We present a new estimator for the self-energy based on a combination of two equations of motion and discuss its benefits for numerical renormalization group (NRG) calculations. In challenging regimes, NRG results from the standard estimator, a ratio of two correlators, often suffer from artifacts: The imaginary part of the retarded self-energy is not properly normalized and, at low energies, overshoots to unphysical values and displays wiggles. We show that the new estimator resolves the artifacts in these properties as they can be determined directly from the imaginary parts of auxiliary correlators and do not involve real parts obtained by Kramers–Kronig transform. Furthermore, we find that the new estimator yields converged results with reduced numerical effort (for a lower number of kept states) and thus is highly valuable when applying NRG to multiorbital systems. Our analysis is targeted at NRG treatments of quantum impurity models, especially those arising within dynamical mean-field theory, but most results can be straightforwardly generalized to other impurity or cluster solvers.

We develop a theory of bound states in the continuum (BICs) in multipolar lattices—periodic arrays of resonant multipoles. We predict that BICs are completely robust to changes in the lattice parameters, remaining pinned to specific directions in the $k$ space. The lack of radiation for BICs in such structures is protected by the symmetry of multipoles forming the lattice. We also show that some multipolar lattices can host BICs forming a continuous line in the $k$ space, and such BICs carry zero topological charge. The developed approach sets a direct fundamental relation between the topological charge of BICs and the asymptotic behavior of the $Q$ factor in its vicinity. We believe that our theory is a significant step towards gaining deeper insight into the physics of BICs and the engineering of high-$Q$ states in all-dielectric metasurfaces.

We construct models hosting classical fractal spin liquids on two realistic three-dimensional (3D) lattices of corner-sharing triangles: trillium and hyperhyperkagome (HHK). Both models involve the same form of three-spin Ising interactions on triangular plaquettes as the Newman-Moore (NM) model on the 2D triangular lattice. However, in contrast to the NM model and its 3D generalizations, their degenerate ground states and low-lying excitations cannot be described in terms of scalar cellular automata (CA), because the corresponding fractal structures lack a simplifying algebraic property, often termed the “freshman's dream.” By identifying a link to matrix CAs—that makes essential use of the crystallographic structure—we show that both models exhibit fractal symmetries of a distinct class to the NM-type models. We devise a procedure to explicitly construct low-energy excitations consisting of finite sets of immobile defects or “fractons,” by flipping arbitrarily large self-similar subsets of spins, whose fractal dimensions we compute analytically. We show that these excitations are associated with energetic barriers which increase logarithmically with system size, leading to “fragile” glassy dynamics, whose existence we confirm via classical Monte Carlo simulations. We also discuss consequences for spontaneous fractal symmetry breaking when quantum fluctuations are introduced by a transverse magnetic field, and propose multispin correlation function diagnostics for such transitions. Our findings suggest that matrix CAs may provide a fruitful route to identifying fractal symmetries and fractonlike behavior in lattice models, with possible implications for the study of fracton topological order.

Structurally similar transition metal systems can have widely differing magnetic properties. A prime example of this is found for bilayers on Ir(111), where the Pd/Fe/Ir(111) ground state has a well-established noncollinear spin texture, while Pd/Co/Ir(111) presents a ferromagnetic (FM) single domain. To unravel the origins of these different magnetic behaviors, an investigation of Pd/Fe and Pd/Co bilayers on Ir(111) is here performed. Based on the obtained ab initio electronic structure, exchange coupling parameters $J_{ij}$, and Dzyaloshinskii-Moriya interactions (DMIs), we demonstrate that, although in Pd/Co/Ir(111) the DMI is significant, two ingredients play a key role in the origin of noncollinearity in Pd/Fe/Ir(111): the presence of magnetic frustrations and a much more in-plane DMI, both with a long-range influence. The $J_{ij}$ and DMI behaviors in both systems can be explained in terms of a simple rigid-band-like model. Also, by performing spin-dynamics simulations with the magnetic parameters tuned from the Pd/Co/Ir(111) to Pd/Fe/Ir(111) ab initio values, we could find conditions for the emergence of skyrmionic phases in the originally FM Pd/Co/Ir(111).

We have studied the charge dynamics for the layered polar Dirac semimetal ${\mathrm{BaMnSb}}_2$ by means of optical spectroscopy and ab initio calculation. A large anisotropy in the optical conductivity spectra is observed for two orthogonal light polarizations $E^{\omega }$, both of which are perpendicular to the stacking direction of monoatomic layers; a clear peak emerges at the absorption edge of the Dirac band for $E^{\omega }$ parallel to the zigzag chain in the distorted Sb square net, but a tiny kink is observed for $E^{\omega }$ perpendicular to the chain. We also found that the effective electron mass is about an order of magnitude larger in the direction perpendicular to the chain than parallel to it. The theoretical calculation suggests that the dispersion of the Dirac band perpendicular to the chain is suppressed due to the reduced orbital hybridization, leading to the large in-plane electronic anisotropy.

