Symmetry-resolved entanglement entropy provides a powerful framework for probing the internal structure of quantum many-body states by decomposing entanglement into contributions from distinct symmetry sectors. In this work, we apply matrix product state techniques to study the bosonic, non-Abelian Moore-Read quantum Hall state, enabling precise numerical evaluation of both the full counting statistics and symmetry-resolved entanglement entropies. Our results reveal an approximate equipartition of entanglement among symmetry sectors, consistent with theoretical expectations and subject to finite-size corrections. The results also show that these expectations for symmetry-resolved entanglement entropy remain valid in the case of a non-Abelian state where the topological sectors cannot be distinguished by the Abelian U(1) symmetry alone, and where neutral and charged modes possess distinct velocities. We additionally perform a detailed comparison of the entanglement spectrum with predictions from the Li-Haldane conjecture, finding remarkable agreement, and enabling a more precise understanding of the effects of the distinct neutral and charged velocities. This not only provides a stringent test of the conjecture but also highlights its explanatory power in understanding the origin and structure of finite-size effects across different symmetry sectors.

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:Σ_d→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 demonstrate that the widely used plasma analogy is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a fundamental difference between quantum Hall droplets and plasmas (Coulomb gases): the former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal transformations. Consequently, the plasma analogy fails at the edge, except in fine-tuned geometries, as it does not account for the emergent local edge velocity. We quantitatively show how the analogy's failure affects physical quantities, such as fluctuations of local observables and absorption rates in microwave spectroscopy, measurable in both solid-state experiments and quantum simulators.

Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this paper, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction 1/2, by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.

We present a general method for calculating R\'enyi entropies in the ground state of a one-dimensional critical system with mixed open boundaries, for an interval starting at one of its ends. In the conformal field theory framework, this computation boils down to the evaluation of the correlation function of one twist field and two boundary condition changing operators in the cyclic orbifold. Exploiting null-vectors of the cyclic orbifold, we derive ordinary differential equations satisfied by these correlation functions. In particular, we obtain an explicit expression for the second R\'enyi entropy valid for any diagonal minimal model, but with a particular set of mixed boundary conditions. In order to compare our results with numerical data for the Ising and three-state Potts critical chains, we also identify and compute the leading finite size corrections.

We study two-dimensional (2D) droplets of noninteracting electrons in a strong magnetic field, placed in a confining potential with arbitrary shape. Using semiclassical methods adapted to the lowest Landau level, we obtain near-Gaussian energy eigenstates that are localized on level curves of the potential and have a position-dependent height. This one-particle insight allows us to deduce explicit formulas for expectation values of local many-body observables, such as density and current, in the thermodynamic limit. In particular, correlations along the edge are long-ranged and inhomogeneous. As we show, this is consistent with the system’s universal low-energy description as a free 1D chiral conformal field theory of edge modes, known from earlier works in simple geometries. A delicate interplay between radial and angular dependencies of eigenfunctions ultimately ensures that the theory is homogeneous in terms of the canonical angle variable of the potential, despite its apparent inhomogeneity in terms of more naïve angular coordinates. Finally, we propose a scheme to measure the anisotropy by subjecting the droplet to microwave radiation; we compute the corresponding absorption rate and show that it depends on the droplet’s shape and the waves’ polarization. These results, both local and global, are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential.

We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.

Outcomes of measurements are characterized by an infinite family of generalized uncertainties, or cumulants, which provide information beyond the mean and variance of the observable. Here, we investigate the cumulants of a conserved charge in a subregion with corners. We derive nonperturbative relations for the area law, and more interestingly, the angle dependence, showing how it is determined by geometric moments of the correlation function. These hold for translation invariant systems under great generality, including strongly interacting ones. We test our findings by using two-dimensional topological quantum Hall states of bosons and fermions at both integer and fractional fillings. We find that the odd cumulants' shape dependence differs from the even ones. For instance, the third cumulant shows nearly universal behavior for integer and fractional Laughlin Hall states in the lowest Landau level. Furthermore, we examine the relation between even cumulants and the R\'enyi entanglement entropy, where we use new results for the fractional state at filling 1/3 to compare these quantities in the strongly interacting regime. We discuss the implications of these findings for other systems, including gapless Dirac fermions, and more general conformal field theories.

