The quantum coherence of electronic quasiparticles underpins many of the emerging transport properties of conductors at small scales. Novel electronic implementations of quantum optics devices are now available with perspectives such as “flying-qubit” manipulations. However, electronic quantum interferences in conductors remained up to now limited to propagation paths shorter than $30\mu \mathrm{m}$ independent of the material. Here we demonstrate strong electronic quantum interferences after a propagation along two 0.1-mm-long pathways in a circuit. Interferences of visibility as high as 80% and 40% are observed on electronic analogues of the Mach-Zehnder interferometer of, respectively, $24\text{−}\mu \mathrm{m}$ and 0.1-mm arm length, consistently corresponding to a 0.25-mm electronic phase coherence length. While such devices perform best in the integer quantum Hall regime at filling factor 2, the electronic interferences are restricted by the Coulomb interaction between copropagating edge channels. We overcome this limitation by closing the inner channel in micron-scale loops of frozen internal degrees of freedom combined with a loop-closing strategy providing an essential isolation from the environment.

Prethermalization has been extensively studied in systems close to integrability. We propose a more general, yet conceptually simpler, setup for this phenomenon. We consider a—possibly nonintegrable—reference dynamics, weakly perturbed so that the perturbation breaks at least one conservation law of the reference dynamics. We argue then that the evolution of the system proceeds via intermediate (generalized) equilibrium states of the reference dynamics. The motion on the manifold of equilibrium states is governed by an autonomous equation, flowing towards global equilibrium in a time of order $g^{-2}$, where $g$ is the perturbation strength. We also describe the leading correction to the time-dependent reference equilibrium state, which is, in general, of order $g$. The theory is well confirmed in numerical calculations of model Hamiltonians, for which we use a numerical linked cluster expansion and full exact diagonalization.

Dipolar coupling plays a fundamental role in the interaction between electrically or magnetically polarized species such as magnetic atoms and dipolar molecules in a gas or dipolar excitons in the solid state. Unlike Coulomb or contactlike interactions found in many atomic, molecular, and condensed-matter systems, this interaction is long-ranged and highly anisotropic, as it changes from repulsive to attractive depending on the relative positions and orientation of the dipoles. Because of this unique property, many exotic, symmetry-breaking collective states have been recently predicted for cold dipolar gases, but only a few have been experimentally detected and only in dilute atomic dipolar Bose-Einstein condensates. Here, we report on the first observation of attractive dipolar coupling between excitonic dipoles using a new design of stacked semiconductor bilayers. We show that the presence of a dipolar exciton fluid in one bilayer modifies the spatial distribution and increases the binding energy of excitonic dipoles in a vertically remote layer. The binding energy changes are explained using a many-body polaron model describing the deformation of the exciton cloud due to its interaction with a remote dipolar exciton. The surprising nonmonotonic dependence on the cloud density indicates the important role of dipolar correlations, which is unique to dense, strongly interacting dipolar solid-state systems. Our concept provides a route for the realization of dipolar lattices with strong anisotropic interactions in semiconductor systems, which open the way for the observation of theoretically predicted new and exotic collective phases, as well as for engineering and sensing their collective excitations.

We investigate the influence of Landau quantization on the superconducting instability for a pure layered superconductor in the magnetic field directed perpendicular to the layers. We demonstrate that the quantization corrections to the Cooper-pairing kernel with finite Zeeman spin splitting promote the formation of the nonuniform state in which the order parameter is periodically modulated along the magnetic field, i.e., between the layers (Fulde-Ferrell-Larkin-Ovchinnikov [FFLO] state). The conventional uniform state experiences such a quantization-induced FFLO instability at low temperatures even in a common case of predominantly orbital suppression of superconductivity when the Zeeman spin splitting is expected to have a relatively weak effect. The maximum relative FFLO temperature is given by the ratio of the superconducting transition temperature and the Fermi energy. This maximum is realized when the ratio of the spin-spitting energy and the Landau-level separation is half-integer. These results imply that the FFLO states may exist not only in the Pauli-limited superconductors but also in very clean materials with small Zeeman spin-splitting energy. We expect that the described quantization-promoted FFLO instability is a general phenomenon, which may be found in materials with different electronic spectra and order-parameter symmetries.

