One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is crucially important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original $\mathcal{O}(\beta )$ to $\mathcal{O}({\beta }^{2/3})$ up to a logarithmic factor, thereby suggesting subballistic propagation of entanglement by imaginary-time evolution. This qualitatively differs from the real-time evolution, which usually induces linear growth of entanglement. We also prove analogous bounds for the Rényi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with $n$ spins, we prove that the Gibbs state is well approximated by a matrix product operator with a sublinear bond dimension for $\beta =o[\log (n)]$. This proof allows us to rigorously establish, for the first time, a quasilinear time classical algorithm for constructing a matrix product state representation of 1D quantum Gibbs states at arbitrary temperatures of $\beta =o[\log (n)]$. Our new technical ingredient is a block decomposition of the Gibbs state that bears a resemblance to the decomposition of real-time evolution given by Haah et al. [Proceedings of the 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2018), pp. 350–360].

Artificial spin ices (ASIs) are interacting arrays of lithographically defined nanomagnets in which novel, frustrated magnetic phases can be intentionally designed. A key emergent description of fundamental excitations in ASIs is that of magnetic monopoles—mobile quasiparticles that carry an effective magnetic charge. Here, we demonstrate that the archetypal square ASI lattice can host, in specific regions of its magnetic phase diagram, plasmalike regimes containing a high density of mobile magnetic monopoles. These regimes result from the magnetic field-tunable tension on the Dirac strings connecting mobile monopoles. By passively “listening” to spontaneous monopole noise under conditions of strict thermal equilibrium, we reveal their intrinsic dynamics and show that monopole kinetics are most diffusive (that is, minimally correlated) in the plasma regime. These results open the door to on-demand monopole regimes having continuously field-tunable densities and dynamic properties, thereby providing a new paradigm for probing the physics of effective magnetic charges in synthetic matter.

Topological superconductors (TSCs) are correlated quantum states with simultaneous off-diagonal long-range order and nontrivial topological invariants. They produce gapless or zero-energy boundary excitations, including Majorana zero modes and chiral Majorana edge states with topologically protected phase coherence essential for fault-tolerant quantum computing. Candidate TSCs are very rare in nature. Here, we propose a novel route toward emergent quasi-one-dimensional (1D) TSCs in naturally embedded quantum structures such as atomic line defects in unconventional spin-singlet $s$-wave and $d$-wave superconductors. We show that inversion symmetry breaking and charge transfer due to the missing atoms lead to the occupation of incipient impurity bands and mixed-parity spin-singlet and -triplet Cooper pairing of neighboring electrons traversing the line defect. Nontrivial topological invariants arise and occupy a large part of the parameter space, including the time-reversal symmetry-breaking Zeeman coupling due to applied magnetic field or defect-induced magnetism, creating TSCs in different topological classes with robust Majorana zero modes at both ends of the line defect. Beyond providing a novel mechanism for the recent discovery of zero-energy bound states at both ends of an atomic line defect in monolayer Fe(Te,Se) superconductors, the findings pave the way for new material realizations of the simplest and most robust 1D TSCs using embedded quantum structures in unconventional superconductors with large pairing energy gaps and high transition temperatures.

The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics. Here, we study how the minimum number of variables needed to accurately describe the important features of a data set—the intrinsic dimension ($I_d$)—behaves in the vicinity of phase transitions. We employ state-of-the-art nearest-neighbors-based $I_d$ estimators to compute the $I_d$ of raw Monte Carlo thermal configurations across different phase transitions: first-order, second-order, and Berezinskii-Kosterlitz-Thouless. For all the considered cases, we find that the $I_d$ uniquely characterizes the transition regime. The finite-size analysis of the $I_d$ allows us to not only identify critical points with an accuracy comparable to methods that rely on a priori identification of order parameters but also to determine the corresponding (critical) exponent $\nu$ in the case of continuous transitions. For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods. Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme and suggest direct parallelism between conventional order parameters in real space and the intrinsic dimension in the data space.

