We present a scalable architecture for the exploration of interacting topological phases of photons in arrays of microwave cavities, using established techniques from cavity and circuit quantum electrodynamics. A time-reversal symmetry-breaking (nonreciprocal) flux is induced by coupling the microwave cavities to ferrites, allowing for the production of a variety of topological band structures including the $\alpha =1/4$ Hofstadter model. To induce photon-photon interactions, the cavities are coupled to superconducting qubits; we find these interactions are sufficient to stabilize a $\nu =1/2$ bosonic Laughlin puddle. Exact diagonalization studies demonstrate that this architecture is robust to experimentally achievable levels of disorder. These advances provide an exciting opportunity to employ the quantum circuit toolkit for the exploration of strongly interacting topological materials.

The interface between the two complex oxides ${\mathrm{LaAlO}}_3$ and ${\mathrm{SrTiO}}_3$ has remarkable properties that can be locally reconfigured between conducting and insulating states using a conductive atomic force microscope. Prior investigations of “sketched” quantum dot devices revealed a phase in which electrons form pairs, implying a strongly attractive electron-electron interaction. Here, we show that these devices with strong electron-electron interactions can exhibit a gate-tunable transition from a pair-tunneling regime to a single-electron (Andreev bound state) tunneling regime where the interactions become repulsive. The electron-electron interaction sign change is associated with a Lifshitz transition where the $d_{xz}$ and $d_{yz}$ bands start to become occupied. This electronically tunable electron-electron interaction, combined with the nanoscale reconfigurability of this system, provides an interesting starting point towards solid-state quantum simulation.

Interfaces between correlated complex oxides are promising avenues to realize new forms of magnetism that arise as a result of charge transfer, proximity effects, and locally broken symmetries. We report on the discovery of a noncollinear magnetic structure in superlattices of the ferromagnetic metallic oxide ${\mathrm{La}}_{2/3}{\mathrm{Sr}}_{1/3}{\mathrm{MnO}}_3$ (LSMO) and the correlated metal ${\mathrm{LaNiO}}_3$ (LNO). The exchange interaction between LSMO layers is mediated by the intervening LNO, such that the angle between the magnetization of neighboring LSMO layers varies in an oscillatory manner with the thickness of the LNO layer. The magnetic field, temperature, and spacer thickness dependence of the noncollinear structure are inconsistent with the bilinear and biquadratic interactions that are used to model the magnetic structure in conventional metallic multilayers. A model that couples the LSMO layers to a helical spin state within the LNO fits the observed behavior. We propose that the spin-helix results from the interaction between a spatially varying spin susceptibility within the LNO and interfacial charge transfer that creates localized ${\mathrm{Ni}}^{2+}$ states. Our work suggests a new approach to engineering noncollinear spin textures in metallic oxide heterostructures.

Entanglement, and, in particular, the entanglement spectrum, plays a major role in characterizing many-body quantum systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to date because of the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of many-body states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian and is based on the ability to produce several copies of the state under investigation, together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) Bose-Hubbard model where such a measurement protocol reveals topological features of the Haldane phase.

Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to study several important features of the unique state and subspace cases. We quantify how subspaces retain information about initial states and show how to use Lindbladians to simulate any quantum channels. We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected subspaces. We derive a Kubo formula governing linear response of subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zero-frequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either Hamiltonian or jump-operator perturbations or corrections to adiabatic evolution, is different from the conventional Lindbladian dissipative gap and, in certain cases, is equivalent to the excitation gap of a related Hamiltonian.

Strongly driving a two-level quantum system with light leads to a ladder of Floquet states separated by the photon energy. Nanoscale quantum devices allow the interplay of confined electrons, phonons, and photons to be studied under strong driving conditions. Here, we show that a single electron in a periodically driven double quantum dot functions as a “Floquet gain medium,” where population imbalances in the double quantum dot Floquet quasienergy levels lead to an intricate pattern of gain and loss features in the cavity response. We further measure a large intracavity photon number $n_c$ in the absence of a cavity drive field, due to equilibration in the Floquet picture. Our device operates in the absence of a dc current—one and the same electron is repeatedly driven to the excited state to generate population inversion. These results pave the way to future studies of nonclassical light and thermalization of driven quantum systems.

