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In twisted van der Waals materials, local atomic relaxation can significantly alter the underlying electronic structure and properties. Characterizing the lattice reconstruction and the impact of strain is essential to better understand and harness the resulting emergent electronic states. Here, we use a scanning single-electron transistor to image spatial modulations in the electronic structure of helical trilayer graphene, demonstrating relaxation into a superstructure of large domains with uniform moire periodicity. We further show that the supermoire domain size is enhanced by strain and can even be altered in subsequent measurements of the same device, while nevertheless maintaining the same local electronic properties within each domain. Finally, we observe higher conductance at the boundaries between domains, consistent with the prediction that they host counter-propagating topological edge modes. Our work confirms that lattice relaxation can produce moire-periodic order in twisted multilayers, demonstrates strain-engineering as a viable path for designing topological networks at the supermoire scale, and paves the way to direct imaging of correlation-driven topological phases and boundary modes.

We propose fractional anomalous Hall crystals (FAHCs) as possible ground states of strongly interacting electrons in parent bands with Berry curvature. FAHCs are exotic states of matter that spontaneously break continuous translation symmetry to form a fractional Chern insulator. We construct a unified family of variational wavefunctions that describe FAHCs and their competing states in the presence of uniform parent Berry curvature. We calculate their variational energy with Coulomb interactions semi-analytically in the thermodynamic limit. Our analysis reveals that FAHCs can be energetically favorable over both Wigner crystals and integer anomalous Hall crystals for sufficiently strong interactions or flat dispersion.

Manipulating solids using the time-periodic drive of a laser pulse is a promising route to generate new phases of matter. Whether such `Floquet-Bloch' manipulation can be achieved in topological magnetic systems with disorder has so far been unclear. In this work, we realize Floquet-Bloch manipulation of the Dirac surface-state mass of the topological antiferromagnet (AFM) MnBi$_2$Te$_4$. Using time- and angle-resolved photoemission spectroscopy (tr-ARPES), we show that opposite helicities of mid-infrared circularly polarized light result in substantially different Dirac mass gaps in the AFM phase, despite the equilibrium Dirac cone being massless. We explain our findings in terms of a Dirac fermion with a random mass. Our results underscore Floquet-Bloch manipulation as a powerful tool for controlling topology even in the presence of disorder, and for uncovering properties of materials that may elude conventional probes.

We study the edges of fractional quantum spin Hall insulators (FQSH) with half-integer spin Hall conductance. These states can be viewed as symmetric combinations of a spin-up and spin-down half-integer fractional quantum Hall state (FQH) that are time-reversal invariant, and conserve the z-component of spin. We consider the non-Abelian states based on the Pfaffian, anti-Pfaffian, PH-Pfaffian, and 221 FQH, and generic Abelian FQH. For strong enough spin-conserving interactions, we find that all the non-Abelian and Abelian edges flow to the same fixed point that consists of a single pair of charged counter-propagating bosonic modes. If spin-conservation is broken, the Abelian edge can be fully gapped in a time-reversal symmetric fashion. The non-Abelian edge with broken spin-conservation remains gapless due to time-reversal symmetry, and can flow to a new fixed point with a helical gapless pair of Majorana fermions. We discuss the possible relevance of our results to the recent observation of a half-integer edge conductance in twisted MoTe2., Comment: 9 pages

Three-dimensional Dirac semimetals can be driven into an insulating state by coupling to a charge density wave (CDW) order. Here, we consider the quantized crystalline responses of such charge-ordered Dirac semimetals, which we dub Dirac-CDW insulators, in which charge is bound to disclination defects of the lattice. Using analytic and numeric methods we show the following. First, when the CDW is lattice-commensurate, disclination-line defects of the lattice have a quantized charge per length. Second, when the CDW is inversion-symmetric, disclinations of the lattice have a quantized electric polarization. Third, when the CDW is lattice-commensurate and inversion-symmetric, disclinations are characterized by a "disclination filling anomaly" -- a quantized difference in the total charge bound to disclination-lines of Dirac-CDW with open and periodic boundaries. We construct an effective response theory that captures the topological responses of the Dirac-CDW insulators in terms of a total derivative term, denoted the $R\wedge F$ term. The $R\wedge F$ term describes the crystalline analog of the axion electrodynamics that are found in Weyl semimetal-CDW insulators. We also use the crystalline responses and corresponding response theories to classify the strongly correlated topological phases of three-dimensions Dirac-semimetals.

