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Abstract Quantum systems evolve in time in one of two ways: through the Schrödinger equation or wavefunction collapse. So far, deterministic control of quantum many-body systems in the lab has focused on the former, due to the probabilistic nature of measurements. This imposes serious limitations: preparing long-range entangled states, for example, requires extensive circuit depth if restricted to unitary dynamics. In this work, we use mid-circuit measurement and feed-forward to implement deterministic non-unitary dynamics on Quantinuum’s H1 programmable ion-trap quantum computer. Enabled by these capabilities, we demonstrate a constant-depth procedure for creating a toric code ground state in real-time. In addition to reaching high stabilizer fidelities, we create a non-Abelian defect whose presence is confirmed by transmuting anyons via braiding. This work clears the way towards creating complex topological orders in the lab and exploring deterministic non-unitary dynamics via measurement and feed-forward.

Electromagnetic response of topological materials is described the so called axion electrodynamics which contains additional relations between the fields. Here the authors extend the theory of axion electrodynamics to general optical frequencies and apply it to a realistic topological antiferromagnet.

In conventional materials, charge carriers are electron-like quasiparticles, but topological bands allow for more exotic possibilities. Here, the authors predict that in the Chern-ferromagnet phase of twisted bilayer graphene charge is carried by spin polarons, bound states of an electron and a spin flip.

XY* transitions represent one of the simplest examples of unconventional quantum criticality, in which fractionally charged excitations condense into a superfluid and display novel features that combine quantum criticality and fractionalization. Nevertheless, their experimental realization is challenging. Here we propose to study the XY* transition in quantum Hall bilayers at filling (ν_{1},ν_{2})=(1/3,2/3) where the exciton condensate (EC) phase plays the role of the superfluid. Supported by exact diagonalization calculation, we argue that there is a continuous transition between an EC phase at small bilayer separation to a pair of decoupled fractional quantum Hall states at large separation. The transition is driven by condensation of a fractional exciton, a bound state of a Laughlin quasiparticle and quasihole, and is in the XY* universality class. The fractionalization is manifested by unusual properties including a large anomalous exponent and fractional universal conductivity, which can be conveniently measured through interlayer tunneling and counterflow transport, respectively. We also show that the edge is likely to realize the newly predicted extraordinary log boundary criticality. Our work highlights the promise of quantum Hall bilayers as an ideal platform for exploring exotic bulk and boundary critical behaviors that are amenable to immediate experimental exploration in dual-gated bilayer systems. The XY* critical theory can be generalized to a bilayer system with an arbitrary Abelian state in one layer and its particle-hole partner in the other layer. Therefore, we anticipate many distinct XY* transitions corresponding to the different Laughlin states and Jain sequences in the single-layer case.

Twisted double bilayer graphene hosts flat bands that can be tuned with an electric field. Here, by using gate-tuned scanning tunneling spectroscopy, the authors demonstrate the tunability of the flat band and reveal spectral signatures of correlated electron states and the topological nature of the flat band.

Motivated by the higher Chern bands of twisted graphene multilayers, we consider flat bands with arbitrary Chern number C with ideal quantum geometry. While C>1 bands differ from Landau levels, we show that these bands host exact fractional Chern insulator (FCI) ground states for short-range interactions. We show how to decompose ideal higher Chern bands into separate ideal bands with Chern number 1 that are intertwined through translation and rotation symmetry. The decomposed bands admit an SU(C) action that combines real space and momentum space translations. Remarkably, they also allow for analytic construction of exact many-body ground states, such as generalized quantum Hall ferromagnets and FCIs, including flavor-singlet Halperin states and Laughlin ferromagnets in the limit of short-range interactions. In this limit, the SU(C) action is promoted to a symmetry on the ground-state subspace. While flavor singlet states are translation symmetric, the flavor ferromagnets correspond to translation broken states and admit charged skyrmion excitations corresponding to a spatially varying density wave pattern. We confirm our analytic predictions with numerical simulations of ideal bands of twisted chiral multilayers of graphene, and discuss consequences for experimentally accessible systems such as monolayer graphene twisted relative to a Bernal bilayer.

Long-range entanglement—the backbone of topologically ordered states—cannot be created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation. Recently, it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases. Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call “shots.” First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class 2 (such as D_{4} or Q_{8}). We show that after the measurement, the wave function always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome. Moreover, the clean quantum double ground state can be deterministically prepared via feedforward—gates that depend on the measurement outcomes. Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group. As a special case, this gives an explicit protocol to prepare twisted quantum doubles for all solvable groups. Third, we argue that certain topological orders, such as nonsolvable quantum doubles or Fibonacci anyons, define nontrivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots. Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones. This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.