We study time dynamics of 1D disordered Heisenberg spin-1/2 chains focusing on a regime of large system sizes and a long-time evolution. This regime is relevant for observation of many-body localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to the few thousands of tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remains fully local at the longest timescales we reach, provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.

We introduce an effective theory for quantum critical points (QCPs) in heavy-fermion systems, involving a change in carrier density without symmetry breaking. Our theory captures a strongly coupled metallic QCP, leading to robust marginal Fermi-liquid transport phenomenology, and associated linear in temperature $\left(T\right)$ “strange metal” resistivity, all within a controlled large-$N$ limit. In the parameter regime of strong damping of emergent bosonic excitations, the QCP also displays a near-universal “Planckian” transport lifetime ${\tau }_{\mathrm{tr}}\sim \hslash /\left(k_{B}T\right)$. This is contrasted with the conventional so-called “slave boson” theory of the Kondo breakdown, where the large-$N$ limit describes a weak coupling fixed point and nontrivial transport behavior may only be obtained through uncontrolled $1/N$ corrections. We also compute the weak-field Hall coefficient within the effective model as the system is tuned across the transition. We then find that, between the two plateaus reflecting the different carrier densities in the two Fermi-liquid phases, the Hall coefficient can develop a peak in the critical crossover regime, like in recent experimental findings, in the parameter regime of weak boson damping.

The precise crystal symmetry, and hence the emergence of the electric polarization, still remains an open question in the multiferroic $R{\mathrm{Mn}}_2{\mathrm{O}}_5$ ($R=$ rare earth, Bi, Y). While previous diffraction studies have indicated the centrosymmetric space group $Pbam$, an atomic displacement allowing for electric polarization would require a noncentrosymmetric crystal symmetry. Our single crystal neutron diffraction experiments on ${\mathrm{TbMn}}_2{\mathrm{O}}_5$ provide direct evidence of a reduced crystallographic symmetry with the polar space group $P{12}_{1}1$ already above the multiferroic phase transition, indicating that a symmetric ${\mathbf{S}}_{\mathbf{i}}·{\mathbf{S}}_{\mathbf{j}}$ spin coupling, i.e. magnetostriction is the dominating mechanism in the commensurate magnetic phase. Furthermore, the commensurate magnetic reflections are in accordance with a quartile step spin spiral along the $c$ axis. Therefore, the antisymmetric ${\mathbf{S}}_{\mathbf{i}}\times {\mathbf{S}}_{\mathbf{j}}$ exchange via the inverse Dzyaloshinskii-Moriya interaction contributes as well and becomes the leading term in the low-temperature incommensurate spin-spiral magnetic phase. These findings provide important information for the understanding of the complex interplay between magnetic and structural order throughout the multiferroic $R{\mathrm{Mn}}_2{\mathrm{O}}_5$ series.

The polar magnet ${\mathrm{CaBaCo}}_4{\mathrm{O}}_7$ is known to exhibit the largest magnetic-field-driven electric polarization change $\left(\mathrm{\Delta }P\right)$ associated with an antiferromagnetic- (AFM-) ferrimagnetic (FIM) transition in a narrow temperature range between 62 and 69 K. In this paper, we investigate the effect of Ni doping on its multiferroic properties by means of magnetization, electric polarization, dielectric constant, and magnetostriction measurements on single crystals of ${\mathrm{CaBaCo}}_{3.9}{\mathrm{Ni}}_{0.1}{\mathrm{O}}_7$ up to 50 T. In the doped material, two kinds of AFM phases appear below 78 K, accompanying negative $\mathrm{\Delta }P$. Upon the application of a magnetic field along any crystallographic axis, giant positive $\mathrm{\Delta }P$ of up to 11 to 12 $\mathrm{mC}/{\mathrm{m}}^2$ is observed along with an AFM-FIM transition in the whole temperature range below 78 K. The giant magnetoelectric effect inherent in ${\mathrm{CaBaCo}}_4{\mathrm{O}}_7$ can be further enhanced just by a small amount of chemical substitution in terms of: (i) increasing the magnitude of $\mathrm{\Delta }P$ and (ii) expanding the temperature range in which giant $\mathrm{\Delta }P$ appears.