We study the effects of quasiperiodicity on the stability of conventional and unconventional superconductors. Quasiperiodicity is modelled using the three-dimensional Aubry-Andre model, a system in which electrons are coupled to a translation-symmetry-breaking potential that is incommensurate with the underlying lattice. Upon increasing the strength of the quasiperiodic potential, the single-particle eigenstates undergo a transition from a ballistic to a diffusive character. Here, we study the instability of the model towards superconductivity. We find that in the ballistic regime, the system is unstable towards both $s$-wave and $p$-wave superconductivity. In contrast, only the conventional $s$-wave instability survives in the intermediate diffusive regime. Our results suggest a version of Anderson's theorem for quasiperiodic systems, relating the normal state dynamics to the stability of conventional and unconventional superconductivity. These findings are relevant vis-a-vis recent studies of superconductivity in quasiperiodic moire structures., Comment: 5 pages, 4 figures

We systematically explore the structural and electronic properties of twisted trilayer graphene systems. In general, these systems are characterized by two twist angles, which lead to two incommensurate moir\'{e} periods. We show that lattice relaxation results in the formation of domains of periodic single-moir\'{e} structures only for twist angles close to the simplest fractions. For the majority of other twist angles, the incommensurate moir\'{e} periods lead to a quasicrystalline structure. We identify experimentally relevant magic angles at which the electronic density of states is sharply peaked and strongly correlated physics is most likely to be realized., Comment: 11+7 pages, 10 figures

Determining the symmetry breaking order of correlated quantum phases is essential for understanding the microscopic interactions in their host systems. The flat bands in magic angle twisted bilayer graphene (MATBG) provide an especially rich arena to investigate such interaction-driven ground states, and while progress has been made in identifying the correlated insulators and their excitations at commensurate moire filling factors, the spin-valley polarizations of the topological states that emerge at high magnetic field remain unknown. Here we introduce a new technique based on twist-decoupled van der Waals layers that enables measurements of their electronic band structure and, by studying the backscattering between counter-propagating edge states, determination of relative spin polarization of the their edge modes. Applying this method to twist-decoupled MATBG and monolayer graphene, we find that the broken-symmetry quantum Hall states that extend from the charge neutrality point in MATBG are spin-unpolarized at even integer filling factors. The measurements also indicate that the correlated Chern insulator emerging from half filling of the flat valence band is spin-unpolarized, but suggest that its conduction band counterpart may be spin-polarized. Our results constrain models of spin-valley ordering in MATBG and establish a versatile approach to study the electronic properties of van der Waals systems.

$GdTe_{3}$ is a layered antiferromagnet which has attracted attention due to its exceptionally high mobility, distinctive unidirectional incommensurate charge density wave (CDW), superconductivity under pressure, and a cascade of magnetic transitions between 7 and 12 K, with as yet unknown order parameters. Here, we use spin-polarized scanning tunneling microscopy to directly image the charge and magnetic orders in $GdTe_{3}$. Below 7 K, we find a striped antiferromagnetic phase with twice the periodicity of the Gd lattice and perpendicular to the CDW. As we heat the sample, we discover a spin density wave with the same periodicity as the CDW between 7 and 12 K; the viability of this phase is supported by our Landau free energy model. Our work reveals the order parameters of the magnetic phases in $GdTe_{3}$ and shows how the interplay between charge and spin can generate a cascade of magnetic orders., Comment: 46 pgs.; 4 main figures, 20 supplementary figures

Topological defects are singularities in an ordered phase that can have a profound effect on phase transitions and serve as a window into the order parameter. In this work we use scanning tunneling microscopy to visualize the role of topological defects in the novel magnetic field induced disappearance of an intertwined charge density wave (CDW) in the heavy fermion superconductor, UTe2. By simultaneously imaging the amplitude and phase of the CDW order, we reveal pairs of topological defects with positive and negative phase winding. The pairs are directly correlated with a zero CDW amplitude and increase in number with increasing magnetic field. These observations can be captured by a Ginzburg Landau model of a uniform superconductor coexisting with a pair density wave. A magnetic field generates vortices of the superconducting and pair density wave order which can create topological defects in the CDW and induce the experimentally observed melting of the CDW at the upper critical field. Our work reveals the important role of magnetic field generated topological defects in the melting the CDW order parameter in UTe2 and provides support for the existence of a parent pair density wave order on the surface of UTe2., Comment: 29 Pages, includes manuscript and supplemental information, 4 Main figures and 8 Supplemental Figures

