Nonequilibrium Criticality at the Onset of Time-Crystalline Order.

We explore the phase transitions at the onset of time-crystalline order in O(N) models driven out of equilibrium. The spontaneous breaking of time translation symmetry and its Goldstone mode are captured by an effective description with O(N)×SO(2) symmetry, where the emergent external SO(2) results from a transmutation of the internal symmetry of time translations. Using the renormalization group and the ε=4-d expansion in a leading two-loop analysis, we identify a new nonequilibrium universality class. Strikingly, it controls the long-distance physics no matter how small the microscopic breaking of equilibrium conditions is. The O(N=2)×SO(2) symmetry group is realized for magnon condensation in pumped yttrium iron garnet films and in exciton-polariton systems with a polarization degree of freedom.

Nonlinear Transport due to Magnetic-Field-Induced Flat Bands in the Nodal-Line Semimetal ZrTe_{5}.

The Dirac material ZrTe_{5} at very low carrier density was recently found to be a nodal-line semimetal, where ultraflat bands are expected to emerge in magnetic fields parallel to the nodal-line plane. Here, we report that in very low carrier-density samples of ZrTe_{5}, when the current and the magnetic field are both along the crystallographic a axis, the current-voltage characteristics presents a pronounced nonlinearity which tends to saturate in the ultra quantum limit. The magnetic-field dependence of the nonlinear coefficient is well explained by the Boltzmann theory for flat-band transport, and we argue that this nonlinear transport is likely due to the combined effect of flat bands and charge puddles; the latter appear due to very low carrier densities.

First-order quantum phase transition in the hybrid metal-Mott insulator transition metal dichalcogenide 4Hb-TaS2.

Coupling together distinct correlated and topologically nontrivial electronic phases of matter can potentially induce novel electronic orders and phase transitions among them. Transition metal dichalcogenide compounds serve as a bedrock for exploration of such hybrid systems. They host a variety of exotic electronic phases, and their Van der Waals nature enables to admix them, either by exfoliation and stacking or by stoichiometric growth, and thereby induce novel correlated complexes. Here, we investigate the compound 4Hb-TaS2 that interleaves the Mott-insulating state of 1T-TaS2 and the putative spin liquid it hosts together with the metallic state of 2H-TaS2 and the low-temperature superconducting phase it harbors using scanning tunneling spectroscopy. We reveal a thermodynamic phase diagram that hosts a first-order quantum phase transition between a correlated Kondo-like cluster state and a depleted flat band state. We demonstrate that this intrinsic transition can be induced by an electric field and temperature as well as by manipulation of the interlayer coupling with the probe tip, hence allowing to reversibly toggle between the Kondo-like cluster and the depleted flat band states. The phase transition is manifested by a discontinuous change of the complete electronic spectrum accompanied by hysteresis and low-frequency noise. We find that the shape of the transition line in the phase diagram is determined by the local compressibility and the entropy of the two electronic states. Our findings set such heterogeneous structures as an exciting platform for systematic investigation and manipulation of Mott-metal transitions and strongly correlated phases and quantum phase transitions therein.

Gigantic Magnetochiral Anisotropy in the Topological Semimetal ZrTe_{5}.

Topological materials with broken inversion symmetry can give rise to nonreciprocal responses, such as the current rectification controlled by magnetic fields via magnetochiral anisotropy. Bulk nonreciprocal responses usually stem from relativistic corrections and are always very small. Here we report our discovery that ZrTe_{5} crystals in proximity to a topological quantum phase transition present gigantic magnetochiral anisotropy, which is the largest ever observed to date. We argue that a very low carrier density, inhomogeneities, and a torus-shaped Fermi surface induced by breaking of inversion symmetry in a Dirac material are central to explain this extraordinary property.

Stability of Floquet Majorana Box Qubits.

In one-dimensional topological superconductors driven periodically with the frequency ω, two types of topological edge modes may appear, the well-known Majorana zero mode and a Floquet Majorana mode located at the quasienergy ℏω/2. We investigate two Josephson-coupled topological quantum wires in the presence of Coulomb interactions, forming a so-called Majorana box qubit. An oscillating gate voltage can induce Floquet Majorana modes in both wires. This allows for the encoding of three qubits in a sector with fixed electron parity. If such a system is prepared by increasing the amplitude of oscillations adiabatically, it is inherently unstable as interactions resonantly create quasiparticles. This can be avoided by using instead a protocol where the oscillation frequency is increased slowly. In this case, one can find a parameter regime where the system remains stable.

