Rectified electric current induced by irradiating light, a so-called photocurrent, is an established phenomenon in optoelectronic physics. In this paper, we present a comprehensive classification of the photocurrent response arising from the parity violation in bulk systems. We clarify the contrasting role of $\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetries and consequently find new types of photocurrent phenomena characteristic of parity-violating magnets—the intrinsic Fermi surface effect and the gyration current. In particular, the gyration current is induced by circularly polarized light, and it is the counterpart of the shift current caused by linearly polarized light. This photocurrent adds a new functionality of materials studied in various fields of condensed matter physics such as multiferroics and spintronics. A list of materials is provided. Furthermore, we show that the gyration current is strongly enhanced by topologically nontrivial band dispersion. On the basis of the microscopic analysis of Dirac models, we demonstrate the divergent photocurrent response and elucidate the importance of the tilting of Dirac cones.

The quantum anomalous Hall (QAH) effect in a magnetic topological insulator (TI) represents a new state of matter originating from the interplay between topology and magnetism. The defining characteristics of the QAH ground state are the quantized Hall resistivity (${\rho }_{\mathrm{yx}}$) and vanishing longitudinal resistivity (${\rho }_{\mathrm{xx}}$) in the absence of an external magnetic field. A fundamental question concerning the QAH effect is whether it is merely a zero-magnetic-field quantum Hall (QH) effect or if it can host unique quantum phases and phase transitions that are unavailable elsewhere. The most dramatic departure of the QAH systems from other QH systems lies in the magnetic disorders that induce spatially random magnetization. Because disorder and magnetism play pivotal roles in the phase diagram of two-dimensional electron systems, the high degree of magnetic disorders in QAH systems may create novel phases and quantum critical phenomena. In this work, we perform systematic transport studies of a series of magnetic TIs with varied strength of magnetic disorders. We find that the ground state can be categorized into two distinct classes: the QAH phase and the anomalous Hall (AH) insulator phase, as the zero-magnetic-field counterparts of the QH liquid and Hall insulator phases in the QH systems. In the low-disorder limit of the QAH phase, we observe a universal quantized longitudinal resistance ${\rho }_{\mathrm{xx}}=h/e^2$ at the coercive field. In the AH insulator regime, we find that a magnetic field can drive it to the QAH phase through a quantum critical point with scaling behaviors that are distinct from those in the QH phase transition. We propose that the transmission between chiral edge states at domain boundaries, tunable by magnetic disorder and magnetic fields, is the key for determining the QAH ground state.

We explore field and temperature scaling of magnetoresistance in underdoped ($x=0$, $x=0.19$) and optimally doped ($x=0.31$) samples of the high-temperature superconductor ${\mathrm{BaFe}}_2({\mathrm{As}}_{1-x}{\mathrm{P}}_x{)}_2$. In all cases, the magnetoresistance is $H$ linear at high fields. We demonstrate that the data can be explained by an orbital model in the presence of strongly anisotropic quasiparticle spectra and scattering time due to antiferromagnetism. In optimally doped samples, the magnetoresistance is controlled by the properties of small regions of the Fermi surface called “hot spots,” where antiferromagnetic excitations induce a large quasiparticle scattering rate. The anisotropic scattering rate results in hyperbolic $H/T$ magnetoresistance scaling, which competes with the more conventional Kohler scaling. We argue that these results constitute a coherent picture of magnetotransport in ${\mathrm{BaFe}}_2({\mathrm{As}}_{1-x}{\mathrm{P}}_x{)}_2$, which links the origin of $H$-linear resistivity to antiferromagnetic hot spots. Implications for the $T$-linear resistivity at zero field are discussed.

We present a theoretical study of the temporal and spatial coherence properties of a topological laser device built by including saturable gain on the edge sites of a Harper-Hofstadter lattice for photons. For small enough lattices, the Bogoliubov analysis applies, and the coherence time is almost determined by the total number of photons in the device in agreement with the standard Schawlow-Townes phase diffusion. In larger lattices, looking at the lasing edge mode in the comoving frame of its chiral motion, the spatiotemporal correlations of long-wavelength fluctuations display a Kardar-Parisi-Zhang (KPZ) scaling. Still, at very long times, when the finite size of the device starts to matter, the functional form of the temporal decay of coherence changes from the KPZ stretched exponential to a Schawlow-Townes-like exponential, while the nonlinear dynamics of KPZ fluctuations remains visible as a broadened linewidth as compared to the Bogoliubov-Schawlow-Townes prediction. While we establish the above behaviors also for nontopological 1D laser arrays, the crucial role of topology in protecting the coherence from static disorder is finally highlighted: Our numerical calculations suggest the dramatically reinforced coherence properties of topological lasers compared to corresponding nontopological devices. These results open exciting possibilities for both fundamental studies of nonequilibrium statistical mechanics and concrete applications to laser devices.