The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics. However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results. Here we show that the problem of compact $\mathrm{U}(1)$ lattice gauge theory coupled to fermionic matter in $(2+1)\mathrm{D}$ is possible to access via sign-problem-free quantum Monte Carlo simulations. One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations. By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetry-breaking phases such as the valence-bond solid (VBS) and Néel antiferromagnet emerge. Deconfined phases with algebraic spin and VBS correlation functions are also observed. Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases. The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality. In particular, for four flavors $N_f=4$, our data suggest a continuous quantum phase transition between the VBS and Néel order. We also provide preliminary theoretical analysis for these quantum phase transitions.

Ultrafast photodoping of the Mott insulators, possessing strong correlation between electronic and magnetic degrees of freedom, holds promise for launching an ultrafast dynamics of spins which cannot be described in terms of conventional models of ultrafast magnetism. Here we study the ultrafast laser-induced dynamics of the magnetic order in a novel spin-orbit Mott insulator ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ featuring an uncompensated pattern of antiferromagnetic spin ordering. Using the transient magneto-optical Kerr effect sensitive to the net magnetization, we reveal that photodoping by femtosecond laser pulses with photon energy above the Mott gap launches melting of the antiferromagnetic order seen as ultrafast demagnetization with a characteristic time of 300 fs followed by a sub-10-ps recovery. Nonequilibrium dynamical mean-field theory calculations based on the single-band Hubbard model confirm that ultrafast demagnetization is primarily governed by the laser-induced generation of electron-hole pairs, although the precise simulated time dependencies are rather different from the experimentally observed ones. To describe the experimental results, here we suggest a phenomenological model which is based on Onsager’s formalism and accounts for the photogenerated electron-hole pairs using the concepts of holons and doublons.

We study the yielding behavior of a model glass under cyclic athermal quasistatic deformation and at finite rate and temperature, computationally, and show that yielding is characterized by the discontinuous appearance of shear bands, whose width is about ten particle diameters at their initiation, in which the strain gets localized. Strain localization is accompanied by a corresponding change in the energies and a decrease in the density in the shear band. We show that the glass remains well annealed outside the shear band, whereas the energies correspond to the highest possible energy minima at the given density within the shear band. Diffusive motion of particles characterizing the yielded state are confined to the shear bands, whose mean positions display movement over repeated cycles. Outside the shear band, particle motions are subdiffusive but remain finite. Despite the discontinuous nature of their appearance, shear bands are reversible in the sense that a reduction in the amplitude of cyclic deformation to values below yielding leads to the healing and disappearance of the shear bands.

We present an extensive overview of the phase diagram, spin-wave excitations, and finite-temperature transitions of the anisotropic-exchange magnets on an ideal nearest-neighbor triangular lattice. We investigate transitions between five principal classical phases of the corresponding model: ferromagnetic, Néel, its dual, and the two stripe phases. Transitions are identified by the spin-wave instabilities and by the Luttinger-Tisza approach, and we highlight the benefits of the former while outlining the shortcomings of the latter. Some of the transitions are direct and others occur via intermediate phases with more complicated forms of ordering. The spin-wave spectrum in the Néel phase is obtained and is shown to be nonreciprocal, ${ϵ}_{\alpha ,\mathbf{k}}\ne {ϵ}_{\alpha ,-\mathbf{k}}$, in the presence of anisotropic bond-dependent interactions. In a portion of the Néel phase, we find spin-wave instabilities to a long-range spiral-like state. This transition boundary is similar to that of the spin-liquid phase of the $S=1/2$ model discovered in our prior work, suggesting a possible connection between the two. Further, in the stripe phases, quantum fluctuations are mostly negligible, leaving the ordered moment nearly saturated even for the $S=1/2$ case. However, for a two-dimensional surface of the full 3D parameter space, the spin-wave spectrum in one of the stripe phases exhibits an enigmatic accidental degeneracy manifested by pseudo-Goldstone modes. As a result, despite the nearly classical ground state, the ordering transition temperature in a wide region of the phase diagram is significantly suppressed from the mean-field expectation. We identify this accidental degeneracy as due to an exact correspondence to an extended Kitaev-Heisenberg model with emergent symmetries that naturally lead to the pseudo-Goldstone modes. There are previously studied dualities within the Kitaev-Heisenberg model on the triangular lattice that are exposed here in a wider parameter space. One important implication of this correspondence for the $S=1/2$ case is the existence of a region of the spin-liquid phase that is dual to the spin-liquid phase discovered recently by us. We complement our studies by the density-matrix renormalization group of the $S=1/2$ model to confirm some of the duality relations and to verify the existence of the dual spin-liquid phase.

