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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 letter, 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 averaged 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., Comment: 4+8 pages, 3+3 figures

Measurements profoundly impact quantum systems, and can be used to create new states of matter out of equilibrium. Here, we investigate the multipartite entanglement structure that emerges in quantum circuits involving unitaries and measurements. We describe how a balance between measurements and unitary evolution can lead to multipartite entanglement spreading to distances far greater than what is found in non-monitored systems, thus evading the usual fate of entanglement. We introduce a graphical representation based on spanning graphs that allows to infer the evolution of genuine multipartite entanglement for general subregions. We exemplify our findings on circuits that realize a 1d measurement-induced dynamical phase transition, where we find genuine 3-party entanglement at all separations. The 2- and 4-party cases are also covered with examples. Finally, we discuss how our approach can provide fundamental insights regarding entanglement dynamics for a wide class of quantum circuits and architectures., Comment: 5+2 pages, 4+2 figures

We develop a protocol, dubbed entanglement microscopy, to obtain the full reduced density matrix associated with subregions in quantum Monte Carlo simulations for bosonic and fermionic manybody systems. Our microscopy allows to perform quantum state tomography, and thus gives access to true entanglement measures, such as the logarithmic negativity (LN). We exemplify our method by studying the phase diagram near quantum critical points (QCP) in 2 spatial dimensions: the transverse field Ising model and a Gross-Neveu-Yukawa transition of Dirac fermions. Our main results are: i) the Ising QCP exhibits short-range entanglement with a finite sudden death of the LN both in space and temperature; ii) the Gross-Neveu QCP has a power-law decaying fermionic LN consistent with conformal field theory (CFT) exponents; iii) going beyond bipartite entanglement, we find no detectable 3-party entanglement in a large parameter window near the Ising QCP in 2d, in contrast to 1d. We also analytically obtain the large-temperature power law scaling of the fermionic LN for general interacting systems. Our approach allows one to perform quantum state tomography locally in a way that is analogous to atomic-scale imaging with a scanning tunneling microscope. Controlled entanglement microscopy opens a new window into quantum matter, with countless systems waiting to be explored., Comment: 8+8 pages, 4+11 figures

Quantum entanglement is a fundamentally non-local correlation between particles. In its simplest realisation, a measurement on one particle is affected by a prior measurement on its partner, irrespective of their separation. For multiple particles, purely collective types of entanglement exist but their detection, even theoretically, remains an outstanding open question. Here, we show that all forms of multipartite entanglement entirely disappear during the typical evolution of a system as it heats up, evolves in time, or as its parts become separated. These results follow from the nature of the entanglement-free continent in the space of physical states, and hold in great generality. We illustrate these phenomena with a frustrated molecular quantum magnet in and out of equilibrium. In contrast, if the particles are fermions, such as electrons, another notion of entanglement exists that protects bipartite quantum correlations. However, truly collective fermionic entanglement disappears during typical evolution, thus sharing the same fate as in bosonic systems. These findings provide fundamental knowledge about the structure of entanglement in quantum matter and architectures, paving the way for its manipulation., Comment: 15+7 pages single column, 3+3 figures, v2: Improved discussion and SM, v3: New results for fermion genuine multipartite entanglement

We study the properties of 2+1d conformal field theories (CFTs) in a background magnetic field. Using generalized particle-vortex duality, we argue that in many cases of interest the theory becomes gapped, which allows us to make a number of predictions for the magnetic response, background monopole operators, and more. Explicit calculations at large N for Wilson-Fisher and Gross-Neveu CFTs support our claim, and yield the spectrum of background (defect) monopole operators. Finally, we point out that other possibilities exist: certain CFTs can become metallic in a magnetic field. Such a scenario occurs for a Dirac fermion coupled to a Chern-Simons gauge field, where a non-Fermi liquid is argued to emerge., Comment: 8+10 pages

We study the entanglement between disjoint subregions in quantum critical systems through the lens of the logarithmic negativity. We work with systems in arbitrary dimensions, including conformal field theories and their corresponding lattice Hamiltonians, as well as resonating valence-bond states. At small separations, the logarithmic negativity is big and displays universal behavior, but we show non-perturbatively that it decays faster than any power at large separations. This can already be seen in the minimal setting of single-spin subregions. The corresponding absence of distillable entanglement at large separations generalizes the 1d result, and indicates that quantum critical groundstates do not possess long-range bipartite entanglement, at least for bosons. For systems with fermions, a more suitable definition of the logarithmic negativity exists that takes into account fermion parity, and we show that it decays algebraically. Along the way we obtain general results for the moments of the partially transposed density matrix., Comment: 7+4 pages. v2: Minor improvements and additions in the SM. v3: Title change, improvement of the discussion and broader scope. v4: Published version, new title

We grow the web of dualities for conformal field theories (CFTs) in 2+1 spacetime dimensions to include quantum critical transitions of Dirac fermions. Our construction uses the seed duality, equating a free Dirac fermion with a complex boson coupled to a Chern-Simons gauge field, to express various Gross-Neveu-Yukawa (GNY) critical points of N fermions in terms of the same number of complex bosons and partner gauge fields. In some instances, monopole operators appear explicitly in the dual action, such as for the superconducting transition of a single Dirac fermion, which is a CFT with emergent supersymmetry. We match phase diagrams, symmetries and their anomalies. By further gauging certain symmetries, we derive offspring dualities. For example, starting with the duality for the regular GNY transition of 2 Dirac fermions, we arrive at the duality between a Neel-to-VBS deconfined quantum critical point and a gauged-Gross-Neveu transition. Finally, we use a large-N expansion for monopole operators, and compare with known scaling dimensions in the simplest single-flavour GNY transition., Comment: 11+2 pages, single-column

