Non-diagonal Boundary Terms Research Articles

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

A novel Bethe ansatz scheme for the one-dimensional Hubbard model

We propose a novel characterization of the exact solution of the one-dimensional Hubbard model with unparallel boundary magnetic fields. The non-diagonal boundary terms break the U(1) symmetry of the system and the number of electrons with fixed spin is not conserved. Thus it is difficult to study the physical quantities in the thermodynamic limit based on the previously obtained inhomogeneous T−Q relation and Bethe ansatz equations. In order to solve this problem, we propose the t−W scheme. The basic idea is that the eigenvalues of the transfer matrix can be expressed by its zero-roots instead of the usual Bethe roots. This method allows us to study thermodynamic properties of the system such as the ground state, surface energy, charge and spin excitations. The t−W scheme is universal and can be applied to the systems either with or without U(1) symmetry.

Exact solution of a quantum spin chain with competing bulk and boundary terms

We obtain the exact solution of an anisotropic quantum spin chain including nearest neighbor (NN), next nearest neighbor (NNN), chiral three-spin interactions and non-diagonal boundary terms. Because the NNN couplings are involved, we find that the off-diagonal boundary reflections can not only induce the anisotropic Dzyloshinsky–Moriya interactions at boundaries but also enhance the anisotropy of first and last bonds. These properties are absent if only the NN spin-exchanging interactions are considered. Due to that the U(1) symmetry is broken, we analytically solve the system by using the off-diagonal Bethe ansatz. The inhomogeneous T − Q relation, energy spectrum and associated Bethe ansatz equations are given explicitly. This paper provides an universal scheme to construct new integrable models with certain interesting interactions.

A note on a boundary sine-Gordon model at the free-Fermion point

We investigate the free-Fermion point of a boundary sine-Gordon model with nondiagonal boundary interactions for the ground state using auxiliary functions obtained from T − Q equations of a corresponding inhomogeneous open spin- XXZ chain with nondiagonal boundary terms. In particular, we obtain the Casimir energy. Our result for the Casimir energy is shown to agree with the result from the TBA approach. The analytical result for the effective central charge in the ultraviolet (UV) limit is also verified from the plots of effective central charge for intermediate values of volume.

The open XXX spin chain in the SoV framework: scalar product of separate states

We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T-Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.

Small polaron with generic open boundary conditions: Exact solution via the off-diagonal Bethe ansatz

The small polaron, a one-dimensional lattice model of interacting spinless fermions, with generic non-diagonal boundary terms is studied by the off-diagonal Bethe ansatz method. The presence of the Grassmann valued non-diagonal boundary fields gives rise to a typical U(1)-symmetry-broken fermionic model. The exact spectra of the Hamiltonian and the associated Bethe ansatz equations are derived by constructing an inhomogeneous T–Q relation.

Exact solution of the alternating XXZ spin chain with generic non-diagonal boundaries

The integrable XXZ alternating spin chain with generic non-diagonal boundary terms specified by the most general non-diagonal K-matrices is studied via the off-diagonal Bethe Ansatz method. Based on the intrinsic properties of the fused R-matrices and K-matrices, we obtain certain closed operator identities and conditions, which allow us to construct an inhomogeneous T−Q relation and the associated Bethe Ansatz equations accounting for the eigenvalues of the transfer matrix.

The solution of an open XXZ chain with arbitrary spin revisited

The Bethe ansatz solutions for an open XXZ spin chain with arbitrary spin with N sites and nondiagonal boundary terms are revisited. The anisotropy parameter, for cases considered here, has values , where r and q are positive integers with q restricted to odd integers. Numerical results are presented to support the solutions.

Direct Proof of Determinant Representation for Scalar Products of the XXZ Gaudin Model with Generic Non-Diagonal Boundary Terms

After constructing the Bethe state of the XXZ Gaudin model with generic non-diagonal boundary terms, we analyze the properties of this state and obtain the determinant representations of the scalar products for this XXZ Gaudin model.

Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms

The Izergin-Korepin model with general non-diagonal boundary terms, a typical integrable model beyond A-type and without U(1)-symmetry, is studied via the off-diagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and the K-matrices, certain operator product identities of the transfer matrix are obtained at some special points of the spectral parameter. These identities and the asymptotic behaviors of the transfer matrix together allow us to construct the inhomogeneous T-Q relation and the associated Bethe ansatz equations. In the diagonal boundary limit, the reduced results coincide exactly with those obtained via other methods.

The complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms

We use the quantum separation of variables (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates for the inhomogeneous case are constructed in terms of solutions of a system of quadratic equations. This SOV representation permits us to compute scalar products and can be used to calculate form factors and correlation functions.

An inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin

An inhomogeneous T-Q equation has recently been proposed by Cao, Yang, Shi and Wang for the open spin-1/2 XXX chain with general (nondiagonal) boundary terms. We argue that a simplified version of this equation describes all the eigenvalues of the transfer matrix of this model. We also propose a generating function for the inhomogeneous T-Q equations of arbitrary spin.

Bethe ansatz solution of the small polaron with nondiagonal boundary terms

The small polaron with generic, nondiagonal boundary terms is investigated within the framework of quantum integrability. The fusion hierarchy of the transfer matrices and its truncation for particular values of the anisotropy parameter are both employed, so that the spectral problem is formulated in terms of a TQ equation. The solution of this equation for generic boundary conditions is based on a deformation of the diagonal case. The eigenvalues of the model are extracted and the corresponding Bethe ansatz equations are presented. Finally, we comment on the eigenvectors of the model and explicitly compute the eigenstate of the model which evolves into the Fock vacuum when the off-diagonal boundary terms are switched off.

Generalized T–Q relations and the open spin-s XXZ chain with nondiagonal boundary terms

We consider the open spin-s XXZ quantum spin chain with nondiagonal boundary terms. By exploiting certain functional relations at roots of unity, we derive a generalized form of T–Q relation involving more than one independent Q(u), which we use to propose the Bethe-ansatz-type expressions for the eigenvalues of the transfer matrix. At most two of the boundary parameters are set to be arbitrary and the bulk anisotropy parameter has values η = iπ/2,iπ/4,…. We also provide numerical evidence for the completeness of the Bethe-ansatz-type solutions derived, using the s = 1 case as an example.

Determinant representations for scalar products of the XXZ Gaudin model with general boundary terms

We obtain the determinant representations of the scalar products for the XXZ Gaudin model with generic non-diagonal boundary terms.

Drinfeld Twist and Symmetric Bethe Vectors of Open XYZ Chain with Non-Diagonal Boundary Terms

With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex solid-on-solid (SOS) model, we find that in the F-basis provided by the twist the two sets of pseudo-particle creation operators simultaneously take completely symmetric and polarization free form. This allows us to obtain the explicit and completely symmetric expressions of the two sets of Bethe states of the model.

Second Reference State and Complete Eigenstates of Open XYZ Chain

The second reference state of the open XYZ spin chain with non-diagonal boundary terms is studied. The associated Bethe states exactly yield the second set of eigenvalues proposed recently by functional Bethe Ansatz.

Scalar products of the open XYZ chain with non-diagonal boundary terms

With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix of the eight-vertex solid-on-solid (SOS) model, we obtain the determinant representations of the scalar products of Bethe states for the open XYZ chain with non-diagonal boundary terms. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.

Determinant representations of scalar products for the open XXZ chain with non-diagonal boundary terms

The determinant representation of the scalar products of the Bethe states of the open XXZ spin chain with non-diagonal boundary terms is studied. Using the vertex-face correspondence, we transfer the problem into the corresponding trigonometric solid-on-solid (SOS) model with diagonal boundary terms. With the help of the Drinfeld twist or factorizing F-matrix, we obtain the determinant representation of the scalar products of the Bethe states of the associated SOS model. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.

Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end

With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representation of the partition function of the six-vertex model with a non-diagonal reflecting end under domain wall boundary condition.