We investigate the thermal-transport properties of the kagome antiferromagnet Cd-kapellasite (Cd-K). We find that a field-suppression effect on the longitudinal thermal conductivity ${\kappa }_{xx}$ sets in below approximately 25 K. This field-suppression effect at 15 T becomes as large as 80% at low temperatures, suggesting a large spin contribution ${\kappa }_{xx}^{\mathrm{sp}}$ in ${\kappa }_{xx}$. We also find clear thermal Hall signals in the spin liquid phase in all Cd-K samples. The magnitude of the thermal Hall conductivity ${\kappa }_{xy}$ shows a significant dependence on the sample’s scattering time, as seen in the rise of the peak ${\kappa }_{xy}$ value in almost linear fashion with the magnitude of ${\kappa }_{xx}$. On the other hand, the temperature dependence of ${\kappa }_{xy}$ is similar in all Cd-K samples; ${\kappa }_{xy}$ shows a peak at almost the same temperature of the peak of the phonon thermal conductivity ${\kappa }_{xx}^{\mathrm{ph}}$ which is estimated by ${\kappa }_{xx}$ at 15 T. These results indicate the presence of a dominant phonon thermal Hall ${\kappa }_{xy}^{\mathrm{ph}}$ at 15 T. In addition to ${\kappa }_{xy}^{\mathrm{ph}}$, we find that the field dependence of ${\kappa }_{xy}$ at low fields turns out to be nonlinear at low temperatures, concomitantly with the appearance of the field suppression of ${\kappa }_{xx}$, indicating the presence of a spin thermal Hall ${\kappa }_{xy}^{\mathrm{sp}}$ at low fields. Remarkably, by assembling the ${\kappa }_{xx}$ dependence of ${\kappa }_{xy}^{\mathrm{sp}}$ data of other kagome antiferromagnets, we find that, whereas ${\kappa }_{xy}^{\mathrm{sp}}$ stays a constant in the low-${\kappa }_{xx}$ region, ${\kappa }_{xy}^{\mathrm{sp}}$ starts to increase as ${\kappa }_{xx}$ does in the high-${\kappa }_{xx}$ region. This ${\kappa }_{xx}$ dependence of ${\kappa }_{xy}^{\mathrm{sp}}$ indicates the presence of both intrinsic and extrinsic mechanisms in the spin thermal Hall effect in kagome antiferromagnets. Furthermore, both ${\kappa }_{xy}^{\mathrm{ph}}$ and ${\kappa }_{xy}^{\mathrm{sp}}$ disappear in the antiferromagnetic ordered phase at low fields, showing that phonons alone do not exhibit the thermal Hall effect. A high field above approximately 7 T induces ${\kappa }_{xy}^{\mathrm{ph}}$, concomitantly with a field-induced increase of ${\kappa }_{xx}$ and the specific heat, suggesting a coupling of the phonons to the field-induced spin excitations as the origin of ${\kappa }_{xy}^{\mathrm{ph}}$.

We present a sign-problem-free quantum Monte Carlo study of a model that exhibits quantum phase transitions without symmetry breaking and associated changes in the size of the Fermi surface. The model is an Ising gauge theory on the square lattice coupled to an Ising matter field and spinful “orthogonal” fermions at half filling, both carrying Ising gauge charges. In contrast to previous studies, our model hosts an electronlike, gauge-neutral fermion excitation providing access to Fermi-liquid phases. One of the phases of the model is a previously studied orthogonal semimetal, which has ${\mathbb{Z}}_2$ topological order and Luttinger-volume-violating Fermi points with gapless orthogonal fermion excitations. We elucidate the global phase diagram of the model: Along with a conventional Fermi-liquid phase with a large Luttinger-volume Fermi surface, we also find a “deconfined” Fermi liquid in which the large Fermi surface coexists with fractionalized excitations. We present results for the electron spectral function, showing its evolution from the orthogonal semimetal with a spectral weight near momenta $\{\pm \pi /2,\pm \pi /2\}$ to a large Fermi surface.

