We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

In Cu-doped ${\mathrm{Co}}_3{\mathrm{O}}_4$ nanoparticles $({\mathrm{Cu}}_x{\mathrm{Co}}_{3\text{−}x}{\mathrm{O}}_4;0\le x\le 0.5)$ Cu occupies both octahedral and tetrahedral sites with a $2+$ oxidation state. As the Cu doping increases, we observe changes in the crystal structure corresponding to a Jahn-Teller distortion of the ${\mathrm{Cu}}^{2+}$ sites. To mediate charge balance with ${\mathrm{Cu}}^{2+}$ entering the octahedral sites, a hole forms in the O $2p$ orbitals bonded to the ${\mathrm{Cu}}^{2+}\left(O_h\right)$. ${\mathrm{Cu}}^{2+}\left(T_d\right)$ is noninteracting and disrupts the existing antiferromagnetic interactions between the ${\mathrm{Co}}^{2+}\left(T_d\right)$ ions, while ${\mathrm{Cu}}^{2+}\left(O_h\right)$ exhibits a ferromagnetic response as a result of a hybrid form of exchange occurring between ${\mathrm{Cu}}^{2+}\left(O_h\right)$ and ${\mathrm{Co}}^{2+}\left(T_d\right)$. Emergence of the $3d^9$ ligand hole is directly responsible for the origin of ferromagnetic Cu in the octahedral sites and this results in the unusual magnetism in ${\mathrm{Cu}}_x{\mathrm{Co}}_{3\text{−}x}{\mathrm{O}}_4$.

We discuss the emergence of an effective low-energy theory for the real-time dynamics of two classical impurity spins within the framework of a prototypical and purely classical model of indirect magnetic exchange: two classical impurity spins are embedded in a host system which consists of a finite number of classical spins localized on the sites of a lattice and interacting via a nearest-neighbor Heisenberg exchange. An effective low-energy theory for the slow impurity-spin dynamics is derived for the regime, where the local exchange coupling between impurity and host spins is weak. To this end, we apply the recently developed adiabatic spin dynamics (ASD) theory. Besides the Hamiltonian-like classical spin torques, the ASD additionally accounts for a novel topological spin torque that originates as a holonomy effect in the close-to-adiabatic-dynamics regime. It is shown that the effective low-energy precession dynamics cannot be derived from an effective Hamilton function and is characterized by a nonvanishing precession frequency even if the initial state deviates only slightly from a ground state. The effective theory is compared to the fully numerical solution of the equations of motion for the whole system of impurity and host spins to identify the parameter regime where the adiabatic effective theory applies. Effective theories beyond the adiabatic approximation must necessarily include dynamic host degrees of freedom and go beyond the idea of a simple indirect magnetic exchange. We discuss an example of a generalized constrained spin dynamics which does improve the description but also fails for certain geometrical setups.

We study in detail the properties of $\pi$ fluxes embedded in a state with a finite density of anyons that form either a Fermi liquid or a Bose-Einstein condensate. By employing a recently developed exact lattice bosonization in 2D, we demonstrate that such $\pi$ flux remains a fully deconfined quasiparticle with a finite energy cost in a Fermi liquid of emergent fermions coupled to a ${\mathbb{Z}}_2$ gauge field. This $\pi$ flux is accompanied by a screening cloud of fermions, which in the case of a Fermi gas with a parabolic dispersion binds exactly $1/8$ of a fermionic hole. In addition, there is a long-ranged power-law oscillatory disturbance of the liquid surrounding the $\pi$ flux akin to Friedel oscillations. These results carry over directly to the $\pi$ flux excitations in orthogonal metals. In sharp contrast, when the $\pi$ flux is surrounded by a Bose-Einstein condensate of particles coupled to a ${\mathbb{Z}}_2$ gauge field, it binds a superfluid half-vortex, becoming a marginally confined excitation with a logarithmic energy cost divergence.

Under the action of coherent periodic driving a generic quantum system will undergo Floquet heating and continuously absorb energy until it reaches a featureless thermal state. The phase-space constraints induced by certain symmetries can, however, prevent this and allow the system to dynamically form robust steady states with off-diagonal long-range order. In this work, we take the Hubbard model on an arbitrary lattice with arbitrary filling and, by simultaneously diagonalizing the two possible SU(2) symmetries of the system, we analytically construct the correlated steady states for different symmetry classes of driving. This construction allows us to make verifiable, quantitative predictions about the long-range particle-hole and spin-exchange correlations that these states can possess. In the case when both SU(2) symmetries are preserved in the thermodynamic limit we show how the driving can be used to form a unique condensate which simultaneously hosts particle-hole and spin-wave order.

The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo $e$, only from calculations on ribbon geometries of two different orientations. We elucidate the dependence of the interior quadrupole and edge dipole contributions upon the gauge used to construct the Wannier functions, finding that while these are individually gauge dependent, their sum is gauge independent. From this we conclude that the edge polarization is not by itself a physical observable and that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint using two Wannier construction procedures, one based on projection and another based on a gauge-consistent nested Wannier construction. We validate our theory by demonstrating the correct prediction of corner charge for several tight-binding models. We comment on the relations between our approach and previous ones that have appeared in the literature.

