The isolated susceptibility ${\chi }_{\mathrm{I}}$ may be defined as a (nonthermodynamic) average over the canonical ensemble, but while it has often been discussed in the literature, it has not been clearly measured. Here, we demonstrate an unambiguous measurement of ${\chi }_{\mathrm{I}}$ at avoided nuclear-electronic level crossings in a dilute spin ice system, containing well-separated holmium ions. We show that ${\chi }_{\mathrm{I}}$ quantifies the superposition of quasiclassical spin states at these points and is a direct measure of state concurrence and populations.

We find the admittance $Y\left(\omega \right)$ of a Josephson junction at or near a topological transition. The dependence of the admittance on frequency and temperature at the critical point is universal and determined by the symmetries of the system. Despite the absence of a spectral gap at the transition, the dissipative response may remain weak at low energies: $\mathrm{Re}Y\left(\omega \right)\propto max{(\omega ,T)}^2$. This behavior is strikingly different from the electromagnetic response of a normal metal. Away from the critical point, the scaling functions for the dependence of the admittance on frequency and temperature are controlled by at most two parameters.

We investigate how the real and imaginary charge density waves interplay at the Van Hove singularity on the hexagonal lattices. A phenomenological analysis indicates the formation of $3Q$ complex orders at all three nesting momenta. Under a total phase condition, unequal phases at the three momenta break the rotation symmetry generally. The $3Q$ complex orders constitute a rich Haldane-model phase diagram. When effective time-reversal symmetries arise under 1-site translations, the Dirac semimetals are protected. The breakdown of these symmetries gaps the Dirac points and leads to the trivial and Chern insulator phases. These phases are deformations of purely real and imaginary orders, which exhibit trivial site and/or bond density and chiral flux orders, respectively. The exotic single-Dirac-point semimetals also appear along the gapless phase boundary. We further show that the theoretical model offers transparent interpretations of experimental observations in the kagome metals $A{\text{V}}_3{\text{Sb}}_5$ with $A=\text{K},\text{Rb},\text{Cs}$. The topological charge density waves may be identified with the complex orders in the Chern insulator phase. Meanwhile, the lower-temperature symmetry-breaking phenomena may be interpreted as the secondary orders from the complex order ground states. Our work sheds light on the nature of the topological charge density waves in the kagome metals $A{\text{V}}_3{\text{Sb}}_5$ and may offer useful indications to the experimentally observed charge orders in the future experiments.

Here, by combining a symmetry-based analysis with numerical computations, we predict a different kind of magnetic ordering—antichiral ferromagnetism. This term aims to reflect that spontaneous modulation of the magnetization direction $\mathbf{m}\left(\mathbf{r}\right)$ appears in a way that both types of chirality (handedness) exist simultaneously, and alternate in space. Without loss of generality, we focus our investigation on crystals with full tetrahedral symmetry where chiral interaction terms—Lifshitz invariants—are forbidden by symmetry. However, we demonstrate that the leading chirality-related term leads to nontrivial smooth magnetic textures exhibiting antichirality. In addition to the unconventional ground state, the revealed ordering gives rise to rich phenomena such as unique magnetic domains and skyrmions.

We have applied nuclear magnetic resonance spectroscopy to study the distinctive network of nodal lines in the Dirac semimetal ZrSiTe. The low-$T$ behavior is dominated by a symmetry-protected nodal line, with NMR providing a sensitive probe of the diamagnetic response of the associated carriers. A sharp low-$T$ minimum in the NMR shift and ${\left(T_{1}T\right)}^{-1}$ provides a quantitative measure of the dispersionless, quasi-two-dimensional behavior of this nodal line. We also identify a Van Hove singularity closely connected to this nodal line, and an associated $T$-induced Lifshitz transition. A disconnect in the NMR shift and linewidth at this temperature indicates the change in electronic behavior associated with this topological change. These features have an orientation-dependent behavior indicating a field-dependent scaling of the associated band energies.