The exploration of quantum spin liquids (QSLs) has been guided by different approaches including the resonating valence bond (RVB) picture, deconfined lattice gauge theories, and the Kitaev model. More recently, a spin-liquid ground state was numerically established on the ruby lattice, inspired by the Rydberg blockade mechanism. Here, we unify these varied approaches in a single parent Hamiltonian, in which local fluctuations of anyons stabilize deconfinement. The parent Hamiltonian is defined on kagomé triangles—each hosting four RVB-like states—and includes only Ising interactions and single-site transverse fields. In the weak-field limit, the ruby spin liquid and exactly soluble kagomé dimer models are recovered, while the strong-field limit reduces to the Kitaev honeycomb model, thereby unifying three seemingly different approaches to QSLs. We similarly obtain the chiral Yao-Kivelson model, the honeycomb toric code, and a new spin-1 quadrupolar Kitaev model. The last is shown to be in a QSL phase by a nonlocal mapping to the kagomé Ising antiferromagnet. We demonstrate various applications of our framework, including (a) an adiabatic deformation of the ruby lattice model to the exactly soluble kagomé dimer model, conclusively establishing the QSL phase in the former and (b) demystifying the dynamical protocol for measuring off-diagonal strings in the Rydberg implementation of the ruby lattice spin liquid. More generally, we find an intimate connection between Kitaev couplings and the repulsive interactions used for emergent dimer models. For instance, we show how a spin-1/2 XXZ model on the ruby lattice encodes a Kitaev honeycomb model, providing a new route toward realizing the latter in cold-atom or solid-state systems.

Twisted bilayer graphene exhibits correlated electronic phases and superconductivity, but its precise nature is under debate. Here, Lee and Khalaf et al. study a twisted double bilayer graphene, where ferromagnetic insulator and spin triplet superconducting phases can be stabilized.

The Dirac spin liquid is a candidate description for the strongly correlated behaviour of some quantum magnets. Song et al. study the symmetry dependence physics of monopole excitations and argue that the lattice-dependent consequences for magnetic ordering may provide a unifying picture for 2D quantum magnetism.

We construct high-quality graphene-based van der Waals devices with narrow superconducting niobium nitride (NbN) electrodes, in which superconductivity and a robust fractional quantum Hall (FQH) state coexist. We find a possible signature for crossed Andreev reflection (CAR) across the superconductor separating two FQH edges. Our observed CAR probabilities in the particlelike fractional fillings are markedly higher than those in the integer and hole-conjugate fractional fillings and depend strongly on temperature and magnetic field unlike the other fillings. Further, we find a filling-independent CAR probability in integer fillings, which we attribute to spin-orbit coupling in NbN allowing for Andreev reflection between spin-polarized edges. These results provide a route to realize novel topological superconducting phases in FQH-superconductor hybrid devices based on graphene and NbN.

The rich phenomenology of twisted bilayer graphene (TBG) near the magic angle is believed to arise from electron correlations in topological flat bands. An unbiased approach to this problem is highly desirable, but also particularly challenging, given the multiple electron flavors, the topological obstruction to defining tight-binding models, and the long-ranged Coulomb interactions. While numerical simulations of realistic models have thus far been confined to zero temperature, typically excluding some spin or valley species, analytic progress has relied on fixed point models away from the realistic limit. Here, we present unbiased Monte Carlo simulations of realistic models of magic-angle TBG at charge neutrality. We establish the absence of a sign problem for this model in a momentum-space approach and describe a computationally tractable formulation that applies even on breaking chiral symmetry and including band dispersion. Our results include (i) the emergence of an insulating Kramers intervalley coherent ground state in competition with a correlated semimetal phase, (ii) detailed temperature evolution of order parameters and electronic spectral functions that reveal a “pseudogap” regime, in which gap features are established at a higher temperature than the onset of order, and (iii) predictions for electronic tunneling spectra and their evolution with temperature. Our results pave the way towards uncovering the physics of magic-angle graphene through exact simulations of over a hundred electrons across a wide temperature range.

The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.