We prove that all the criteria proposed in the literature to identify a Chern band hosting exact fractional Chern insulating ground states, in fact, describe an equivalence with a lowest Landau level defined in curved space under a non-uniform magnetic field. In addition, we design an operational test for the most general instance of such lowest Landau level mapping, which only relies on the computationally inexpensive evaluation of Bloch wavefunctions' derivatives. Our work clarifies the common origin of various Chern-idealness criteria, proves that these criteria exhaust all possible lowest Landau levels, and hints at classes of Chern bands that may posses interesting phases beyond Landau level physics.

We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The operator content and operator product expansion of the cyclic orbifolds are revisited in terms of this algebra. The fusion rules and fusion numbers are computed via the Verlinde formula. This allows one to predict which conformal blocks appear in a given four-point function of twisted or untwisted operators, which is relevant for the computation of various entanglement measures in one-dimensional critical systems.

We consider area-preserving deformations of the plane, acting on electronic wave functions through 'quantomorphisms' that change both the underlying metric and the confining potential. We show that adiabatic sequences of such transformations produce Berry phases that can be written in closed form in terms of the many-body current and density, even in the presence of interactions. For a large class of deformations that generalize squeezing and shearing, the leading piece of the phase is a super-extensive AharonovBohm term (proportional to N2 for N electrons) in the thermodynamic limit. Its gauge-invariant subleading partner only measures the current, whose dominant contribution to the phase stems from a jump at the edge in the limit of strong magnetic fields. This results in a finite Berry curvature per unit area, reminiscent of the Hall viscosity. We show that the latter is in fact included in our formalism, bypassing its standard derivation on a torus and suggesting realistic experimental setups for its observation in quantum simulators.

We study the cumulants of the charge distribution of a subregion for two-dimensional quantum Hall states of bosons and fermions at both integer and fractional fillings, focusing on subregions with corners. Even cumulants, which include the variance, satisfy an area law with subleading corrections sensitive to finer geometric details of the subregion such as corner contributions, while at the same time probing non-trivial sum rules for general correlation functions. We perform a systematic study of such corner terms, by a combination of analytic, numerically exact, and Monte Carlo computations. We also study odd charge cumulants, for which the area-law term vanishes and distinct corner contributions appear. The observed shape dependence of the third cumulant shows nearly universal behavior for integer and fractional Laughlin Hall states in the lowest Landau level. While these states serve as our main example, many of our finding are expected to hold in considerable generality. As an illustration, we discuss properties of gapless Dirac fermions, and more general conformal field theories.

Understanding the fluctuations of observables is one of the main goals in science, be it theoretical or experimental, quantum or classical. We investigate such fluctuations in a subregion of the full system, focusing on geometries with sharp corners. We report that the angle dependence is super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. The prefactor contains important physical information: we show in particular that it gives access to the long-wavelength limit of the structure factor. We exemplify our findings with fractional quantum Hall states, topological insulators, scale invariant quantum critical theories, and metals. We suggest experimental tests, and anticipate that our findings can be generalized to other spatial dimensions or geometries. In addition, we highlight the similarities of the fluctuation shape dependence with findings relating to quantum entanglement measures.

We consider the entanglement entropy in critical one-dimensional quantum systems with open boundary conditions. We show that the second Rényi entropy of an interval away from the boundary can be computed exactly, provided the same conformal boundary condition is applied on both sides. The result involves the annulus partition function. We compare our exact result with numerical computations for the critical quantum Ising chain with open boundary conditions. We find excellent agreement, and we analyse in detail the finite-size corrections, which are known to be much larger than for a periodic system.

We study the full counting statistics (FCS) and symmetry-resolved entanglement entropies of integer and fractional quantum Hall states. For the filled lowest Landau level of spin-polarized electrons on an infinite cylinder, we compute exactly the charged moments associated with a cut orthogonal to the cylinder's axis. This yields the behavior of FCS and entropies in the limit of large perimeters: in a suitable range of fluctuations, FCS is Gaussian and entanglement spreads evenly among different charge sectors. Subleading charge-dependent corrections to equipartition are also derived. We then extend the analysis to Laughlin wave functions, where entanglement spectroscopy is carried out assuming the Li-Haldane conjecture. The results confirm equipartition up to small charge-dependent terms, and are then matched with numerical computations based on exact matrix product states.