The results of the renormalization group are commonly advertised as the existence of power-law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions. We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.

By combining theory and experiments, we demonstrate that dipolar quantum gases of both ${}^{166}\mathrm{Er}$ and ${}^{164}\mathrm{Dy}$ support a state with supersolid properties, where a spontaneous density modulation and a global phase coherence coexist. This paradoxical state occurs in a well-defined parameter range, separating the phases of a regular Bose-Einstein condensate and of an insulating droplet array, and is rooted in the roton mode softening, on the one side, and in the stabilization driven by quantum fluctuations, on the other side. Here, we identify the parameter regime for each of the three phases. In the experiment, we rely on a detailed analysis of the interference patterns resulting from the free expansion of the gas, quantifying both its density modulation and its global phase coherence. Reaching the phases via a slow interaction tuning, starting from a stable condensate, we observe that ${}^{166}\mathrm{Er}$ and ${}^{164}\mathrm{Dy}$ exhibit a striking difference in the lifetime of the supersolid properties, due to the different atom loss rates in the two systems. Indeed, while in ${}^{166}\mathrm{Er}$ the supersolid behavior survives only a few tens of milliseconds, we observe coherent density modulations for more than 150 ms in ${}^{164}\mathrm{Dy}$. Building on this long lifetime, we demonstrate an alternative path to reach the supersolid regime, relying solely on evaporative cooling starting from a thermal gas.

We introduce a class of gapped three-dimensional models, dubbed “cage-net fracton models,” which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility. Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional “flux strings” built out of pointlike excitations. Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton-ordered (equivalently, flux-string-condensed) phase, such as its ground-state wave function and spectrum of excitations. Through the examples of doubled-Ising and ${\mathrm{SU}(2)}_k$ cage-net models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension. In the doubled-Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states among two-dimensional non-Abelian anyons and Abelian particles. We further show that the ground-state wave function of such phases can be understood as a fluctuating network of extended objects—cages—and strings, which we dub a cage-net condensate. Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.

When an extended system is coupled at its opposite boundaries to two reservoirs at different temperatures or chemical potentials, it cannot achieve a global thermal equilibrium and is instead driven to a set of current-carrying nonequilibrium states. Despite the broad relevance of such a scenario to metallic systems, there have been limited investigations of the entanglement structure of the resulting long-time states, in part due to the fundamental difficulty in solving realistic models for disordered, interacting electrons. We investigate this problem by carefully analyzing two “toy” models for coherent quantum transport of diffusive fermions: the celebrated three-dimensional, noninteracting Anderson model, and a class of random quantum circuits acting on a chain of qubits, which exactly maps to a diffusive, interacting fermion problem. Crucially, the random circuit model can also be tuned to have no interactions between the fermions, similar to the Anderson model. We show that the long-time states of driven noninteracting fermions exhibit volume-law mutual information and entanglement, both for our random circuit model and for the nonequilibrium steady state of the Anderson model. With interactions, the random circuit model is quantum chaotic and approaches local equilibrium, with only short-range entanglement. These results provide a generic picture for the emergence of local equilibrium in current-driven, quantum-chaotic systems, and also provide examples of stable, highly entangled, many-body states out of equilibrium. We discuss experimental techniques to probe these effects in low-temperature mesoscopic wires or ultracold atomic gases.