The physical realization of Z_{2} topological order as encountered in the paradigmatic toric code has proven to be an elusive goal. We predict that this phase of matter can be realized in a two-dimensional array of Rydberg atoms placed on the ruby lattice, at specific values of the Rydberg blockade radius. First, we show that the blockade model—also known as a “PXP” model—realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term. This model can be interpreted as a Z_{2} gauge theory whose dynamics is generated by monomer fluctuations. We obtain its phase diagram using the numerical density matrix renormalization group method and find a topological quantum liquid (TQL) as evidenced by multiple measures including (i) a continuous transition between two featureless phases, (ii) a topological entanglement entropy of ln2 as measured in various geometries, (iii) degenerate topological ground states, and (iv) the expected modular matrix from ground state overlap. Next, we show that the TQL persists upon including realistic, algebraically decaying van der Waals interactions V(r)∼1/r^{6} for a choice of lattice parameters. Moreover, we can directly access topological loop operators, including the Fredenhagen-Marcu order parameter. We show how these can be measured experimentally using a dynamic protocol, providing a “smoking gun” experimental signature of the TQL phase. Finally, we show how to trap an emergent anyon and realize different topological boundary conditions, and we discuss the implications for exploring fault-tolerant quantum memories.

Understanding the role of topology in determining electronic structure can lead to the discovery, or appreciation, of materials with exotic properties such as protected surface states. Here, the authors present a framework for identifying topologically distinct band-structures for all 3D space groups.

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in d≥1 dimensions. Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries. In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries. Specifically, in 2D, the monopole is a source of 2π flux, and the polarization is determined by the crystal momentum of the 2π flux. In 3D, the polarization is determined by the projective representation of translation symmetries on Dirac monopoles. Our approach also leads to a concrete scheme to calculate polarization in 2D, which in principle can be applied even to strongly interacting systems. For open boundary conditions, the bulk polarization leads to an altered boundary Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.

In this paper and its upcoming sequel, we study nonequilibrium dynamics in driven (1+1)-dimensional conformal field theories (CFTs) with periodic, quasiperiodic, and random driving. We study a soluble family of drives in which the Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength. The resulting time evolution is then captured by a Möbius coordinate transformation. In this paper, we establish the general framework and focus on the first two classes. In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement and energy evolution in different phases, i.e., the heating and nonheating phases and the phase transition between them. In quasiperiodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence. We find that (i) the nonheating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the nonheating fixed point, where the entanglement entropy and energy oscillate at the Fibonacci numbers, but grow logarithmically and polynomially at the non-Fibonacci numbers; (v) for certain choices of driving Hamiltonians, the nonheating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasicrystal. In addition, another quasiperiodically driven CFT with an Aubry-André–type sequence is also studied. We compare the CFT results to lattice calculations and find remarkable agreement.

We report on a fully self-consistent Hartree-Fock calculation of interaction effects on the moiré flat bands of twisted bilayer graphene, assuming that valley U(1) symmetry is respected. We use realistic band structures and interactions and focus on the charge neutrality point, where experiments have variously reported either insulating or semimetallic behavior. Restricting the search to orders for which the valley U(1) symmetry remains unbroken, we find three types of self-consistent solutions with competitive ground-state energy (i) insulators that break C_{2}T symmetry, including valley Chern insulators; (ii) spin- or valley-polarized insulators; and (iii) rotation C_{3} symmetry-breaking semimetals the gaplessness of which is protected by the topology of the moiré flat bands. We find that the relative stability of these states can be tuned by weak strains that break C_{3} rotation. The nematic semimetal and also, somewhat unexpectedly, the C_{2}T breaking insulators are stabilized by weak strain. These ground states may be related to the semimetallic and insulating behaviors seen at charge neutrality, and the sample variability of their observation. We also compare with the results of scanning tunneling microscopy measurements near charge neutrality.

We study the energy and entanglement dynamics of (1+1)D conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown that this model supports both a nonheating and a heating phase. Here, we analytically establish several robust and “superuniversal” features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and an antichiral peak, which leads to an exponential growth in the total energy. The peak locations are set by fixed points of the Möbius transformation. Second, all of the quantum entanglement is shared between these two peaks. In each driving period, a number of Bell pairs are generated, with one member pumped to the chiral peak and the other member pumped to the antichiral peak. These Bell pairs are localized, accumulate at these two peaks, and can serve as a source of quantum entanglement. Third, in both the heating and nonheating phases, we find that the total energy is related to the half system entanglement entropy by a simple relation E(t)∝cexp[(6/c)S(t)] with c being the central charge. In addition, we show that the nonheating phase, in which the energy and entanglement oscillate in time, is unstable to small fluctuations of the driving frequency in contrast to the heating phase. Finally, we point out an analogy to the periodically driven harmonic oscillator which allows us to understand global features of the phases and introduce a quasiparticle picture to explain the spatial structure, which can be generalized to setups beyond the SSD construction.