Detection of Topological Spin Textures via Nonlinear Magnetic Responses.

Topologically nontrivial spin textures, such as skyrmions and dislocations, display emergent electrodynamics and can be moved by spin currents over macroscopic distances. These unique properties and their nanoscale size make them excellent candidates for the development of next-generation race-track memory and unconventional computing. A major challenge for these applications and the investigation of nanoscale magnetic structures in general is the realization of suitable detection schemes. We study magnetic disclinations, dislocations, and domain walls in FeGe and reveal pronounced responses that distinguish them from the helimagnetic background. A combination of magnetic force microscopy (MFM) and micromagnetic simulations links the response to the local magnetic susceptibility, that is, characteristic changes in the spin texture driven by the MFM tip. On the basis of the findings, which we explain using nonlinear response theory, we propose a read-out scheme using superconducting microcoils, presenting an innovative approach for detecting topological spin textures and domain walls in device-relevant geometries.

Dicke Transition in Open Many-Body Systems Determined by Fluctuation Effects.

We develop an approach to describe the Dicke transition of interacting many-particle systems strongly coupled to the light of a lossy cavity. A mean-field approach is combined with a perturbative treatment of fluctuations beyond mean field, which becomes exact in the thermodynamic limit. These fluctuations completely change the nature of the steady state, determine the thermal character of the transition, and lead to universal properties of the emerging self-organized states. We validate our results by comparing them with time-dependent matrix-product-state calculations.

Quantum confinement of the Dirac surface states in topological-insulator nanowires.

The non-trivial topology of three-dimensional topological insulators dictates the appearance of gapless Dirac surface states. Intriguingly, when made into a nanowire, quantum confinement leads to a peculiar gapped Dirac sub-band structure. This gap is useful for, e.g., future Majorana qubits based on TIs. Furthermore, these sub-bands can be manipulated by a magnetic flux and are an ideal platform for generating stable Majorana zero modes, playing a key role in topological quantum computing. However, direct evidence for the Dirac sub-bands in TI nanowires has not been reported so far. Here, using devices fabricated from thin bulk-insulating (Bi1-xSbx)2Te3 nanowires we show that non-equidistant resistance peaks, observed upon gate-tuning the chemical potential across the Dirac point, are the unique signatures of the quantized sub-bands. These TI nanowires open the way to address the topological mesoscopic physics, and eventually the Majorana physics when proximitized by an s-wave superconductor.

Publisher Correction: Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Critical Behavior near the Many-Body Localization Transition in Driven Open Systems.

Coupling a many-body localized system to a thermal bath breaks local conservation laws and washes out signatures of localization. When the bath is nonthermal or when the system is also weakly driven, local conserved quantities acquire a highly nonthermal stationary value. We demonstrate how this property can be used to study the many-body localization phase transition in weakly open systems. Here, the strength of the coupling to the nonthermal baths plays a similar role as a finite temperature in a T=0 quantum phase transition. By tuning this parameter, we can detect key features of the many-body localization (MBL) transition: the divergence of the dynamical exponent due to Griffiths effects in one dimension and the critical disorder strength. We apply these ideas to study the MBL critical point numerically. The possibility to observe critical signatures of the MBL transition in an open system allows for new numerical approaches that overcome the limitations of exact diagonalization studies. Here, we propose a scalable numerical scheme to study the MBL critical point using matrix-product operator solution to the Lindblad equation.

Band Bending and Valence Band Quantization at Line Defects in MoS2.