It is now well established that photonic systems can exhibit topological energy bands. Similar to their electronic counterparts, this leads to the formation of chiral edge modes which can be used to transmit light in a manner that is protected against backscattering. While it is understood how classical signals can propagate under these conditions, it is an outstanding important question how the quantum vacuum fluctuations of the electromagnetic field get modified in the presence of a topological band structure. We address this challenge by exploring a setting where a nonzero topological invariant guarantees the presence of a parametrically unstable chiral edge mode in a system with boundaries, even though there are no bulk-mode instabilities. We show that one can exploit this to realize a topologically protected, quantum-limited traveling wave parametric amplifier. The device is naturally protected against both internal losses and backscattering; the latter feature is in stark contrast to standard traveling wave amplifiers. This adds a new example to the list of potential quantum devices that profit from topological transport.

The physics of nematic liquid crystals has been the subject of intensive research since the late 19th century. However, the focus of this pursuit has been centered around uniaxial and biaxial nematics associated with constituents bearing a $D_{\infty h}$ or $D_{2h}$ symmetry, respectively. In view of general symmetries, however, these are singularly special since nematic order can in principle involve any point-group symmetry. Given the progress in tailoring nanoparticles with particular shapes and interactions, this vast family of “generalized nematics” might become accessible in the laboratory. Little is known because the order parameter theories associated with the highly symmetric point groups are remarkably complicated, involving tensor order parameters of high rank. Here, we show that the generic features of the statistical physics of such systems can be studied in a highly flexible and efficient fashion using a mathematical tool borrowed from high-energy physics: discrete non-Abelian gauge theory. Explicitly, we construct a family of lattice gauge models encapsulating nematic ordering of general three-dimensional point-group symmetries. We find that the most symmetrical generalized nematics are subjected to thermal fluctuations of unprecedented severity. As a result, novel forms of fluctuation phenomena become possible. In particular, we demonstrate that a vestigial phase carrying no more than chiral order becomes ubiquitous departing from high point-group symmetry chiral building blocks, such as $I$, $O$, and $T$ symmetric matter.

Dirac and Weyl semimetals form an ideal platform for testing ideas developed in high-energy physics to describe massless relativistic particles. One such quintessentially field-theoretic idea of the chiral anomaly already resulted in the prediction and subsequent observation of the pronounced negative magnetoresistance in these novel materials for parallel electric and magnetic fields. Here, we predict that the chiral anomaly occurs—and has experimentally observable consequences—when real electromagnetic fields $\mathit{E}$ and $\mathit{B}$ are replaced by strain-induced pseudo-electromagnetic fields $\mathit{e}$ and $\mathit{b}$. For example, a uniform pseudomagnetic field $\mathit{b}$ is generated when a Weyl semimetal nanowire is put under torsion. In accordance with the chiral anomaly equation, we predict a negative contribution to the wire resistance proportional to the square of the torsion strength. Remarkably, left- and right-moving chiral modes are then spatially segregated to the bulk and surface of the wire forming a “topological coaxial cable.” This produces hydrodynamic flow with potentially very long relaxation time. Another effect we predict is the ultrasonic attenuation and electromagnetic emission due to a time-periodic mechanical deformation causing pseudoelectric field $\mathit{e}$. These novel manifestations of the chiral anomaly are most striking in the semimetals with a single pair of Weyl nodes but also occur in Dirac semimetals such as ${\mathrm{Cd}}_3{\mathrm{As}}_2$ and ${\mathrm{Na}}_3\mathrm{Bi}$ and Weyl semimetals with unbroken time-reversal symmetry.

We use a combination of experimental techniques to demonstrate a general occurrence of spin-orbit interaction (SOI) in graphene on transition metal dichalcogenide (TMD) substrates. Our measurements indicate that SOI is ultrastrong and extremely robust, despite it being merely interfacially induced, with neither graphene nor the TMD substrates changing their structure. This is found to be the case irrespective of the TMD material used, of the transport regime, of the carrier type in the graphene band, or of the thickness of the graphene multilayer. Specifically, we perform weak antilocalization (WAL) measurements as the simplest and most general diagnostic of SOI, and we show that the spin relaxation time is very short (approximately 0.2 ps or less) in all cases regardless of the elastic scattering time, whose value varies over nearly 2 orders of magnitude. Such a short spin-relaxation time strongly suggests that the SOI originates from a modification of graphene band structure. We confirmed this expectation by measuring a gate-dependent beating, and a corresponding frequency splitting, in the low-field Shubnikov–de Haas magnetoresistance oscillations in high-quality bilayer graphene devices on ${\mathrm{WSe}}_2$. These measurements provide an unambiguous diagnostic of a SOI-induced splitting in the electronic band structure, and their analysis allows us to determine the SOI coupling constants for the Rashba term and the so-called spin-valley coupling term, i.e., the terms that were recently predicted theoretically for interface-induced SOI in graphene. The magnitude of the SOI splitting is found to be on the order of 10 meV, more than 100 times greater than the SOI intrinsic to graphene. Both the band character of the interfacially induced SOI and its robustness and large magnitude make graphene-on-TMD a promising system to realize and explore a variety of spin-dependent transport phenomena, such as, in particular, spin-Hall and valley-Hall topological insulating states.

