Neural-network quantum states have recently been introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between neural-network quantum states in the form of restricted Boltzmann machines and some classes of tensor-network states in arbitrary dimensions. In particular, we demonstrate that short-range restricted Boltzmann machines are entangled plaquette states, while fully connected restricted Boltzmann machines are string-bond states with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of restricted Boltzmann machines and their efficiency at representing many-body quantum states. String-bond states also provide a generic way of enhancing the power of neural-network quantum states and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of tensor networks and the efficiency of neural-network quantum states into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional tensor networks, we show that, because of their nonlocal geometry, neural-network quantum states and their string-bond-state extension can describe a lattice fractional quantum Hall state exactly. In addition, we provide numerical evidence that neural-network quantum states can approximate a chiral spin liquid with better accuracy than entangled plaquette states and local string-bond states. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of string-bond states as a tool in more traditional machine-learning applications.

Multiferroism can originate from the breaking of inversion symmetry caused by magnetic-spiral order. The usual mechanism for stabilizing a magnetic spiral is competition between magnetic exchange interactions differing by their range and sign, such as nearest-neighbor and next-nearest-neighbor interactions. In insulating compounds, it is unusual for these interactions to be both comparable in magnitude and of a strength that can induce magnetic ordering at room temperature. Therefore, the onset temperatures for multiferroism through this mechanism are typically low. By considering a realistic model for multiferroic ${\mathrm{YBaCuFeO}}_5$, we propose an alternative mechanism for magnetic-spiral order, and hence for multiferroism, that occurs at much higher temperatures. We show, using Monte Carlo simulations and electronic structure calculations based on density functional theory, that the Heisenberg model on a geometrically nonfrustrated lattice with only nearest-neighbor interactions can have a spiral phase up to high temperature when frustrating bonds are introduced randomly along a single crystallographic direction as caused, e.g., by a particular type of chemical disorder. This long-range correlated pattern of frustration avoids ferroelectrically inactive spin-glass order. Finally, we provide an intuitive explanation for this mechanism and discuss its generalization to other materials.

We construct entanglement renormalization schemes that provably approximate the ground states of noninteracting-fermion nearest-neighbor hopping Hamiltonians on the one-dimensional discrete line and the two-dimensional square lattice. These schemes give hierarchical quantum circuits that build up the states from unentangled degrees of freedom. The circuits are based on pairs of discrete wavelet transforms, which are approximately related by a “half-shift”: translation by half a unit cell. The presence of the Fermi surface in the two-dimensional model requires a special kind of circuit architecture to properly capture the entanglement in the ground state. We show how the error in the approximation can be controlled without ever performing a variational optimization.

In this work, we present a model of the surface states of nonsymmorphic semimetals. These are derived from surface mass terms that lift the high degeneracy imposed on the band structure by the nonsymmorphic bulk symmetries. Reflecting the reduced symmetry at the surface, the bulk bands are strongly modified. This leads to the creation of two-dimensional floating or unpinned bands, which are distinct from Shockley states, quantum well states, or topologically protected surface states. We focus on the layered semimetal ZrSiS to clarify the origin of its surface states. We demonstrate an excellent agreement between density functional theory calculations and angle-resolved photoemission spectroscopy measurements and present an effective four-band model in which similar surface bands appear. Finally, we emphasize the role of the surface chemical potential by comparing the surface density of states in samples with and without potassium coating. Our findings can be extended to related compounds and generalized to other crystals with nonsymmorphic symmetries.

