The ability to prepare a physical system in a desired quantum state is central to many areas of physics such as nuclear magnetic resonance, cold atoms, and quantum computing. Yet, preparing states quickly and with high fidelity remains a formidable challenge. In this work, we implement cutting-edge reinforcement learning (RL) techniques and show that their performance is comparable to optimal control methods in the task of finding short, high-fidelity driving protocol from an initial to a target state in nonintegrable many-body quantum systems of interacting qubits. RL methods learn about the underlying physical system solely through a single scalar reward (the fidelity of the resulting state) calculated from numerical simulations of the physical system. We further show that quantum-state manipulation viewed as an optimization problem exhibits a spin-glass-like phase transition in the space of protocols as a function of the protocol duration. Our RL-aided approach helps identify variational protocols with nearly optimal fidelity, even in the glassy phase, where optimal state manipulation is exponentially hard. This study highlights the potential usefulness of RL for applications in out-of-equilibrium quantum physics.

We speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix $\widehat{\rho }=e^{-\beta \widehat{H}}$ onto itself. We refer to this scheme of doubling $\beta$ in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve $\widehat{\rho }$ on a linear quasicontinuous grid in inverse temperature $\beta \equiv 1/T$. As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase-transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. In general, XTRG can reach low temperatures exponentially fast and, thus, not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs), which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground-state properties. The entanglement properties, as well as the renormalization-group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately $\ln \beta$) are shown in both spin chains and square-lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg $XXZ$ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.

Excursions far from their equilibrium structures can bring crystalline solids through collective transformations including transitions into new phases that may be transient or long-lived. The direct spectroscopic observation of far-from-equilibrium rearrangements provides fundamental mechanistic insight into chemical and structural transformations and a potential route to practical applications, including ultrafast optical control over material structure and properties. However, in many cases, photoinduced transitions are irreversible or only slowly reversible, or the light fluence required exceeds material damage thresholds. This requirement precludes conventional ultrafast spectroscopy, in which optical excitation and probe pulses irradiate the sample many times, each measurement providing information about the sample response at just one probe delay time following excitation, with each measurement at a high repetition rate and with the sample fully recovering its initial state in between measurements. Using a single-shot, real-time measurement method, we are able to observe the photoinduced phase transition from the semimetallic, low-symmetry phase of crystalline bismuth into a high-symmetry phase whose existence at high electronic excitation densities is predicted based on earlier measurements at moderate excitation densities below the damage threshold. Our observations indicate that coherent lattice vibrational motion launched upon photoexcitation with an incident fluence above $10\mathrm{mJ}/{\mathrm{cm}}^2$ in bulk bismuth brings the lattice structure directly into the high-symmetry configuration for several picoseconds, after which carrier relaxation and diffusion restore the equilibrium lattice configuration.

While Hermiticity lies at the heart of quantum mechanics, recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. Examples include parity-time-symmetric optical systems with gain and loss, dissipative Bose-Einstein condensates, exciton-polariton systems, and biological networks. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum-Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi)edge modes. We then apply the $K$ theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer (AZ) classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a ${\mathbb{Z}}_2$ topological index for arbitrary quantum channels (completely positive trace-preserving maps). Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems.

We use scanning optical magnetometry to study the broadband frequency spectra of spontaneous magnetization fluctuations, or “magnetization noise,” in an archetypal ferromagnetic film that can be smoothly tuned through a spin-reorientation transition (SRT). The SRT is achieved by laterally varying the magnetic anisotropy across an ultrathin $\mathrm{Pt}/\mathrm{Co}/\mathrm{Pt}$ trilayer, from the perpendicular to in-plane direction, via graded ${\mathrm{Ar}}^{+}$ irradiation. In regions exhibiting perpendicular anisotropy, the power spectrum of the magnetization noise $S(\nu )$ exhibits a remarkably robust ${\nu }^{-3/2}$ power law over frequencies $\nu$ from 1 kHz to 1 MHz. As the SRT region is traversed, however, $S(\nu )$ spectra develop a steadily increasing critical frequency ${\nu }_0$, below which the noise power is spectrally flat, indicating an evolving low-frequency cutoff for magnetization fluctuations. The magnetization noise depends strongly on applied in- and out-of-plane magnetic fields, revealing local anisotropies and also a field-induced emergence of fluctuations in otherwise stable ferromagnetic films. Finally, we demonstrate that higher-order correlators can be computed from the noise. These results highlight broadband spectroscopy of thermodynamic fluctuations as a powerful tool to characterize the interplay between thermal and magnetic energy scales, and as a means of characterizing phase transitions in ferromagnets.