We investigate separability and entanglement of Rokhsar-Kivelson (RK) states and resonating valence-bond (RVB) states. These states play a prominent role in condensed matter physics, as they can describe quantum spin liquids and quantum critical states of matter, depending on their underlying lattices. For dimer RK states on arbitrary tileable graphs, we prove the exact separability of the reduced density matrix of $k$ disconnected subsystems, implying the absence of bipartite and multipartite entanglement between the subsystems. For more general RK states with local constraints, we argue separability in the thermodynamic limit, and show that any local RK state has zero logarithmic negativity, even if the density matrix is not exactly separable. In the case of adjacent subsystems, we find an exact expression for the logarithmic negativity in terms of partition functions of the underlying statistical model. For RVB states, we show separability for disconnected subsystems up to exponentially small terms in the distance $d$ between the subsystems, and that the logarithmic negativity is exponentially suppressed with $d$. We argue that separability does hold in the scaling limit, even for arbitrarily small ratio $d/L$, where $L$ is the characteristic size of the subsystems. Our results hold for arbitrary lattices, and encompass a large class of RK and RVB states, which include certain gapped quantum spin liquids and gapless quantum critical systems., Comment: 18 pages, 8 figures, v2: new discussion on multipartite entanglement and separability, v3: minor modifications

Multi-orbital quantum materials with strong interactions can host a variety of novel phases. In this work we study the possibility of interaction-driven superconductivity in the iridate compound Sr$_2$IrO$_4$ under strain and doping. We find numerous regimes of strain-induced superconductivity in which the pairing structure depends on model parameters. Spin-fluctuation mediated superconductivity is modeled by a Hubbard-Kanamori model with an effective particle-particle interaction, calculated via the random phase approximation. Magnetic orders are found using the Stoner criterion. The most likely superconducting order we find has $d$-wave pairing, predominantly in the total angular momentum, $J=1/2$ states. Moreover, an $s_{\pm}$-order which mixes different bands is found at high Hund's coupling, and at high strain anisotropic $s$- and $d$-wave orders emerge. Finally, we show that in a fine-tuned region of parameters a spin-triplet $p$-wave order exists. The combination of strong spin-orbit coupling, interactions, and a sensitivity of the band structure to strain proves a fruitful avenue for engineering new quantum phases., Comment: 26 pages, 21 figures

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., Comment: 5+10 pages, 3+4 figures; v2: title changed to reflect the generality of our results, manuscript reorganized and discussions improved

We give evidence for 3d bosonization in Conformal Field Theories (CFTs) by computing monopole operator scaling dimensions in 2+1 dimensional quantum electrodynamics (QED3) with Chern-Simons level $k$ and $N$ complex bosons in a large $N,k$ expansion. We first consider the $k=0$ case, where we show that scaling dimensions previously computed to subleading order in $1/N$ can be extrapolated to $N=1$ and matched to $O(2)$ Wilson-Fisher CFT scaling dimensions with around 5\% error, which is evidence for particle-vortex duality. We then generalize the subleading calculation to large $N,k$ and fixed $k/N$, extrapolate to $N=k=1$, and consider monopole operators that are conjectured to be dual to non-degenerate scalar operators in a theory of a single Dirac fermion. We find matches typically with 1\% error or less, which is strong evidence of this so-called `seed' duality that implies a web of 3d bosonization dualities among CFTs., Comment: 4 pages plus appendices, no figures. v2 minor typos corrected, submitted for publication

We study two entanglement measures in a large family of systems including incompressible quantum Hall states: the logarithmic negativity (LN), and mutual information (MI). For pure states, obtained for example from a bipartition at zero temperature, these provide distinct characterizations of the entanglement present between two spatial subregions, while for mixed states (such as at finite temperature) only the LN remains a good entanglement measure. Our focus is on regions that have corners, either adjacent or tip-touching. We first obtain non-perturbative properties regarding the geometrical dependence of the LN and MI in a large family of isotropic states, including fractional quantum Hall states. A close similarity is observed with mutual charge fluctuations, where super-universal angle dependence holds. For the MI, we make stronger statements due to strong subadditivity. We also give ramifications of our general analysis to conformal field theories (CFTs) in two spatial dimensions. We then explicitly verify these properties with integer quantum Hall states. To do so we develop two independent approaches to obtain the fermionic LN, which takes into account Fermi statistics: an overlap-matrix method, and a real-space lattice discretization. At finite temperature, we find a rapid decrease of the LN well inside the cyclotron gap at integer fillings. We further show that the LN decays faster compared to the MI at high temperatures., Comment: 15+7 pages, 16+2 figures