The coupling between lattice and charge degrees of freedom in condensed matter materials is ubiquitous and can often result in interesting properties and ordered phases, including conventional superconductivity, charge-density wave order, and metal-insulator transitions. Angle-resolved photoemission spectroscopy and both neutron and nonresonant x-ray scattering serve as effective probes for determining the behavior of appropriate, individual degrees of freedom—the electronic structure and lattice excitation, or phonon dispersion, respectively. However, each provides less direct information about the mutual coupling between the degrees of freedom, usually through self-energy effects, which tend to renormalize and broaden spectral features precisely where the coupling is strong, impacting one’s ability to quantitatively characterize the coupling. Here, we demonstrate that resonant inelastic x-ray scattering, or RIXS, can be an effective tool to directly determine the relative strength and momentum dependence of the electron-phonon coupling in condensed matter systems. Using a diagrammatic approach for an eight-band model of copper oxides, we study the contributions from the lowest-order diagrams to the full RIXS intensity for a realistic scattering geometry, accounting for matrix element effects in the scattering cross section, as well as the momentum dependence of the electron-phonon coupling vertex. A detailed examination of these maps offers a unique perspective into the characteristics of electron-phonon coupling, which complements both neutron and nonresonant x-ray scattering, as well as Raman and infrared conductivity.

Polar catastrophe at the interface of oxide materials with strongly correlated electrons has triggered a flurry of new research activities. The expectations are that the design of such advanced interfaces will become a powerful route to engineer devices with novel functionalities. Here, we investigate the initial stages of growth and the electronic structure of the spintronic ${\mathrm{Fe}}_3{\mathrm{O}}_4/\mathrm{MgO}(001)$ interface. Using soft x-ray absorption spectroscopy, we have discovered that the so-called A-sites are completely missing in the first ${\mathrm{Fe}}_3{\mathrm{O}}_4$ monolayer. This discovery allows us to develop an unexpected but elegant growth principle in which, during deposition, the Fe atoms are constantly on the move to solve the divergent electrostatic potential problem, thereby ensuring epitaxy and stoichiometry at the same time. This growth principle provides a new perspective for the design of interfaces.

Thermal conductivity in dielectric crystals is the result of the relaxation of lattice vibrations described by the phonon Boltzmann transport equation. Remarkably, an exact microscopic definition of the heat carriers and their relaxation times is still missing: Phonons, typically regarded as the relevant excitations for thermal transport, cannot be identified as the heat carriers when most scattering events conserve momentum and do not dissipate heat flux. This is the case for two-dimensional or layered materials at room temperature, or three-dimensional crystals at cryogenic temperatures. In this work, we show that the eigenvectors of the scattering matrix in the Boltzmann equation define collective phonon excitations, which are termed here “relaxons”. These excitations have well-defined relaxation times, directly related to heat-flux dissipation, and they provide an exact description of thermal transport as a kinetic theory of the relaxon gas. We show why Matthiessen’s rule is violated, and we construct a procedure for obtaining the mean free paths and relaxation times of the relaxons. These considerations are general and would also apply to other semiclassical transport models, such as the electronic Boltzmann equation. For heat transport, they remain relevant even in conventional crystals like silicon, but they are of the utmost importance in the case of two-dimensional materials, where they can revise, by several orders of magnitude, the relevant time and length scales for thermal transport in the hydrodynamic regime.

Herbertsmithite, ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$, is a two-dimensional kagome lattice realization of a spin liquid, with evidence for fractionalized excitations and a gapped ground state. Such a quantum spin liquid has been proposed to underlie high-temperature superconductivity and is predicted to produce a wealth of new states, including a Dirac metal at $1/3$ electron doping. Here, we report the topochemical synthesis of electron-doped ${\mathrm{ZnLi}}_x{\mathrm{Cu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ from $x=0$ to $x=1.8$ ($3/5$ per ${\mathrm{Cu}}^{2+}$). Contrary to expectations, no metallicity or superconductivity is induced. Instead, we find a systematic suppression of magnetic behavior across the phase diagram. Our results demonstrate that significant theoretical work is needed to understand and predict the role of doping in magnetically frustrated narrow band insulators, particularly the interplay between local structural disorder and tendency toward electron localization, and pave the way for future studies of doped spin liquids.