Pursuing complementary field-theoretic and numerical methods, we here paint the global phase diagram of a three-dimensional dirty Weyl system. The generalized Harris criterion, augmented by a perturbative renormalization-group analysis shows that weak disorder is an irrelevant perturbation at the Weyl semimetal (WSM)-insulator quantum-critical point. But, a metallic phase sets in through a quantum phase transition (QPT) at strong disorder across a multicritical point. The field-theoretic predictions for the correlation length exponent $\nu =2$ and dynamic scaling exponent $z=5/4$ at this multicritical point are in good agreement with the ones extracted numerically, yielding $\nu =1.98\pm 0.10$ and $z=1.26\pm 0.05$, from the scaling of the average density of states (DOS). Deep inside the WSM phase, generic disorder is also an irrelevant perturbation, while a metallic phase appears at strong disorder through a QPT. We here demonstrate that in the presence of generic but strong disorder, the WSM-metal QPT is ultimately always characterized by the exponents $\nu =1$ and $z=3/2$ (to one-loop order), originating from intranode or chiral-symmetric (e.g., regular and axial potential) disorder. We here anchor such emergent chiral superuniversality through complementary renormalization-group calculations, controlled via $\epsilon$ expansions, and numerical analysis of average DOS across WSM-metal QPT. In addition, we also discuss a subsequent QPT (at even stronger disorder) of a Weyl metal into an Anderson insulator by numerically computing the typical DOS at zero energy. The scaling behavior of various physical observables, such as residue of quasiparticle pole, dynamic conductivity, specific heat, Grüneisen ratio, inside various phases as well as across various QPTs in the global phase diagram of a dirty Weyl liquid, are discussed.

The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures. A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity. However, this transition impedes experimental explorations of real-world TLLs. Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths. Quantum simulation can provide a powerful workaround. Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity. Measurements reveal the renormalization group “beta function” for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling. Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored. With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.

The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs), featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with “hinge” surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.

Topological semimetals are under intensive theoretical and experimental studies. The first step of these studies is always the theoretical (numerical) predication of one or several candidate materials, based on first-principles numerics. In these calculations, it is crucial that all topological band crossings, including their types and positions in the Brillouin zone, are found. While band crossings along high-symmetry lines, which are routinely scanned in numerics, are simple to locate, the ones at generic momenta are notoriously time-consuming to find and may be easily missed. In this paper, we establish a theoretical scheme of diagnosis for topological semimetals where all band crossings are at generic momenta in systems with time-reversal symmetry and negligible spin-orbital coupling. The scheme uses only the symmetry (inversion and rotation) eigenvalues of the valence bands at high-symmetry points in the Brillouin zone as input and provides the types (lines or points), topological charges, numbers, and configurations of all robust topological band crossings, if any, at generic momenta. The nature of the new diagnosis scheme allows for full automation and parallelization and paves the way to high-throughput numerical predictions of topological semimetals.

Recent discoveries of topological phases realized in electronic states in solids have revealed an important role of topology, which ubiquitously appears in various materials in nature. Many well-known materials have turned out to be topological materials, and this new viewpoint of topology has opened a new horizon in material science. In this paper, we find that electrides are suitable for achieving various topological phases, including topological insulating and topological semimetal phases. In the electrides, in which electrons serve as anions, the bands occupied by the anionic electrons lie near the Fermi level, because the anionic electrons are weakly bound by the lattice. This property of the electrides is favorable for achieving band inversions needed for topological phases, and, thus, the electrides are prone to topological phases. From such a point of view, we find many topological electrides, ${\mathrm{Y}}_2\mathrm{C}$ [nodal-line semimetal (NLS)], ${\mathrm{Sc}}_2\mathrm{C}$ (insulator with a $\pi$ Zak phase), ${\mathrm{Sr}}_2\mathrm{Bi}$ (NLS), HfBr (quantum spin Hall system), and LaBr (quantum anomalous Hall insulator), by using ab initio calculation. The close relationship between the electrides and the topological materials is useful in material science in both fields.