We explore the magnetically ordered ground state of the isovalently substituted Mott insulator ${\mathrm{Y}}_{1-x}{\mathrm{La}}_x\mathrm{Ti}{\mathrm{O}}_3$ for $x\le$ 0.3 via single-crystal growth, magnetometry, neutron diffraction, x-ray magnetic circular dichroism, muon spin rotation, and small-angle neutron scattering (SANS). We find that the decrease in the magnetic transition temperature on approaching the ferromagnetic-antiferromagnetic phase boundary at the La concentration $x_c\approx 0.3$ is accompanied by a strong suppression of both bulk and local ordered magnetic moments, along with a volumewise separation into magnetically ordered and paramagnetic regions. The thermal phase transition does not show conventional second-order behavior since neither a clear signature of dynamic critical behavior nor a power-law divergence of the magnetic correlation length is found for the studied substitution range; this finding becomes increasingly obvious with increasing La substitution. We find no evidence for a spin-glass phase. Finally, from SANS and magnetometry measurements, we discern a crossover from easy-axis to easy-plane magnetocrystalline anisotropy with increasing La substitution. These results indicate complex changes in magnetic structure upon approaching the phase boundary.

One of the major puzzles in condensed matter physics has been the observation of a Mott-insulating state away from half-filling. Several theoretical proposals aimed to elucidate this phenomenon have been put forth, a notable one being phase separation and an associated percolation-induced Mott insulator-metal transition. In the present work we study the prototypical doped Mott-insulating rare-earth titanate ${\mathrm{YTiO}}_3$, in which the insulating state survives up to a large hole concentration of 35%. Single crystals of ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{TiO}}_3$ with $0\le x\le 0.5$, spanning the insulator-metal transition, are grown and investigated. Using x-ray absorption spectroscopy, a powerful technique capable of probing element-specific electronic states, we find that the primary effect of hole doping is to induce electronic phase separation into hole-rich and hole-poor regions. The data reveal the formation of electronic states within the Mott-Hubbard gap, near the Fermi level, which increase in spectral weight with increasing doping. From a comparison with $\mathrm{DFT}+U$ calculations, we infer that the hole-poor and hole-rich components have charge densities that correspond to the insulating $x=0$ and metallic $x\sim 0.5$ states, respectively, and that the new electronic states arise from the metallic component. Our results indicate that the doping-induced insulator-metal transition in ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{TiO}}_3$ is indeed percolative in nature, and thus of inherent first-order character.

We introduce and explore low-energy configurations in two-dimensional arrays consisting of Ising-type dipolar coupled nanomagnets lithographically defined onto three-nanomagnet vertices arranged in a triangular coordination. Thus, the system is dubbed the trimerized triangular lattice. Employing synchrotron-based photoemission electron microscopy, we perform temperature-dependent magnetic imaging of moment configurations. These states are then characterized in terms of spin correlations and magnetic structure factors. The results reveal a competition between ferromagnetic and vortex dominated orders, which can be controlled by varying the relevant lattice parameter and the corresponding competing interactions.

We investigate the impact of two types of disorder, bond randomness and site dilution, on the spin dynamics in the Kitaev model on a honeycomb lattice. The ground state of this model is a canonical quantum spin liquid with spin fractionalization into two types of quasiparticles, itinerant Majorana fermions and localized fluxes, for which the spin dynamics provides a good probe of the fractionalization. Using unbiased quantum Monte Carlo simulations, we calculate the temperature evolution of the dynamical spin structure factor, the magnetic susceptibility, and the nuclear magnetic resonance (NMR) relaxation rate while changing the strength of disorder systematically. In the dynamical spin structure factor, we find that the two types of disorder seriously affect the low-energy peak originating dominantly from the flux excitations, rather than the high-energy continuum from the Majorana excitations, in a different way: The bond randomness softens the peak to the lower energy with broadening, which suggests the closing of the spin gap, whereas the site dilution smears the peak and in addition develops the other sharp peaks inside the spin gap including the zero energy. We show that the zero-energy spin excitations, which originate from the Majorana zero modes induced around the site vacancies, survive up to the temperature comparable to the energy scale of the Kitaev interaction. We also find that the two types of disorder affect the low-temperature behavior of the magnetic susceptibility and the NMR relaxation rate in a different way. For the bond randomness, the low-temperature susceptibility does not show any qualitative change against the weak disorder but it changes to divergent behavior while increasing the strength of disorder. We find that this crossover corresponds to the softening of the low-energy peak in the dynamical structure factor. Similar distinct behaviors for the weak and strong disorders are also observed in the NMR relaxation rate; an exponential decay changes into a power-law decay. In contrast, for the site dilution, we find no such crossover; divergent behavior in the susceptibility and a power-law decay in the NMR relaxation rate immediately appear with the introduction of the site dilution, which is also attributed to the emergence of the Majorana zero modes. We discuss the relevance of our results to experiments for the Kitaev candidate materials with disorders. The peculiar magnetic responses found by the present systematic analysis would be helpful to not only identify the dominant type of disorder in real materials but also examine the experimental realization of the Kitaev spin liquid by introducing disorder.