This work investigates the out-of-equilibrium dynamics of dipole and higher-moment conserving systems with long-range interactions, drawing inspiration from trapped ion experiments in strongly tilted potentials. We introduce a hierarchical sequence of multipole-conserving models characterized by power-law decaying couplings. Although the moments are always globally conserved, adjusting the power-law exponents of the couplings induces various regimes in which only a subset of multipole moments are effectively locally conserved. We examine the late-time hydrodynamics analytically and numerically using an effective classical framework, uncovering a rich dynamical phase diagram that includes subdiffusion, conventional diffusion, and L\'evy flights. Our results are unified in an analytic reciprocal relationship that captures the nested hierarchy of hydrodynamics in multipole conserving systems where only a subset of the moments are locally conserved. Moreover, we extend our findings to higher dimensions and explore the emergence of long-time scales, reminiscent of pre-thermal regimes, in systems with low charge density. Lastly, we corroborate our results through state-of-the-art numerical simulations of a fully quantum long-range dipole-conserving system and discuss their relevance to trapped-ion experimental setups., Comment: 16+3 pages, 9 figures

Quantized responses are important tools for understanding and characterizing the universal features of topological phases of matter. In this work, we consider a class of topological crystalline insulators in $3$D with $C_n$ lattice rotation symmetry along a fixed axis, in addition to either mirror symmetry or particle-hole symmetry. These insulators can realize quantized mixed geometry-charge responses. When the surface of these insulators is gapped, disclinations on the surface carry a fractional charge that is half the minimal amount that can occur in purely $2$D systems. Similarly, disclination lines in the bulk carry a fractionally quantized electric polarization. These effects, and other related phenomena, are captured by a $3$D topological response term that couples the lattice curvature to the electromagnetic field strength. Additionally, mirror symmetric insulators with this response can be smoothly deformed into a higher-order octopole insulator with quantized corner charges. We also construct a symmetry indicator form for the topological invariant that describes the quantized response of the mirror symmetric topological crystalline insulators, and discuss an unusual response quantization in time-reversal breaking systems., Comment: 26 pages + appendices, 12 figures

The intense interest in triplet superconductivity partly stems from theoretical predictions of exotic excitations such as non-abelian Majorana modes, chiral supercurrents, and half-quantum vortices. However, fundamentally new, and unexpected states may emerge when triplet superconductivity appears in a strongly correlated system. In this work we use scanning tunneling microscopy to reveal an unusual charge density wave (CDW) order in the heavy fermion triplet superconductor, UTe2. Our high-resolution maps reveal a multi-component incommensurate CDW whose intensity get weaker with increasing field, eventually disappearing at the superconducting critical field, Hc2. To explain the origin and phenomenology of this unusual CDW, we construct a Ginzburg-Landau theory for a uniform triplet superconductor coexisting with three triplet pair density wave (PDW) states. This theory gives rise to daughter CDWs which would be sensitive to magnetic field due to their origin in a triplet PDW state, and naturally explains our data. Our discovery of a CDW sensitive to magnetic fields and strongly intertwined with superconductivity, provides important new information for understanding the order parameter of UTe2 and uncovers the possible existence of a new kind of triplet PDW order which has not been previously explored., Comment: 12 pages, 4 figures, Supplementary information