In magic angle twisted bilayer graphene (TBG), electron-electron interactions play a central role, resulting in correlated insulating states at certain integer fillings. Identifying the nature of these insulators is a central question, and it is potentially linked to the relatively high-temperature superconductivity observed in the same devices. Here, we address this question using a combination of analytical strong-coupling arguments and a comprehensive Hartree-Fock numerical calculation, which includes the effect of remote bands. The ground state we obtain at charge neutrality is an unusual ordered state, which we call the Kramers intervalley-coherent (K-IVC) insulator. In its simplest form, the K-IVC order exhibits a pattern of alternating circulating currents that triples the graphene unit cell, leading to an “orbital magnetization density wave.” Although translation and time-reversal symmetry are broken, a combined “Kramers” time-reversal symmetry is preserved. Our analytic arguments are built on first identifying an approximate U(4)×U(4) symmetry, resulting from the remarkable properties of the TBG band structure, which helps select a low-energy manifold of states that are further split to favor the K-IVC state. This low-energy manifold is also found in the Hartree-Fock numerical calculation. We show that symmetry-lowering perturbations can stabilize other insulators and the semimetallic state, and we discuss the ground state at half-filling and give a comparison with experiments.

The recent observation of superconductivity at relatively high temperatures in hole-doped NdNiO_{2} has generated considerable interest, particularly due to its similarity with the infinite layer cuprates. Building on the observation that the Ni^{2+} ions resulting from hole doping are commonly found in the spin-triplet state, we introduce and study a variant of the t-J model in which the holes carry S=1. We name this model the type-II t-J model. We find two distinct mechanisms for d-wave superconductivity. In both scenarios the pairing is driven by the spin coupling J. However, coherence is gained in distinct ways in these two scenarios. In the first case, the spin-one holes condense, leading to a d-wave superconductor along with spin-symmetry breaking. Different orders, including spin-nematic orders, are possible. This scenario is captured by a spin-one slave boson theory. In the second scenario, a coherent and symmetric d-wave superconductor is achieved from “Kondo resonance”: spin-one holes contribute two electrons to form a large Fermi surface together with the spin-1/2 singly occupied sites. The large Fermi surface then undergoes d-wave pairing because of spin coupling J, similar to heavy fermion superconductors. We propose a three-fermion parton theory to treat these two different scenarios in one unified framework and calculate its doping phase diagram within a self-consistent mean-field approximation. Our study shows that a combination of “cuprate physics” and “heavy fermion physics” may emerge in the type-II t-J model.

Recent experiments on bilayer graphene twisted near the magic angle have observed spontaneous integer quantum Hall states in the presence of an aligned hexagonal boron nitride substrate. These states arise from the complete filling of Chern bands. A natural question is whether fractional filling of the same bands would lead to fractional Chern insulators (FCIs), i.e., fractional quantum Hall states realized in the absence of a magnetic field. Here, we argue that the magic angle graphene bands have favorable quantum geometry for realizing FCIs. We show that in the tractable “chiral” limit, the flatbands wave functions are an analytic function of the crystal momentum. This remarkable property fixes the quantum metric up to an overall momentum-dependent scale factor, the local Berry curvature, whose variation is itself small. Thus the three conditions associated with FCI stability—(i) narrow bands, (ii) “ideal” quantum metric, and (iii) relatively uniform Berry curvature—are all satisfied. Our work emphasizes continuum real-space approaches to FCIs, in contrast to earlier works, which mostly focused on tight-binding models. This enables us to construct a Laughlin wave function on a real-space torus that is a zero-energy ground state of the Coulomb interaction in the limit of very short screening length. Finally, we discuss the evolution of the band geometry on tuning away from the chiral limit and show numerically that some of the desirable properties continue to hold at a quantitative level for realistic parameter values.

The interplay of symmetry and topology has been at the forefront of recent progress in quantum matter. Here, we uncover an unexpected connection between band topology and the description of competing orders in a quantum magnet. Specifically, we show that aspects of band topology protected by crystalline symmetries determine key properties of the Dirac spin liquid (DSL), which can be defined on the honeycomb, square, triangular, and kagome lattices. At low energies, the DSL on all of these lattices is described by an emergent quantum electrodynamics (QED_{3}) with N_{f}=4 flavors of Dirac fermions coupled to a U(1) gauge field. However, the symmetry properties of the magnetic monopoles, an important class of critical degrees of freedom, behave very differently on different lattices. In particular, we show that the lattice momentum and angular momentum of monopoles can be determined from the charge (or Wannier) centers of the corresponding spinon insulator. We also show that for DSLs on bipartite lattices, there always exists a monopole that transforms trivially under all microscopic symmetries owing to the existence of a parent SU(2) gauge theory. We connect our results to generalized Lieb-Schultz-Mattis theorems and also derive the time-reversal and reflection properties of monopoles. Our results indicate that recent insights into free-fermion band topology can also guide the description of strongly correlated quantum matter.

