Given a critical quantum spin chain, we show how universal information about its quantum critical point can be extracted from wavefunction overlaps. More specifically, we consider overlap between low-energy eigenstates of the spin chain Hamiltonian with different boundary conditions, namely periodic boundary conditions and open boundary conditions. We show that such overlaps decay polynomially with the system size, where the exponent only depends on the central charge. Furthermore, the bulk-to-boundary operator product expansion (OPE) coefficients can be extracted from the overlaps involving excited states. We illustrate the proposal with the Ising model and the three-state Potts model.
Given a critical quantum spin chain, we show how universal information about its quantum critical point can be extracted from wavefunction overlaps. More specifically, we consider overlap between low-energy eigenstates of the spin chain Hamiltonian with different boundary conditions, namely periodic boundary conditions and open boundary conditions. We show that such overlaps decay polynomially with the system size, where the exponent only depends on the central charge. Furthermore, the bulk-to-boundary operator product expansion (OPE) coefficients can be extracted from the overlaps involving excited states. We illustrate the proposal with the Ising model and the three-state Potts model.
Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c\ensuremath{\approx}2.1$ and the lowest spin-1 operator in the twisted sector has scaling dimension $1<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.4$.
Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.
Universal properties of a critical quantum spin chain are encoded in the underlying conformal field theory (CFT). This underlying CFT is fully characterized by its conformal data. We propose a method to extract the conformal data from a critical quantum spin chain with both periodic and antiperiodic boundary conditions (PBC and APBC) based on low-energy eigenstates, generalizing previous work on spin chains with only PBC. First, scaling dimensions and conformal spins are extracted from the energies and momenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasoro generators is generalized to APBC and used to identify conformal towers. Third, local and string operators on the lattice are identified with CFT operators with PBC and APBC, respectively. Finally, operator product expansion coefficients are extracted by computing matrix elements of lattice primary operators in the low-energy subspaces with PBC and APBC. To go beyond exact diagonalization, tensor network methods based on periodic uniform matrix product states are used. We illustrate our approach with critical and tricritical Ising quantum spin chains. In the latter case, we propose lattice operators that correspond to super-Virasoro generators and verify their action on low-energy eigenstates. In this way, we explore the emergence of superconformal symmetry in the quantum spin chain.
Boundary conformal field theory (BCFT) contains universal data that are usually accessed microscopically by imposing spatial boundaries on the lattice. In non-Hermitian many-body systems, however, changing boundary conditions can qualitatively reorganize spectra and eigenstates, and their boundary criticality remains elusive. Here, we introduce a periodic-chain spectroscopy to extract universal boundary quantities, such as the Affleck-Ludwig g-factor, directly from non-Hermitian bulk-critical quantum chains, avoiding the need to engineer microscopic open boundaries and circumventing subtle boundary effects in non-Hermitian systems. We illustrate our method with a PT-symmetric Ising realization of the real nonunitary Yang-Lee CFT and reveal a universal negative excited-to-ground ratio, which has no counterpart in unitary critical theories and provides a microscopic signature of nonunitarity. For the genuinely complex fixed points of the non-Hermitian five-state Potts chain, we extract intrinsically complex boundary coefficients, verify the exact Kramers-Wannier duality relation, and select the consistent analytic-continuation branch of the boundary states. Our results establish a route to nonunitary BCFT universal data using only knowledge of the bulk critical system, opening a window into non-Hermitian boundary criticality.
At a quantum critical point, the low-energy physics of a quantum spin chain is described by conformal field theory (CFT). Given the Hamiltonian of a critical spin chain, we propose a variational method to build an approximate lattice representation ϕ_{α} of the corresponding primary CFT operators ϕ_{α}^{CFT}. We then show how to numerically compute the operator product expansion coefficients C_{αβγ}^{CFT} governing the fusion of two primary fields. In this way, we complete the implementation of Cardy's program, outlined in the 1980s, which aims to extract the universality class of a phase transition, as encoded in the so-called conformal data of the underlying CFT, starting from a microscopic description. Our approach, demonstrated here for the critical quantum Ising model, only requires a generic (i.e., in general, nonintegrable) critical lattice Hamiltonian as its input.
Some quantum critical states cannot be smoothly deformed into each other without either crossing some multicritical points or explicitly breaking certain symmetries even if they belong to the same universality class. This brings up the notion of "symmetry-enriched" quantum criticality. While recent works in the literature focused on critical states with robust degenerate edge modes, we propose that the conformal boundary condition (B.C.) is a more generic characteristic of such quantum critical states. We show that in two families of quantum spin chains, which generalize the Ising and the three-state Potts models, the quantum critical point between a symmetry-protected topological phase and a symmetry-breaking order realizes a conformal B.C. distinct from the simple Ising and Potts chains. Furthermore, we argue that the conformal B.C. can be derived from the bulk effective field theory, which realizes a novel bulk-boundary correspondence in symmetry-enriched quantum critical states.