Integer quantum Hall effect for bosons

Floquet Flux Attachment in Cold Atomic Systems

Flux attachment provides a powerful conceptual framework for understanding certain forms of topological order, including most notably the fractional quantum Hall effect. Despite its ubiquitous use as a theoretical tool, directly realizing flux attachment in a microscopic setting remains an open challenge. Here, we propose a simple approach to realizing flux attachment in a periodically driven (Floquet) system of either spins or hard-core bosons. We demonstrate that such a system naturally realizes correlated hopping interactions and provides a sharp connection between such interactions and flux attachment. Starting with a simple, nearest-neighbor, free boson model, we find evidence-from both a coupled-wire analysis and large-scale density matrix renormalization group simulations-that Floquet flux attachment stabilizes the bosonic integer quantum Hall state at 1/4 filling (on a square lattice), and the Halperin-221 fractional quantum Hall state at 1/6 filling (on a honeycomb lattice). At 1/2 filling on the square lattice, time-reversal symmetry is instead spontaneously broken and bosonic integer quantum Hall states with opposite Hall conductances are degenerate. Finally, we propose an optical-lattice-based implementation of our model on a square lattice and discuss prospects for adiabatic preparation as well as effects of Floquet heating.

Bose Metals, from Prediction to Realization

Bose metals are metals made of Cooper pairs, which form at very low temperatures in superconducting films and Josephson junction arrays as an intermediate phase between superconductivity and superinsulation. We predicted the existence of this 2D metallic phase of bosons in the mid 1990s, showing that they arise due to topological quantum effects. The observation of Bose metals in perfectly regular Josephson junction arrays fully confirms our prediction and rules out alternative models based on disorder. Here, we review the basic mechanism leading to Bose metals. The key points are that the relevant vortices in granular superconductors are core-less, mobile XY vortices which can tunnel through the system due to quantum phase slips, that there is no charge-phase commutation relation preventing such vortices from being simultaneously out of condensate with charges, and that out-of-condensate charges and vortices are subject to topological mutual statistics interactions, a quantum effect that dominates at low temperatures. These repulsive mutual statistics interactions are sufficient to increase the energy of the Cooper pairs and lift them out of condensate. The result is a topological ground state in which charge conduction along edges and vortex movement across them organize themselves so as to generate the observed metallic saturation at low temperatures. This state is known today as a bosonic topological insulator.

Unveiling correlated two-dimensional topological insulators through fermionic tensor network states-classification, edge theories and variational wavefunctions

The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations andnon-chiraltopological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

Spin-resolved topology and partial axion angles in three-dimensional insulators

Symmetry-protected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in time-reversal- ([Formula: see text]-) invariant (helical) 3D TCIs-termed higher-order TCIs (HOTIs)-the boundary signatures can manifest as a sample-dependent network of 1D hinge states. We here introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the intrinsic bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), "spin-Weyl" semimetals, and [Formula: see text]-doubled axion insulator (T-DAXI) states with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and half-quantized 2D TI surface states originating from a partial parity anomaly. Using ab-initio calculations, we demonstrate that β-MoTe2 realizes a spin-Weyl state and that α-BiBr hosts both 3D QSHI and T-DAXI regimes.