Twisted bilayer graphene (TBLG) has emerged as an important platform for studying correlated phenomena, including unconventional superconductivity, in two-dimensional systems. The complexity of the atomic-scale structures in TBLG has made even the study of single-particle physics at low energies around the Fermi level, quite challenging. Our goal here is to provide a convenient and physically motivated picture of single-particle physics in TBLG using reduced models with the smallest possible number of localized orbitals. The reduced models exactly reproduce the low-energy bands of ab initio tight-binding models, including the effects of atomic relaxations. Furthermore, we obtain for the first time the corresponding Wannier orbitals that incorporate all symmetries of TBLG, which are also calculated as a function of angle, a requisite first step towards incorporating electron interaction effects. We construct eight-band and five-band models for the low-energy states for twist angles between 1.3^{∘} and 0.6^{∘}. The models are created using a multistep Wannier projection technique starting with appropriate ab initio k·p continuum Hamiltonians. Our procedure can also readily capture the perturbative effects of substrates and external displacement fields while offering a significant reduction in complexity for studying electron-electron correlation phenomena in realistic situations.

Band engineering in twisted bilayers of the five generic two-dimensional Bravais networks is demonstrated. We first derive constraints on the interlayer coupling that determines an effective potential for low-energy physics. The constraints are originated from symmetry and fixed without reference to microscopic details. Having the constraints on the potential helps us to predict and understand electron behavior under the influence of the moiré structure, like formation of flat bands. It is also pointed out that the generic constraints become most relevant when the typical length scale of the microscopic interlayer coupling is moderate. The concepts are numerically tested in simple tight-binding models, showing the band flattening compatible with the potential profile fixed by the generic constraints. On the basis of the generic theory, we propose a possibility of anisotropic band flattening, in which quasi-one-dimensional band dispersion is generated from relatively isotropic original band dispersion. In the strongly correlated regime, the anisotropic band flattening leads to a spin-orbital model where intertwined magnetic and orbital ordering can give rise to rich physics. Overall, our theory and demonstration give useful information for screening out interesting materials from the material database.

We study a possible deconfined quantum phase transition in a realistic model of a two-dimensional Shastry-Sutherland quantum magnet, using both numerical and field theoretic techniques. Using the infinite density matrix renormalization group (IDMRG) method, we verify the existence of an intermediate plaquette valence bond solid (PVBS) order, with twofold degeneracy, between the dimer and Néel ordered phases. We argue that the quantum phase transition between the Néel and PVBS orders may be described by a deconfined quantum critical point (DQCP) with an emergent O(4) symmetry. By analyzing the correlation length spectrum obtained from IDMRG, we provide evidence for the DQCP and emergent O(4) symmetry in the lattice model. Such a phase transition has been reported in the recent pressure-tuned experiments in the Shastry-Sutherland lattice material SrCu_{2}(BO_{3})_{2} [Nat. Phys. 13, 962 (2017)NPAHAX1745-247310.1038/nphys4190]. The nonsymmorphic lattice structure of the Shastry-Sutherland compound leads to extinction points in the scattering, where we predict sharp signatures of a DQCP in both the phonon and magnon spectra associated with the spinon continuum. The effect of weak interlayer couplings present in the three-dimensional material is also discussed. Our results should help guide the experimental study of DQCP in quantum magnets.

We study a two-dimensional tight-binding model of a topological crystalline insulator (TCI) protected by rotation symmetry. The model is built by stacking two Chern insulators with opposite Chern numbers which transform under conjugate representations of the rotation group, e.g., p_{±} orbitals. Despite its apparent similarity to the Kane-Mele model, it does not host stable gapless surface states. Nevertheless, the model exhibits topological responses including the appearance of quantized fractional charge bound to rotational defects (disclinations) and the pumping of angular momentum in response to threading an elementary magnetic flux, which are described by a mutual Chern-Simons coupling between the electromagnetic gauge field and an effective gauge field corresponding to the rotation symmetry. In addition, we show that although the filled bands of the model do not admit a symmetric Wannier representation, this obstruction is removed upon the addition of appropriate atomic orbitals, which implies “fragile” topology. As a result, the response of the model can be derived by representing it as a superposition of atomic orbitals with positive and negative integer coefficients. Following the analysis of the model, which serves as a prototypical example of 2D TCIs protected by rotation, we show that all TCIs protected by point group symmetries which do not have protected surface states are either atomic insulators or fragile phases. Remarkably, this implies that gapless surface states exist in free-electron systems if and only if there is a stable Wannier obstruction. We then use dimensional reduction to map the problem of classifying 2D TCIs protected by rotation to a zero-dimensional problem which is then used to obtain the complete noninteracting classification of such TCIs as well as the reduction of this classification in the presence of interactions.