Quantum spin liquids (QSLs) are intriguing phases of matter possessing fractionalized excitations. Several quasi-two-dimensional materials have been proposed as candidate QSLs, but direct evidence for fractionalization in these systems is still lacking. In this paper, we show that the interplane thermal conductivity in layered QSLs carries a unique signature of fractionalization. We examine several types of gapless QSL phases—a $Z_2$ QSL with either a Dirac spectrum or a spinon Fermi surface, and a $U(1)$ QSL with a Fermi surface—and consider both clean and disordered systems. In all cases, the in-plane and $c$-axis thermal conductivities have a different power-law dependence on temperature because of the different mechanisms of transport in the two directions: In the planes, the thermal current is carried by fractionalized excitations, whereas the interplane current is carried by integer (nonfractional) excitations. In layered $Z_2$ and $U(1)$ QSLs with a Fermi surface, and in the disordered $Z_2$ QSL with a Dirac dispersion, the $c$-axis thermal conductivity is parametrically smaller than the in-plane one but parametrically larger than the phonon contribution at low temperatures.

We provide experimental and theoretical evidence for a pressure-induced Mott insulator-metal transition in ${\mathrm{Fe}}_2{\mathrm{O}}_3$ characterized by site-selective delocalization of the electrons. Density functional plus dynamical mean-field theory ($\mathrm{DFT}+\mathrm{DMFT}$) calculations, along with Mössbauer spectroscopy, x-ray diffraction, and electrical transport measurements on ${\mathrm{Fe}}_2{\mathrm{O}}_3$ up to 100 GPa, reveal this site-selective Mott transition between 50 and 68 GPa, such that the metallization can be described by ${({}^{\mathrm{VI}}{\mathrm{Fe}}^{3+\mathrm{HS}})}_2{\mathrm{O}}_3$ $[R\overline{3}c\text{structure}]\underset{50\mathrm{GPa}}{\to }({}^{\mathrm{VIII}}{\mathrm{Fe}}^{3+\mathrm{HS}}{{}^{\mathrm{VI}}\mathrm{Fe}}^{\mathrm{M}}){\mathrm{O}}_3$ $[P{2}_1/n\text{structure}]\underset{68\mathrm{GPa}}{\to }{({}^{\mathrm{VI}}{\mathrm{Fe}}^{\mathrm{M}})}_2{\mathrm{O}}_3[Aba2/PPv\text{structure}]$. Within the $P{2}_1/n$ crystal structure, characterized by two distinct coordination sites (VI and VIII), we observe equal abundances of ferric ions (${\mathrm{Fe}}^{3+}$) and ions having delocalized electrons (${\mathrm{Fe}}^{\mathrm{M}}$), and only at higher pressures is a fully metallic high-pressure structure obtained, all at room temperature. Thereby, the transition is characterized by delocalization/metallization of the $3d$ electrons on half the Fe sites, with a site-dependent collapse of local moments. Above approximately 50 GPa, ${\mathrm{Fe}}_2{\mathrm{O}}_3$ is a strongly correlated metal with reduced electron mobility (large band renormalizations) of $m^{*}/m\sim 4$ and 6 near the Fermi level. Importantly, upon decompression, we observe a site-selective (metallic) to conventional Mott insulator phase transition $({}^{\mathrm{VIII}}{\mathrm{Fe}}^{3+\mathrm{HS}}{{}^{\mathrm{VI}}\mathrm{Fe}}^{\mathrm{M}}){\mathrm{O}}_3\underset{50\mathrm{GPa}}{\to }({}^{\mathrm{VIII}}{\mathrm{Fe}}^{3+\mathrm{HS}}{{}^{\mathrm{VI}}\mathrm{Fe}}^{3+\mathrm{HS}}){\mathrm{O}}_3$ within the same $P{2}_1/n$ structure, indicating a decoupling of the electronic and lattice degrees of freedom. Our results offer a model for understanding insulator-metal transitions in correlated electron materials, showing that the interplay of electronic correlations and crystal structure may result in rather complex behavior of the electronic and magnetic states of such compounds.