At sufficiently large transport currents $I_{\mathrm{tr}}$, a defect at the edge of a superconducting strip acts as a gate for the vortices entering into it. These vortices form a jet, which is narrow near the defect and expands due to the repulsion of vortices as they move to the opposite edge of the strip, giving rise to a transverse voltage $V_{\perp }$. Here, relying upon the equation of vortex motion under competing vortex-vortex and $I_{\mathrm{tr}}$-vortex interactions, we derive the vortex jet shapes in narrow ($\xi \ll w\lesssim {\lambda }_{\mathrm{eff}}$) and wide ($w\gg {\lambda }_{\mathrm{eff}}$) strips [$\xi$: coherence length, $w$: strip width, ${\lambda }_{\mathrm{eff}}$: effective penetration depth]. We predict a nonmonotonic dependence $V_{\perp }\left(I_{\mathrm{tr}}\right)$ which can be measured with Hall voltage leads placed on the line $V_1{V}_2$ at a small distance $l$ apart from the edge defect and which changes its sign upon $l\to -l$ reversal. For narrow strips, we compare the theoretical predictions with experiment, by fitting the $V_{\perp }(I_{\mathrm{tr}},l)$ data for $1\text{−}\mu \mathrm{m}$-wide MoSi strips with single edge defects milled by a focused ion beam at distances $l=16\text{–}80$ nm from the line $V_1{V}_2$. For wide strips, the derived magnetic field dependence of the vortex jet shape is in line with the recent experimental observations for vortices moving in Pb bridges with a narrowing. Our findings are augmented with the time-dependent Ginzburg-Landau simulations which reproduce the calculated vortex jet shapes and the $V_{\perp }(I_{\mathrm{tr}},l)$ maxima. Furthermore, with increase of $I_{\mathrm{tr}}$, the numerical modeling unveils the evolution of vortex jets to vortex rivers, complementing the analytical theory in the entire range of $I_{\mathrm{tr}}$.

When two graphene sheets are twisted relative to each other by a small angle, enhanced correlations lead to superconductivity whose origin remains under debate. Here, we derive some general constraints on superconductivity in twisted bilayer graphene (TBG), independent of its underlying mechanism. Neglecting weak coupling between valleys, the global symmetry group of TBG consists of independent spin rotations in each valley in addition to valley charge rotations, $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$. This symmetry is further enhanced to a full SU(4) in the idealized chiral limit. In both cases, we show that any charge $2e$ pairing must break the global symmetry. Additionally, if the pairing is unitary the resulting superconductor admits fractional vortices. This leads to the following general statement. Any superconducting condensate in either symmetry class has to satisfy one of three possibilities: (i) the superconducting pairing is nonunitary, (ii) the superconducting condensate has charge $2e$ but admits at least half quantum vortices which carry a flux of $h/4e$, or (iii) the superconducting condensate has charge $2me, m>1$, with vortices carrying $h/2me$ flux. The latter possibility can be realized by a symmetric charge $4e$ superconductor ($m=2$). Nonunitary pairing (i) is expected for superconductors observed in the vicinity of flavor polarized states. On the other hand, in the absence of flavor polarization, e.g., in the vicinity of charge neutrality, one of the two exotic possibilities (ii) and (iii) is expected. We sketch how all three scenarios can be realized in different limits within a strong coupling theory of superconductivity based on skyrmions. Finally we discuss the effect of symmetry lowering anisotropies and experimental implications of these scenarios.

Magnetic impurities on a superconductor induce subgap Yu-Shiba-Rusinov (YSR) bound states, localized at the impurity site and fading away from it for distances up to several nanometers. In this paper, we present a theoretical method to calculate the spatial distribution of the YSR spectrum of a two-dimensional superconductor with arbitrary Fermi contours (FCs) in the presence of magnetic impurities. Based on the Green's function (GF) formalism, we obtain a general analytical expression by approximating an arbitrary contour shape to a regular polygon. This method allows us to show the connection between the spatial decay (and, hence, the extension) of YSR states and the shape of the FC of the host superconductor. We demonstrate the accuracy of this approximation by comparing the results with those obtained from an exact numerical calculation based on a tight-binding Hamiltonian. We further apply the analytical formalism to compute the evolution of YSR states in the presence of a nearby impurity atom, and compare the results with scanning tunneling microscopy measurements on interacting manganese dimers on the $\beta \text{−}{\mathrm{Bi}}_2\mathrm{Pd}$ superconductor. The method can be easily extended to any arbitrary number of magnetically coupled impurities, thus providing a useful tool for simulating the spectral properties of interacting YSR states in artificial atomic nanostructures.