We report the observation of a linear magnetoresistance in single crystals and epitaxial thin films of the pyrochlore iridate Bi _2 Ir _2 O _7 . The linear magnetoresistance is positive and isotropic at low temperatures, without any sign of saturation up to 35 T. As temperature increases, the linear field dependence gradually evolves to a quadratic field dependence. The temperature and field dependence of magnetoresistance of Bi _2 Ir _2 O _7 bears strikingly resemblance to the scale invariant magnetoresistance observed in the strange metal phase in high T _c cuprates. However, the residual resistivity of Bi _2 Ir _2 O _7 is more than two orders of magnitude higher than the curpates. Our results suggest that the correlation between linear magnetoresistance and quantum fluctuations may exist beyond high temperature superconductors.

We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process. We obtain our results in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the “reservoir” that facilitates the dissipation. In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.

The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs), featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with “hinge” surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.

A remarkable recent experiment has observed Mott insulator and proximate superconductor phases in twisted bilayer graphene when electrons partly fill a nearly flat miniband that arises a “magic” twist angle. However, the nature of the Mott insulator, the origin of superconductivity, and an effective low-energy model remain to be determined. We propose a Mott insulator with intervalley coherence that spontaneously breaks U(1) valley symmetry and describe a mechanism that selects this order over the competing magnetically ordered states favored by the Hund’s coupling. We also identify symmetry-related features of the nearly flat band that are key to understanding the strong correlation physics and constrain any tight-binding description. First, although the charge density is concentrated on the triangular-lattice sites of the moiré pattern, the Wannier states of the tight-binding model must be centered on different sites which form a honeycomb lattice. Next, spatially localizing electrons derived from the nearly flat band necessarily breaks valley and other symmetries within any mean-field treatment, which is suggestive of a valley-ordered Mott state, and also dictates that additional symmetry breaking is present to remove symmetry-enforced band contacts. Tight-binding models describing the nearly flat miniband are derived, which highlight the importance of further neighbor hopping and interactions. We discuss consequences of this picture for superconducting states obtained on doping the valley-ordered Mott insulator. We show how important features of the experimental phenomenology may be explained and suggest a number of further experiments for the future. We also describe a model for correlated states in trilayer graphene heterostructures and contrast it with the bilayer case.

Recent experiments in graphene heterostructures have observed Chern insulators—integer and fractional quantum Hall states made possible by a periodic substrate potential. Here, we study theoretically that the competition between different Chern insulators, which can be tuned by the amplitude of the periodic potential, leads to a new family of quantum critical points described by QED_{3}-Chern-Simons theory. At these critical points, N_{f} flavors of Dirac fermions interact through an emergent U(1) gauge theory at Chern-Simons level K, and remarkably, the entire family (with any N_{f} or K) can be realized at special values of the external magnetic field. Transitions between particle-hole conjugate Jain states realize “pure” QED_{3}, in which multiple flavors of Dirac fermions interact with a Maxwell U(1) gauge field. The multiflavor nature of the critical point leads to an emergent SU(N_{f}) symmetry. Specifically, at the transition from a ν=1/3 to 2/3 quantum Hall state, the emergent SU(3) symmetry predicts an octet of charge density waves with enhanced susceptibilities, which is verified by DMRG numerical simulations on microscopic models applicable to graphene heterostructures. We propose experiments on Chern insulators that could resolve open questions in the study of (2+1)-dimensional conformal field theories and test recent duality inspired conjectures.

Massless (2+1)D Dirac fermions arise in a variety of systems from graphene to the surfaces of topological insulators, where generating a mass is typically associated with breaking a symmetry. However, with strong interactions, a symmetric gapped phase can arise for multiples of eight Dirac fermions. A continuous quantum phase transition from the massless Dirac phase to this massive phase, which we term symmetric mass generation, is necessarily beyond the Landau paradigm and is hard to describe even at the conceptual level. Nevertheless, such transition has been consistently observed in several numerical studies recently. Here, we propose a theory for the symmetric mass generation transition which is reminiscent of deconfined criticality and involves emergent non-Abelian gauge fields coupled both to Dirac fermions and to critical Higgs bosons. We motivate the theory using an explicit parton construction and discuss predictions for numerics. Additionally, we show that the fermion Green’s function is expected to undergo a zero-to-pole transition across the critical point.