We show that the insulating states of magic-angle twisted bilayer graphene support a series of collective modes corresponding to local particle-hole excitations on triangular lattice sites. Our theory is based on a continuum model of the magic angle flat bands. When the system is insulating at moiré band filling $\nu =-3$, our calculations show that the ground state supports seven low-energy modes that lie well below the charge gap throughout the moiré Brillouin zone, one of which couples strongly to THz photons. The low-energy collective modes are faithfully described by a model with a local $SU\left(8\right)$ degree of freedom in each moiré unit cell that we identify as the direct product of spin, valley, and an orbital pseudospin. Apart from spin and valley-wave modes, the collective mode spectrum includes a low-energy intraflavor exciton mode associated with transitions between flat valence and conduction band orbitals.

As one of the most fundamental physical phenomena, the anomalous Hall effect (AHE) typically occurs in ferromagnetic materials but is not expected in the conventional superconductors. Here, we have observed a giant AHE in kagome superconductor $\mathrm{Cs}{\mathrm{V}}_3{\mathrm{Sb}}_5$ with transition temperature ($T_c$) of 2.7 K. The anomalous Hall conductivity reaches up to $2.1\times {10}^4{\mathrm{\Omega }}^{–1}\mathrm{c}{\mathrm{m}}^{–1}$ which is larger than those observed in most of the ferromagnetic metals. Strikingly, the emergence of AHE exactly follows the higher-temperature charge-density-wave (CDW) transition with $T_{\mathrm{CDW}} \sim 94\mathrm{K}$, indicating a strong correlation between the CDW state and AHE. Furthermore, AHE disappears when the CDW transition is completely suppressed at high pressure. The origin for AHE is attributed to enhanced skew scattering in the CDW state and large Berry curvature arising from the kagome lattice. These discoveries make $\mathrm{Cs}{\mathrm{V}}_3{\mathrm{Sb}}_5$ as an ideal platform to study the interplay among nontrivial band topology, CDW, and unconventional superconductivity.

In this paper we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space and relate the cumulants of this distribution to the properties of the quasilocal integrals of motion in the MBL phase. We demonstrate nonself-averaging property of the many-body fractal dimension $D_q$ and directly relate it to the jump of $D_q$ as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of $D_q$ in the whole range of parameters. Introducing a simple toy model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to nonergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the noninteracting case.

Flat energy bands of model lattice Hamiltonians provide a key ingredient in designing dispersionless wave excitations and have become a versatile platform to study various aspects of interacting many-body systems. Their essential merit lies in hosting compactly localized eigenstates which originate from destructive interference induced by the lattice geometry, in turn often based on symmetry principles. We here show that flat bands can be generated from a hidden symmetry of the lattice unit cell, revealed as a permutation symmetry upon reduction of the cell over two sites governed by an effective dimer Hamiltonian. This so-called latent symmetry is intimately connected to a symmetry between possible walks of a particle along the cell sites, starting and ending on each of the effective dimer sites. The summed amplitudes of any eigenstate with odd parity on the effective dimer sites vanish on special site subsets called walk multiplets. We exploit this to construct flat bands by using a latently symmetric unit cell coupled into a lattice via walk multiplet interconnections. We demonstrate that the resulting flat bands are tunable by different parametrizations of the lattice Hamiltonian matrix elements which preserve the latent symmetry. The developed framework may offer fruitful perspectives to analyze and design flat band structures.