The thermal Hall effect, which arises when heat flows transverse to an applied thermal gradient, has become an important observable in the study of quantum materials. Recent experiments found a large thermal Hall conductivity $\kappa_{xy}$ in many high-temperature cuprate superconductors, including deep inside the Mott insulator, but the underlying mechanism remains unknown. Here, we uncover a surprising linear temperature dependence for the inverse thermal Hall resistivity, $1/\rho_H=-\kappa_{xx}^2/\kappa_{xy}$, in the Mott insulating cuprates $\mathrm{La_2CuO_4}$ and $\mathrm{Sr_2CuO_2Cl_2}$. We also find this linear scaling in the pseudogap state of Nd-LSCO in the out-of-plane direction, highlighting the importance of phonons. On the electron-doped side, the linear inverse thermal Hall signal emerges in NCCO and PCCO at various dopings, including in the strange metal. Although such dependence arises in the simple Drude model for itinerant electrons, its origin is unclear in strongly correlated Mott insulating or pseudogap states. We perform a Boltzmann analysis for phonons that incorporates skew-scattering, and we are able to identify regimes where a linear $T$ inverse Hall resistivity appears. Finally, we suggest future experiments that would further our fundamental understanding of heat transport in the cuprates, and other quantum materials., Comment: 6+5 pages

We show how the tracer motion of tagged, distinguishable particles can effectively describe transport in various homogeneous quantum many-body systems with constraints. We consider systems of spinful particles on a one-dimensional lattice subjected to constrained spin interactions, such that some or even all multipole moments of the effective spin pattern formed by the particles are conserved. On the one hand, when all moments - and thus the entire spin pattern - are conserved, dynamical spin correlations reduce to tracer motion identically, generically yielding a subdiffusive dynamical exponent $z=4$. This provides a common framework to understand the dynamics of several constrained lattice models, including models with XNOR or $tJ_z$ - constraints. We consider random unitary circuit dynamics with such a conserved spin pattern and use the tracer picture to obtain exact expressions for their late-time dynamical correlations. Our results can also be extended to integrable quantum many-body systems that feature a conserved spin pattern but whose dynamics is insensitive to the pattern, which includes for example the folded XXZ spin chain. On the other hand, when only a finite number of moments of the pattern are conserved, the dynamics is described by a convolution of the internal hydrodynamics of the spin pattern with a tracer distribution function. As a consequence, we find that the tracer universality is robust in generic systems if at least the quadrupole moment of the pattern remains conserved. In cases where only total magnetization and dipole moment of the pattern are constant, we uncover an intriguing coexistence of two processes with equal dynamical exponent but different scaling functions, which we relate to phase coexistence at a first order transition., Comment: 17 pages, 10 figures

The entanglement entropy (EE) encodes key properties of quantum many-body systems. It is usually calculated for subregions of finite volume (or area in 2d). In this work, we study the EE of skeletal regions that have \textit{no} volume, such as a line in 2d. We show that skeletal entanglement displays new behavior compared to its bulk counterpart, and leads to distinct universal quantities. We provide non-perturbative bounds for the skeletal area-law coefficient of a large family of quantum states. We then explore skeletal scaling for the toric code, conformal bosons and Dirac fermions, Lifshitz critical points, and Fermi liquids. We discover signatures including skeletal topological EE, novel corner terms, and strict area-law scaling for metals. These findings suggest that skeletal entropy serves as a measure for the range of entanglement. We discuss the possibility of a continuum description involving the fusion of defect operators. Finally, we outline open questions relating to other systems, and measures such as the logarithmic negativity., Comment: 5+8 pages, 3+2 figures; v2: discussion and presentation of our results improved; v3: match published version

We study a vast family of continuum Rokhsar-Kivelson (RK) states, which have their groundstate encoded by a local quantum field theory. These describe certain quantum magnets, and are also important in quantum information. We prove the separability of the reduced density matrix of two disconnected subsystems, implying the absence of entanglement between the two subsystems -- a stronger statement than the vanishing of logarithmic negativity. As a particular instance, we investigate the case where the groundstate is described by a relativistic boson, which is relevant for certain magnets or Lifshitz critical points with dynamical exponent $z=2$, and we propose nontrivial deformations that preserve their RK structure. Specializing to 1D systems, we study a deformation that maps the groundstate to the quantum harmonic oscillator, leading to a gap for the boson. We study the resulting correlation functions, and find that cluster decomposition is restored. We analytically compute the $c$-function for the entanglement entropy along a renormalization group flow for the wavefunction, which is found to be strictly decreasing as in CFTs. Finally, we comment on the relations to certain stoquastic quantum spin chains. We show that the Motzkin and Fredkin chains possess unusual entanglement properties not properly captured by previous studies., Comment: 15+5 pages, 10 figures; v2: changed title, manuscript reorganized, discussions improved; v3: match published version

