Abstract In Can et al (2016 Phys. Rev. Lett. 117 ), quantum Hall states on singular surfaces were shown to possess an emergent conformal symmetry. In this paper, we develop this idea further and flesh out details on the emergent conformal symmetry in holomorphic adiabatic states, which we define in the paper. We highlight the connection between the universal features of geometric transport of quantum Hall states and holomorphic dimension of primary fields in conformal field theory. In parallel we compute the universal finite-size corrections to the free energy of a critical system on a hyperbolic sphere with conical and cusp singularities, thus extending the result of Cardy and Peschel for critical systems on a flat cone (Cardy and Peschel 1988 Nucl. Phys . B 300 377–92), and the known results for critical systems on polyhedra and flat branched Riemann surfaces.
Abstract In Can et al (2016 Phys. Rev. Lett. 117 ), quantum Hall states on singular surfaces were shown to possess an emergent conformal symmetry. In this paper, we develop this idea further and flesh out details on the emergent conformal symmetry in holomorphic adiabatic states, which we define in the paper. We highlight the connection between the universal features of geometric transport of quantum Hall states and holomorphic dimension of primary fields in conformal field theory. In parallel we compute the universal finite-size corrections to the free energy of a critical system on a hyperbolic sphere with conical and cusp singularities, thus extending the result of Cardy and Peschel for critical systems on a flat cone (Cardy and Peschel 1988 Nucl. Phys . B 300 377–92), and the known results for critical systems on polyhedra and flat branched Riemann surfaces.
Abstract We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.
We study the emergence of 2D conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. Using only the lattice Hamiltonian H=∑jhj as an input, we construct operators Hn (Fourier modes of the local Hamiltonian terms hj) that transform the low-energy eigenstates of H in the same way as certain combinations of the Virasoro generators do in the CFT. These operators were introduced for certain integrable lattice models by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we propose and demonstrate the use of operators Hn to extract conformal data in nonintegrable quantum spin chains. In particular, we explain how to estimate the central charge from a simple ground-state expectation value and to systematically identify the low-energy states of H corresponding to Virasoro primary operators, as must be done in order to identify the CFT.
This note presents a purely geometric construction of the so-called twist-field correlation functions in Conformal Field Theory (CFT), derived from conical singularities. This approach provides a purely mathematical interpretation of the seminal results in physics by Cardy and Calabrese on the entanglement entropy of quantum systems. Specifically, we begin by defining CFT partition functions on surfaces with conical singularities, using a ``Cauchy-Hadamard renormalization'' of the Polyakov anomaly integral. Next, we demonstrate that for a branched cover f:Σd→Σ with d sheets, where the cover inherits the pullback of a smooth metric from the base, a specific ratio of partition functions on the cover to the base transforms under conformal changes of the base metric in the same way as a correlation function of CFT primary fields with specific conformal weights. We also provide a discussion of the physical background and motivation for entanglement entropy, focusing on path integrals and the replica trick, which serves as an introduction to these ideas for a mathematical audience.
We numerically obtain the conformal spectrum of several classical spin models on a two-dimensional lattice with open boundaries for every boundary fixed point obtained by the Cardy's derivation [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)]. In order to extract accurate conformal data, we implement the tensor network renormalization algorithm [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] extended so as to be applicable to a square lattice with open boundaries. We successfully compute the boundary conformal spectrum consistent with the underlying boundary conformal field theories (BCFTs) for the Ising, tricritical Ising, and three-state Potts models on the lattice, which allows us to confirm the validity of the BCFT analyses for the surface critical behaviors of those lattice models.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, RP2). Through a cut-and-sew process, RP2 is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term 21lnk related to a nonorientable genus, with k a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and c the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.