Blurring the boundary between bosons and fermions lies at the heart of a wide range of intriguing quantum phenomena in multiple disciplines, ranging from condensed matter physics and atomic, molecular, and optical physics to high-energy physics. One such example is a multicomponent Fermi gas with $\mathrm{SU}(N)$ symmetry that is expected to behave like spinless bosons in the large-$N$ limit, where the large number of internal states weakens constraints from the Pauli exclusion principle. However, bosonization in $\mathrm{SU}(N)$ fermions has never been established in high dimensions where exact solutions are absent. Here, we report direct evidence for bosonization in a $\mathrm{SU}(N)$ fermionic ytterbium gas with tunable $N$ in three dimensions (3D). We measure contacts, the central quantity controlling dilute quantum gases, from the momentum distribution and find that the contact per spin approaches a constant with a $1/N$ scaling in the low-fugacity regime consistent with our theoretical prediction. This scaling signifies the vanishing role of the fermionic statistics in thermodynamics and allows us to verify bosonization through measuring a single physical quantity. Our work delivers a highly controllable quantum simulator to exchange the bosonic and fermionic statistics through tuning the internal degrees of freedom in any generic dimensions. It also suggests a new route toward exploring multicomponent quantum systems and their underlying symmetries with contacts.

Superconductivity with a remarkably high $T_c$ has recently been observed in hole-doped ${\mathrm{NdNiO}}_2$, a material that shares similarities with the high-$T_c$ cuprates. This discovery promises new insights into the mechanism of unconventional superconductivity, but at the modeling level, there are fundamental issues that need to be resolved. While it is generally agreed that the low-energy properties of cuprates can, to a large extent, be captured by a single-band model, there has been a controversy in the recent literature about the importance of a multiband description of the nickelates. Here, we use a multisite extension of the recently developed $GW+\mathrm{EDMFT}$ method, which is free of adjustable parameters, to self-consistently compute the interaction parameters and electronic structure of hole-doped ${\mathrm{NdNiO}}_2$. This full ab initio simulation demonstrates the importance of a multiorbital description, even for the undoped compound, and it produces results for the resistivity and Hall conductance in qualitative agreement with experiment.

We study the low-frequency properties of the bulk photovoltaic effect in topological semimetals. The bulk photovoltaic effect is a nonlinear optical effect that generates dc photocurrents under uniform irradiation, which is allowed by noncentrosymmetry. It is a promising mechanism for a terahertz photodetection based on topological semimetals. Here, we systematically investigate the low-frequency behavior of the second-order optical conductivity in point-node semimetals. Through symmetry and power-counting analysis, we show that Dirac and Weyl points with tilted cones show the leading low-frequency divergence. In particular, we find new divergent behaviors of the conductivity of Dirac and Weyl points under circularly polarized light, where the conductivity scales as ${\omega }^{-2}$ and ${\omega }^{-1}$ near the gap-closing point in two and three dimensions, respectively. We provide a further perspective on the low-frequency bulk photovoltaic effect by revealing the complete quantum geometric meaning of the second-order optical conductivity tensor. The bulk photovoltaic effect has two origins, which are the transition of electron position and the transition of electron velocity during the optical excitation, and the resulting photocurrents are, respectively, called the shift current and the injection current. Based on an analysis of two-band models, we show that the injection current is controlled by the quantum metric and Berry curvature, whereas the shift current is governed by the Christoffel symbols near the gap-closing points in semimetals. Finally, for further demonstrations of our theory beyond simple two-band models, we perform first-principles calculations on the shift and injection photocurrent conductivities as well as geometric quantities of antiferromagnetic ${\mathrm{MnGeO}}_3$ and ferromagnetic PrGeAl, respectively, as representatives of real magnetic Dirac and Weyl semimetals. Our calculations reveal gigantic peaks in many nonvanishing elements of photoconductivity tensors below a photon energy of about 0.2 eV in both ${\mathrm{MnGeO}}_3$ and PrGeAl. In particular, we show the ${\omega }^{-1}$ enhancement of the shift conductivity tensors due to the divergent behavior of the geometric quantities near the Dirac and Weyl points as well as slightly gapped topological nodes. Moreover, the low-frequency bulk photovoltaic effect is tunable by carrier doping and magnetization orientation rotation. Our work brings new insights into the structure of nonlinear optical responses as well as the design of semimetal-based terahertz photodetectors.