Crystallization of bosonic quantum Hall states in a rotating quantum gas

The dominance of interactions over kinetic energy lies at the heart of strongly correlated quantum matter, from fractional quantum Hall liquids¹, to atoms in optical lattices² and twisted bilayer graphene³. Crystalline phases often compete with correlated quantum liquids, and transitions between them occur when the energy cost of forming a density wave approaches zero. A prime example occurs for electrons in high-strength magnetic fields, where the instability of quantum Hall liquids towards a Wigner crystal^4-9 is heralded by a roton-like softening of density modulations at the magnetic length^7,10-12. Remarkably, interacting bosons in a gauge field are also expected to form analogous liquid and crystalline states^13-21. However, combining interactions with strong synthetic magnetic fields has been a challenge for experiments on bosonic quantum gases^18,21. Here we study the purely interaction-driven dynamics of a Landau gauge Bose-Einstein condensate²² in and near the lowest Landau level. We observe a spontaneous crystallization driven by condensation of magneto-rotons^7,10, excitations visible as density modulations at the magnetic length. Increasing the cloud density smoothly connects this behaviour to a quantum version of the Kelvin-Helmholtz hydrodynamic instability, driven by the sheared internal flow profile of the rapidly rotating condensate. At long times the condensate self-organizes into a persistent array of droplets separated by vortex streets, which are stabilized by a balance of interactions and effective magnetic forces.

Symmetry-Protected Topological Phases in a Rydberg Glass

Recent theoretical studies predict that structural disorder, serving as a bridge connecting a crystalline material to an amorphous material, can induce a topological insulator from a trivial phase. However, to experimentally observe such a topological phase transition is very challenging due to the difficulty in controlling structural disorder in a quantum material. Given experimental realization of randomly positioned Rydberg atoms, such a system is naturally suited to studying structural disorder induced topological phase transitions and topological amorphous phases. Motivated by the development, we study topological phases in an experimentally accessible one-dimensional amorphous Rydberg atom chain with random atom configurations. In the single-particle level, we find symmetry-protected topological amorphous insulators and a structural disorder induced topological phase transition, indicating that Rydberg atoms provide an ideal platform to experimentally observe the phenomenon using state-of-the-art technologies. Furthermore, we predict the existence of a gapless symmetry-protected topological phase of interacting bosons in the experimentally accessible system. The resultant many-body topological amorphous phase is characterized by a Z_{2} invariant.

Signatures of bosonic Landau levels in a finite-momentum superconductor

Charged particles subjected to magnetic fields form Landau levels (LLs). Originally studied in the context of electrons in metals¹, fermionic LLs continue to attract interest as hosts of exotic electronic phenomena^2,3. Bosonic LLs are also expected to realize novel quantum phenomena^4,5, but, apart from recent advances in synthetic systems^6,7, they remain relatively unexplored. Cooper pairs in superconductors-composite bosons formed by electrons-represent a potential condensed-matter platform for bosonic LLs. Under certain conditions, an applied magnetic field is expected to stabilize an unusual superconductor with finite-momentum Cooper pairs^8,9 and exert control over bosonic LLs^10-13. Here we report thermodynamic signatures, observed by torque magnetometry, of bosonic LL transitions in the layered superconductor Ba₆Nb₁₁S₂₈. By applying an in-plane magnetic field, we observe an abrupt, partial suppression of diamagnetism below the upper critical magnetic field, which is suggestive of an emergent phase within the superconducting state. With increasing out-of-plane magnetic field, we observe a series of sharp modulations in the upper critical magnetic field that are indicative of distinct vortex states and with a structure that agrees with predictions for Cooper pair LL transitions in a finite-momentum superconductor^10-14. By applying Onsager's quantization rule¹⁵, we extract the momentum. Furthermore, study of the fermionic LLs shows evidence for a non-zero Berry phase. This suggests opportunities to study bosonic LLs, topological superconductivity, and their interplay via transport¹⁶, scattering¹⁷, scanning probe¹⁸ and exfoliation techniques¹⁹.

Statistical Interactions and Boson-Anyon Duality in Fractional Quantum Hall Fluids

We present an exact scheme of bosonization for anyons (including fermions) in the two-dimensional manifold of the quantum Hall fluid. This gives every fractional quantum Hall phase of the electrons one or more dual bosonic descriptions. For interacting electrons, the statistical transmutation from anyons to bosons allows us to explicitly derive the microscopic statistical interaction between the anyons, in the form of the effective two-body and few-body interactions. This also leads to a number of unexpected topological phases of the single component bosonic fractional quantum Hall effect that may be experimentally accessible. Numerical analysis of the energy spectrum and ground state entanglement properties are carried out for simple examples.