We have investigated the Kondo physics of a single magnetic impurity embedded in multi-Dirac (Weyl) node fermionic systems. By using a generic effective model for the host material and employing a numerical renormalization group approach we access the low temperature behavior of the system, identifying the existence of Kondo screening in single-, double-, and triple-Dirac (Weyl) node models. We find that in any multi-Dirac node systems the low-energy regime lies within one of the known classes of pseudogap Kondo problem, extensively studied in the literature. Kondo screening is also observed for time reversal symmetry broken Weyl systems. This is, however, possible only in the particle-hole symmetry broken regime obtained for finite chemical potential $\mu$. Although weakly, breaking time-reversal symmetry suppresses the Kondo resonance, especially in the single-node Weyl semimetals. More interesting Kondo screening regimes are obtained for inversion symmetry broken multi-Weyl fermions. In these systems the Kondo regimes of double- and triple-Weyl node models are much richer than in the single-Weyl node model. While in the single-Weyl node model the Kondo temperature increases monotonically with $\left|\mu \right|$ regardless the value of the inversion symmetry breaking parameter $Q_0$, in double- and triple-Weyl node models there are two distinct regimes: (i) For $Q_0<\left|\mu \right|$ the Kondo temperature depends strongly on $\mu$, while (ii) for $Q_0>\left|\mu \right|$ the Kondo temperature depends very weakly on $\mu$, resembling the flat-band single impurity Anderson model.

We present triple-axis neutron scattering studies of static and dynamic magnetic stripes in an optimally oxygen-doped cuprate superconductor, ${\mathrm{La}}_2{\mathrm{CuO}}_{4+y}$, which exhibits a clean superconducting transition at $T_c=42\mathrm{K}$. Polarization analysis reveals that the magnetic stripe structure is equally represented along both of the tetragonal crystal axes and that the fluctuating stripes display significant weight for in-plane as well as out-of-plane spin components. Both static magnetic order as well as low-energy fluctuations are fully developed in zero applied magnetic field and the low-energy spin fluctuations at $\hslash \omega =0.3\text{–}10\mathrm{meV}$ intensify on cooling. We interpret this as an indication that superconductivity and low-energy spin fluctuations coexist microscopically in spatial regions which are separated from domains with static magnetic order.

We investigate the effect of long-range pairing interactions on the phase diagram and the quantum transport properties in a semiconductor-superconductor hybrid nanowire. Both the power-law and exponential decay rates of long-range pairing are considered for comparison. It is found that a long-range topological phase hosting massive edge modes is induced when the exponent of the power-law decay is less than one. By diagonalizing the tight-binding model, we find that near-zero-energy Andreev bound states (ABSs) could be induced in the topologically trivial phase for slowly decayed long-range pairing interactions. For a one-dimensional long-range Kitaev chain, the shot noise is suggested to serve as a probe to discriminate Majorana bound states (MBSs) and topological massive modes. Our numerical results indicate that for realistic device parameters, the Fano factor of the shot noise is irrelevant to the topological property of the system, and it fails to distinguish the MBSs and ABSs. However, the noise Fano factor is generally consistent with the variation of the bound state energy, which can be used to detect the energy splitting of these bound states.

Spin-dependent scattering at the nonmagnet/ferromagnet interface plays a key role in determining the amplitude of unidirectional spin Hall magnetoresistance (USMR), similar to giant magnetoresistance (GMR). We report the enhancement of USMR by inserting a thin Cu interlayer into the Pt/Co interface, where the Cu/Co system is well known to exhibit a large GMR. A measurement of the spin-orbit torque shows that the spin current injection into the Co layer is not modulated by the Cu interlayer. In addition, USMR increases with the Cu thickness for the ultrathin regime as the Cu/Co interface is formed, reaching a peak before decreasing, owing to the shunting effect. Our results suggest an interfacial origin of the enhanced USMR and highlight the close similarities between USMR and GMR.

An investigation of the structural, magnetic, thermodynamic, and charge transport properties of noncentrosymmetric hexagonal ScFeGe reveals it to be an anisotropic metal with a transition to a weak itinerant incommensurate helimagnetic state below $T_N=36$ K. Neutron diffraction measurements discovered a temperature and field independent helical wave vector $\mathit{k}$ = (0 0 0.193) with magnetic moments of 0.53 ${\mu }_B$ per Fe confined to the $ab$ plane. Density functional theory calculations are consistent with these measurements and find several bands that cross the Fermi level along the $c$ axis with a nearly degenerate set of flat bands just above the Fermi energy. The anisotropy found in the electrical transport is reflected in the calculated Fermi surface, which consists of several warped flat sheets along the $c$ axis with two regions of significant nesting, one of which has a wave vector that closely matches that found in the neutron diffraction. The electronic structure calculations, along with a strong anomaly in the $c$ -axis conductivity at $T_N$, signal a Fermi surface driven magnetic transition, similar to that found in spin density wave materials. Magnetic fields applied in the $ab$ plane result in a metamagnetic transition with a threshold field of $\approx 6.7$ T along with a sharp, strongly temperature dependent discontinuity and a change in sign of the magnetoresistance for in-plane currents. Thus, ScFeGe is an ideal system to investigate the effect of in-plane magnetic fields on a helimagnet with a $c$ -axis propagation vector, where the relative strength of the magnetic interactions and anisotropies determine the topology and magnetic structure.