An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer. It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones. Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits $N$ or the depth $D$ of the circuit. This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing. These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum. They are characterized by an exponentially decaying fidelity $\mathcal{F}\sim (1-\epsilon {)}^{ND}$ with an error rate $\epsilon$ per operation as small as $\approx 1\%$ for current devices with several dozen qubits or even smaller for smaller devices. In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer. Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately. This compression introduces a finite error rate $\epsilon$ so that the algorithms closely mimic the behavior of real quantum computing devices. The computing time of our algorithm increases only linearly with $N$ and $D$ in sharp contrast with exact simulation algorithms. We illustrate our algorithms with simulations of random circuits for qubits connected in both one- and two-dimensional lattices. We find that $\epsilon$ can be decreased at a polynomial cost in computing power down to a minimum error ${\epsilon }_{\infty }$. Getting below ${\epsilon }_{\infty }$ requires computing resources that increase exponentially with ${\epsilon }_{\infty }/\epsilon$. For a two-dimensional array of $N=54$ qubits and a circuit with control-$Z$ gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours. For more complex gates such as a swap gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time. Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction $(\sim {10}^{-8})$ of the system Hilbert space is actually being exploited.

Thouless’ quantization of adiabatic particle transport permits one to associate an integer topological charge with each atom of an electronically gapped material. If these charges are additive and independent of atomic positions, they provide a rigorous definition of atomic oxidation states and atoms can be identified as integer-charge carriers in ionic conductors. Whenever these conditions are met, charge transport is necessarily convective; i.e., it cannot occur without substantial ionic flow, a transport regime that we dub trivial. We show that the topological requirements that allow these conditions to be broken are the same that would determine a Thouless’ pump mechanism if the system were subject to a suitably defined time-periodic Hamiltonian. The occurrence of these requirements determines a nontrivial transport regime whereby charge can flow without any ionic convection, even in electronic insulators. These results are first demonstrated with a couple of simple molecular models that display a quantum-pump mechanism upon introduction of a fictitious time dependence of the atomic positions along a closed loop in configuration space. We finally examine the impact of our findings on the transport properties of nonstoichiometric alkali-halide melts, where the same topological conditions that would induce a quantum-pump mechanism along certain closed loops in configuration space also determine a nontrivial transport regime such that most of the total charge current results to be uncorrelated from the ionic ones.

The light-matter interaction can be utilized to qualitatively alter physical properties of materials. Recent theoretical and experimental studies have explored this possibility of controlling matter by light based on driving many-body systems via strong classical electromagnetic radiation, leading to a time-dependent Hamiltonian for electronic or lattice degrees of freedom. To avoid inevitable heating, pump-probe setups with ultrashort laser pulses have so far been used to study transient light-induced modifications in materials. Here, we pursue yet another direction of controlling quantum matter by modifying quantum fluctuations of its electromagnetic environment. In contrast to earlier proposals on light-enhanced electron-electron interactions, we consider a dipolar quantum many-body system embedded in a cavity composed of metal mirrors and formulate a theoretical framework to manipulate its equilibrium properties on the basis of quantum light-matter interaction. We analyze hybridization of different types of the fundamental excitations, including dipolar phonons, cavity photons, and plasmons in metal mirrors, arising from the cavity confinement in the regime of strong light-matter interaction. This hybridization qualitatively alters the nature of the collective excitations and can be used to selectively control energy-level structures in a wide range of platforms. Most notably, in quantum paraelectrics, we show that the cavity-induced softening of infrared optical phonons enhances the ferroelectric phase in comparison with the bulk materials. Our findings suggest an intriguing possibility of inducing a superradiant-type transition via the light-matter coupling without external pumping. We also discuss possible applications of the cavity-induced modifications in collective excitations to molecular materials and excitonic devices.

We present a novel, nonparametric form for compactly representing entangled many-body quantum states, which we call a “Gaussian process state.” In contrast to other approaches, we define this state explicitly in terms of a configurational data set, with the probability amplitudes statistically inferred from this data according to Bayesian statistics. In this way, the nonlocal physical correlated features of the state can be analytically resummed, allowing for exponential complexity to underpin the ansatz, but efficiently represented in a small data set. The state is found to be highly compact, systematically improvable, and efficient to sample, representing a large number of known variational states within its span. It is also proven to be a “universal approximator” for quantum states, able to capture any entangled many-body state with increasing data-set size. We develop two numerical approaches which can learn this form directly—a fragmentation approach and direct variational optimization—and apply these schemes to the fermionic Hubbard model. We find competitive or superior descriptions of correlated quantum problems compared to existing state-of-the-art variational ansatzes, as well as other numerical methods.