Geometric squeezing into the lowest Landau level

The equivalence between particles under rotation and charged particles in a magnetic field relates phenomena as diverse as spinning atomic nuclei, weather patterns, and the quantum Hall effect. For such systems, quantum mechanics dictates that translations along different directions do not commute, implying a Heisenberg uncertainty relation between spatial coordinates. We implement squeezing of this geometric quantum uncertainty, resulting in a rotating Bose-Einstein condensate occupying a single Landau gauge wave function. We resolve the extent of zero-point cyclotron orbits and demonstrate geometric squeezing of the orbits' centers 7 decibels below the standard quantum limit. The condensate attains an angular momentum exceeding 1000 quanta per particle and an interatomic distance comparable to the cyclotron orbit. This offers an alternative route toward strongly correlated bosonic fluids.

Fractional Quantum Hall Effect in Weyl Semimetals

Weyl semimetal may be thought of as a gapless topological phase protected by the chiral anomaly, where the symmetries involved in the anomaly are the U(1) charge conservation and the crystal translational symmetry. The absence of a band gap in a weakly interacting Weyl semimetal is mandated by the electronic structure topology and is guaranteed as long as the symmetries and the anomaly are intact. The nontrivial topology also manifests in the Fermi arc surface states and topological response, in particular taking the form of an anomalous Hall effect in magnetic Weyl semimetals, whose magnitude is only determined by the location of the Weyl nodes in the Brillouin zone. Here we consider the situation when the interactions are not weak and ask whether it is possible to open a gap in a magnetic Weyl semimetal while preserving its nontrivial electronic structure topology along with the translational and the charge conservation symmetries. Surprisingly, the answer turns out to be yes. The resulting topologically ordered state provides a nontrivial realization of the fractional quantum Hall effect in three spatial dimensions in the absence of an external magnetic field, which cannot be viewed as a stack of two dimensional states. Our state contains loop excitations with nontrivial braiding statistics when linked with lattice dislocations.

Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

Bosonic symmetry protected topological (BSPT) states, the bosonic analogue of topological insulators, have attracted enormous theoretical interest in the last few years. Although BSPT states have been classified by various approaches, there is so far no successful experimental realization of any BSPT state in two or higher dimensions. In this paper, we propose that a two-dimensional BSPT state with U(1)×U(1) symmetry can be realized in bilayer graphene in a magnetic field. Here the two U(1) symmetries represent total spin S^{z} and total charge conservation, respectively. The Coulomb interaction plays a central role in this proposal-it gaps out all the fermions at the boundary, so that only bosonic charge and spin degrees of freedom are gapless and protected at the edge. Based on the above conclusion, we propose that the bulk quantum phase transition between the BSPT and trivial phase, which can be driven by applying both magnetic and electric fields, can become a "bosonic phase transition" with interactions. That is, only bosonic modes close their gap at the transition, which is fundamentally different from all the well-known topological insulator to trivial insulator transitions that occur for free fermion systems. We discuss various experimental consequences of this proposal.

Semisynthetic zigzag optical lattice for ultracold bosons

We propose a cold-atom realization of a zigzag ladder. The two legs of the ladder correspond to a "synthetic" dimension given by two internal (spin) states of the atoms, so that tunneling between them can be realized as a laser-assisted process. The zigzag geometry is achieved by employing a spin-dependent optical lattice with the site position depending on the internal atomic state, i.e., on the ladder's leg. The lattice offers a possibility to tune the single-particle dispersion from a double-well to a single-minimum configuration. In contrast to previously considered semisynthetic lattices with a square geometry, the tunneling in the synthetic dimension is accompanied by spatial displacements of atoms. Therefore, the atom-atom interactions are nonlocal and act along the diagonal (semisynthetic) direction. We investigate the ground-state properties of the system for the case of strongly interacting bosons. In particular, we find that the interplay between the frustration induced by the magnetic field and the interactions gives rise to an interesting gapped phase at fractional filling factors corresponding to one particle per magnetic unit cell.