We introduce a method, dubbed the flux-fusion anomaly test, to detect certain anomalous symmetry fractionalization patterns in two-dimensional symmetry-enriched topological (SET) phases. We focus on bosonic systems with ${\mathbb{Z}}_2$ topological order and a symmetry group of the form $G=\mathrm{U}(1)⋊G^{\prime }$, where $G^{\prime }$ is an arbitrary group that may include spatial symmetries and/or time reversal. The anomalous fractionalization patterns we identify cannot occur in strictly $d=2$ systems but can occur at surfaces of $d=3$ symmetry-protected topological (SPT) phases. This observation leads to examples of $d=3$ bosonic topological crystalline insulators (TCIs) that, to our knowledge, have not previously been identified. In some cases, these $d=3$ bosonic TCIs can have an anomalous superfluid at the surface, which is characterized by nontrivial projective transformations of the superfluid vortices under symmetry. The basic idea of our anomaly test is to introduce fluxes of the U(1) symmetry and to show that some fractionalization patterns cannot be extended to a consistent action of $G^{\prime }$ symmetry on the fluxes. For some anomalies, this can be described in terms of dimensional reduction to $d=1$ SPT phases. We apply our method to several different symmetry groups with nontrivial anomalies, including $G=\mathrm{U}(1)\times {\mathbb{Z}}_2^T$ and $G=\mathrm{U}(1)\times {\mathbb{Z}}_2^P$, where ${\mathbb{Z}}_2^T$ and ${\mathbb{Z}}_2^P$ are time-reversal and $d=2$ reflection symmetry, respectively.

Topological phases of matter are a potential platform for the storage and processing of quantum information with intrinsic error rates that decrease exponentially with inverse temperature and with the length scales of the system, such as the distance between quasiparticles. However, it is less well understood how error rates depend on the speed with which non-Abelian quasiparticles are braided. In general, diabatic corrections to the holonomy or Berry’s matrix vanish at least inversely with the length of time for the braid, with faster decay occurring as the time dependence is made smoother. We show that such corrections will not affect quantum information encoded in topological degrees of freedom, unless they involve the creation of topologically nontrivial quasiparticles. Moreover, we show how measurements that detect unintentionally created quasiparticles can be used to control this source of error.

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 $\tilde{G}$, 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 $H^2\mathbf{(}\tilde{G},U(1)\mathbf{)}$. 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 ${\mathbb{Z}}_2$ bosonic FSPT may be realized by very simple Hamiltonians of the type currently available in existing cold atoms and trapped ion experiments.

We study the effects of strain on the properties and dynamics of Wannier excitons in monolayer (phosphorene) and few-layer black phosphorus (BP), a promising two-dimensional material for optoelectronic applications due to its high mobility, mechanical strength, and strain-tunable direct band gap. We compare the results to the case of molybdenum disulphide (${\mathrm{MoS}}_2$) monolayers. We find that the so-called funnel effect, i.e., the possibility of controlling exciton motion by means of inhomogeneous strains, is much stronger in few-layer BP than in ${\mathrm{MoS}}_2$ monolayers and, crucially, is of opposite sign. Instead of excitons accumulating isotropically around regions of high tensile strain like in ${\mathrm{MoS}}_2$, excitons in BP are pushed away from said regions. This inverse funnel effect is moreover highly anisotropic, with much larger funnel distances along the armchair crystallographic direction, leading to a directional focusing of exciton flow. A strong inverse funnel effect could enable simpler designs of funnel solar cells and offer new possibilities for the manipulation and harvesting of light.

Recent improvements in the control of quantum systems make it seem feasible to finally build a quantum computer within a decade. While it has been shown that such a quantum computer can in principle solve certain small electronic structure problems and idealized model Hamiltonians, the highly relevant problem of directly solving a complex correlated material appears to require a prohibitive amount of resources. Here, we show that by using a hybrid quantum-classical algorithm that incorporates the power of a small quantum computer into a framework of classical embedding algorithms, the electronic structure of complex correlated materials can be efficiently tackled using a quantum computer. In our approach, the quantum computer solves a small effective quantum impurity problem that is self-consistently determined via a feedback loop between the quantum and classical computation. Use of a quantum computer enables much larger and more accurate simulations than with any known classical algorithm, and will allow many open questions in quantum materials to be resolved once a small quantum computer with around 100 logical qubits becomes available.

We show how particle-vortex duality in $d=2+1$ dimensions arises as part of an intricate web of relationships between different field theories. The starting point is “bosonization,” a conjectured duality that uses flux attachment to transmute the statistics of relativistic particles. From this seed, we derive many old and new dualities. These include particle-vortex duality for bosons as well as the recently discovered counterpart for fermions.