Finite size spectrum of the staggered six-vertex model with Uq( sl $$ \mathfrak{sl} $$ (2))-invariant boundary conditions

Published for SISSA by Springer Received: November 3, 2021 Revised: December 16, 2021 Accepted: December 28, 2021 Published: January 14, 2022 Holger Frahm and Sascha Gehrmann Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, Hannover 30167, Germany E-mail: , 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 noncompact 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. Keywords: Bethe Ansatz, Lattice Integrable Models, Conformal Field Theory ArXiv ePrint: 2111.00850 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2022)070 JHEP01(2022)070 Finite size spectrum of the staggered six-vertex model with Uq(sl(2))-invariant boundary conditions Contents 1 2 Definition of the model 2 3 The Hamiltonian limit 7 4 Numerical study of small number of lattice sites 12 5 The root density approach for the ground state 12 6 Analysis of the finite size spectrum 6.1 Continuous part 6.2 Discrete part 17 17 23 7 Summary 27 1 Introduction The analysis of the finite size spectra of two-dimensional vertex models and the corresponding (1 + 1)-dimensional quantum spin chains has long been used e.g. to identify the effective field theories describing the low energy behaviour of correlated many-body systems in the presence of strong quantum fluctuations. Recent studies of (super) spin chains related to network models for quantum Hall transitions, the anti-ferromagnetic Potts model, intersecting loops or two-dimensional polymers have shown that the continuum descriptions may involve conformal field theories (CFTs) with a non-compact target space leading to a continuous component to the spectrum of conformal weights in the thermodynamic limit [1–8]. In this paper we study the Z2 -staggered six-vertex model with anisotropy 0 < γ < π/2 and staggering parameter γ < α < π − γ. At the ‘self-dual’ point, α = π/2, this model is equivalent to the critical anti-ferromagnetic Potts model [9]. At low energies it can be described effectively in terms of the SL(2, R)k /U(1) sigma model, a CFT on the twodimensional Euclidean black hole background [10–12], at level k = π/γ [4, 13–16]: for periodic boundary conditions the observed finite size spectrum of the lattice model and the density of states in the continua emerging in the thermodynamic limit have been found to agree with what is known for this CFT. Moreover, the quantum number describing the states in the continuum has been related to a conserved quasi-momentum operator in the lattice model. The construction of this operator relies on the existence of the staggering of the vertex model in the vertical direction: the two-row transfer matrix of the periodic model generating conserved quantities such as the Hamiltonian factorizes into a product of two commuting single-row transfer matrices taken at spectral parameters shifted by the staggering parameter. The quasi-momentum operator, on the other hand, is obtained in an expansion of the ratio of these single-row operators. –1– JHEP01(2022)070 1 Introduction 2 Definition of the model We use the following convention for the R-matrix for the XXZ-model,   sinh (u + iγ) 0 0 0     0 sinh (u) sinh (iγ) 0  . R(u) =   0 sinh (iγ) sinh (u) 0   0 0 0 sinh (u + iγ) –2– (2.1) JHEP01(2022)070 More recently, the effect of boundary conditions on the spectrum of this model has been studied: at the self-dual point the staggered six-vertex model has been shown to (2) be related to the R-matrix of the D2 affine Lie algebra [17]. This has motivated the (2) construction of a D2 spin chain with a particular choice of integrable open boundary conditions which also possesses a continuous spectrum of conformal weights related to the (2) Euclidean black hole CFT [18]. Similar to the periodic case, the transfer matrix of this D2 model can be factorized into products of transfer matrices of the six-vertex model [19]. This (2) procedure maps the boundary terms of the D2 chain to Uq (sl(2)) quantum group invariant open boundary conditions of the six-vertex model [20, 21]. In the latter formulation an integrable model with these boundary conditions can be extended to generic values of the staggering parameter α. Furthermore, it allows for the definition of an analog of the quasi-momentum operator for the open boundary model. This turns out to be particularly useful for the identification of states from the discrete part of the CFT spectrum which are not present in the periodic model (although these states do appear under a twist, see e.g. [5, 16, 22]). Below we recall the construction of the double-row transfer matrix of the inhomogeneous six-vertex model with quantum group invariant boundary conditions and its Bethe ansatz solution for Z2 -staggered inhomogeneities ±iα/2. Introducing the same staggering in the auxiliary direction, we obtain commuting four-row transfer matrices, pairs of which can be related by a duality transformation changing the staggering parameter as α → π−α. As for the periodic staggered six-vertex model another family of commuting integrals of motion is generated by a quotient of the double-row transfer matrices. Representative members of these families are the Hamiltonian and the so-called quasi-momentum operator of the staggered XXZ spin chain constructed in section 3. The Temperley-Lieb representation of the Hamiltonian and its relation to other models in certain limiting cases is shown. Based on our numerical diagonalization of the Hamiltonian and the quasi-momentum operator for small lattice sizes, we identify the solutions of the Bethe equations relevant for the low energy part of the spectrum. Using the root density formalism [23] the ground state of the system in the thermodynamic limit is characterized. For the analysis of the finite size spectrum, we solve the Bethe equations numerically for large system sizes. This uncovers the role of the quasi-momentum in the characterization of the continuous part of the conformal spectrum and the emergence of discrete states as the anisotropy γ is varied. The paper ends with a summary of our findings. This four by four matrix can be interpreted as an oper (...truncated)