Monopole operators are studied in a large family of quantum critical points between Dirac and topological quantum spin liquids (QSLs): chiral and Z$_{2}$ QSLs. These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2+1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction. For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED$_{3}$-GN), while for the transitions to Z$_{2}$ QSLs it is a superconducting pairing term with general spin/valley structure (generalized QED$_{3}$-Z$_{2}$GN). Using the state-operator correspondence, we obtain monopole scaling dimensions to sub-leading order in 1/N. For monopoles with a minimal topological charge q=1/2, the scaling dimension is 2N*0.26510 at leading-order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z$_{2}$ case (the expression is also given for a general pairing term). Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins. The analogous result in QED$_{3}$ is also obtained and we find a sub-leading contribution of -0.038138(5), which differs slightly from the value first obtained in the literature. The scaling dimension of a QED$_{3}$-GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED$_{3}$-GN with 2N=2 flavors and the CP$^{1}$ model. Additionally, non-minimally charged monopoles on both sides of the duality have similar scaling dimensions. By studying the large-q asymptotics of the scaling dimensions in QED$_{3}$, QED$_{3}$-GN, and QED$_{3}$-Z$_{2}$GN we verify that the constant O(q$^{0}$) coefficient precisely matches the universal non-perturbative prediction for CFTs with a global U(1) symmetry., Comment: 37 pages

We study a quantum spin-1/2 chain that is dual to the canonical problem of non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain (dubbed Ising-Kawasaki) is stoquastic, and depends on the Ising couplings normalized by the bath's temperature. We give its exact ground states. Proceeding with uniform couplings, we study the one- and two-magnon excitations. Solutions for the latter are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding their hybridization with the continuum. We find that that the gapless chain hosts multiple dynamics at low energy as seen through the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic., Comment: 11+3 pages, 7+2 figures. v3: Minor changes including more detailed description of low energy excited states

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 when only a subregion of the full system can be observed, focusing on geometries with sharp corners. We report that the dependence on the opening angle 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 illustrate our findings with several examples, including fractional quantum Hall states, scale invariant quantum critical theories, and metals. Finally, we discuss connections with quantum entanglement, extensions to three dimensions, as well as experiments to probe the geometry of fluctuations., Comment: 7+7 pages, 2+0 figures. v2: New results regarding the volume and area law terms for general geometries and dimensions

Quantum spin liquids host novel emergent excitations, such as monopoles of an emergent gauge field. Here, we study the hierarchy of monopole operators that emerges at quantum critical points (QCPs) between a two-dimensional Dirac spin liquid and various ordered phases. This is described by a confinement transition of quantum electrodynamics in two spatial dimensions (QED3 Gross-Neveu theories). Focusing on a spin ordering transition, we get the scaling dimension of monopoles at leading order in a large-N expansion, where 2N is the number of Dirac fermions, as a function of the monopole's total magnetic spin. Monopoles with a maximal spin have the smallest scaling dimension while monopoles with a vanishing magnetic spin have the largest one, the same as in pure QED3. The organization of monopoles in multiplets of the QCP's symmetry group SU(2) x SU(N) is shown for general N., Comment: 46 pages. Paper submitted to a special issue of Annals of Physics dedicated to Philip W. Anderson

We present a comprehensive study of a three-orbital lattice model suitable for the layered iridate Sr2IrO4. Our analysis includes various on-site interactions (including Hubbard and Hund's) as well as compressive strain, and a Zeeman magnetic field. We use a self-consistent mean field approach with multiple order parameters to characterize the resulting phases. While in some parameter regimes the compound is well described by an effective J=1/2 model, in other regimes the full multiorbital description is needed. As a function of the compressive strain, we uncover two quantum phase transitions: first a continuous metal-insulator transition, and subsequently a first order magnetic melting of the antiferromagnetic order. Crucially, bands of both J=1/2 and J=3/2 nature play important roles in these transitions. Our results qualitatively agree with experiments of Sr2IrO4 under strain induced by a substrate, and motivate the study of higher strains.

We study the quantum entanglement structure of integer quantum Hall states via the reduced density matrix of spatial subregions. In particular, we examine the eigenstates, spectrum and entanglement entropy (EE) of the density matrix for various ground and excited states, with or without mass anisotropy. We focus on an important class of regions that contain sharp corners or cusps, leading to a geometric angle-dependent contribution to the EE. We unravel surprising relations by comparing this corner term at different fillings. We further find that the corner term, when properly normalized, has nearly the same angle dependence as numerous conformal field theories (CFTs) in two spatial dimensions, which hints at a broader structure. In fact, the Hall corner term is found to obey bounds that were previously obtained for CFTs. In addition, the low-lying entanglement spectrum and the corresponding eigenfunctions reveal "excitations" localized near corners. Finally, we present an outlook for fractional quantum Hall states., Comment: 16+3 pages, 14+2 figures, 1+1 tables

EuB6 has for a long time captured the attention of the physics community, as it shows a ferromagnetic phase transition leading to a insulator the metal transition together with colossal magnetoresistance (CMR). EuB6 has a very low carrier density, which is known to drastically change the interaction between the localized Eu moments and the conduction electrons. One of early triumphs of the quantum theory in condensed matter was the presence of Fermi surface, which is intimately linked to the symmetry of the underlying crystal lattice. This symmetry can be probed by angle resolved magnetoresistance (AMRO) measurements. Here, we present angle resolved magnetoresistance (AMRO) measurements that show a that in EuB6 this symmetry is broken, possibly indicating the presence of a quantum nematic phase. We identify the region in the temperature-magnetic field phase diagram where the magnetoresistance shows two-fold oscillations instead of the expected fourfold pattern. Quantum nematic phases are analogous to classical liquid crystals. Like liquid crystals, which break the rotational symmetry of space, their quantum analogs break the point-group symmetry of the crystal due to strong electron-electron interactions, as in quantum Hall states, Sr3Ru2O7, and high temperature superconductors. This is the same region where magnetic polarons were previously observed, suggesting that they drive the nematicity in EuB6. This is also the region of the phase diagram where EuB6 shows a colossal magnetoresistance (CMR). This novel interplay between magnetic and electronic properties could thus be harnessed for spintronic applications., Comment: 6+5 pages. v2: added demagnetization analysis