The variation of the electronic structure normal to 1D defects in quasi-freestanding MoS2, grown by molecular beam epitaxy, is investigated through high resolution scanning tunneling spectroscopy at 5 K. Strong upward bending of valence and conduction bands toward the line defects is found for the 4|4E mirror twin boundary and island edges but not for the 4|4P mirror twin boundary. Quantized energy levels in the valence band are observed wherever upward band bending takes place. Focusing on the common 4|4E mirror twin boundary, density functional theory calculations give an estimate of its charging, which agrees well with electrostatic modeling. We show that the line charge can also be assessed from the filling of the boundary-localized electronic band, whereby we provide a measurement of the theoretically predicted quantized polarization charge at MoS2 mirror twin boundaries. These calculations elucidate the origin of band bending and charging at these 1D defects in MoS2. The 4|4E mirror twin boundary not only impairs charge transport of electrons and holes due to band bending, but holes are additionally subject to a potential barrier, which is inferred from the independence of the quantized energy landscape on either side of the boundary.

Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase.

The phase transition most commonly observed is probably melting, a transition from ordered crystalline solids to disordered isotropic liquids. In three dimensions, melting is a single, first-order phase transition. In two-dimensional systems, however, theory predicts a general scenario of two continuous phase transitions separated by an intermediate, oriented liquid state, the so-called hexatic phase with short-range translational and quasi-long-range orientational orders. Such hexatic phases occur in colloidal systems, Wigner solids and liquid crystals, all composed of real-matter particles. In contrast, skyrmions are countable soliton configurations with non-trivial topology and these quasi-particles can form two-dimensional lattices. Here we show, by direct imaging with cryo-Lorentz transmission electron microscopy, that magnetic field variations can tune the phase of the skyrmion ensembles in Cu2OSeO3 from a two-dimensional solid through the long-speculated skyrmion hexatic phase to a liquid. The local spin order persists throughout the process. Remarkably, our quantitative analysis demonstrates that the aforementioned topological-defect-induced crystal melting scenario well describes the observed phase transitions.

Thermally condensing photons into a coherently split state of light.

The quantum state of light plays a crucial role in a wide range of fields, from quantum information science to precision measurements. Whereas complex quantum states can be created for electrons in solid-state materials through mere cooling, optical manipulation and control builds on nonthermodynamic methods. Using an optical dye microcavity, we show that photon wave packets can be split through thermalization within a potential with two minima subject to tunnel coupling. At room temperature, photons condense into a quantum-coherent bifurcated ground state. Fringe signals upon recombination show the relative coherence between the two wells, demonstrating a working interferometer with the nonunitary thermodynamic beam splitter. Our energetically driven optical-state preparation method provides a route for exploring correlated and entangled optical many-body states.

Suppression of Quasiparticle Scattering Signals in Bilayer Graphene Due to Layer Polarization and Destructive Interference.

We study chemically gated bilayer graphene using scanning tunneling microscopy and spectroscopy complemented by tight-binding calculations. Gating is achieved by intercalating Cs between bilayer graphene and Ir(111), thereby shifting the conduction band minima below the chemical potential. Scattering between electronic states (both intraband and interband) is detected via quasiparticle interference. However, not all expected processes are visible in our experiment. We uncover two general effects causing this suppression: first, intercalation leads to an asymmetrical distribution of the states within the two layers, which significantly reduces the scanning tunneling spectroscopy signal of standing waves mainly present in the lower layer; second, forward scattering processes, connecting points on the constant energy contours with parallel velocities, do not produce pronounced standing waves due to destructive interference. We present a theory to describe the interference signal for a general n-band material.

Activating Many-Body Localization in Solids by Driving with Light.

Because of the presence of phonons, many-body localization (MBL) does not occur in disordered solids, even if disorder is strong. Local conservation laws characterizing an underlying MBL phase decay due to the coupling to phonons. We show that this decay can be compensated when the system is driven out of equilibrium. The resulting variations of the local temperature provide characteristic fingerprints of an underlying MBL phase. We consider a one-dimensional disordered spin chain, which is weakly coupled to a phonon bath and weakly irradiated by white light. The irradiation has weak effects in the ergodic phase. However, if the system is in the MBL phase, irradiation induces strong temperature variations despite the coupling to phonons. Temperature variations can be used similar to an order parameter to detect MBL phases, the phase transition, and a MBL correlation length.

Planar Hall effect from the surface of topological insulators.