A method to calculate the one-body Green’s function for ground states of correlated electron materials is formulated by extending the variational Monte Carlo method. We benchmark against the exact diagonalization (ED) for the one- and two-dimensional Hubbard models of 16-site lattices, which proves high accuracy of the method. The application of the method to a larger-sized Hubbard model on the square lattice correctly reproduces the Mott insulating behavior at half-filling and gap structures of the $d$-wave superconducting state of the hole-doped Hubbard model in the ground state optimized by enforcing the charge uniformity, evidencing a wide applicability to strongly correlated electron systems. From the obtained $d$-wave superconducting gap of the charge-uniform state, we find that the gap amplitude at the antinodal point is several times larger than the experimental value when we employ a realistic parameter as a model of the cuprate superconductors. The effective attractive interaction of carriers in the $d$-wave superconducting state inferred for an optimized state of the Hubbard model is as large as the order of the nearest-neighbor transfer, which is far beyond the former expectation in the cuprates. We discuss the nature of the superconducting state of the Hubbard model in terms of the overestimate of the gap and the attractive interaction in comparison to the cuprates.

When the ground state of a localized ion is a non-Kramers doublet, such localized ions may carry multipolar moments. For example, ${\mathrm{Pr}}^{3+}$ ions in a cubic environment would possess quadrupolar and octupolar, but no magnetic dipole, moments. When such multipolar moments are placed in a metallic host, unusual interactions between these local moments and conduction electrons arise, in contrast to the familiar magnetic dipole interactions in the classic Kondo problem. In this work, we consider the interaction between a single quadrupolar-octupolar local moment and conduction electrons with $p$-orbital symmetry as a concrete model for the multipolar Kondo problem. We show that this model can be written most naturally in the spin-orbital entangled basis of conduction electrons. Using this basis, the perturbative renormalization-group (RG) fixed points are readily identified. There are two kinds of fixed points, one for the two-channel Kondo and the other for a novel fixed point. We investigate the nature of the novel fixed point nonperturbatively using non-Abelian bosonization, current algebra, and conformal field theory approaches. It is shown that the novel fixed point leads to a, previously unidentified, non-Fermi liquid state with entangled spin and orbital degrees of freedom, which shows resistivity $\rho \sim T^{\mathrm{\Delta }}$ and diverging specific heat coefficient $C/T\sim T^{-1+2\mathrm{\Delta }}$, with $\mathrm{\Delta }=1/5$. Our results open up the possibility of myriads of non-Fermi liquid states, depending on the choices of multipolar moments and conduction electron orbitals, which would be relevant for many rare-earth metallic systems.

Continuously monitoring the environment of a quantum many-body system reduces the entropy of (purifies) the reduced density matrix of the system, conditional on the outcomes of the measurements. We show that, for mixed initial states, a balanced competition between measurements and entangling interactions within the system can result in a dynamical purification phase transition between (i) a phase that locally purifies at a constant system-size-independent rate and (ii) a “mixed” phase where the purification time diverges exponentially in the system size. The residual entropy density in the mixed phase implies the existence of a quantum error-protected subspace, where quantum information is reliably encoded against the future nonunitary evolution of the system. We show that these codes are of potential relevance to fault-tolerant quantum computation as they are often highly degenerate and satisfy optimal trade-offs between encoded information densities and error thresholds. In spatially local models in $1+1$ dimensions, this phase transition for mixed initial states occurs concurrently with a recently identified class of entanglement phase transitions for pure initial states. The purification transition studied here also generalizes to systems with long-range interactions, where conventional notions of entanglement transitions have to be reformulated. We numerically explore this transition for monitored random quantum circuits in $1+1$ dimensions and all-to-all models. Unlike in pure initial states, the mutual information of an initially completely mixed state in $1+1$ dimensions grows sublinearly in time due to the formation of the error-protected subspace. Purification dynamics is likely a more robust probe of the transition in experiments, where imperfections generically reduce entanglement and drive the system towards mixed states. We describe the motivations for studying this novel class of nonequilibrium quantum dynamics in the context of advanced quantum computing platforms and fault-tolerant quantum computation.

