation”, because it involves half of the steps of the reflection equation. It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations. We provide explicit solutions in a number of cases characterized by special symmetries. These correspond to the “symmetric pairs” (SU(N),SO(N)) and (SO(N),SO(D)\otimes⊗SO(N-D)), where in each pair the first and second elements are the symmetry groups of the spin chain and the integrable state, respectively. These solutions can be considered as explicit representations of the corresponding twisted Yangians, that are new in a number of cases. Examples include certain concrete MPS relevant for the computation of one-point functions in defect AdS/CFT.

Gauge theory and boundary integrability

We study the mixed topological/holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ×ℂ)/ℤ2, obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order ℏ calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The ℤ2-action on Σ × ℂ fixes a line L, and line operators on L are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the RTT presentation of the boundary Yang-Baxter equation.

Open spin chains from determinant like operators in ABJM theory

We study the mixing problem of the determinant like operators in ABJM theory to two loop order in the scalar sector. The gravity duals of these operators are open strings attached to the maximal giant graviton, which is a D4-brane wrapping a $\mathbb{CP}^2$ inside $\mathbb{CP}^3$ in our case. The anomalous dimension matrix of these operators can be regarded as an open spin chain Hamiltonian. We provide strong evidence of its integrability based on coordinate Bethe ansatz method and boundary Yang-Baxter equation.

New K-matrices with quantum group symmetry

We propose new families of solutions of the boundary Yang–Baxter equation. The open spin-chain transfer matrices constructed with these K-matrices have quantum group symmetry corresponding to removing one node from the Dynkin diagram, namely, , where . These transfer matrices also have a duality symmetry. These symmetries help to account for the degeneracies in the spectrum of the transfer matrix.

Reflection matrices with Uq [osp(2) (2|2m)] symmetry

We propose a classification of the reflection K-matrices (solutions of the boundary Yang–Baxter equation) for the vertex-model. We found four families of solutions, namely, the complete solutions, in which no elements of the reflection K-matrix is null, the block-diagonal solutions, the X-shape solutions and the diagonal solutions. We highlight that these diagonal K-matrices also hold for the vertex-model.

On integrable boundaries in the 2 dimensional O(N) σ-models

We make an attempt to map the integrable boundary conditions for 2 dimensional non-linear σ-models. We do it at various levels: classically, by demanding the existence of infinitely many conserved local charges and also by constructing the double row transfer matrix from the Lax connection, which leads to the spectral curve formulation of the problem; at the quantum level, we describe the solutions of the boundary Yang–Baxter equation and derive the Bethe–Yang equations. We then show how to connect the thermodynamic limit of the boundary Bethe–Yang equations to the spectral curve.

Integrable open spin chains from flavored ABJM theory

We compute the two-loop anomalous dimension matrix in the scalar sector of planar mathcal{N}=3 flavored ABJM theory. Using coordinate Bethe ansatz, we obtain the reflection matrices and confirm that the boundary Yang-Baxter equations are satisfied. This establishes the integrability of this theory in the scalar sector at the two-loop order.

Reflection K-matrices for a nineteen vertex model with [formula omitted] symmetry

We derive the solutions of the boundary Yang–Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra Uq[osp(2|2)(2)]. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection K-matrix are different from zero. In the type II solution, the reflection K-matrix is even (every element of the K-matrix with an odd parity is null) and it has only one boundary free-parameter. Finally, the type III solution corresponds to a diagonal reflection K-matrix with two boundary free-parameters.

Logarithmic minimal models with Robin boundary conditions

We consider general logarithmic minimal models , with coprime, on a strip of N columns with the (r, s) Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. On the lattice, these models are Yang–Baxter integrable loop models that are described algebraically by the one-boundary Temperley–Lieb algebra. The (r, s) Robin boundary conditions are a class of integrable boundary conditions satisfying the boundary Yang–Baxter equations which allow loop segments to either reflect or terminate on the boundary. The associated conformal boundary conditions are organized into infinitely extended Kac tables labelled by the Kac labels and . The Robin vacuum boundary condition, labelled by , is given as a linear combination of Neumann and Dirichlet boundary conditions. The general (r, s) Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an (r, s)-type seam consisting of an r-type seam of width w columns and an s-type seam of width d = s − 1 columns. The r-type seam admits an arbitrary boundary field which we fix to the special value where is the crossing parameter. The s-type boundary introduces d defects into the bulk. We consider the commuting double-row transfer matrices and their associated quantum Hamiltonians and calculate analytically the boundary free energies of the (r, s) Robin boundary conditions. Using finite-size corrections and sequence extrapolation out to system sizes , the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of N for and restricting to the ground state sequences , with the inverse , we find that the conformal weights take the values where is given by the usual Kac formula. The (r, s) Robin boundary conditions are thus conjugate to scaling operators with half-integer values for the Kac label . Level degeneracies support the conjecture that the characters of the associated (reducible or irreducible) representations are given by Virasoro Verma characters.

Anomalous dimensions of heavy operators from magnon energies

We study spin chains with boundaries that are dual to open strings suspended between systems of giant gravitons and dual giant gravitons. The anomalous dimensions computed in the gauge theory are in complete quantitative agreement with energies computed in the dual string theory. The comparison makes use of a description in terms of magnons, generalizing results for a single maximal giant graviton. The symmetries of the problem determine the structure of the magnon boundary reflection/scattering matrix up to a phase. We compute a reflection/scattering matrix element at weak coupling and verify that it is consistent with the answer determined by symmetry. We find the reflection/scattering matrix does not satisfy the boundary Yang-Baxter equation so that the boundary condition on the open spin chain spoils integrability. We also explain the interpretation of the double coset ansatz in the magnon language.

Integrable open spin-chains inAdS3/CFT2correspondences

We study integrable open boundary conditions for d(2,1;\alpha)^2 and psu(1,1|2)^2 spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type IIB open superstrings ending on D-branes in the AdS_3 x S^3 x S^3 x S^1 and AdS_3 x S^3 x T^4 supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1- and D5-brane maximal giant gravitons.

Q-operators for the open Heisenberg spin chain

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.

On the multiparametric vertex model

In this paper we consider families of multiparametric R-matrices to make a systematic study of the boundary Yang–Baxter equations in order to discuss the corresponding families of multiparametric K-matrices. Our results are indeed a non-trivial generalization of the K-matrix solutions of the vertex model when distinct reflections and extra free parameters are admissible.