A prominent feature of topological insulators (TIs) is the surface states comprising of spin-nondegenerate massless Dirac fermions. Recent technical advances have made it possible to address the surface transport properties of TI thin films by tuning the Fermi levels of both top and bottom surfaces. Here we report our discovery of a novel planar Hall effect (PHE) from the TI surface, which results from a hitherto-unknown resistivity anisotropy induced by an in-plane magnetic field. This effect is observed in dual-gated devices of bulk-insulating Bi2-x Sb x Te3 thin films, where the field-induced anisotropy presents a strong dependence on the gate voltage with a characteristic two-peak structure near the Dirac point. The origin of PHE is the peculiar time-reversal-breaking effect of an in-plane magnetic field, which anisotropically lifts the protection of surface Dirac fermions from backscattering. The observed PHE provides a useful tool to analyze and manipulate the topological protection of the TI surface.

Entropy-limited topological protection of skyrmions.

Magnetic skyrmions are topologically protected whirls that decay through singular magnetic configurations known as Bloch points. We used Lorentz transmission electron microscopy to infer the energetics associated with the topological decay of magnetic skyrmions far from equilibrium in the chiral magnet Fe1-x Co x Si. We observed that the lifetime τ of the skyrmions depends exponentially on temperature, [Formula: see text]. The prefactor τ0 of this Arrhenius law changes by more than 30 orders of magnitude for small changes of the magnetic field, reflecting a substantial reduction of the lifetime of skyrmions by entropic effects and, thus, an extreme case of enthalpy-entropy compensation. Such compensation effects, being well known across many different scientific disciplines, affect topological transitions and, thus, topological protection on an unprecedented level.

Pumping approximately integrable systems.

Weak perturbations can drive an interacting many-particle system far from its initial equilibrium state if one is able to pump into degrees of freedom approximately protected by conservation laws. This concept has for example been used to realize Bose-Einstein condensates of photons, magnons and excitons. Integrable quantum systems, like the one-dimensional Heisenberg model, are characterized by an infinite set of conservation laws. Here, we develop a theory of weakly driven integrable systems and show that pumping can induce large spin or heat currents even in the presence of integrability breaking perturbations, since it activates local and quasi-local approximate conserved quantities. The resulting steady state is qualitatively captured by a truncated generalized Gibbs ensemble with Lagrange parameters that depend on the structure but not on the overall amplitude of perturbations nor the initial state. We suggest to use spin-chain materials driven by terahertz radiation to realize integrability-based spin and heat pumps.

Gigantic negative magnetoresistance in the bulk of a disordered topological insulator.

With the recent discovery of Weyl semimetals, the phenomenon of negative magnetoresistance (MR) is attracting renewed interest. Large negative MR is usually related to magnetism, but the chiral anomaly in Weyl semimetals is a rare exception. Here we report a mechanism for large negative MR which is also unrelated to magnetism but is related to disorder. In the nearly bulk-insulating topological insulator TlBi0.15Sb0.85Te2, we observed gigantic negative MR reaching 98% in 14 T at 10 K, which is unprecedented in a nonmagnetic system. Supported by numerical simulations, we argue that this phenomenon is likely due to the Zeeman effect on a barely percolating current path formed in the disordered bulk. Since disorder can also lead to non-saturating linear MR in Ag2+δSe, the present finding suggests that disorder engineering in narrow-gap systems is useful for realizing gigantic MR in both positive and negative directions.

Magnetic Skyrmions and Skyrmion Clusters in the Helical Phase of Cu_{2}OSeO_{3}.

Skyrmions are nanometric spin whirls that can be stabilized in magnets lacking inversion symmetry. The properties of isolated Skyrmions embedded in a ferromagnetic background have been intensively studied. We show that single Skyrmions and clusters of Skyrmions can also form in the helical phase and investigate theoretically their energetics and dynamics. The helical background provides natural one-dimensional channels along which a Skyrmion can move rapidly. In contrast to Skyrmions in ferromagnets, the Skyrmion-Skyrmion interaction has a strong attractive component and thus Skyrmions tend to form clusters with characteristic shapes. These clusters are directly observed in transmission electron microscopy measurements in thin films of Cu_{2}OSeO_{3}. Topological quantization, high mobility, and the confinement of Skyrmions in channels provided by the helical background may be useful for future spintronics devices.