arXiv:2310.18499

Open boundary conditions of the $D^{(2)}_3$ spin chain and sectors of conformal field theories

Pete Rigas

Published Oct 27, 2023• 0 citations

Abstract

We study open boundary conditions for the $D^{(2)}_3$ spin chain, which shares connections with the six-vertex model, under staggering, and also to the antiferromagnetic Potts model. By formulating a suitable transfer matrix that is related to $K$ matrices and to the Jimbo $R$-matrix, we obtain an analytical expression for the Hamiltonian, as a logarithmic derivative of the transfer matrix, under open boundary conditions. Such computations have connections with several objects studied in Integrable Probability, which underlie exactly solvable structures.

DOI: 10.48550/arxiv.2310.18499

Related papers

Exact physical quantities of the $$ {D}_2^{(2)} $$ spin chain model with generic open boundary conditions

Pengcheng Lu, Junpeng Cao, Wen‐Li Yang +2 more

A bstract We study the quantum integrable spin chain model associated with the twisted $$ {D}_2^{(2)} $$ D 2 2 algebra (or simply the $$ {D}_2^{(2)} $$ D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the $$ {D}_2^{(2)} $$ D 2 2 model in different regimes of boundary parameters.

arXiv:2502.09262Journal of High Energy PhysicsMay 15, 20251 citation

Bethe equations for the critical three-state Potts spin chain with toroidal boundary conditions

Martins, M. J.

arXiv:2512.07763Dec 8, 20250 citations

Integrable boundary conditions in the antiferromagnetic Potts model

Niall Robertson, Michal Pawelkiewicz, Jesper Lykke Jacobsen +1 more

Abstract We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine $$ {D}_2^2 $$ D 2 2 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the $$ {D}_2^2 $$ D 2 2 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.

arXiv:2003.03261Journal of High Energy PhysicsMay 27, 202018 citations

Bethe equations for the critical three-state Potts spin chain with toroidal boundary conditions

Marcio Jose Martins

Nuclear Physics BJan 27, 20260 citations

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

Christian Hagendorf, Jean Liénardy

arXiv:2008.03220Journal of Statistical Mechanics Theory and ExperimentJan 1, 20211 citation

TheD(D3)-anyon chain: integrable boundary conditions and excitation spectra

Peter E. Finch, Holger Frahm

arXiv:1211.4449New Journal of PhysicsMay 22, 201315 citations

arXiv:2310.18499

Open boundary conditions of the $D^{(2)}_3$ spin chain and sectors of conformal field theoriesOpen boundary conditions of the $D^{(2)}_3$ spin chain and sectors of conformal field theories

Pete Rigas

Published Oct 27, 2023• 0 citations

arXiv HTML PDF

Abstract

We study open boundary conditions for the

D^{(2)}_3

spin chain, which shares connections with the six-vertex model, under staggering, and also to the antiferromagnetic Potts model. By formulating a suitable transfer matrix that is related to

K

matrices and to the Jimbo

R

-matrix, we obtain an analytical expression for the Hamiltonian, as a logarithmic derivative of the transfer matrix, under open boundary conditions. Such computations have connections with several objects studied in Integrable Probability, which underlie exactly solvable structures.

DOI: 10.48550/arxiv.2310.18499

Related papers

Exact physical quantities of the ${D}_2^{(2)}$ spin chain model with generic open boundary conditionsExact physical quantities of the $$ {D}_2^{(2)} $$ spin chain model with generic open boundary conditions

Pengcheng Lu, Junpeng Cao, Wen‐Li Yang +2 more

A bstract We study the quantum integrable spin chain model associated with the twisted ${D}_2^{(2)}$ D 2 2 algebra (or simply the ${D}_2^{(2)}$ D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the ${D}_2^{(2)}$ D 2 2 model in different regimes of boundary parameters.A bstract We study the quantum integrable spin chain model associated with the twisted $$ {D}_2^{(2)} $$ D 2 2 algebra (or simply the $$ {D}_2^{(2)} $$ D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the $$ {D}_2^{(2)} $$ D 2 2 model in different regimes of boundary parameters.

arXiv:2502.09262Journal of High Energy PhysicsMay 15, 20251 citation

Bethe equations for the critical three-state Potts spin chain with toroidal boundary conditions

Martins, M. J.

In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrix operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrix operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.

arXiv:2512.07763Dec 8, 20250 citations

Integrable boundary conditions in the antiferromagnetic Potts model

Niall Robertson, Michal Pawelkiewicz, Jesper Lykke Jacobsen +1 more

Abstract We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine ${D}_2^2$ D 2 2 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the ${D}_2^2$ D 2 2 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.Abstract We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine $$ {D}_2^2 $$ D 2 2 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the $$ {D}_2^2 $$ D 2 2 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.

arXiv:2003.03261Journal of High Energy PhysicsMay 27, 202018 citations

Bethe equations for the critical three-state Potts spin chain with toroidal boundary conditions

Marcio Jose Martins

In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrices operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrices operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.

Nuclear Physics BJan 27, 20260 citations

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

Christian Hagendorf, Jean Liénardy

Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

arXiv:2008.03220Journal of Statistical Mechanics Theory and ExperimentJan 1, 20211 citation

TheD(D3)-anyon chain: integrable boundary conditions and excitation spectra

Peter E. Finch, Holger Frahm

Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group D 3 are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the Bethe equations for the spin chain. The low-energy effective theories and operator content of the models (in both the spin chain and fusion path formulation) are identified from analytical and numerical studies of the finite-size spectra. For all boundary conditions considered the continuum theory is found to be a product of two conformal field theories. Depending on the coupling constants the factors can be a Z 4 parafermion or a minimal model.Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group D 3 are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the Bethe equations for the spin chain. The low-energy effective theories and operator content of the models (in both the spin chain and fusion path formulation) are identified from analytical and numerical studies of the finite-size spectra. For all boundary conditions considered the continuum theory is found to be a product of two conformal field theories. Depending on the coupling constants the factors can be a Z 4 parafermion or a minimal model.

arXiv:1211.4449New Journal of PhysicsMay 22, 201315 citations

Open boundary conditions of the

D^{(2)}_3

spin chain and sectors of conformal field theories

We study open boundary conditions for the

D^{(2)}_3

spin chain, which shares connections with the six-vertex model, under staggering, and also to the antiferromagnetic Potts model. By formulating a suitable transfer matrix that is related to

K

matrices and to the Jimbo

R

Exact physical quantities of the

{D}_2^{(2)}

spin chain model with generic open boundary conditions

A bstract We study the quantum integrable spin chain model associated with the twisted

{D}_2^{(2)}

D 2 2 algebra (or simply the

{D}_2^{(2)}

D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the

{D}_2^{(2)}

D 2 2 model in different regimes of boundary parameters.

Abstract We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine

{D}_2^2

D 2 2 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the

{D}_2^2

D 2 2 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.