The gapless modes on the edge of four-dimensional (4D) quantum Hall droplets are known to be anisotropic: they only propagate in one direction, foliating the 3D boundary into independent 1D conduction channels. This foliation is extremely sensitive to the confining potential and generically yields chaotic flows. Here we study the quantum correlations and entanglement of such edge modes in 4D droplets confined by harmonic traps, whose boundary is a squashed three-sphere. Commensurable trapping frequencies lead to periodic trajectories of electronic guiding centers; the corresponding edge modes propagate independently along S-1 fibers, forming a bundle of 1D conformal field theories over a 2D base space. By contrast, incommensurable frequencies produce quasi-periodic, ergodic trajectories, each of which covers its invariant torus densely; the corresponding correlation function of edge modes has fractal features. This wealth of behaviors highlights the sharp differences between 4D Hall droplets and their 2D peers; it also exhibits the dependence of 4D edge modes on the choice of trap, suggesting the existence of observable bifurcations due to droplet deformations.

In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay algebraically with system size, with exponents related to the operators' scaling dimensions. In contrast, in the case of a U(1) symmetry group, the corrections only decay logarithmically with system size, with model-dependent prefactors. We show that the determination of these prefactors boils down to the computation of twisted overlaps.

We investigate the entanglement entropy (EE) and charge fluctuations in models where the low-energy physics is governed by massless Dirac fermions. We focus on the response to flux insertion which, for the EE, is widely assumed to be universal, i.e., independent of the microscopic details. We provide an analytical derivation of the EE and charge fluctuations for the seminal example of graphene, using the dimensional reduction of its tight-binding model to the one-dimensional Su-Schrieffer-Heeger model. Our asymptotic expression for the EE matches the conformal field theory prediction. We show that the charge variance has the same asymptotic behavior, up to a constant prefactor. To check the validity of universality arguments, we numerically consider several models, with different geometries and numbers of Dirac cones, and either for strictly two-dimensional models or for a gapless surface mode of three-dimensional topological insulators. We also show that the flux response does not depend on the entangling surface geometry as long as it encloses the flux. Finally we consider the universal corner contributions to the EE. We show that in the presence of corners, the Kitaev-Preskill subtraction scheme provides nonuniversal, geometry-dependent results.

Coupled-wire constructions offer particularly simple and powerful models to capture the essence of strongly correlated topological phases of matter. They often rely on effective theories valid in the low-energy and strong-coupling limits, which impose severe constraints on the physical systems where they could be realized. We investigate the microscopic relevance of a class of coupled-wire models and their possible experimental realization in cold-atom experiments. We connect with earlier results and prove the emergence of fractional quantum Hall states in the limit of strong interwire tunneling. Contrary to previous studies relying on renormalization group arguments, our microscopic approach exposes the connection between coupled-wire constructions and model wave functions in continuum Landau levels. Then, we use exact-diagonalization methods to investigate the appearance of these fractional quantum Hall states in more realistic settings. We examine the parameter regimes where these strongly correlated phases arise, and provide a way to detect their appearance in cold-atom experiments through standard time-of-flight measurements. Motivated by this experimental probe, we finally propose a realization of our model with cold atoms in spin-dependent optical lattices. Our estimates show that the previous fractional quantum Hall phases lie within experimentally accessible parameter regimes, giving a viable route toward their experimental study.

We investigate the entanglement entropy in the Integer Quantum Hall effect in the presence of an edge, performing an exact calculation directly from the microscopic two-dimensional wavefunction. The edge contribution is shown to coincide exactly with that of a chiral Dirac fermion, and this correspondence holds for an arbitrary collection of intervals. In particular for a single interval the celebrated conformal formula is recovered with left and right central charges $c+\bar{c} =1$. Using Monte-Carlo techniques we establish that this behavior persists for strongly interacting systems such as Laughlin liquids. This illustrates how entanglement entropy is not only capable of detecting the presence of massless degrees of freedom, but also of pinpointing their position in real space, as well as elucidating their nature.