We investigate the response of 3D Luttinger semimetals to localized charge and spin impurities as a function of doping. The strong spin-orbit coupling of these materials strongly influences the Friedel oscillations and RKKY interactions. This can be seen at short distances with an $1/r^4$ divergence of the responses, and anisotropic behavior. Certain of the spin-orbital signatures are robust to temperature, even if the charge and spin oscillations are smeared out, and give an unusual diamagnetic Pauli susceptibility. We compare our results to the experimental literature on the bismuth-based half-Heuslers such as YPtBi and on the pyrochlore iridate Pr$_2$Ir$_2$O$_7$., Comment: 10 pages, 4 figures

The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a $\mathbb{Z}_2$ quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques., Comment: 9+4 pages, 4+3 figures. v3: New results regarding emergent dynamical self-duality due to visons. New figure of vison-pair spectra. Extended discussion and appendices

Boundaries constitute a rich playground for quantum many-body systems because they can lead to novel degrees of freedom such as protected boundary states in topological phases. Here, we study the groundstate of integer quantum Hall systems in the presence of boundaries through the reduced density matrix of a spatial region. We work in the lowest Landau level and choose our region to intersect the boundary at arbitrary angles. The entanglement entropy (EE) contains a logarithmic contribution coming from the chiral edge modes, and matches the corresponding conformal field theory prediction. We uncover an additional contribution due to the boundary corners. We characterize the angle-dependence of this boundary corner term, and compare it to the bulk corner EE. We further analyze the spatial structure of entanglement via the eigenstates associated with the reduced density matrix, and construct a spatially-resolved EE. The influence of the physical boundary and the region's geometry on the reduced density matrix is thus clarified. Finally, we discuss the implications of our findings for other topological phases, as well as quantum critical systems such as conformal field theories in 2 spatial dimensions., Comment: 10+5 pages, 9+3 figures, 1+1 tables

Monopole operators are topological disorder operators in 2+1 dimensional compact gauge field theories appearing notably in quantum magnets with fractionalized excitations. For example, their proliferation in a spin-1/2 kagome Heisenberg antiferromagnet triggers a quantum phase transition from a Dirac spin liquid phase to an antiferromagnet. The quantum critical point (QCP) for this transition is described by a conformal field theory: Compact quantum electrodynamics (QED3) with a fermionic self-interaction, a type of QED3-Gross-Neveu model. We obtain the scaling dimensions of monopole operators at the QCP using a state-operator correspondence and a large-N expansion, where 2N is the number of fermion flavors. We characterize the hierarchy of monopole operators at this SU(2) x SU(N) symmetric QCP., Comment: Submitted to the proceedings volume for the Quantum Theory and Symmetry XI conference held at the Centre de Recherches Math\'ematiques, Montr\'eal

We investigate the superconductivity of 3D Luttinger semimetals, such as YPtBi, where Cooper pairs are constituted of spin-3/2 quasiparticles. Various pairing mechanisms have already been considered for these semimetals, such as from polar phonons modes, and in this work we explore pairing from the screened electron-electron Coulomb repulsion. In these materials, the small Fermi energy and the spin-orbit coupling strongly influence how charge fluctuations can mediate pairing. We find the superconducting critical temperature as a function of doping for an s-wave order parameter, and determine its sensitivity to changes in the dielectric permittivity. Also, we discuss how order parameters other than s-wave may lead to a larger critical temperature, due to spin-orbit coupling., Comment: 8 pages, 2 figures. Submitted to the proceedings volume for the Quantum Theory and Symmetry XI conference held at the Centre de Recherches Math\'ematiques in Universit\'e de Montr\'eal

We study superconductivity driven by screened Coulomb repulsion in three-dimensional Luttinger semimetals with a quadratic band touching and strong spin-orbit coupling. In these semimetals, the Cooper pairs are formed by spin-3/2 fermions with non-trivial wavefunctions. We numerically solve the linear Eliashberg equation to obtain the critical temperature of a singlet s-wave gap function as a function of doping, with account of spin-orbit and self-energy corrections. In order to understand the underlying mechanism of superconductivity, we compute the sensitivity of the critical temperature to changes in the dielectric function $\epsilon(i\Omega_n,q)$. We relate our results to the plasmon and Kohn-Luttinger mechanisms. Finally, we discuss the validity of our approach and compare our results to the litterature. We find good agreement with some bismuth-based half-Heuslers, such as YPtBi, and speculate on superconductivity in the iridate Pr2Ir2O7., Comment: 17 pages, 6 figures ; published version