We study the spin-excitation spectrum (dynamic structure factor) of the spin-$1/2$ square-lattice Heisenberg antiferromagnet and an extended model (the $J\text{−}Q$ model) including four-spin interactions $Q$ in addition to the Heisenberg exchange $J$. Using an improved method for stochastic analytic continuation of imaginary-time correlation functions computed with quantum Monte Carlo simulations, we can treat the sharp ($\delta$-function) contribution to the structure factor expected from spin-wave (magnon) excitations, in addition to resolving a continuum above the magnon energy. Spectra for the Heisenberg model are in excellent agreement with recent neutron-scattering experiments on $\mathrm{Cu}(\mathrm{DCOO}{)}_2·4{\mathrm{D}}_2\mathrm{O}$, where a broad spectral-weight continuum at wave vector $q=(\pi ,0)$ was interpreted as deconfined spinons, i.e., fractional excitations carrying half of the spin of a magnon. Our results at $(\pi ,0)$ show a similar reduction of the magnon weight and a large continuum, while the continuum is much smaller at $q=(\pi /2,\pi /2)$ (as also seen experimentally). We further investigate the reasons for the small magnon weight at $(\pi ,0)$ and the nature of the corresponding excitation by studying the evolution of the spectral functions in the $J\text{−}Q$ model. Upon turning on the $Q$ interaction, we observe a rapid reduction of the magnon weight to zero, well before the system undergoes a deconfined quantum phase transition into a nonmagnetic spontaneously dimerized state. Based on these results, we reinterpret the picture of deconfined spinons at $(\pi ,0)$ in the experiments as nearly deconfined spinons—a precursor to deconfined quantum criticality. To further elucidate the picture of a fragile $(\pi ,0)$-magnon pole in the Heisenberg model and its depletion in the $J\text{−}Q$ model, we introduce an effective model of the excitations in which a magnon can split into two spinons that do not separate but fluctuate in and out of the magnon space (in analogy to the resonance between a photon and a particle-hole pair in the exciton-polariton problem). The model can reproduce the reduction of magnon weight and lowered excitation energy at $(\pi ,0)$ in the Heisenberg model, as well as the energy maximum and smaller continuum at $(\pi /2,\pi /2)$. It can also account for the rapid loss of the $(\pi ,0)$ magnon with increasing $Q$ and the remarkable persistence of a large magnon pole at $q=(\pi /2,\pi /2)$ even at the deconfined critical point. The fragility of the magnons close to $(\pi ,0)$ in the Heisenberg model suggests that various interactions that likely are important in many materials—e.g., longer-range pair exchange, ring exchange, and spin-phonon interactions—may also destroy these magnons and lead to even stronger spinon signatures than in $\mathrm{Cu}(\mathrm{DCOO}{)}_2·4{\mathrm{D}}_2\mathrm{O}$.

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented match the mathematical structure underlying the topological classification of these crystals in terms of $K$-theory and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify all allowed topological phases of spinless particles in crystals in class $A$. Employing this classification, we study transitions between topological phases within class $A$ that are driven by band inversions at high-symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure and to identify whether or not they give rise to intermediate Weyl semimetallic phases.

We study broken symmetry states at integer Landau-level fillings in multivalley quantum Hall systems whose low-energy dispersions are anisotropic. When the Fermi surface of individual pockets lacks twofold rotational symmetry, like in bismuth (111) [Feldman et al. , Observation of a Nematic Quantum Hall Liquid on the Surface of Bismuth, Science 354, 316 (2016)] and in ${\mathrm{Sn}}_{1-x}{\mathrm{Pb}}_x\mathrm{Se}$ (001) [Dziawa et al., Topological Crystalline Insulator States in ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$, Nat. Mater. 11, 1023 (2012)] surfaces, interactions tend to drive the formation of quantum Hall ferroelectric states. We demonstrate that the dipole moment in these states has an intimate relation to the Fermi surface geometry of the parent metal. In quantum Hall nematic states, like those arising in AlAs quantum wells, we demonstrate the existence of unusually robust Skyrmion quasiparticles.