In this work, we present a collection of three-dimensional higher-order symmetry protected topological phases (HOSPTs) with gapless hinge modes that exist only in strongly interacting systems subject to subsystem symmetry constraints. We use a coupled wire construction to generate three families of microscopic lattice models: insulators with helical hinge modes, superconductors with chiral Majorana hinge modes, and fractionalized insulators with helical hinge modes that carry fractional charge. In particular, these HOSPTs do not require spatial symmetry protection, but are instead protected by subsystem symmetries, and support "fractonic" quasiparticle excitations that move within only a low-dimensional sub-manifold of the system. We analyze the anomaly structure for the boundary theory and the entanglement Hamiltonian, and show that the side surfaces of these HOSPTs, despite being partially gapped, exhibit symmetry anomalies, and can only be realized as the boundary of three-dimensional HOSPT phases., Comment: 16 pages, 9 figures

Two-dimensional higher-order topological insulators can display a number of exotic phenomena such as half-integer charges localized at corners or disclination defects. In this paper, we analyze these phenomena, focusing on the paradigmatic example of the quadrupole insulator with $C_4$ rotation symmetry, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We extend this analysis to interacting systems, and predict the response of fractional quadrupole insulators, which exhibit charge $e/2(2k+1)$ bound to corners and disclinations., Comment: 11 pages, 2 figures

Higher order topological insulators (HOTIs) are a novel form of insulating quantum matter, which are characterized by having gapped boundaries that are separated by gapless corner or hinge states. Recently, it has been proposed that the essential features of a large class of HOTIs are captured by topological multipolar response theories. In this work, we show that these multipolar responses can be realized in interacting lattice models, which conserve both charge and dipole. In this work we study several models in both the strongly interacting and mean-field limits. In $2$D we consider a ring-exchange model which exhibits a quadrupole response, and can be tuned to a $C_4$ symmetric higher order topological phase with half-integer quadrupole moment, as well as half-integer corner charges. We then extend this model to develop an analytic description of adiabatic dipole pumping in an interacting lattice model. The quadrupole moment changes during this pumping process, and if the process is periodic, we show the total change in the quadrupole moment is quantized as an integer. We also consider two interacting $3$D lattice models with chiral hinge modes. We show that the chiral hinge modes are heralds of a recently proposed "dipolar Chern-Simons" response, which is related to the quadrupole response by dimensional reduction. Interestingly, we find that in the mean field limit, both the $2$D and $3$D interacting models we consider here are equivalent to known models of non-interacting HOTIs (or boundary obstructed versions). The self-consistent mean-field theory treatment provides insight into the connection between free-fermion (mean-field) theories having vanishing polarization and interacting models where dipole moments are microscopically conserved., Comment: 28 pages, 16 figures

The Kondo-Heinsberg chain is an interesting model of a strongly correlated system which has a broad superconducting state with pair-density wave (PDW) order. Some of us have recently proposed that this PDW state is a symmetry-protected topological (SPT) state, and the gapped spin sector of the model supports Majorana zero modes. In this work, we reexamine this problem using a combination of numeric and analytic methods. In extensive density matrix renormalization group calculations, we find no evidence of a topological ground state degeneracy or the previously proposed Majorana zero modes in the PDW phase of this model. This result motivated us to reexamine the original arguments for the existence of the Majorana zero modes. A careful analysis of the effective continuum field theory of the model shows that the Hilbert space of the spin sector of the theory does not contain any single Majorana fermion excitations. This analysis shows that the PDW state of the doped 1D Kondo-Heisenberg model is not an SPT with Majorana zero modes., Comment: 13 pages, 8 figures

Non-perturbative constraints on many body physics--such as the famous Lieb-Schultz-Mattis theorem--are valuable tools for studying strongly correlated systems. To this end, we present a number of non-perturbative results that constrain the low-energy physics of systems having conserved dipole moments. We find that for these systems, a unique translationally invariant gapped ground state is only possible if the polarization of the system is integer. Furthermore, if a lattice system also has $U(1)$ subsystem charge conservation symmetry, a unique gapped ground state is only possible if the particle filling along these subsystems is integer. We also apply these methods to spin systems, and determine criteria for the existence of a new type of magnetic response plateau in the presence of a non-uniform magnetic field. Finally, we formulate a version of Luttinger's theorem for 1D systems consisting of dipoles., Comment: 14 pages, 1 figure