Quantum many-body systems have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in $2+1$ dimensions, we show that the corner-induced logarithmic terms of free CFTs and certain Lifshitz theories are simply related to those that arise when the corner touches the boundary. We test our findings on the lattice, including a numerical implementation of Neumann boundary conditions. We also propose an ansatz, the boundary Extensive Mutual Information model, for a CFT with a boundary whose entanglement entropy is purely geometrical. This model shows the same bulk-boundary connection as Dirac fermions and certain supersymmetric CFTs that have a holographic dual. Finally, we discuss how our results can be generalized to all dimensions as well as to massive quantum field theories., Comment: 13+6 pages, 8+1 figures, 2+1 tables; v2: match published version

We study the quantum phase transition from a Dirac spin liquid to an antiferromagnet driven by condensing monopoles with spin quantum numbers. We describe the transition in field theory by tuning a fermion interaction to condense a spin-Hall mass, which in turn allows the appropriate monopole operators to proliferate and confine the fermions. We compute various critical exponents at the quantum critical point (QCP), including the scaling dimensions of monopole operators by using the state-operator correspondence of conformal field theory. We find that the degeneracy of monopoles in QED3 is lifted and yields a non-trivial monopole hierarchy at the QCP. In particular, the lowest monopole dimension is found to be smaller than that of QED3 using a large $N_f$ expansion where $2N_f$ is the number of fermion flavors. For the minimal magnetic charge, this dimension is $0.39N_f$ at leading order. We also study the QCP between Dirac and chiral spin liquids, which allows us to test a conjectured duality to a bosonic CP$^1$ theory. Finally, we discuss the implications of our results for quantum magnets on the Kagome lattice., Comment: 22+7 pages, 9+4 figures, 7 tables. v2: Additional discussion about the transition to a chiral spin liquid allowing us to test a conjectured duality to a bosonic CP1 theory

We study the structure of divergences and universal terms of the entanglement and R\'enyi entropies for singular regions. First, we show that for $(3+1)$-dimensional free conformal field theories (CFTs), entangling regions emanating from vertices give rise to a universal contribution $S_n^{\rm univ}= -\frac{1}{8\pi}f_b(n) \int_{\gamma} k^2 \log^2(R/\delta)$, where $\gamma$ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and $k$ the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vanishes for polyhedral corners. For those, we argue that the universal contribution, which is logarithmic, is not controlled by a local integral, but rather it depends on details of the CFT in a complicated way. We also study the angle dependence for the entanglement entropy of wedge singularities in 3+1 dimensions. This is done for general CFTs in the smooth limit, and using free and holographic CFTs at generic angles. In the latter case, we show that the wedge contribution is not proportional to the entanglement entropy of a corner region in the $(2+1)$-dimensional holographic CFT. Finally, we show that the mutual information of two regions that touch at a point is not necessarily divergent, as long as the contact is through a sufficiently sharp corner. Similarly, we provide examples of singular entangling regions which do not modify the structure of divergences of the entanglement entropy compared with smooth surfaces., Comment: 53 pages, 7 figures; v2: references added

Due to strong spin-orbit coupling, charge carriers in three-dimensional quadratic band touching Luttinger semimetals have non-trivial wavefunctions characterized by a pseudospin of 3/2. We compute the dielectric permittivity of such semimetals at finite doping, within the random phase approximation. Because of interband coupling, the dielectric screening shows a reduced plasma frequency and an increased spectral weight at small wavevectors, compared to a regular quadratic band. This weakens the effective Coulomb potential, modifying the single-particle self-energy for which we present both analytical and numerical results. At a low carrier density, the quasiparticle properties of this model strongly deviate from that of a single quadratic band. We compare our findings with experimental results for alpha-Sn, HgSe, HgTe, YPtBi and Pr2Ir2O7., Comment: 14 pages, 6 figures ; published version

The entanglement entropy (EE) can measure the entanglement between a spatial subregion and its complement, which provides key information about quantum states. Here, rather than focusing on specific regions, we study how the entanglement entropy changes with small deformations of the entangling surface. This leads to the notion of entanglement susceptibilities. These relate the variation of the EE to the geometric variation of the subregion. We determine the form of the leading entanglement susceptibilities for a large class of scale invariant states, such as groundstates of conformal field theories, and systems with Lifshitz scaling, which includes fixed points governed by disorder. We then use the susceptibilities to derive the universal contributions that arise due to non-smooth features in the entangling surface: corners in 2d, as well as cones and trihedral vertices in 3d. We finally discuss the generalization to Renyi entropies., Comment: 11 pages (single-column), 3 figures, 1 table

Quantum spin liquids (QSL) are exotic phases of matter that host fractionalized excitations. It is difficult for local probes to characterize QSL, whereas quantum entanglement can serve as a powerful diagnostic tool due to its non-locality. The kagome antiferromagnetic Heisenberg model is one of the most studied and experimentally relevant models for QSL, but its solution remains under debate. Here, we perform a numerical Aharonov-Bohm experiment on this model and uncover universal features of the entanglement entropy. By means of the density-matrix renormalization group, we reveal the entanglement signatures of emergent Dirac spinons, which are the fractionalized excitations of the QSL. This scheme provides qualitative insights into the nature of kagome QSL, and can be used to study other quantum states of matter. As a concrete example, we also benchmark our methods on an interacting quantum critical point between a Dirac semimetal and a charge ordered phase., Comment: 10 pages with 6 figures and 1 table