Topological magnets are a new family of quantum materials providing great potential to realize emergent phenomena, such as the quantum anomalous Hall effect and the axion-insulator state. Here, we present our discovery that the stoichiometric ferromagnet ${\mathrm{MnBi}}_8{\mathrm{Te}}_{13}$ with natural heterostructure ${\mathrm{MnBi}}_2{\mathrm{Te}}_4/({\mathrm{Bi}}_2{\mathrm{Te}}_3{)}_3$ is an unprecedented “half-magnetic topological insulator,” with the magnetization existing at the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ surface but not at the opposite surface terminated by triple ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ layers. Our angle-resolved photoemission spectroscopy measurements unveil a massive Dirac gap at the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ surface and a gapless Dirac cone on the other side. Remarkably, the Dirac gap (about 28 meV) at the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ surface decreases monotonically with increasing temperature and closes right at the Curie temperature, thereby representing the first smoking-gun spectroscopic evidence of a magnetization-induced topological surface gap among all known magnetic topological materials. We further demonstrate theoretically that the half-magnetic topological insulator is desirable to realize the surface anomalous Hall effect, which serves as direct proof of the general concept of axion electrodynamics in condensed matter systems.

Phase transitions share the universal feature of enhanced fluctuations near the transition point. Here, we show that density fluctuations reveal how a Bose-Einstein condensate of dipolar atoms spontaneously breaks its translation symmetry and enters the supersolid state of matter—a phase that combines superfluidity with crystalline order. We report on the first direct in situ measurement of density fluctuations across the superfluid-supersolid phase transition. This measurement allows us to introduce a general and straightforward way to extract the static structure factor, estimate the spectrum of elementary excitations, and image the dominant fluctuation patterns. We observe a strong response in the static structure factor and infer a distinct roton minimum in the dispersion relation. Furthermore, we show that the characteristic fluctuations correspond to elementary excitations such as the roton modes, which are theoretically predicted to be dominant at the quantum critical point, and that the supersolid state supports both superfluid as well as crystal phonons.

The Kitaev honeycomb model has attracted significant attention due to its exactly solvable spin-liquid ground state with fractionalized Majorana excitations and its possible materialization in magnetic Mott insulators with strong spin-orbit couplings. Recently, the $5d$-electron compound ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ has shown to be a strong candidate for Kitaev physics considering the absence of any signs of a long-range ordered magnetic state. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes and reproduces the apparent power-law upturn in the specific heat measurements of ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$. Physically, the vacancies can originate from various sources such as missing magnetic moments or the presence of nonmagnetic impurities (true vacancies), or from local weak couplings of magnetic moments due to strong but rare bond randomness (quasivacancies). We show numerically that the vacancy effect is readily detectable even at low vacancy concentrations and that it is not very sensitive either to the nature of vacancies or to different flux backgrounds. We also study the response of the site-diluted Kitaev spin liquid to the three-spin interaction term, which breaks time-reversal symmetry and imitates an external magnetic field. We propose a field-induced flux-sector transition where the ground state becomes flux-free for larger fields, resulting in a clear suppression of the low-temperature specific heat. Finally, we discuss the effect of dangling Majorana fermions in the case of true vacancies and show that their coupling to an applied magnetic field via the Zeeman interaction can also account for the scaling behavior in the high-field limit observed in ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$.

Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate. This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements. We find that, surprisingly, EPTs are possible even in the absence of scrambling unitary dynamics, where they are best understood as arising from measurements alone. This finding motivates us to introduce measurement-only models, in which the “scrambling” and “unscrambling” effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes. These models represent a novel form of an EPT, conceptually distinct from that in hybrid unitary-projective circuits. We explore the entanglement phase diagrams, critical points, and quantum code properties of some of these measurement-only models. We find that the principle driving the EPTs in these models is frustration, or mutual incompatibility, of the measurements. Surprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local ($\gtrsim 3$-body) operators. We identify a class of exceptions to this behavior (“bipartite ensembles”) which cannot sustain an entangling phase but display dual area-law phases, possibly with different kinds of quantum order, separated by self-dual critical points. Finally, we introduce a measure of information spreading in dynamics with measurements and use it to demonstrate the emergence of a statistical light cone, despite the nonlocality inherent to quantum measurements.