Topological orders describe a new class of gapped quantum phases of matter at zero temperature, that has various patterns of many-body quantum entanglement. Previously, topological orders in one and two spatial dimensions have been systematically understood and classified. This paper [together with Phys. Rev. X 8, 021074 (2018)] develops a systematic and classifying understanding of topological orders in three-dimensional bosonic systems, where the number of topological types for the pointlike and stringlike excitations is assumed to be finite. Our systematic understanding comes from the unique canonical boundary for each $3+1\mathrm{D}$ topological order. We find that the pointlike and stringlike excitations on the canonical boundary are described fully by a mathematical theory—the so-called fusion 2-categories. This theory allows us to classify $3+1\mathrm{D}$ topological orders in bosonic systems in terms of a subset of fusion 2-categories. This systematic understanding further leads to a systematic understanding of $3+1\mathrm{D}$ topological orders in bosonic and fermionic systems with arbitrary finite unitary symmetry.

We study the spreading of initially local operators under unitary time evolution in a one-dimensional random quantum circuit model that is constrained to conserve a $U(1)$ charge and also the dipole moment of this charge. These constraints are motivated by the quantum dynamics of fracton phases. We discover that the charge remains localized at its initial position, providing a crisp example of a nonergodic dynamical phase of random circuit dynamics. This localization can be understood as a consequence of the return properties of low-dimensional random walks, through a mechanism reminiscent of weak localization, but insensitive to dephasing. The charge dynamics is well described by a system of coupled hydrodynamic equations, which makes several nontrivial predictions that are all in good agreement with numerics in one dimension. Importantly, these equations also predict localization in two-dimensional fractonic random circuits. We further find that the immobile fractonic charge emits nonconserved operators, whose spreading is governed by exponents that are distinct from those observed in nonfractonic circuits. These nonstandard exponents are also explained by our coupled hydrodynamic equations. Where entanglement properties are concerned, we find that fractonic operators exhibit a short time linear growth of observable entanglement with saturation to an area law, as well as a subthermal volume law for operator entanglement. The entanglement spectrum is found to follow semi-Poisson statistics, similar to eigenstates of many-body localized systems. The nonergodic phenomenology is found to persist to initial conditions containing nonzero density of dipolar or fractonic charge, including states near the sector of maximal charge. Our work implies that low-dimensional fracton systems should preserve forever a memory of their initial conditions in local observables under noisy quantum dynamics, thereby constituting ideal memories. It also implies that one- and two-dimensional fracton systems should realize true many-body localization (MBL) under Hamiltonian dynamics, even in the absence of disorder, with the obstructions to MBL in translation-invariant systems and in spatial dimensions greater than one being evaded by the nature of the mechanism responsible for localization. We also suggest a possible route to new nonergodic phases in high dimensions.

Strong magnetic field induces at least two phase transitions in graphite beyond the quantum limit where many-body effects are expected. We report on a study using a state-of-the-art nondestructive magnet allowing us to attain 90.5 T at 1.4 K, which reveals a new field-induced phase and evidence that the insulating state destroyed at 75 T is an excitonic condensate of electron-hole pairs. By monitoring the angle dependence of in-plane and out-of-plane magnetoresistance, we distinguish between the role of cyclotron and Zeeman energies in driving various phase transitions. We find that, with the notable exception of the transition field separating the two insulating states, the threshold magnetic field for all other transitions displays an exact cosine angular dependence. Remarkably, the threshold field for the destruction of the second insulator (phase B) is temperature independent with no detectable Landau-level crossing nearby. We conclude that the field-induced insulator starts as a weak-coupling density wave but ends as a strong-coupling excitonic insulator of spin-polarized electron-hole pairs.

Two- or three-dimensional metals are usually well described by weakly interacting, fermionic quasiparticles. This concept breaks down in one dimension due to strong Coulomb interactions. There, low-energy electronic excitations are expected to be bosonic collective modes, which fractionalize into independent spin- and charge-density waves. Experimental research on one-dimensional metals is still hampered by their difficult realization, their limited accessibility to measurements, and by competing or obscuring effects such as Peierls distortions or zero bias anomalies. Here we overcome these difficulties by constructing a well-isolated, one-dimensional metal of finite length present in ${\mathrm{MoS}}_2$ mirror-twin boundaries. Using scanning tunneling spectroscopy we measure the single-particle density of the interacting electron system as a function of energy and position in the 1D box. Comparison to theoretical modeling provides unambiguous evidence that we are observing spin-charge separation in real space.