We study gapless quantum spin chains with spin 1/2 and 1: the Fredkin and Motzkin models. Their entangled groundstates are known exactly but not their excitation spectra. We first express the groundstates in the continuum which allows for the calculation of spin and entanglement properties in a unified fashion. Doing so, we uncover an emergent conformal-type symmetry, thus consolidating the connection to a widely studied family of Lifshitz quantum critical points in 2d. We then obtain the low lying excited states via large-scale DMRG simulations and find that the dynamical exponent is z = 3.2 in both cases. Other excited states show a different z, indicating that these models have multiple dynamics. Moreover, we modify the spin-1/2 model by adding a ferromagnetic Heisenberg term, which changes the entire spectrum. We track the resulting non-trivial evolution of the dynamical exponents using DMRG. Finally, we exploit an exact map from the quantum Hamiltonian to the non-equilibrium dynamics of a classical spin chain to shed light on the quantum dynamics., Comment: For a special issue of Journal of Physics A celebrating John Cardy's 70th birthday. 19+6 pages, 7+3 figures, 2+1 tables. v2: final version with minor modifications

Many-body systems with multiple emergent time scales arise in various contexts, including classical critical systems, correlated quantum materials, and ultra-cold atoms. We investigate such non-trivial quantum dynamics in a new setting: a spin-1 bilinear-biquadratic chain. It has a solvable entangled groundstate, but a gapless excitation spectrum that is poorly understood. By using large-scale DMRG simulations, we find that the lowest excitations have a dynamical exponent $z$ that varies from 2 to 3.2 as we vary a coupling in the Hamiltonian. We find an additional gapless mode with a continuously varying exponent $2\leq z <2.7$, which establishes the presence of multiple dynamics. In order to explain these striking properties, we construct a continuum wavefunction for the groundstate, which correctly describes the correlations and entanglement properties. We also give a continuum parent Hamiltonian, but show that additional ingredients are needed to capture the excitations of the chain. By using an exact mapping to the non-equilibrium dynamics of a classical spin chain, we find that the large dynamical exponent is due to subdiffusive spin motion. Finally, we discuss the connections to other spin chains and to a family of quantum critical models in 2d., Comment: 4+5 pages, 2+4 figures, 1 table. v2: minor changes to reflect the published version

We discuss dynamical response functions near quantum critical points, allowing for both a finite temperature and detuning by a relevant operator. When the quantum critical point is described by a conformal field theory (CFT), conformal perturbation theory and the operator product expansion can be used to fix the first few leading terms at high frequencies. Knowledge of the high frequency response allows us then to derive non-perturbative sum rules. We show, via explicit computations, how holography recovers the general results of CFT, and the associated sum rules, for any holographic field theory with a conformal UV completion -- regardless of any possible new ordering and/or scaling physics in the IR. We numerically obtain holographic response functions at all frequencies, allowing us to probe the breakdown of the asymptotic high-frequency regime. Finally, we show that high frequency response functions in holographic Lifshitz theories are quite similar to their conformal counterparts, even though they are not strongly constrained by symmetry., Comment: 45+14 pages, 9 figures. v2: small clarifications, added references

We study the von Neumann and R\'enyi entanglement entropy (EE) of scale-invariant theories defined on tori in 2+1 and 3+1 spacetime dimensions. We focus on spatial bi-partitions of the torus into two cylinders, and allow for twisted boundary conditions along the non-contractible cycles. Various analytical and numerical results are obtained for the universal EE of the relativistic boson and Dirac fermion conformal field theories (CFTs), and for the fermionic quadratic band touching and the boson with $z=2$ Lifshitz scaling. The shape dependence of the EE clearly distinguishes these theories, although intriguing similarities are found in certain limits. We also study the evolution of the EE when a mass is introduced to detune the system from its scale-invariant point, by employing a renormalized EE that goes beyond a naive subtraction of the area law. In certain cases we find non-monotonic behavior of the torus EE under RG flow, which distinguishes it from the EE of a disk., Comment: 47 pages, 13 figures

We study the universal contributions to the entanglement entropy (EE) of 2+1d and 3+1d holographic conformal field theories (CFTs) on topologically non-trivial manifolds, focusing on tori. The holographic bulk corresponds to AdS-soliton geometries. We characterize the properties of these regulator-independent EE terms as a function of both the size of the cylindrical entangling region, and the shape of the torus. In 2+1d, in the simple limit where the torus becomes a thin 1d ring, the EE reduces to a shape-independent constant $2\gamma$. This is twice the EE obtained by bipartitioning an infinite cylinder into equal halves. We study the RG flow of $\gamma$ by defining a renormalized EE that 1) is applicable to general QFTs, 2) resolves the failure of the area law subtraction, and 3) is inspired by the F-theorem. We find that the renormalized $\gamma$ decreases monotonically when the holographic CFT is deformed by a relevant operator for all allowed scaling dimensions. We also discuss the question of non-uniqueness of such renormalized EEs both in 2+1d and 3+1d., Comment: 22 pages, 11 figures, v2: minor changes, refs. added

We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with O($N$) symmetry in the limit of large $N$ for general entanglement geometries. We show that the leading large $N$ result can be obtained from the entanglement entropy of $N$ Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of $N$ massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at $N=\infty$. Notably, for a semi-infinite cylindrical region it scales as $N^0$, in stark contrast to the $N$-linear result of the Gaussian fixed point., Comment: (v3) 23 pages, 3 figures. Published version. Added references, minor corrections and clarifications

