The action of the Virasoro algebra in quantum spin chains. I. The non-rational case

Abstract

We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra ("Koo-Saleur generators"[arXiv:hep-th/9312156]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [arXiv:2007.11539] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules ${\mathcal{W}}_{j,1}$ for $j\ne 0$ contains pairs of "conjugate states" with conformal weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ that give rise to dual structures: Verma or co-Verma modules. The limit of ${\mathcal{W}}_{0,{\mathfrak{q}}^{\pm 2}}$ contains diagonal fields $(h_{r,1},h_{r,1})$ and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in ${\mathfrak{q}}^{\pm 2}$. In order to obtain matrix elements of Koo-Saleur generators at large system size $N$ we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

arXiv comment61 pages, 13 figures, 12 tables v2: Minor edits. Corrected typo in eq (188)

References in corpus (36)

• Form factors of the XXZ Heisenberg spin-1/2 finite chain
• Topological Defects on the Lattice I: The Ising model
• The Blob Algebra and the Periodic Temperley-Lieb Algebra
• Associative-algebraic approach to logarithmic conformal field theories
• Long-distance asymptotics of spin-spin correlation functions for the XXZ spin chain
• On an algebraic approach to higher dimensional statistical mechanics
• Conformal Algebras of 2D Disordered Systems
• Correlation amplitude for the XXZ spin chain in the disordered regime
• Entanglement in non-unitary quantum critical spin chains
• Representations of the Virasoro algebra from lattice models
• A conformal bootstrap approach to critical percolation in two dimensions
• Three-point functions in c <= 1 Liouville theory and conformal loop ensembles
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states
• Continuum limit and symmetries of the periodic gl(1|1) spin chain
• Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap
• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• Conformal fields and operator product expansion in critical quantum spin chains
• The puzzle of bulk conformal field theories at central charge c=0
• Next Nearest-Neighbor Correlation Functions of the Spin-1/2 XXZ Chain at Critical Region
• Bimodule structure in the periodic gl(1|1) spin chain
• On Generalized $Q$-systems
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• A lattice approach to the conformal $\OSp(2S+2|2S)$ supercoset sigma model. Part I: Algebraic structures in the spin chain. The Brauer algebra
• Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• On zero-remainder conditions in the Bethe ansatz
• Logarithmic Superconformal Minimal Models
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Dimer correlation amplitudes and dimer excitation gap in spin-1/2 XXZ and Heisenberg chains
• Lattice W-algebras and logarithmic CFTs
• Scale invariant transfer matrices and Hamiltionians
• Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models
• The modified XXZ Heisenberg chain, conformal invariance, surface exponents of c<1 systems, and hidden symmetries of the finite chains
• Chiral SU(2)_k currents as local operators in vertex models and spin chains
• Bethe state norms for the Heisenberg spin chain in the scaling limit

Cited by in corpus (4)

• Conformal field theory from lattice fermions
• The Quantum Gravity Disk: Discrete Current Algebra
• On correlation functions in models related to the Temperley-Lieb algebra
• Irreducible Virasoro representations in Liouville conformal field theory