Layered van der Waals 2D magnetic materials are of great interest in fundamental condensed-matter physics research, as well as for potential applications in spintronics and device physics. We present neutron powder diffraction data using new ultrahigh-pressure techniques to measure the magnetic structure of Mott-insulating 2D honeycomb antiferromagnet ${\mathrm{FePS}}_3$ at pressures up to 183 kbar and temperatures down to 80 K. These data are complemented by high-pressure magnetometry and reverse Monte Carlo modeling of the spin configurations. As pressure is applied, the previously measured ambient-pressure magnetic order switches from an antiferromagnetic to a ferromagnetic interplanar interaction and from 2D-like to 3D-like character. The overall antiferromagnetic structure within the $ab$ planes, ferromagnetic chains antiferromagnetically coupled, is preserved, but the magnetic propagation vector is altered from $\mathbf{k}=(0,1,\frac{1}{2})$ to $\mathbf{k}=(0,1,0)$, a halving of the magnetic unit cell size. At higher pressures, coincident with the second structural transition and the insulator-metal transition in this compound, we observe a suppression of this long-range order and emergence of a form of magnetic short-range order which survives above room temperature. Reverse Monte Carlo fitting suggests this phase to be a short-ranged version of the original ambient-pressure structure—with the Fe moment size remaining of similar magnitude and with a return to antiferromagnetic interplanar correlations. The persistence of magnetism well into the HP-II metallic state is an observation in contradiction with previous x-ray spectroscopy results which suggest a spin-crossover transition.

An anapole state that breaks inversion and time-reversal symmetries while preserving translation symmetry of an underlying lattice has aroused great interest as a new quantum state, but only a few candidate materials have been reported. Recently, in a spin-orbit coupled Mott insulator ${\mathrm{Sr}}_2({\mathrm{Ir}}_{1-x}{\mathrm{Rh}}_x){\mathrm{O}}_4$, the emergence of a possible hidden-order phase with broken inversion symmetry has been suggested at $T_{\mathrm{\Omega }}$ above the Néel temperature by optical second-harmonic-generation measurements. Moreover, polarized neutron diffraction measurements revealed broken time-reversal symmetry below $T_{\mathrm{\Omega }}$, which was supported by subsequent muon spin relaxation experiments. However, the nature of this mysterious phase remains largely elusive. Here, we investigate the hidden-order phase through the combined measurements of the in-plane magnetic anisotropy with exceptionally high-precision magnetic torque and the nematic susceptibility with elastoresistance. A distinct twofold in-plane magnetic anisotropy along the [110] Ir-O-Ir bond direction sets in below about $T_{\mathrm{\Omega }}$, providing thermodynamic evidence for a nematic phase transition with broken $C_4$ rotational symmetry. However, in contrast to the even-parity nematic transition reported in other correlated electron systems, the nematic susceptibility exhibits no divergent behavior towards $T_{\mathrm{\Omega }}$. These results provide bulk evidence for an odd-parity order parameter with broken rotational symmetry in the hidden-order state. We discuss the hidden order in terms of an anapole state, in which the polar toroidal moment is induced by two current loops in each ${\mathrm{IrO}}_6$ octahedron of opposite chirality. Contrary to the simplest loop-current pattern previously suggested, the present results are consistent with a pattern in which the intra-unit cell loop current flows along only one of the diagonal directions in the ${\mathrm{IrO}}_4$ square.

Microcavity exciton polariton systems can have a wide range of macroscopic quantum effects that may be turned into better photonic technologies. Polariton Bose-Einstein condensation and photon lasing have been widely accepted in the limits of low and high carrier densities, but identification of the expected Bardeen-Cooper-Schrieffer (BCS) state at intermediate densities remains elusive, as the optical-gain mechanism cannot be directly inferred from existing experiments. Here, using a microcavity with strong polarization selectivity, we gain direct experimental access to the reservoir absorption in the presence of polariton condensation and lasing, which reveals a fermionic gain mechanism underlying the polariton laser. A microscopic many-particle theory shows that this polariton lasing state is consistent with an open-dissipative-pumped system analog of a polaritonic BCS state.

Topological semimetals have revealed a wide array of novel transport phenomena, including electron hydrodynamics, quantum field theoretic anomalies, and extreme magnetoresistances and mobilities. However, the scattering mechanisms central to the fundamental transport properties remain largely unexplored. Here, we reveal signatures of significant phonon-electron scattering in the type-II Weyl semimetal ${\mathrm{WP}}_2$ via temperature-dependent Raman spectroscopy. Over a large temperature range, we find that the decay rates of the lowest energy $A_1$ modes are dominated by phonon-electron rather than phonon-phonon scattering. In conjunction with first-principles calculations, a combined analysis of the momentum, energy, and symmetry-allowed decay paths indicates this results from finite momentum interband and intraband scattering of the electrons. The excellent agreement with theory further suggests that such results could be true for the acoustic modes. We thus provide evidence for the importance of phonons in the transport properties of topological semimetals and identify specific properties that may contribute to such behavior in other materials.

