Abstract The finite size spectrum of the critical ℤ2-staggered spin-1/2 XXZ model with quantum group invariant boundary conditions is studied. For a particular (self-dual) choice of the staggering the spectrum of conformal weights of this model has been recently been shown to have a continuous component, similar as in the model with periodic boundary conditions whose continuum limit has been found to be described in terms of the non-compact SU(2, ℝ)/U(1) Euclidean black hole conformal field theory (CFT). Here we show that the same is true for a range of the staggering parameter. In addition we find that levels from the discrete part of the spectrum of this CFT emerge as the anisotropy is varied. The finite size amplitudes of both the continuous and the discrete levels are related to the corresponding eigenvalues of a quasi-momentum operator which commutes with the Hamiltonian and the transfer matrix of the model.
Finite size spectrum of the staggered six-vertex model with Uq(sl(2))-invariant boundary conditions
Abstract The finite size spectrum of the critical ℤ2-staggered spin-1/2 XXZ model with quantum group invariant boundary conditions is studied. For a particular (self-dual) choice of the staggering the spectrum of conformal weights of this model has been recently been shown to have a continuous component, similar as in the model with periodic boundary conditions whose continuum limit has been found to be described in terms of the non-compact SU(2, ℝ)/U(1) Euclidean black hole conformal field theory (CFT). Here we show that the same is true for a range of the staggering parameter. In addition we find that levels from the discrete part of the spectrum of this CFT emerge as the anisotropy is varied. The finite size amplitudes of both the continuous and the discrete levels are related to the corresponding eigenvalues of a quasi-momentum operator which commutes with the Hamiltonian and the transfer matrix of the model.
The finite-size spectrum of the critical staggered six-vertex model with antidiagonal boundary conditions is studied. Similar to the case of periodic boundary conditions, we identify three different phases. In two of those, the underlying conformal field theory can be identified to be related to the twisted U(1) Kac-Moody algebra. In contrast, the finite size scaling in the third regime, whose critical behaviour with the (quasi-)periodic BCs is related to the 2d black hole CFTs possessing a non-compact degree of freedom, is more subtle. Here with antidiagonal BCs imposed, the corrections to the scaling of the ground state grow logarithmically with the system size, while the energy gaps appear to close logarithmically. Moreover, we obtain an explicit formula for the Q-operator which is useful for numerical implementation.
The finite-size spectrum of the critical staggered six-vertex model with antidiagonal boundary conditions is studied. Similar to the case of periodic boundary conditions, we identify three different phases. In two of those, the underlying conformal field theory can be identified to be related to the twisted U(1) Kac-Moody algebra. In contrast, the finite size scaling in the third regime, whose critical behaviour with the (quasi-)periodic BCs is related to the 2d black hole CFTs possessing a non-compact degree of freedom, is more subtle. Here with antidiagonal BCs imposed, the corrections to the scaling of the ground state grow logarithmically with the system size, while the energy gaps appear to close logarithmically. Moreover, we obtain an explicit formula for the Q-operator which is useful for numerical implementation.
Quantum-group-invariant Dn+1(2) models: Bethe ansatz and finite-size spectrum
A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra Dn+1(2) D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra D3(2) D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 D2(2) D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the D2(2) D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.
We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses Uq(sl(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter Q operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the Q operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the z-projection of the total spin operator being equal to zero.
Scaling limit of the staggered six-vertex model with Uq(sl(2)) invariant boundary conditions
We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses U_q\big(\mathfrak{sl}(2)\big) U q ( 𝔰 𝔩 ( 2 ) ) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter Q Q operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the Q Q operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the z z -projection of the total spin operator being equal to zero.
Abstract Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable D22 D 2 2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.