The scrambling of quantum information in closed many-body systems, as measured by out-of-time-ordered correlation functions (OTOCs), has received considerable attention lately. Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits. Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, “generate their own noise” and look effectively random on sufficient length scales and timescales. In this paper, we extend this approach to systems with locally conserved quantities (e.g., energy). We do this by considering local random unitary circuits with a conserved U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time-ordered and out-of-time-ordered correlation functions; both can have a diffusively relaxing component or “hydrodynamic tail” at late times. We verify the presence of such tails also in a deterministic, periodically driven system. We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific asymmetric shape, decaying as a power law behind the front. These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation. Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential $\mu$, and apply perturbative arguments to show that for $\mu \gg 1$ the ballistic front of information spreading can only develop at times exponentially large in $\mu$—with the information traveling diffusively at earlier times. We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.

We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process. We obtain our results in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the “reservoir” that facilitates the dissipation. In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.

Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some “topological” features: They support fractional bulk excitations (dubbed fractons) and a ground-state degeneracy that is robust to local perturbations. However, because previous fracton models have been defined and analyzed only on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this paper, we demonstrate that the $X$-cube model, a prototypical type-I fracton model, can be defined on general three-dimensional manifolds. Our construction revolves around the notion of a singular compact total foliation of the spatial manifold, which constructs a lattice from intersecting stacks of parallel surfaces called leaves. We find that the ground-state degeneracy depends on the topology of the leaves and the pattern of leaf intersections. We further show that such a dependence can be understood from a renormalization group transformation for the $X$-cube model, wherein the system size can be changed by adding or removing 2D layers of topological states. Our results lead to an improved definition of fracton phase and bring to the fore the topological nature of fracton orders.

Exchange coupling is a key ingredient for spin-based quantum technologies since it can be used to entangle spin qubits and create logical spin qubits. However, the influence of the electronic valley degree of freedom in silicon on exchange interactions is presently the subject of important open questions. Here we investigate the influence of valleys on exchange in a coupled donor–quantum-dot system, a basic building block of recently proposed schemes for robust quantum information processing. Using a scanning tunneling microscope tip to position the quantum dot with sub-nm precision, we find a near monotonic exchange characteristic where lattice-aperiodic modulations associated with valley degrees of freedom comprise less than 2% of exchange. From this we conclude that intravalley tunneling processes that preserve the donor’s $\pm x$ and $\pm y$ valley index are filtered out of the interaction with the $\pm z$ valley quantum dot, and that the $\pm x$ and $\pm y$ intervalley processes where the electron valley index changes are weak. Complemented by tight-binding calculations of exchange versus donor depth, the demonstrated electrostatic tunability of donor–quantum-dot exchange can be used to compensate the remaining intravalley $\pm z$ oscillations to realize uniform interactions in an array of highly coherent donor spins.

Symmetry-protected topological (SPT) states have boundary ’t Hooft anomalies that obstruct the effective boundary theory realized in its own dimension with UV completion and with an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$-symmetry along the boundary becomes on-site when viewed as an extended $H$-symmetry, via a suitable group extension $1\to K\to H\to G\to 1$. Namely, a nonperturbative global (gauge or gravitational) anomaly in $G$ becomes anomaly free in $H$. This guides us to construct an exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries applicable to any SPT state of any finite symmetry group, including on-site unitary and antiunitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary in any spacetime dimension or, more naturally, by a topological emergent $K$-gauge theory with a global symmetry $G$ on a $3+1\mathrm{D}$ bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Applying our approach to a $1+1\mathrm{D}$ boundary of $2+1\mathrm{D}$ bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped boundaries or interfaces obtained via symmetry breaking (either global symmetry breaking or the Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT states, topologically ordered gauge theories, and symmetry enriched topologically ordered (SET) states leads to generic boundaries or interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