In this work, we investigate the ability of the cumulant expansion (CE) to capture one-particle spectral information in electron-phonon coupled systems at both zero and finite temperatures. In particular, we present a comprehensive study of the second- and fourth-order CEs for the one-dimensional Holstein model as compared with numerically exact methods. We investigate both finite sized systems as well as the approach to the thermodynamic limit, drawing distinctions, and connections between the behavior of systems in and away from the thermodynamic limit that enable a greater understanding of the ability of the CE to capture real-frequency information across the full range of wave vectors. We find that for zero electronic momentum, the spectral function is well described by the second-order CE at low and high temperatures. However, for nonzero electronic momenta, the CE is only accurate at high temperature. We analyze the fourth-order cumulant and find that while it improves the description of the short-time dynamics encoded in the one-particle Green's function, it can introduce divergences in the time domain as well as unphysical negative spectral weight in the spectral function. When well-behaved, the fourth-order CE does provide notable accurate corrections to the second-order CE. Finally, we use our results to comment on the use of the CE as a tool for calculating transport behavior in the realistic ab initio modeling of materials.

A detailed analysis of the ferrimagnetic ground state of ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$has been performed using inelastic neutron scattering. Although the proposed valence of the nominal ${\mathrm{Mn}}^{2+}$ ions would have quenched orbital angular momentum, a significant exchange anisotropy exists in ${\mathrm{Mn}}_3{\mathrm{Si}}_2{\mathrm{Te}}_6$. This apparent exchange anisotropy is a manifestation of a weak spin-orbit coupling in the layered material. We employ a detailed simulation of the spin-wave spectrum coupling traditional refinement of dispersion parameters to image analysis techniques, while including Monte Carlo simulations of the instrumental resolution to accurately identify the exchange couplings to the third nearest neighbor. An independent validation of our results is made by comparing our final Hamiltonian to heat capacity measurements.

We report high-resolution single-crystal inelastic neutron scattering measurements on the spin-1/2 antiferromagnet $\mathrm{Ba}\left(\mathrm{TiO}\right){\mathrm{Cu}}_4({\mathrm{PO}}_4{)}_4$. This material is formed from layers of four-site “cupola” structures, oriented alternately upwards and downwards, which constitute a rather special realization of two-dimensional (2D) square-lattice magnetism. The strong Dzyaloshinskii-Moriya (DM) interaction within each cupola, or plaquette, unit has a geometry largely unexplored among the numerous studies of magnetic properties in 2D Heisenberg models with spin and spatial anisotropies. We have measured the magnetic excitations at zero field and in fields up to 5 T, finding a complex mode structure with multiple characteristic features that allow us to extract all the relevant magnetic interactions by modeling within the linear spin-wave approximation. We demonstrate that $\mathrm{Ba}\left(\mathrm{TiO}\right){\mathrm{Cu}}_4({\mathrm{PO}}_4{)}_4$ is a checkerboard system with almost equal intra- and interplaquette couplings, in which the intraplaquette DM interaction is instrumental both in enforcing robust magnetic order and in opening a large gap at the Brillouin-zone center. We place our observations in the perspective of generalized phase diagrams for spin-1/2 square-lattice models and materials, where exploring anisotropies and frustration as routes to quantum disorder remains a frontier research problem.

A consequence of the disorder and Coulomb interaction competition is the electron-glass phase observed in several Anderson insulators. The disorder in these systems, typically degenerate semiconductors, is stronger than the interaction, more so the higher is the carrier concentration $N$ of the system. Here we report on a feature observed in the electron-glass phase of ${\mathrm{In}}_{\text{x}}\mathrm{O}$ with the lowest $N$ yet studied. The feature, resolved as a broad peak in field-effect measurements, has not been recognized in previously studied Anderson insulators. Several empirical facts associated with the phenomenon are consistent with the conjecture that it reflects a correlated charge distribution. In particular, the feature may be turned on and off by gate-voltage maneuvering, suggesting the relevance of charge arrangements. It may also be suppressed by either temperature, non-Ohmic field, or exposure to infrared illumination. After being washed out, the feature reappears when the system is allowed to relax for sufficiently long time. A puzzling aspect that arises is the apparent absence of the phenomenon when the carrier concentration increases above a certain value. This is reminiscent of the glass-transition conundrum, except that the role of temperature in the latter is played by disorder. Analysis of these findings highlights several issues that challenge our understanding of the disorder-interaction interplay in Anderson insulators.

