Finite size spectrum of the staggered six-vertex model with Uq($$ \mathfrak{sl} $$(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.

Integrable boundary conditions for staggered vertex models

“…From our discussion above we know that only the alternating or quasi periodic staggering leads to a local Hamiltonian. Both cases have been studied extensively in [27,28] and [26] respectively. Using the vertex representation of the Temperley-Lieb generators e i,i+1 :…”

“…Remarkably, the choice of boundary conditions has a profound influence on the low energy properties of the staggered models: the effective theory of (5.9) with anisotropy γ < 2δ 0 < π − γ has been identified to be the SL(2, R)/U(1) sigma model at level k = π/γ with a noncompact spectrum of conformal weights [27,28]. On the contrary, Robertson et al found that the continuum limit of the model (5.10) is compact [47].…”

“…The spectral flow starting from the alternating model (5.9), ϑ = 0, can be studied in a similar way: here a class of low energy states (including the ground state) is known [27,28] to Rescaled real parts Ẽ of the eigenenergies of the staggered six-vertex Hamiltonian with 2L = 8 sites, anisotropy γ = 0.9 for the charge sector Sz = 2 for the spectral flow (5.13). Energies have been multiplied with − cos(ϑ + 2γ) sin 5 (ϑ + γ) to regularize the singularities as described in the main text.…”

“…Among these the most studied example is the periodically staggered six-vertex model whose low energy effective theory has been identified to be the SL(2, R)/U(1) black hole conformal field theory (CFT) [18,[21][22][23][24][25]. More recently, the influence of open U q (sl(2))-invariant boundary conditions in this model has been studied, both for the self-dual staggering related to the Potts model [26,27] and also away from the self-dual line [28]. These studies have shown that boundary conditions have a profound effect: depending on their choice the symmetry of the ground state may be spontaneously broken or the continuous component of the conformal spectrum disappears completely.…”

“…Its role in staggered models without a non-compact continuum limit, however, has not been studied yet. The definition of the quasi momentum relies on the possibility to introduce a staggering in the vertical direction of the vertex model which is compatible with the horizontal one [21,23,28,29]. That such an operator cannot be defined in the homogeneous case has impeded progress in the analysis of the spectrum other models where indications for a continuous spectrum of conformal weights have been observed, namely the a (2) N−1 models and a family of orthosymplectic superspin chains [30][31][32][33][34][35][36].…”