While designing the energy-momentum relation of photons is key to many linear, nonlinear, and quantum optical phenomena, a new set of light-matter properties may be realized by employing the topology of the photonic bath itself. In this work we experimentally investigate the properties of superconducting qubits coupled to a metamaterial waveguide based on a photonic analog of the Su-Schrieffer-Heeger model. We explore topologically induced properties of qubits coupled to such a waveguide, ranging from the formation of directional qubit-photon bound states to topology-dependent cooperative radiation effects. Addition of qubits to this waveguide system also enables direct quantum control over topological edge states that form in finite waveguide systems, useful for instance in constructing a topologically protected quantum communication channel. More broadly, our work demonstrates the opportunity that topological waveguide-QED systems offer in the synthesis and study of many-body states with exotic long-range quantum correlations.

Twisted bilayer graphene (TBG) provides a unique framework to elucidate the interplay between strong correlations and topological phenomena in two-dimensional systems. The existence of multiple electronic degrees of freedom—charge, spin, and valley—gives rise to a plethora of possible ordered states and instabilities. Identifying which of them are realized in the regime of strong correlations is fundamental to shed light on the nature of the superconducting and correlated insulating states observed in the TBG experiments. Here, we use unbiased, sign-problem-free quantum Monte Carlo simulations to solve an effective interacting lattice model for TBG at charge neutrality. Besides the usual cluster Hubbard-like repulsion, this model also contains an assisted-hopping interaction that emerges due to the nontrivial topological properties of TBG. Such a nonlocal interaction fundamentally alters the phase diagram at charge neutrality, gapping the Dirac cones even for infinitesimally small interactions. As the interaction strength increases, a sequence of different correlated insulating phases emerge, including a quantum valley Hall state with topological edge states, an intervalley-coherent insulator, and a valence bond solid. The charge-neutrality correlated insulating phases discovered here provide the sought-after reference states needed for a comprehensive understanding of the insulating states at integer fillings and the proximate superconducting states of TBG.

The bond-dependent Kitaev model on the honeycomb lattice with anyonic excitations has recently attracted considerable attention. However, in solid-state materials other spin interactions are present, and among such additional interactions, the off-diagonal symmetric Gamma interaction, another type of bond-dependent term, has been particularly challenging to fully understand. A minimal Kitaev-Gamma model has been investigated by various numerical techniques under a magnetic field, but definite conclusions about field-induced spin liquids remain elusive. One reason for this may lie in the limited sizes of the two-dimensional geometry it is possible to access numerically, and missed incommensurately ordered states may be interpreted as a spin liquid. Here we focus on the Kitaev-Gamma ladder model as a guide to the phase space of disordered states which could potentially become a spin liquid in the two-dimensional limit. We determine the entire phase diagram in the presence of a magnetic field along the [111] direction. Because of the competition between the interactions and the field, an extremely rich phase diagram emerges with 15 distinct phases. Focusing near the antiferromagnetic Kitaev region, we identify nine different phases solely within this region: several incommensurate magnetically ordered phases, spin-nematic, and two chiral phases with enhanced entanglement. Of particular interest is a highly entangled phase with staggered chirality with zero-net flux occurring at intermediate field, which along with its companion phases outlines a heart-shaped region of high entanglement, the heart of entanglement. We compare our results for the ladder with a $C_3$ symmetric cluster of the two-dimensional honeycomb lattice, and offer insight into possible spin liquids in the two-dimensional limit.

The heavy-fluxonium circuit is a promising building block for superconducting quantum processors due to its long relaxation and dephasing time at the flux-frustration point. However, the suppressed charge matrix elements and low transition frequency make it challenging to perform fast single-qubit gates using standard protocols. We report on new protocols for reset, fast coherent control, and readout that allow high-quality operation of the qubit with a 14 MHz transition frequency, an order of magnitude lower in energy than the ambient thermal energy scale. We utilize higher levels of the fluxonium to read out the qubit state and to initialize the qubit with 97% fidelity corresponding to cooling it to $190\mu \mathrm{K}$. Instead of using standard microwave pulses, we control the qubit only with fast-flux pulses, generating control fields much larger than the qubit frequency. We develop a universal set of gates based on nonadiabatic Landau-Zener transitions that act in 20–60 ns, less than the single-qubit Larmor period. We measure qubit coherence of $T_1,T_{2e}\sim 300\mu \mathrm{s}$ for a fluxonium in a 2D architecture and realize single-qubit gates with an average gate fidelity of 99.8% as characterized by randomized benchmarking.