In recent years, there have been numerous examples of twisted bilayer systems that host remarkable physical properties that are not found in their untwisted counterparts. Motivated by this, we study the properties of twisted bilayers of the Kitaev honeycomb model in the Abelian spin liquid phase. We show that for strong, short-ranged, interlayer interactions, a super-lattice of non-Abelian defects forms in the twisted bilayer system. These non-Abelian defects are wormhole-like genons that allow anyons from one layer to tunnel to the other layer. We find that when a magnetic field is applied to the system, the low energy dynamics of the twisted bilayer system can be mapped onto four quantum Ising models arising from the degrees of freedom localized on the genon defects. At small twist angles, the Ising models are in a trivial paramagnetic phase, and at large twist angles, they are in a ferromagnetic phase., Comment: 12 pages (10 figures)

Insulating systems are characterized by their insensitivity to twisted boundary conditions as quantified by the charge stiffness and charge localization length. The latter quantity was shown to be related to the expectation value of the many-body position operator and serves as a universal criterion to distinguish between metals and insulators. In this work we extend these concepts to a new class of quantum systems having conserved charge and dipole moments. We refine the concept of a charge insulator by introducing notions of multipolar insulators, e.g., a charge insulator could be a dipole insulator or dipole metal. We develop a universal criterion to distinguish between these phases by extending the concept of charge stiffness and localization to analogous versions for multipole moments, but with our focus on dipoles. We are able relate the dipole localization scale to the expectation value of a recently introduced many-body quadrupole operator. This refined structure allows for the identification of phase transitions where charge remains localized but, e.g., dipoles delocalize. We illustrate the proposed criterion using several exactly solvable models that exemplify these concepts, and discuss a possible realization in cold-atom systems., Comment: 11+4 pages, 2 figures

Subsystem symmetries are intermediate between global and gauge symmetries. One can treat these symmetries either like global symmetries that act on subregions of a system, or gauge symmetries that act on the regions transverse to the regions acted upon by the symmetry. We show that this latter interpretation can lead to an understanding of global, topology-dependent features in systems with subsystem symmetries. We demonstrate this with an exactly-solvable lattice model constructed from a 2D system of bosons coupled to a vector field with a 1D subsystem symmetry. The model is shown to host a robust ground state degeneracy that depends on the spatial topology of the underlying manifold, and localized zero energy modes on corners of the system. A continuum field theory description of these phenomena is derived in terms of an anisotropic, modified version of the Abelian K-matrix Chern-Simons field theory. We show that this continuum description can lead to geometric-type effects such as corner states and edge states whose character depends on the orientation of the edge., Comment: 14 pages, 7 figures

Some interfaces between two different topologically ordered systems can be gapped. In earlier work it has been shown that such gapped interfaces can themselves be effective one dimensional topological systems that possess localized topological modes in open boundary geometries. Here we focus on how this occurs in the context of an interface between two, single-component Laughlin states of opposite chirality, and with filling fractions $\nu_1=1/p$ and $\nu_2=1/pn^2$. While one type of interface in such systems has been previously studied, we show that allowing for edge reconstruction effects opens up a wide variety of possible gapped interfaces depending on the number of divisors of $n.$ We apply a complementary description of the $\nu_2=1/pn^2$ system in terms of Laughlin states coupled to a discrete gauge $\mathbb{Z}_n$ field. This enables us to identify possible interfaces to the $\nu_1$ system based on complete or partial confinement of this gauge field. We determine the tunneling properties, ground state degeneracy, and the nature of the non-Abelian zero modes of each interface in order to physically distinguish them., Comment: 9 pages, 4 figures

Motivated in part by the numerical simulations [ky,kosterlitz1,kosterlitz2] which reveal that the energy to create a defect in a gauge or phase glass scales as $L^{\theta}$ with $\theta<0$ for 2D, thereby implying a vanishing stiffness, we re-examine the relevance of these kinds of models to the Bose metal in light of the new experiments [kapsym,armitage] which reveal that the Hall conductance is zero in the metallic state that disrupts the transition from the superconductor to the insulator in 2D samples. Because of the particle-hole symmetry in the phase glass model, we find that bosonic excitations in a phase glass background generate no Hall conductance at the Gaussian level. Furthermore, this result persists to any order in perturbation theory in the interactions. We show that when particle-hole symmetry is broken, the Hall conductance turns on with the same power law as does the longitudinal conductance. This prediction can be verified experimentally by applying a ground plane to the 2D samples.