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Topological Dirac and Weyl semimetals have an energy spectrum that hosts Weyl nodes appearing in pairs of opposite chirality. Topological stability is ensured when the nodes are separated in momentum space and unique spectral and transport properties follow. In this work, we study the effect of a space-dependent Weyl node separation, which we interpret as an emergent background axial-vector potential, on the electromagnetic response and the energy spectrum of Weyl and Dirac semimetals. This situation can arise in the solid state either from inhomogeneous strain or nonuniform magnetization and can also be engineered in cold atomic systems. Using a semiclassical approach, we show that the resulting axial magnetic field B_{5} is observable through an enhancement of the conductivity as σ∼B_{5}^{2} due to an underlying chiral pseudomagnetic effect. We then use two lattice models to analyze the effect of B_{5} on the spectral properties of topological semimetals. We describe the emergent pseudo-Landau-level structure for different spatial profiles of B_{5}, revealing that (i) the celebrated surface states of Weyl semimetals, the Fermi arcs, can be reinterpreted as n=0 pseudo-Landau levels resulting from a B_{5} confined to the surface, (ii) as a consequence of position-momentum locking, a bulk B_{5} creates pseudo-Landau levels interpolating in real space between Fermi arcs at opposite surfaces, and (iii) there are equilibrium bound currents proportional to B_{5} that average to zero over the sample, which are the analogs of bound currents in magnetic materials. We conclude by discussing how our findings can be probed experimentally.

We construct and classify chiral topological phases in driven (Floquet) systems of strongly interacting bosons, with finite-dimensional site Hilbert spaces, in two spatial dimensions. The construction proceeds by introducing exactly soluble models with chiral edges, which in the presence of many-body localization (MBL) in the bulk are argued to lead to stable chiral phases. These chiral phases do not require any symmetry and in fact owe their existence to the absence of energy conservation in driven systems. Surprisingly, we show that they are classified by a quantized many-body index, which is well defined for any MBL Floquet system. The value of this index, which is always the logarithm of a positive rational number, can be interpreted as the entropy per Floquet cycle pumped along the edge, formalizing the notion of quantum-information flow. We explicitly compute this index for specific models and show that the nontrivial topology leads to edge thermalization, which provides an interesting link between bulk topology and chaos at the edge. We also discuss chiral Floquet phases in interacting fermionic systems and their relation to chiral bosonic phases.

The Lieb-Schultz-Mattis theorem and its higher-dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases with on-site unitary symmetries, enables us to develop a framework for understanding the structure of symmetry-enriched topological phases with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a “spinon” excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of “anyonic spin-orbit coupling,” which may arise from the interplay of translational and on-site symmetries. These include the possibility of on-site symmetry defect branch lines carrying topological charge per unit length and lattice dislocations inducing degeneracies protected by on-site symmetry.

Periodic driving of a quantum system can enable new topological phases with no analog in static systems. In this paper, we systematically classify one-dimensional topological and symmetry-protected topological (SPT) phases in interacting fermionic and bosonic quantum systems subject to periodic driving, which we dub Floquet SPTs (FSPTs). For physical realizations of interacting FSPTs, many-body localization by disorder is a crucial ingredient, required to obtain a stable phase that does not catastrophically heat to infinite temperature. We demonstrate that 1D bosonic and fermionic FSPT phases are classified by the same criteria as equilibrium phases but with an enlarged symmetry group G[over ˜], which now includes discrete time translation symmetry associated with the Floquet evolution. In particular, 1D bosonic FSPTs are classified by projective representations of the enlarged symmetry group H^{2}(G[over ˜],U(1)). We construct explicit lattice models for a variety of systems and then formalize the classification to demonstrate the completeness of this construction. We advocate that a prototypical Z_{2} bosonic FSPT may be realized by very simple Hamiltonians of the type currently available in existing cold atoms and trapped ion experiments.