Chromium dioxide ${\mathrm{CrO}}_2$ belongs to a class of materials called ferromagnetic half-metals, whose peculiar aspect is that they act as a metal in one spin orientation and as a semiconductor or insulator in the opposite one. Despite numerous experimental and theoretical studies motivated by technologically important applications of this material in spintronics, its fundamental properties such as momentum-resolved electron dispersions and the Fermi surface have so far remained experimentally inaccessible because of metastability of its surface, which instantly reduces to amorphous ${\mathrm{Cr}}_2{\mathrm{O}}_3$. In this work, we demonstrate that direct access to the native electronic structure of ${\mathrm{CrO}}_2$ can be achieved with soft-x-ray angle-resolved photoemission spectroscopy whose large probing depth penetrates through the ${\mathrm{Cr}}_2{\mathrm{O}}_3$ layer. For the first time, the electronic dispersions and Fermi surface of ${\mathrm{CrO}}_2$ are measured, which are fundamental prerequisites to solve the long debate on the nature of electronic correlations in this material. Since density functional theory augmented by a relatively weak local Coulomb repulsion gives an exhaustive description of our spectroscopic data, we rule out strong-coupling theories of ${\mathrm{CrO}}_2$. Crucial for the correct interpretation of our experimental data in terms of the valence-band dispersions is the understanding of a nontrivial spectral response of ${\mathrm{CrO}}_2$ caused by interference effects in the photoemission process originating from the nonsymmorphic space group of the rutile crystal structure of ${\mathrm{CrO}}_2$.

One of the central questions in the cuprate research is the nature of the normal state that develops into high-temperature superconductivity (HTSC). In the normal state of hole-doped cuprates, the existence of a charge density wave (CDW) is expected to shed light on the mechanism of HTSC. With evidence emerging for CDW order in the electron-doped cuprates, the CDW is thought to be a universal phenomenon in high-$T_{\mathrm{c}}$ cuprates. However, the CDW phenomena in electron-doped cuprates are quite different than those in hole-doped cuprates. Here, we study the nature of the putative CDW in an electron-doped cuprate through direct comparisons between as-grown and postannealed ${\mathrm{Nd}}_{1.86}{\mathrm{Ce}}_{0.14}{\mathrm{CuO}}_4$ (NCCO) single crystals using Cu $L_3$-edge resonant soft x-ray scattering (RSXS) and angle-resolved photoemission spectroscopy (ARPES). The RSXS result reveals that the nonsuperconducting NCCO shows the same reflections at the wave vector ($\sim 1/4$, 0, $l$) as the reported superconducting NCCO. This superconductivity-insensitive signal is quite different from the CDW reflection in hole-doped cuprates. Moreover, the ARPES result suggests that the fermiology cannot account for such wave vectors. These results call into question the universality of the CDW phenomenon in the cuprates.

Understanding grain growth is key for controlling the microstructure and the mechanical properties of most polycrystalline materials, including metals, alloys, and ceramics. However, the precise mechanisms and kinetics of grain growth remain poorly understood both at the theoretical level and experimentally as direct observation is cumbersome in atomic systems. Here, we study the grain growth process in a polycrystalline monolayer of colloidal hard spheres. We find that the bond-orientational correlation function satisfies the dynamic scaling hypothesis and has the general scaling form predicted for systems containing random domain walls. However, the associated correlation length grows slower than $\sim t^{1/2}$, which corresponds to normal curvature-driven grain growth. To understand the origin of this anomalous grain growth, we directly monitor the evolution of the grain boundary network by measuring the so-called grain boundary character distribution. We show that there is a strong annihilation of large-angle grain boundaries while small-angle grain boundaries become relatively more present. Using scaling arguments, we derive the time dependence of the correlation length and show its good agreement with the data. We conclude that the origin of anomalous grain growth is the curvature-driven coarsening of the large-angle grain boundaries at a rate that depends on their relative length in the total grain boundary network.

Coherent many-body quantum dynamics lies at the heart of quantum simulation and quantum computation. Both require coherent evolution in the exponentially large Hilbert space of an interacting many-body system. To date, trapped ions have defined the state of the art in terms of achievable coherence times in interacting spin chains. Here, we establish an alternative platform by reporting on the observation of coherent, fully interaction-driven quantum revivals of the magnetization in Rydberg-dressed Ising spin chains of atoms trapped in an optical lattice. We identify partial many-body revivals at up to about ten times the characteristic time scale set by the interactions. At the same time, single-site-resolved correlation measurements link the magnetization dynamics with interspin correlations appearing at different distances during the evolution. These results mark an enabling step towards the implementation of Rydberg-atom-based quantum annealers, quantum simulations of higher-dimensional complex magnetic Hamiltonians, and itinerant long-range interacting quantum matter.