Topological defects are crucial to the thermodynamics and structure of condensed matter systems. For instance, when incorporated into crystalline membranes like graphene, disclinations with positive and negative topological charge elastically buckle the material into conical and saddlelike shapes, respectively. A recently uncovered mapping between the interelement spacing in 2D columnar structures and the metric properties of curved surfaces motivates basic questions about the interplay between defects in the cross section of a columnar bundle and its 3D shape. Such questions are critical to the structure of a broad class of filamentous materials, from biological assemblies like protein fibers to nanostructured or microstructured synthetic materials like carbon nanotube bundles. Here, we explore the buckling behavior for elementary disclinations in hexagonal bundles using a combination of continuum elasticity theory and numerical simulations of discrete filaments. We show that shape instabilities are controlled by a single material-dependent parameter that characterizes the ratio of interfilament to intrafilament elastic energies. Along with a host of previously unknown shape equilibria—the filamentous analogs to the conical and saddlelike shapes of defective membranes—we find a profoundly asymmetric response to positive and negative topologically charged defects in the infinite length limit that is without parallel to the membrane analog. The highly nonlinear dependence on the sign of the disclination charge is shown to have a purely geometric origin, stemming from the distinct compatibility (or incompatibility) of effectively positive- (or negative-)curvature geometries with lengthwise-constant filament spacing.

There has been great interest in experimental studies of charged particles in artificial gauge fields. Here, we perform the first cold-atom explorations of the combination of artificial gauge fields and disorder. Using synthetic lattice techniques based on parametrically coupled atomic momentum states, we engineer zigzag chains with a tunable homogeneous flux. The breaking of time-reversal symmetry by the applied flux leads to analogs of spin-orbit coupling and spin-momentum locking, which we observe directly through the chiral dynamics of atoms initialized to single lattice sites. We additionally introduce precisely controlled disorder in the site-energy landscape, allowing us to explore the interplay of disorder and large effective magnetic fields. The combination of correlated disorder and controlled intrarow and interrow tunneling in this system naturally supports energy-dependent localization, relating to a single-particle mobility edge. We measure the localization properties of the extremal eigenstates of this system, the ground state and the most-excited state, and demonstrate clear evidence for a flux-dependent mobility edge. These measurements constitute the first direct evidence for energy-dependent localization in a lower-dimensional system, as well as the first explorations of the combined influence of artificial gauge fields and engineered disorder. Moreover, we provide direct evidence for interaction shifts of the localization transitions for both low- and high-energy eigenstates in correlated disorder, relating to the presence of a many-body mobility edge. The unique combination of strong interactions, controlled disorder, and tunable artificial gauge fields present in this synthetic lattice system should enable myriad explorations into intriguing correlated transport phenomena.

The topological nodal-line semimetal state, serving as a fertile ground for various topological quantum phases, where a topological insulator, Dirac semimetal, or Weyl semimetal can be realized when the certain protecting symmetry is broken, has only been experimentally studied in very few materials. In contrast to discrete nodes, nodal lines with rich topological configurations can lead to more unusual transport phenomena. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, here, we provide compelling evidence of nodal-line fermions in centrosymmetric semimetal ${\mathrm{TiB}}_2$ with a negligible spin-orbit coupling effect. With the band crossings just below the Fermi energy, two groups of Dirac nodal rings are clearly observed without any interference from other bands, one surrounding the Brillouin zone (BZ) corner in the horizontal mirror plane ${\sigma }_h$ and the other surrounding the BZ center in the vertical mirror plane ${\sigma }_v$. The linear dispersions forming Dirac nodal rings are as wide as 2 eV. We further observe that the two groups of nodal rings link together along the $\mathrm{\Gamma }‐K$ direction, composing a nodal-link configuration. The simple electronic structure with Dirac nodal links mainly constituting the Fermi surfaces suggests ${\mathrm{TiB}}_2$ as a remarkable platform for studying and applying the novel physical properties related to nodal-line fermions.