One of the central issues concerning the mechanism of high-temperature superconductivity in cuprates is the nature of the ubiquitous charge order and its implications to superconductivity. Here, we use scanning tunneling microscopy to investigate the evolution of charge order from the optimally doped to strongly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_{6+\delta }$ cuprates. We find that, with increasing hole concentration, the long-range checkerboard order gradually evolves into short-range glassy patterns consisting of diluted charge puddles. Each charge puddle has a unidirectional nematic internal structure and exhibits clear pair density modulations as revealed by the spatial variations of the superconducting coherence peak and gap depth. Both the charge puddles and the nematicity vanish completely in the strongly overdoped nonsuperconducting regime, when another type of short-range order with $\sqrt{2}\times \sqrt{2}$ periodicity emerges. These results shed important new light on the intricate interplay between the intertwined orders and the superconducting phase of cuprates.

Atomic diffusion is a spontaneous process and significantly influences properties of materials, such as fracture toughness, creep-fatigue properties, thermal conductivity, thermoelectric properties, etc. Here, using extensive molecular dynamics simulations based on both ab initio and machine-learning potentials, we demonstrate that an atomic one dimensional cooperative diffusion exists in the simple cubic high-pressure finite-temperature phase of calcium in the premelting regime, where some atoms diffuse cooperatively as chains or even rings, while others remain in the solid state. This intermediate regime is triggered by anharmonicity of the system at high temperature and is stabilized by the competition between the internal energy minimization and the entropy maximization, and has close connections with the unique electronic structures of simple cubic Ca as an electride with a pseudogap. This cooperative diffusion regime explains the abnormal enhancement of the melting line of Ca under high pressure and suggests that the cooperative chain melting is a much more common high-temperature feature among metals under extreme conditions than hitherto thought. The microscopic electronic investigations of these systems combining ab initio and machine-learning data point out the direction for further understanding of other metallic systems such as the glass transition, liquid metals, etc.

Recently reported with nontrivial topological properties and magnetic orders, ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ is an intrinsic, magnetic topological insulator which holds promise for exploring exotic quantum phenomena such as the quantum anomalous Hall effect. However, the layer-dependent magnetism of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, which is fundamental and crucial for further exploration of related quantum phenomena in this system, remains elusive. Here, by using polar reflective magnetic circular dichroism spectroscopy, we show that few-layered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ exhibits an evident odd-even layer-number effect, i.e., the oscillations of the coercivity of the hysteresis loop (at ${\mu }_0{H}_c$) and the spin-flop transition (at ${\mu }_0{H}_1$), concerning the Zeeman energy and magnetic anisotropy energy. Noticeably, an anomalous magnetic hysteresis loop is observed in the even-number septuple-layered ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, which might be attributed to the thickness-independent surface-related magnetization. A linear-chain model is applied to elucidate this odd-even layer-number effect of the spin-flop field and to determine the evolution of the magnetic states when subjected to an external magnetic field. A mean-field method further allows us to fully map the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flake’s magnetic phase diagrams in the parameter space of the magnetic field, layer number, and, especially, temperature. By harnessing the unusual layer-dependent magnetic properties, our work paves the way for further study of quantum phenomena of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$.

Herein we show that nonresonant inelastic x-ray scattering involving an $s$ core level is a powerful spectroscopic method to characterize the excited states of transition metal compounds. The spherical charge distribution of the $s$ core hole allows the orientational dependence of the intensities of the various spectral features to produce a spatial charge image of the associated multiplet states in a straightforward manner, thereby facilitating the identification of their orbital character. In addition, the $s$ core hole does not add an extra orbital angular momentum component to the multiplet structure so that the well-established Sugano-Tanabe-Kamimura diagrams can be used for the analysis of the spectra. For $\alpha$-MnS we observe the spherical charge density corresponding to its high-spin $3d^5$ (${}^6{A}_1$) ground state configuration and we were able to selectively image its excited states and identify them as $t_{2g}$ (${}^5{T}_2$) and $e_g$ (${}^{5}E$) with an energy splitting $10Dq$ of 0.78 eV.