We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures, the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This nonequilibrium dynamical phenomenon is a form of prethermalization and occurs because of obstructions to the equilibration of edge or defect degrees of freedom with the bulk. We analyze the ramifications for ordered and topological phases in one, two, and three dimensions, with examples including Majorana and parafermionic zero modes in interacting spin chains. Our results are based on a nonperturbative analysis valid in any dimension, and they are illustrated by numerical simulations in one dimension. We discuss the implications for experiments on quantum-dot chains tuned into a regime supporting end Majorana zero modes and on trapped ion chains.

Recently, two reports [Krivanek et al. Nature (London) 514, 209 (2014), Lagos et al. Nature (London) 543, 529 (2017)] have demonstrated the amazing possibility to probe vibrational excitations from nanoparticles with a spatial resolution much smaller than the corresponding free-space phonon wavelength using electron-energy-loss spectroscopy (EELS). While Lagos et al. evidenced a strong spatial and spectral modulation of the EELS signal over a nanoparticle, Krivanek et al. did not. Here, we show that discrepancies among different EELS experiments as well as their relation to optical near- and far-field optical experiments [Dai et al. Science 343, 1125 (2014)] can be understood by introducing the concept of confined bright and dark surface phonon modes, whose density of states is probed by EELS. Such a concise formalism is the vibrational counterpart of the broadly used formalism for localized surface plasmons [Ouyang and Isaacson Philos. Mag. B 60, 481 (1989), García de Abajo and Aizpurua Phys. Rev. B 56, 15873 (1997), García de Abajo and Kociak Phys. Rev. Lett. 100, 106804 (2008), Boudarham and Kociak Phys. Rev. B 85, 245447 (2012)]; it makes it straightforward to predict or interpret phenomena already known for localized surface plasmons such as environment-related energy shifts or the possibility of 3D mapping of the related surface charge densities [Collins et al. ACS Photonics 2, 1628 (2015)].

Frustration in magnetic interactions can give rise to disordered ground states with subtle and beautiful properties. The spin ices ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ exemplify this phenomenon, displaying a classical spin-liquid state, with fractionalized magnetic-monopole excitations. Recently, there has been great interest in closely related “quantum spin-ice” materials, following the realization that anisotropic exchange interactions could convert spin ice into a massively entangled, quantum spin liquid, where magnetic monopoles become the charges of an emergent quantum electrodynamics. Here we show that even the simplest model of a quantum spin ice, the XXZ model on the pyrochlore lattice, can realize a still-richer scenario. Using a combination of classical Monte Carlo simulation, semiclassical molecular-dynamics simulation, and analytic field theory, we explore the properties of this model for frustrated transverse exchange. We find not one, but three competing forms of spin liquid, as well as a phase with hidden, spin-nematic order. We explore the experimental signatures of each of these different states, making explicit predictions for inelastic neutron scattering. These results show an intriguing similarity to experiments on a range of pyrochlore oxides.

The layered transition-metal dichalcogenide $1\mathrm{T}\text{−}{\mathrm{TaS}}_2$ has been recently found to undergo a Mott-insulator-to-superconductor transition induced by high pressure, charge doping, or isovalent substitution. By combining scanning tunneling microscopy measurements and first-principles calculations, we investigate the atomic scale electronic structure of the $1\mathrm{T}\text{−}{\mathrm{TaS}}_2$ Mott insulator and its evolution to the metallic state upon isovalent substitution of S with Se. We identify two distinct types of orbital textures—one localized and the other extended—and demonstrate that the interplay between them is the key factor that determines the electronic structure. In particular, we show that the continuous evolution of the charge gap visualized by scanning tunneling microscopy is due to the immersion of the localized-orbital-induced Hubbard bands into the extended-orbital-spanned Fermi sea, featuring a unique evolution from a Mott gap to a charge-transfer gap. This new mechanism of Mottness collapse revealed here suggests an interesting route for creating novel electronic states and designing future electronic devices.