Topological Varma Superfluid in Optical Lattices

Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence. While noninteracting fermionic topological insulators are well established by now and have been classified according to a tenfold scheme, the possible realization of topological states for bosons has not been explored much yet. Furthermore, the role of interactions is far from being understood. Here, we show that a topological state of matter exclusively driven by interactions may occur in the p band of a Lieb optical lattice filled with ultracold bosons. The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature. Although the system is neither topological at the single-particle level nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a nonzero Chern number. Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.

Bosonic Analogue of Dirac Composite Fermi Liquid

We introduce a particle-hole-symmetric metallic state of bosons in a magnetic field at odd-integer filling. This state hosts composite fermions whose energy dispersion features a quadratic band touching and corresponding 2π Berry flux protected by particle-hole and discrete rotation symmetries. We also construct an alternative particle-hole symmetric state-distinct in the presence of inversion symmetry-without Berry flux. As in the Dirac composite Fermi liquid introduced by Son [Phys. Rev. X 5, 031027 (2015)], breaking particle-hole symmetry recovers the familiar Chern-Simons theory. We discuss realizations of this phase both in 2D and on bosonic topological insulator surfaces, as well as signatures in experiments and simulations.

Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model

We study the two-dimensional (2D) Hubbard model using exact diagonalization for spin-1/2 fermions on the triangular and honeycomb lattices decorated with a single hexagon per site. In certain parameter ranges, the Hubbard model maps to a quantum compass model on those lattices. On the triangular lattice, the compass model exhibits collinear stripe antiferromagnetism, implying d-density wave charge order in the original Hubbard model. On the honeycomb lattice, the compass model has a unique, quantum disordered ground state that transforms nontrivially under lattice reflection. The ground state of the Hubbard model on the decorated honeycomb lattice is thus a 2D fermionic symmetry-protected topological phase. This state-protected by time-reversal and reflection symmetries-cannot be connected adiabatically to a free-fermion topological phase.

Kagome Chiral Spin Liquid as a Gauged U(1) Symmetry Protected Topological Phase

While the existence of a chiral spin liquid (CSL) on a class of spin-1/2 kagome antiferromagnets is by now well established numerically, a controlled theoretical path from the lattice model leading to a low-energy topological field theory is still lacking. This we provide via an explicit construction starting from reformulating a microscopic model for a CSL as a lattice gauge theory and deriving the low-energy form of its continuum limit. A crucial ingredient is the realization that the bosonic spinons of the gauge theory exhibit a U(1) symmetry protected topological (SPT) phase, which upon promoting its U(1) global symmetry to a local gauge structure ("gauging"), yields the CSL. We suggest that such an explicit lattice-based construction involving gauging of a SPT phase can be applied more generally to understand topological spin liquids.

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number

The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν=r/(r|C|+1) for bosons, or ν=r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.

Bosonic integer quantum Hall effect in optical flux lattices

In two dimensions strongly interacting bosons in a magnetic field can realize a bosonic integer quantum Hall state, the simplest two-dimensional example of a symmetry-protected topological phase. We propose a realistic implementation of this phase using an optical flux lattice. Through exact diagonalization calculations, we show that the system exhibits a clear bulk gap and the topological signature of the bosonic integer quantum Hall state. In particular, the calculation of the many-body Chern number leads to a quantized Hall conductance in agreement with the analytical predictions. We also study the stability of the phase with respect to some of the experimentally relevant parameters.