We study high frequency response functions, notably the optical conductivity, in the vicinity of quantum critical points (QCPs) by allowing for both detuning from the critical coupling and finite temperature. We consider general dimensions and dynamical exponents. This leads to a unified understanding of sum rules. In systems with emergent Lorentz invariance, powerful methods from conformal field theory allow us to fix the high frequency response in terms of universal coefficients. We test our predictions analytically in the large-N O(N) model and using the gauge-gravity duality, and numerically via Quantum Monte Carlo simulations on a lattice model hosting the interacting superfluid-insulator QCP. In superfluid phases, interacting Goldstone bosons qualitatively change the high frequency optical conductivity, and the corresponding sum rule., Comment: 4+12 pages. 2+2 figures. v2: Added explanations throughout, and new results in Appendix C for the ordered phase. Published version

A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions when the partition involves a corner of angle $\theta$, the subleading term is logarithmic with coefficient $a_\alpha(\theta)$ for the $\alpha$-R\'enyi entropy. In the smooth limit $\theta\!\to\!\pi$, $a_1(\theta)$ yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general R\'enyi indices and angles, $a_\alpha(\theta)$ is richer and few general results exist. We study $a_\alpha(\theta)$ focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in $\theta,\alpha$ regimes hitherto unexplored. We derive field theory estimates for $a_\alpha(\theta)$, including new exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of $a_\alpha(\theta)$ for various theories, including interacting $O(N)$ models., Comment: 13+3 pages, 11+1 figures, 2+2 tables

The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems put on tori in 2d/3d, denoted by $\chi$. Focusing on scale invariant systems, we derive general non-perturbative properties for the shape dependence of $\chi$, and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for $\chi$ in 2d/3d within a model that arises in the study of conformal field theories (CFTs), and use them to obtain ansatzes without fitting parameters for the 2d/3d free boson CFTs. Our numerical lattice calculations show that the ansatzes are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g. Kitaev's honeycomb model., Comment: 4+7 pages, 4+4 figures. v3: improvements throughout to improve clarity; Appendices have been expanded

We analyze the dynamical response functions of strongly interacting quantum critical states described by conformal field theories (CFTs). We construct a self-consistent holographic model that incorporates the relevant scalar operator driving the quantum critical phase transition. Focusing on the finite temperature dynamical conductivity $\sigma(\omega,T)$, we study its dependence on our model parameters, notably the scaling dimension of the relevant operator. It is found that the conductivity is well-approximated by a simple ansatz proposed by Katz et al [1] for a wide range of parameters. We further dissect the conductivity at large frequencies $\omega >> T$ using the operator product expansion, and show how it reveals the spectrum of our model CFT. Our results provide a physically-constrained framework to study the analytic continuation of quantum Monte Carlo data, as we illustrate using the O(2) Wilson-Fisher CFT. Finally, we comment on the variation of the conductivity as we tune away from the quantum critical point, setting the stage for a comprehensive analysis of the phase diagram near the transition., Comment: 25+18 pages; 8+2 figures; 1 table. v3: published version

The entanglement entropy in many gapless quantum systems receives a contribution from corners in the entangling surface in 2+1d. It is characterized by a universal function $a(\theta)$ depending on the opening angle $\theta$, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit $\theta \to \pi$ yields the stress tensor 2-point function coefficient $C_T$ . Little is known about $a(\theta)$ beyond that limit. Here, we show that the next term in the smooth limit expansion contains information beyond the 2- and 3-point correlators of the stress tensor. We conjecture that it encodes 4-point data, making it much richer. Further, we establish strong constraints on this and higher order smooth-limit coefficients. We also show that $a(\theta)$ is lower-bounded by a non-trivial function multiplied by the central charge $C_T$ , e.g. $a(\pi/2) \geq (\pi^2 \ln 2)C_T /6$. This bound for 90-degree corners is nearly saturated by all known results, including recent numerics for the interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given for the R\'enyi entropies. We illustrate our findings using O(N) QCPs, free boson and Dirac fermion CFTs, strongly coupled holographic ones, and other models. Exact results are also given for Lifshitz quantum critical points, and for conical singularities in 3+1d., Comment: 10 + 8 pages, 6 figures, 1 + 2 tables. v2: refs added, minor changes

Topological states of electrons present new avenues to explore the rich phenomenology of correlated quantum matter. Topological insulators (TIs) in particular offer an experimental setting to study novel quantum critical points (QCPs) of massless Dirac fermions, which exist on the sample's surface. Here, we obtain exact results for the zero- and finite-temperature optical conductivity at the semimetal-superconductor QCP for these topological surface states. This strongly interacting QCP is described by a scale invariant theory with emergent supersymmetry, which is a unique symmetry mixing bosons and fermions. We show that supersymmetry implies exact relations between the optical conductivity and two otherwise unrelated properties: the shear viscosity and the entanglement entropy. We discuss experimental considerations for the observation of these signatures in TIs., Comment: 5 pages, 1 figure; 12 pages of supplemental material. v2: presentation substantially simplified; published version. v3: included missing supplemental material