Nontrivial topology and unconventional pairing are two central guiding principles in the contemporary search for and analysis of superconducting materials and heterostructure compounds. Previously, a topological superconductor has been predominantly conceived to result from a topologically nontrivial band subject to an intrinsic or external superconducting proximity effect. Here, we propose a new class of topological superconductors that are uniquely induced by unconventional pairing. They exhibit a boundary-obstructed higher-order topological character and, depending on their dimensionality, feature unprecedently robust Majorana bound states or hinge modes protected by chiral symmetry. We predict the 112 family of iron pnictides, such as ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$, to be highly suited material candidates for our proposal, which can be tested by edge spectroscopy. Because of the boundary obstruction, the topologically nontrivial feature of the 112 pnictides does not reveal itself for a bulk-only torus band analysis without boundaries, and as such, it had evaded previous investigations. Our proposal not only opens a new arena for highly stable Majorana modes in high-temperature superconductors but also provides the smoking gun for extended $s$-wave order in the iron pnictides.

Pumping graphene with circularly polarized light is the archetype of light-tailoring topological bands. Realizing the induced Floquet-Chern-insulator state and demonstrating clear experimental evidence for its topological nature has been a challenge, and it has become clear that scattering effects play a crucial role. We tackle this gap between theory and experiment by employing microscopic quantum kinetic calculations including realistic electron-electron and electron-phonon scattering. Our theory provides a direct link to the build up of the Floquet-Chern-insulator state in light-driven graphene and its detection in time- and angle-resolved photoemission spectroscopy (ARPES). This approach allows us to study the robustness of the Floquet features against dephasing and thermalization effects. We also discuss the ultrafast Hall response in the laser-heated state. Furthermore, the induced pseudospin texture and the associated Berry curvature give rise to momentum-dependent orbital magnetization, which is reflected in circular dichroism in ARPES (CD-ARPES). Combining our nonequilibrium calculations with an accurate one-step theory of photoemission allows us to establish a direct link between the build up of the topological state and the dichroic pump-probe photoemission signal. The characteristic features in CD-ARPES are shown to be stable against heating and dephasing effects. Thus, tracing circular dichroism in time-resolved photoemission provides new insights into transient topological properties.

The existence of fundamentally identical particles represents a foundational distinction between classical and quantum mechanics. Because of their exchange symmetry, identical particles can appear to be entangled—another uniquely quantum phenomenon with far-reaching practical implications. However, a long-standing debate has questioned whether identical particle entanglement is physical or merely a mathematical artifact. In this work, we provide such particle entanglement with a consistent theoretical description as a quantum resource in processes frequently encountered in optical and cold atomic systems. This leads to a plethora of applications of immediate practical impact. On the one hand, we show that the metrological advantage for estimating phase shifts in systems of identical bosons amounts to a measure of their particle entanglement, with a clear-cut operational meaning. On the other hand, we demonstrate in general terms that particle entanglement is the property resulting in directly usable mode entanglement when distributed to separated parties, with particle conservation laws in play. Application of our tools to an experimental implementation with Bose-Einstein condensates leads to the first quantitative estimation of identical particle entanglement. Further connections are revealed between particle entanglement and other resources such as optical nonclassicality and quantum coherence. Overall, this work marks a resolutive step in the ongoing debate by delivering a unifying conceptual and practical understanding of entanglement between identical particles.

There are several FeSe based superconductors, including the bulk FeSe, monolayer FeSe thin film, intercalated ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ and ${\mathrm{Li}}_{1-x}{\mathrm{Fe}}_x\mathrm{OHFeSe}$, etc. Their normal states all show metallic behavior. The key player here is the FeSe layer, which exhibits the highest superconducting transition temperature in the form of monolayer thin film. Recently, a new FeSe based compound, ${\mathrm{CsFe}}_{4-x}{\mathrm{Se}}_4$, with the space group of $Bmmm$ was found. Interestingly, the system shows a strong insulatorlike behavior, although it shares the same FeSe planes as other relatives. Density functional theory calculations indicate that it should be a metal, in sharp contrast with the experimental observations. Here, we report the emergence of unconventional superconductivity by applying pressure to suppress this insulatorlike behavior. At ambient pressure, the insulatorlike behavior cannot be modeled as a band insulator, but it can be described by the variable-range-hopping model for correlated systems. Furthermore, the specific heat down to 400 mK has been measured, and a significant residual coefficient ${\gamma }_0=C/T{{|}_T}_{\to 0}$ is observed, which contrasts the insulatorlike state and suggests some quantum freedom of spin dynamics. By applying pressure, the insulatorlike behavior is gradually suppressed, and the system becomes a metal; finally, superconductivity is achieved at about 5.1 K. The superconducting transition strongly depends on magnetic field and applied current, indicating a fragile superfluid density. Our results suggest that the superconductivity is established by diluted Cooper pairs on top of a strong correlation background in ${\mathrm{CsFe}}_{4-x}{\mathrm{Se}}_4$.