We explore higher-order topological superconductivity in an artificial Dirac material with intrinsic spin-orbit coupling, which is a doped ${\mathbb{Z}}_2$ topological insulator in the normal state. A mechanism for superconductivity due to repulsive interactions, pseudospin pairing, has recently been shown to naturally result in higher-order topology in Dirac systems past a minimum chemical potential [T. Li et al., 2D Mater. 9, 015031 (2022)]. Here we apply this theory through microscopic modeling of a superlattice potential imposed on an inversion-symmetric hole-doped semiconductor heterostructure, known as hole-based semiconductor artificial graphene, and extend previous work to include the effects of spin-orbit coupling. We find that spin-orbit coupling enhances interaction effects, providing an experimental handle to increase the efficiency of the superconducting mechanism. We show that the phase diagram of these systems, as a function of chemical potential and interaction strength, contains three superconducting states: a first-order topological $p+ip$ state, a second-order topological spatially modulated $p+i\tau p$ state, and a second-order topological extended $s$-wave state $s_{\tau }$. We calculate the symmetry-based indicators for the $p+i\tau p$ and $s_{\tau }$ states, which prove these states possess second-order topology. Exact diagonalization results are presented which illustrate the interplay between the boundary physics and spin-orbit interaction. We argue that this class of systems offers an experimental platform to engineer and explore first- and higher-order topological superconducting states.

The $\mathrm{SU}\left(4\right)$ Heisenberg model on a honeycomb lattice is expected to host a quantum spin-orbital liquid at low temperature with a surprising candidate material, $\alpha \text{−}\mathrm{Zr}{\mathrm{Cl}}_3$. We employed the canonical thermal pure quantum state method to investigate the finite-temperature phase of this model. Exploiting the full symmetry of $\mathrm{SU}\left(4\right)$, calculation up to a 24-site cluster, which is equivalent to 48 sites in the spin-$\frac{1}{2}$ language, is possible. This state-of-the-art computation with large-scale parallelization enables us to capture the thermodynamic properties of the $\mathrm{SU}\left(4\right)$ Heisenberg model on a honeycomb lattice. In particular, the specific heat shows a characteristic peak-and-shoulder structure, which should be related to the nature of the low-temperature quantum spin-orbital liquid phase. We also discuss what can be concluded from the assumption that the ground state is gapped and symmetric in view of the generalized Lieb-Schultz-Mattis theorem.

We measure the full elastic tensors of ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$ as a function of temperature through their respective antiferromagnetic phase transitions. Large discontinuities in the bulk moduli at the Néel transitions indicate strong magnetoelastic coupling in both compounds. Strikingly, the discontinuities are nearly a factor of 10 larger in ${\mathrm{Mn}}_3\mathrm{Ge}$ than in ${\mathrm{Mn}}_3\mathrm{Sn}$. We use the magnitudes of the discontinuities to calculate the pressure derivatives of the Néel temperature, which are 39 K/GPa 14.3 K/GPa for ${\mathrm{Mn}}_3\mathrm{Ge}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, respectively. We measured the in-plane shear modulus $c_{66}$, which couples strongly to the magnetic order, in magnetic fields up to 18 T and found quantitatively similar behavior in both compounds. Recent measurements have demonstrated strong piezomagnetism in ${\mathrm{Mn}}_3\mathrm{Sn}$ : Our results suggest that ${\mathrm{Mn}}_3\mathrm{Ge}$ may be an even better candidate for this effect.

Two-photon coherent states are one of the main building pillars of nonlinear and quantum optics. They are the basis for the generation of minimum-uncertainty quantum states and entangled photon pairs, applications not obtainable from standard coherent states or one-photon lasers. Here, we describe a fully resonant optomechanical parametric amplifier involving a polariton condensate in a trap lattice quadratically coupled to mechanical modes. The quadratic coupling derives from nonresonant virtual transitions to extended discrete excited states induced by the optomechanical coupling. Nonresonant continuous-wave laser excitation leads to striking experimental consequences, including the emergence of optomechanically induced intersite parametric oscillations and intersite tunneling of polaritons at discrete intertrap detunings corresponding to sums of energies of the two involved mechanical oscillations (20- and 60-GHz confined vibrations). We show that the coherent mechanical oscillations correspond to parametric resonances with a threshold condition different from that of standard linear optomechanical self-oscillation. The associated Arnold tongues display a complex scenario of states within the instability region. The observed phenomena can have applications for the generation of entangled phonon pairs and squeezed mechanical states relevant in sensing and quantum computation and for the bidirectional frequency conversion of signals in a technologically relevant range.