Bosonic integer quantum Hall effect in an interacting lattice model

We study a bosonic model with correlated hopping on a honeycomb lattice, and show that its ground state is a bosonic integer quantum Hall (BIQH) phase, a prominent example of a symmetry-protected topological (SPT) phase. By using the infinite density matrix renormalization group method, we establish the existence of the BIQH phase by providing clear numerical evidence: (i) a quantized Hall conductance with |σ_{xy}|=2, (ii) two counterpropagating gapless edge modes. Our simple model is an example of a novel class of systems that can stabilize SPT phases protected by a continuous symmetry on lattices and opens up new possibilities for the experimental realization of these exotic phases.

Entanglement monotonicity and the stability of gauge theories in three spacetime dimensions

We employ the recent results on the generalization of the central charge theorem to three spacetime dimensions to derive nonperturbative results for several strongly interacting quantum field theories, including quantum electrodynamics (QED-3), and the theory corresponding to certain quantum phase transitions in condensed matter systems. In particular, by demanding that the universal constant part of the entanglement entropy decreases along the renormalization group flow (F theorem), we find sufficient conditions for the stability of QED-3 against chiral symmetry breaking and confinement. Using similar ideas, we derive strong constraints on the nature of quantum critical points in condensed matter systems with topological order.

Topological response theory of Abelian symmetry-protected topological phases in two dimensions

It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.

Symmetry-protected topological phases from decorated domain walls

Symmetry-protected topological phases generalize the notion of topological insulators to strongly interacting systems of bosons or fermions. A sophisticated group cohomology approach has been used to classify bosonic symmetry-protected topological phases, which however does not transparently predict their properties. Here we provide a physical picture that leads to an intuitive understanding of a large class of symmetry-protected topological phases in d=1,2,3 dimensions. Such a picture allows us to construct explicit models for the symmetry-protected topological phases, write down ground state wave function and discover topological properties of symmetry defects both in the bulk and on the edge of the system. We consider symmetries that include a Z2 subgroup, which allows us to define domain walls. While the usual disordered phase is obtained by proliferating domain walls, we show that symmetry-protected topological phases are realized when these domain walls are decorated, that is, are themselves symmetry-protected topological phases in one lower dimension. This construction works both for unitary Z2 and anti-unitary time reversal symmetry.

Bosonic Mott insulator with Meissner currents

We introduce a generic bosonic model exemplifying that (spin) Meissner currents can persist in insulating phases of matter. We consider two species of interacting bosons on a lattice. Our model exhibits separation of charge (total density) and spin (relative density): the charge sector is gapped in a bosonic Mott insulator phase with total density one, while the spin sector remains superfluid due to interspecies conversion. Coupling the spin sector to the gauge fields yields a spin Meissner effect reflecting the long-range spin superfluid coherence. We investigate the resulting phase diagram and describe other possible spin phases of matter in the Mott regime possessing chiral currents as well as a spin-density wave phase. The model presented here is realizable in Josephson junction arrays and in cold atom experiments.

Integer quantum Hall state in two-component Bose gases in a synthetic magnetic field

We study two-component (or pseudospin-1/2) Bose gases in a strong synthetic magnetic field. Using exact diagonalization, we show that a bosonic analog of an integer quantum Hall state with no intrinsic topological order appears at the total filling factor ν=1+1 when the strengths of intracomponent and intercomponent interactions are comparable with each other. This provides a prime example of a symmetry-protected topological phase in a controlled setting of quantum gases. The real-space entanglement spectrum of this state is found to be comprised of counterpropagating chiral modes consistent with the edge theory derived from an effective Chern-Simons theory.

Nonperturbative effects of a topological theta term on principal chiral nonlinear sigma models in 2 + 1 dimensions

We study the effects of a topological Theta term on 2+1-dimensional principal chiral models, which are nonlinear sigma models defined on Lie group manifolds. We find that when Θ=π, the nature of the disordered phase of the principal chiral model is strongly affected by the topological term: it is either a gapless conformal field theory, or it is gapped and twofold degenerate. The result of our Letter can be used to analyze the boundary states of three-dimensional symmetry protected topological phases.