Low-energy spin excitations in any long-range ordered magnetic system in the absence of magnetocrystalline anisotropy are gapless Goldstone modes emanating from the ordering wave vectors. In helimagnets, these modes hybridize into the so-called helimagnon excitations. Here we employ neutron spectroscopy supported by theoretical calculations to investigate the magnetic excitation spectrum of the isotropic Heisenberg helimagnet ${\mathrm{ZnCr}}_2{\mathrm{Se}}_4$ with a cubic spinel structure, in which spin-$3/2$ magnetic ${\mathrm{Cr}}^{3+}$ ions are arranged in a geometrically frustrated pyrochlore sublattice. Apart from the conventional Goldstone mode emanating from the $(00q_{\mathrm{h}})$ ordering vector, low-energy magnetic excitations in the single-domain proper-screw spiral phase show soft helimagnon modes with a small energy gap of $\sim 0.17\mathrm{meV}$, emerging from two orthogonal wave vectors $(q_{\mathrm{h}}00)$ and $(0q_{\mathrm{h}}0)$ where no magnetic Bragg peaks are present. We term them pseudo-Goldstone magnons, as they appear gapless within linear spin-wave theory and only acquire a finite gap due to higher-order quantum-fluctuation corrections. Our results are likely universal for a broad class of symmetric helimagnets, opening up a new way of studying weak magnon-magnon interactions with accessible spectroscopic methods.

We introduce exactly solvable gapless quantum systems in $d$ dimensions that support symmetry-protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls “decorated” with dimension ($d-1$) SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, these edge modes are robust against arbitrary symmetry-preserving local perturbations near the edges. In two dimensions, we construct wave functions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.

In a many-body localized (MBL) quantum system, the ergodic hypothesis breaks down, giving rise to a fundamentally new many-body phase. Whether and under which conditions MBL can occur in higher dimensions remains an outstanding challenge both for experiments and theory. Here, we experimentally explore the relaxation dynamics of an interacting gas of fermionic potassium atoms loaded in a two-dimensional optical lattice with different quasiperiodic potentials along the two directions. We observe a dramatic slowing down of the relaxation for intermediate disorder strengths. Furthermore, beyond a critical disorder strength, we see negligible relaxation on experimentally accessible time scales, indicating a possible transition into a two-dimensional MBL phase. Our experiments reveal a distinct interplay of interactions, disorder, and dimensionality and provide insights into regimes where controlled theoretical approaches are scarce.

We study the quantum propagation of a Skyrmion in chiral magnetic insulators by generalizing the micromagnetic equations of motion to a finite-temperature path integral formalism, using field theoretic tools. Promoting the center of the Skyrmion to a dynamic quantity, the fluctuations around the Skyrmionic configuration give rise to a time-dependent damping of the Skyrmion motion. From the frequency dependence of the damping kernel, we are able to identify the Skyrmion mass, thus providing a microscopic description of the kinematic properties of Skyrmions. When defects are present or a magnetic trap is applied, the Skyrmion mass acquires a finite value proportional to the effective spin, even at vanishingly small temperature. We demonstrate that a Skyrmion in a confined geometry provided by a magnetic trap behaves as a massive particle owing to its quasi-one-dimensional confinement. An additional quantum mass term is predicted, independent of the effective spin, with an explicit temperature dependence which remains finite even at zero temperature.

We report on the design and performance of an on-chip microwave circulator with a widely (GHz) tunable operation frequency. Nonreciprocity is created with a combination of frequency conversion and delay, and requires neither permanent magnets nor microwave bias tones, allowing on-chip integration with other superconducting circuits without the need for high-bandwidth control lines. Isolation in the device exceeds 20 dB over a bandwidth of tens of MHz, and its insertion loss is small, reaching as low as 0.9 dB at select operation frequencies. Furthermore, the device is linear with respect to input power for signal powers up to hundreds of fW ($\approx {10}^3$ circulating photons), and the direction of circulation can be dynamically reconfigured. We demonstrate its operation at a selection of frequencies between 4 and 6 GHz.