Recently, we employed electronic polarization-resolved Raman spectroscopy to reveal the strongly correlated excitonic insulator (EI) nature of ${\mathrm{Ta}}_2\mathrm{Ni}{\mathrm{Se}}_5$ [Volkov et al., npj Quant. Mater. 6, 52 (2021)], and also showed that for ${\mathrm{Ta}}_2\mathrm{Ni}{\left({\mathrm{Se}}_{1-x}{\mathrm{S}}_x\right)}_5$ alloys the critical excitonic fluctuations diminish with sulfur concentration $x$, exposing a cooperating lattice instability that takes over for large $x$ [Volkov et al., arXiv:2104.07032]. Here we focus on the lattice dynamics of the EI family ${\mathrm{Ta}}_2\mathrm{Ni}{\left({\mathrm{Se}}_{1-x}{\mathrm{S}}_x\right)}_5$ ($x=0$,..., 1). We identify all Raman-active optical phonons of $A_g$ (fully symmetric) and $B_{2g}$ ($ac$-quadrupole-like) symmetries ($D_{2h}$ point group) and study their evolution with temperature and sulfur concentration. We demonstrate the change of selection rules at temperatures below the orthorhombic-to-monoclinic transition at $T_c\left(x\right)$ that is related to the EI phase. We find that $T_c\left(x\right)$ decrease monotonically from 328 K for ${\mathrm{Ta}}_2\mathrm{Ni}{\mathrm{Se}}_5$ to 120 K for ${\mathrm{Ta}}_2\mathrm{Ni}{\mathrm{S}}_5$ and that the magnitude of lattice distortion also decreases with the sulfur concentration $x$. For $x<0.7$, the two lowest frequency $B_{2g}$  phonon modes show strongly asymmetric line shapes at high temperatures due to Fano interference with the broad excitonic continuum present in a semimetallic state. Within the framework of extended Fano model, we develop a quantitative description of the interacting exciton-phonon excitation line shape, enabling us to derive the intrinsic phonon parameters and determine the exciton-phonon interaction strength that affects the transition temperature $T_c\left(x\right)$ . While at low temperatures the intrinsic phonon parameters are in good agreement with the ab initio calculations and the anharmonic decay model, their temperature dependencies show several anomalous behaviors: (i) Frequencies of $B_{2g}$ phonons harden pronouncedly upon cooling in vicinity of $T_c\left(x\right)$ for $x<0.7$ semimetals and, in contrast, soften monotonically for ${\mathrm{Ta}}_2\mathrm{Ni}{\mathrm{S}}_5$ semiconductor; (ii) the lifetime of certain phonons increases strongly below $T_c\left(x\right)$ for $x<0.7$, revealing the gap opening in the broken symmetry phase; and (iii) for most modes, the intensity shows rather strong temperature dependence that we relate to the interplay between electronic and phononic degrees of freedom. For ${\mathrm{Ta}}_2\mathrm{Ni}{\mathrm{Se}}_5$ we also observe signatures of the acoustic mode scattered assisted by the structural domain walls formed below $T_c$. Based on our results, we additionally present a consistent interpretation of the origin of oscillations observed in time-resolved pump-probe experiments.

We report on a study of the optical properties of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ for a large number of temperatures above and below the charge-density-wave (CDW) transition. Above the CDW transition, the low-frequency optical conductivity reveals two Drude components with distinct widths. An examination of the band structure allows us to ascribe the narrow Drude component to a light-electron-like and multiple Dirac bands, and the broad Drude component to heavy-hole bands near the $M$ points which form saddle points near the Fermi level. Upon entering the CDW state, the opening of the CDW gap is clearly observed. A large portion of the broad Drude component is removed by the gap, whereas the narrow Drude component is not affected. Meanwhile, an absorption peak associated with interband transitions involving the saddle points shifts to higher energy and grows in weight. These observations attest to the importance of saddle point nesting in driving the CDW instability in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$.

By analyzing the Lyapunov exponent (LE), we develop a rigorous, fundamental scheme for the study of general non-Hermitian quasicrystals with both a complex phase factor and nonreciprocal hopping. Specially, the localization-delocalization transition point, the $\mathcal{PT}$-symmetry-breaking point, and the winding number transition points are determined by LEs of its dual Hermitian model. The analysis was based on Avila's global theory, and we found that the winding number is directly related to the acceleration and the slope of the LE, while quantization of acceleration is the crucial ingredient of Avila's global theory. This result applies as well to models with higher winding, not only the simplest Aubry-André model. As typical examples, we obtain the analytical phase boundaries of the localization transition for the non-Hermitian Aubry-André model in the whole parameter space, and the complete phase diagram is straightforwardly determined. For the non-Hermitian Soukoulis-Economou model, a high winding model, we show how the phase boundaries of the localization transition and winding number transitions are related to the LEs of its dual Hermitian model. Moreover, we discover an intriguing feature of the robust spectrum, i.e., the spectrum remains invariant when one changes the complex phase parameter $h$ or the nonreciprocal parameter $g$ in the region of $h<|h_c|$ or $g<|g_c|$ if the system is in the extended or localized state, respectively.

