Boundary critical phenomena in the three-state Potts model

Boundary critical phenomena are studied in the three-state Potts model in two dimensions using conformal field theory, duality and renormalization group methods. A presumably complete set of boundary conditions is obtained using both fusion and orbifold methods. Besides the previously known free, fixed and mixed boundary conditions a new one is obtained. This illustrates the necessity of considering fusion with operators that do not occur in the bulk spectrum, to obtain all boundary conditions. It is shown that this new boundary condition is dual to the mixed ones. The phase diagram for the quantum chain version of the Potts model is analysed using duality and renormalization group arguments.Boundary critical phenomena are studied in the three-state Potts model in two dimensions using conformal field theory, duality and renormalization group methods. A presumably complete set of boundary conditions is obtained using both fusion and orbifold methods. Besides the previously known free, fixed and mixed boundary conditions a new one is obtained. This illustrates the necessity of considering fusion with operators that do not occur in the bulk spectrum, to obtain all boundary conditions. It is shown that this new boundary condition is dual to the mixed ones. The phase diagram for the quantum chain version of the Potts model is analysed using duality and renormalization group arguments.

arXiv:cond-mat/9804117Journal of Physics A Mathematical and GeneralJul 17, 199873 citations

Boundary critical phenomena in the quantum Ashkin-Teller model

Yifan Liu, Natalia Chepiga, Yoshiki Fukusumi +1 more

We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the $\mathbb{Z}_2$ -orbifold of the $c=1$ compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying $\mathrm{SU}(2)$ symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with $D_4$ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the $\mathbb{Z}_2$-orbifold of the $c=1$ compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying $\mathrm{SU}(2)$ symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with $D_4$ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.

cond-mat.str-elCross-listedarXiv:2601.16951SciPost PhysicsJan 26, 20261 citation

Critical properties of quantum three- and four-state Potts models with boundaries polarized along the transverse field

Natalia Chepiga

By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we show that boundaries polarized in the direction of the transverse field lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of the excitation spectra we phenomenologically establish the duality between the transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point. Finally, for completeness, we verify that in the quantum three-state Potts model the "new" boundary conditions dual to the mixed ones can be realized by polarizing edge spins along the transverse field.By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we show that boundaries polarized in the direction of the transverse field lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of the excitation spectra we phenomenologically establish the duality between the transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point. Finally, for completeness, we verify that in the quantum three-state Potts model the "new" boundary conditions dual to the mixed ones can be realized by polarizing edge spins along the transverse field.

arXiv:2107.08899SciPost Physics CoreMay 30, 202211 citations

'Duality twisted' boundary conditions in n-state Potts models

Gunter Markus Schutz

The author discusses a new class of toroidal boundary conditions for one-dimensional quantum Hamiltonians with S n symmetry which are related to two-dimensional n-state Potts models in the extreme anisotropic Hamiltonian limit. At their self-dual point (a point where a second-order phase transition occurs for n=2, 3, 4) the duality transformation is shown to be an additional symmetry giving rise to a new class of 'duality twisted' toroidal boundary conditions. The corresponding Hamiltonians are given in terms of generators of the periodic Temperley-Lieb algebra with an odd number of generators. The author discusses as an example the critical Ising model. Here the complete spectrum for the new boundary conditions can be obtained from a projection mechanism in the spin-1/2 XXZ Heisenberg chain.The author discusses a new class of toroidal boundary conditions for one-dimensional quantum Hamiltonians with S n symmetry which are related to two-dimensional n-state Potts models in the extreme anisotropic Hamiltonian limit. At their self-dual point (a point where a second-order phase transition occurs for n=2, 3, 4) the duality transformation is shown to be an additional symmetry giving rise to a new class of 'duality twisted' toroidal boundary conditions. The corresponding Hamiltonians are given in terms of generators of the periodic Temperley-Lieb algebra with an odd number of generators. The author discusses as an example the critical Ising model. Here the complete spectrum for the new boundary conditions can be obtained from a projection mechanism in the spin-1/2 XXZ Heisenberg chain.

Journal of Physics A Mathematical and GeneralSep 21, 199325 citations

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

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

\mathbb{Z}_2

-orbifold of the

c=1

compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying

\mathrm{SU}(2)

symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with

D_4

symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.