We study theoretically and experimentally the emergence of supersolid properties in a dipolar Bose-Einstein condensate. The theory reveals a ground state phase diagram with three distinct regimes—a regular Bose-Einstein condensate and incoherent and coherent arrays of quantum droplets. The coherent droplets are connected by a background condensate, which leads—in addition to the periodic density modulation—to a robust phase coherence throughout the whole system. We further theoretically demonstrate that we are able to dynamically approach the ground state in our experiment and that its lifetime is limited only by three-body losses. Experimentally we probe and confirm the signatures of the phase diagram by observing the in situ density modulation as well as the phase coherence using matter wave interference.

Light interacting with high-electron-density materials elicits an ultrafast coherent many-body screening response on sub- to few-femtosecond timescales, which makes its experimental observation challenging. Here, we describe the coherent two-dimensional (2D) multiphoton photoemission (mPP; $m=2–5$) study of the Shockley surface (SS) state of Ag(111) as a benchmark for spectroscopy of the coherent nonlinear responses of metals to intense optical fields in the perturbative regime; similar 2D signatures can be expected for coherent responses in other materials like low-dimensional semiconductors and strongly correlated materials. Employing interferometric time-resolved multiphoton photoemission spectroscopy (ITR-mPP), we correlate the coherent polarizations excited in the sample with photoelectron energy distributions where the interaction terminates and photoelectrons carry away the information on their excitation. By measuring the nonresonant three- and four-photon photoemission of the SS state, as well as its replicas in above-threshold photoemission (ATP), we record the coherent response of the Ag(111) surface by 2D photoemission spectroscopy and relate it to its band structure. A 2D analysis of the SS state and its ATP replicas shows similar behavior, indicating that they are $m$th- and $m\mathrm{th}+1$-order coherent processes in a contradiction of the common attribution of ATP as a sequential process where a photoelectron excited above the vacuum level absorbs one or more additional photons. We interpret the mPP process by an optical Bloch equation model, which reproduces the main features of the surface state coherent polarization dynamics in ITR-mPP experiments: The distributions of spectroscopic components in 2D photoelectron spectra of coherent mPP are shown to follow systematically the $n/m$ ratio, where $n$ and $m$ are orders of the induced coherence and the photoemission process contributing to the signal.

Weyl and Dirac (semi)metals in three dimensions have robust gapless electronic band structures. Their massless single-body energy spectra are protected by symmetries such as lattice translation, (screw) rotation, and time reversal. In this paper, we discuss many-body interactions in these systems. We focus on strong interactions that preserve symmetries and are outside the single-body mean-field regime. By mapping a Dirac (semi)metal to a model based on a three-dimensional array of coupled Dirac wires, we show (1) the Dirac (semi)metal can acquire a many-body excitation energy gap without breaking the relevant symmetries, and (2) interaction can enable an anomalous Weyl (semi)metallic phase that is otherwise forbidden by symmetries in the single-body setting and can only be present holographically on the boundary of a four-dimensional weak topological insulator. Both of these topological states support fractional gapped (gapless) bulk (respectively, boundary) quasiparticle excitations.

The temperature–magnetic field phase diagram of the mixed honeycomb-triangular lattice system ${\mathrm{K}}_2{\mathrm{Mn}}_3({\mathrm{VO}}_4{)}_2{\mathrm{CO}}_3$ is investigated by means of magnetization, heat-capacity, and neutron-scattering measurements. The results indicate that triangular and honeycomb magnetic layers undergo sequential magnetic orderings and act as nearly independent magnetic sublattices. The honeycomb sublattice orders at about 85 K in a Neél-type antiferromagnetic structure, while the triangular sublattice displays two consecutive ordered states at much lower temperatures, 3 and 2.2 K. The ground state of the triangular sublattice consists of a planar “$Y$” magnetic structure that emerges from an intermediate collinear “up-up-down” state. Applied magnetic fields parallel or perpendicular to the $c$ axis induce exotic ordered phases characterized by various spin-stacking sequences of triangular layers that yield bilayer, three-layer, or four-layer magnetic superstructures. The observed superstructures cannot be explained in the framework of quasiclassical theory based only on nearest-neighbor interlayer coupling and point towards the presence of effective second-nearest-neighbor interactions mediated by fluctuations of the magnetic moments in the honeycomb sublattice.