We investigate measurement-induced phase transitions in the quantum Ising chain coupled to a monitoring environment. We compare two different limits of the measurement problem: the stochastic quantum-state diffusion protocol corresponding to infinite small jumps per unit of time and the no-click limit, corresponding to postselection and described by a non-Hermitian Hamiltonian. In both cases we find a remarkably similar phenomenology as the measurement strength $\gamma$ is increased, namely, a sharp transition from a critical phase with logarithmic scaling of the entanglement to an area-law phase, which occurs at the same value of the measurement rate in the two protocols. An effective central charge, extracted from the logarithmic scaling of the entanglement, vanishes continuously at the common transition point, although with different critical behavior possibly suggesting different universality classes for the two protocols. We interpret the central charge mismatch near the transition in terms of noise-induced disentanglement, as suggested by the entanglement statistics which displays emergent bimodality upon approaching the critical point. The non-Hermitian Hamiltonian and its associated subradiance spectral transition provide a natural framework to understand both the extended critical phase, emerging here for a model which lacks any continuous symmetry, and the entanglement transition into the area law.

By combining neutron scattering and magnetization measurements down to 80 mK, we determine the $(H,T)$ phase diagram of the ${\mathrm{Nd}}_2{\left({\mathrm{Zr}}_{1-x}{\mathrm{Ti}}_x\right)}_2{\mathrm{O}}_7$ pyrochlore magnet compounds. In those samples, Zr is partially substituted by Ti, hence tuning the exchange parameters and testing the robustness of the various phases. In all samples, the ground state remains all in/all out, while the field induces phase transitions toward new states characterized by two in–two out or one out–three in/one in–three out configurations. These transitions manifest as metamagnetic singularities in the magnetization versus field measurements. Strikingly, it is found that moderate substitution reinforces the stability of the all in/all out phase: the Néel temperature, the metamagnetic fields along with the ordered magnetic moment, are higher in substituted samples with $x< 10\%$.

We demonstrate how quantum entanglement can be directly witnessed in the quasi-1D Heisenberg antiferromagnet ${\mathrm{KCuF}}_3$. We apply three entanglement witnesses—one tangle, two tangle, and quantum Fisher information—to its inelastic neutron spectrum and compare with spectra simulated by finite-temperature density matrix renormalization group (DMRG) and classical Monte Carlo methods. We find that each witness provides direct access to entanglement. Of these, quantum Fisher information is the most robust experimentally and indicates the presence of at least bipartite entanglement up to at least 50 K, corresponding to around 10% of the spinon zone-boundary energy. We apply quantum Fisher information to higher spin-$S$ Heisenberg chains and show theoretically that the witnessable entanglement gets suppressed to lower temperatures as the quantum number increases. Finally, we outline how these results can be applied to higher dimensional quantum materials to witness and quantify entanglement.

We describe and benchmark the quasi-four-component (Q4C) approach to relativistic density functional simulations of molecules and solids, using precise, numerically tabulated atom-centered orbital (NAO) basis sets to discretize Dirac's equation. The Q4C approach initially projects the atomic solution to (electron-only) positive-energy states and eventually deals with only two components but retains the precision of traditional four-component (4C) relativistic methods. While Q4C inherently reduces the dimension of the Hamiltonian matrix in diagonalization, the adoption of localized NAO basis functions in solids further limits the computational demand in real space operations, promising a pathway to investigate large and complex systems containing heavy elements with the precision of a 4C method. Here, we first perform validation and benchmark calculations for cohesive properties of a set of diatomic molecules and of previously established periodic model systems (i.e., silver halides). Then we report Q4C relativistic energy band structure benchmarks for a series of 103 periodic materials, including chemical elements up to Bi, and providing quantitative comparisons with more approximate scalar-relativistic and spin-orbit coupled treatments. Finally, we demonstrate the applicability of the method to band structure calculations of simple and complex hybrid organic-inorganic perovskites containing Pb and Bi, i.e., ${\mathrm{Cs}}_2\mathrm{AgBi}{\mathrm{Cl}}_6$ and a larger system (containing 94 atoms per unit cell), ${\left(4\text{−}\mathrm{FPEA}\right)}_2{\mathrm{PbI}}_4$. The effect of full Q4C, compared with scalar relativity, on binding energies can be significant even for relatively light $p$-orbital bonded main group elements such as Br and I—i.e., 0.3 and 0.6 eV for ${\mathrm{Br}}_2$ and ${\mathrm{I}}_2$ binding energies, respectively.