The half-filled Landau level is expected to be approximately particle-hole symmetric, which requires an extension of the Halperin-Lee-Read (HLR) theory of the compressible state observed at this filling. Recent work indicates that, when particle-hole symmetry is preserved, the composite fermions experience a quantized π-Berry phase upon winding around the composite Fermi surface, analogous to Dirac fermions at the surface of a 3D topological insulator. In contrast, the effective low-energy theory of the composite fermion liquid originally proposed by HLR lacks particle-hole symmetry and has vanishing Berry phase. In this paper, we explain how thermoelectric transport measurements can be used to test the Dirac nature of the composite fermions by quantitatively extracting this Berry phase. First, we point out that longitudinal thermopower (Seebeck effect) is nonvanishing because of the unusual nature of particle-hole symmetry in this context and is not sensitive to the Berry phase. In contrast, we find that off-diagonal thermopower (Nernst effect) is directly related to the topological structure of the composite Fermi surface, vanishing for zero Berry phase and taking its maximal value for π Berry phase. In contrast, in purely electrical transport signatures, the Berry phase contributions appear as small corrections to a large background signal, making the Nernst effect a promising diagnostic of the Dirac nature of composite fermions.

It is well known that unitary symmetries can be “gauged,” i.e., defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge-conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an antiunitary symmetry. Here, we discuss a route to gauging time-reversal symmetry that applies to gapped quantum ground states that admit a tensor network representation. The tensor network representation of quantum states provides a notion of locality for the wave function coefficient and hence a notion of locality for the action of complex conjugation in antiunitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time-reversal symmetry twists that act as gauge fluxes through nontrivial loops in the system. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain time-reversal symmetric topological phases in D=1, 2 are readily extracted using these ideas.

In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain “anomalous” SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase. In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H^{4}(G,U(1)), which also precisely labels the set of 3D SPT phases, with symmetry group G. An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid [U(1)_{2}] topological order with a reduced symmetry Z_{2}×Z_{2}⊂SO(3), which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics. Possible extensions to antiunitary symmetries are also discussed.

Three-dimensional (3D) topological superconductors (TScs) protected by time-reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free-fermion phases with this symmetry (class DIII) are indexed by an integer ν, of which ν=1 is realized by the B phase of superfluid ^{3}He. Previously, it was believed that the surface must be gapless unless time-reversal symmetry is broken. Here, we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd ν TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electronlike excitations, SO(3)_{6}, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance that, however, is broken in any two-dimensional realization. By explicitly constructing an exactly soluble Walker-Wang model, we show that it can be realized at the surface of a short-ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons transforming as Kramers pairs, i.e. T^{2}=-1 under time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even ν TSc surface, for which an explicit model is derived using a coupled-layer construction. We argue that this is related to the ν=2 TSc, and we use this to build candidate surface topological orders for ν=4 and ν=8 TScs. The latter is equivalent to the three-fermion state, which is the surface topological order of a Z_{2} bosonic topological phase protected by T invariance. One particular consequence of this equivalence is that a ν=16 TSc admits a trivially gapped T-symmetric surface.

We discuss physical properties of “integer” topological phases of bosons in D=3+1 dimensions, protected by internal symmetries like time reversal and/or charge conservation. These phases invoke interactions in a fundamental way but do not possess topological order; they are bosonic analogs of free-fermion topological insulators and superconductors. While a formal cohomology-based classification of such states was recently discovered, their physical properties remain mysterious. Here, we develop a field-theoretic description of several of these states and show that they possess unusual surface states, which, if gapped, must either break the underlying symmetry or develop topological order. In the latter case, symmetries are implemented in a way that is forbidden in a strictly two-dimensional theory. While these phases are the usual fate of the surface states, exotic gapless states can also be realized. For example, tuning parameters can naturally lead to a deconfined quantum critical point or, in other situations, to a fully symmetric vortex metal phase. We discuss cases where the topological phases are characterized by a quantized magnetoelectric response θ, which, somewhat surprisingly, is an odd multiple of 2π. Two different surface theories are shown to capture these phenomena: The first is a nonlinear sigma model with a topological term. The second invokes vortices on the surface that transform under a projective representation of the symmetry group. We identify a bulk-field theory consistent with these properties, which is a multicomponent background-field theory supplemented, crucially, with a topological term. We also provide bulk sigma-model field theories of these phases and discuss a possible topological phase characterized by the thermal analog of the magnetoelectric effect.

Quantum spin liquids (QSLs) are phases of interacting spins that do not order even at the absolute zero temperature, making it impossible to characterize them by a local order parameter. In this paper, we review the unique view provided by the quantum entanglement on QSLs. We illustrate the crucial role of topological entanglement entropy in diagnosing the non-local order in QSLs, using specific examples such as the chiral spin liquid. We also demonstrate the detection of anyonic quasi-particles and their braiding statistics using quantum entanglement. In the context of gapless QSLs, we discuss the detection of emergent fermionic spinons in a bosonic wavefunction, by studying the size dependence of entanglement entropy.