In ferromagnetic materials, the rich dynamics of magnetic domain walls (DWs) under a magnetic field or current has been successfully described using the well-known $q-\phi$ analytical model. We demonstrate here that this simple unidimensional model holds for multiple-sublattice materials such as ferrimagnetic alloys or synthetic antiferromagnets (SAFs) by using effective parameters, and it is in excellent agreement with double-lattice micromagnetic simulations. We obtain analytical laws for the DW velocity and internal precession angle as a function of net magnetization for different driving forces (magnetic field, spin transfer, and spin-orbit torques) and different propagation regimes in ferrimagnetic alloys and SAFs. The model predicts that several distinctive dynamical features occur near or at the magnetic and the angular compensation points when the net magnetization or the net angular momentum of the system vanishes, and we discuss the experimental observations that have been reported for some of them. Using a higher degree-of-freedom analytical model that accounts for inter-sublattice distortions, we give analytical expressions for these distortions that agree with the micromagnetic simulations. This model shows that the DW velocity and precession rate are independent of the strength of the intersublattice exchange coupling, and it justifies the use of the simpler effective parameters model.

In an amorphous CoFeB layer, coercivity becomes anisotropic with fourfold symmetry when the CoFeB layer exchange couples to an FeRh layer. The angular dependence of coercivity of the CoFeB layer coincides with the in-plane easy-axis direction of the FeRh layer and experiences a ${45}^{\circ }$ shift with the occurrence of a metamagnetic phase transition of the FeRh layer from antiferromagnetism at room temperature to ferromagnetism at 400 K. The intriguing phenomena are well reproduced by our unbiased Monte Carlo simulation. The interfacial exchange and anisotropy energies, as well as the interfacial magnetization in the CoFeB/FeRh bilayer, are disentangled to demonstrate the strong dependence of the imprinting of anisotropy in the CoFeB layer on the interfacial exchange coupling. The evolution of the easy-axis direction of the induced anisotropy arises from the reconstruction of the interfacial exchange energy profile accompanied with the change of the magnetic state of FeRh, which governs the magnetization reversal of the CoFeB layer at both branches. Moreover, the imprinting is further applicable for the uniaxial magnetocrystalline anisotropy. This work not only presents the possibility of directly duplicating anisotropy between dissimilar materials, but it also provides a powerful tool to probe the hidden magnetic structures and/or the properties of materials that have weak magnetism, such as antiferromagnetic materials.

We have investigated the magnetic and charge transport properties of single crystals of the Nowotny chimney ladder compound ${\mathrm{Cr}}_{11}{\mathrm{Ge}}_{19}$ and mapped out a comprehensive phase diagram reflecting the complicated interplay between the Dzyaloshinskii-Moriya (DM) interaction, the dipolar interaction, and the magnetic anisotropy. We have identified a set of interesting magnetic phases and attributed a finite topological Hall effect to the recently discovered biskyrmion phase. These data also suggest the existence of an antiskyrmion state at finite fields for temperatures just below the magnetic ordering temperature $T_c$ as indicated by a distinct change in sign of the topological Hall effect. Above $T_c$, we discovered a region of enhanced magnetic response corresponding to a disordered phase likely existing near the ferromagnetic critical point under small magnetic fields. Strong spin chirality fluctuations are demonstrated by the large value of the topological Hall resistivity persisting up to 1 T, which is most likely due to the existence of the DM interaction. We argue that changes to the topological Hall effect correspond to different topological spin textures that are controlled by magnetic dipolar and DM interactions that vary in importance with temperature.

A site-diluted classical Heisenberg model on the bcc lattice, with only isotropic nearest-neighbor exchange interactions, is used to describe the phase diagram of ${\mathrm{Fe}}_p-{\mathrm{Al}}_q$ alloys ($p+q=1$) in the structurally disordered phase. Due to the larger size of Al atoms, the Fe-Fe exchange interactions along the border of the Al clusters are assumed to increase since the distance between Fe atoms becomes smaller in this region and remains constant for the rest of the lattice. However, for higher Al concentrations the Fe-Fe exchange interaction should decrease as a result of the increase of the lattice size parameter. Considering these assumptions, we have employed Monte Carlo simulations to obtain the thermodynamic properties of the model and obtain quite good fits to the experimental data of the transition temperature as a function of $q$. The results are still better than those recently obtained by assuming superexchange interactions induced by Al between next-nearest-neighbor Fe atoms.

The quantum spin systems ${\mathrm{Cu}}_2{M}^{\prime }{\mathrm{BO}}_5$ $(M^{\prime }=\mathrm{Al},\mathrm{Ga})$ with the ludwigite crystal structure consist of a structurally ordered ${\mathrm{Cu}}^{2+}$ sublattice in the form of three-leg ladders, interpenetrated by a structurally disordered sublattice with a statistically random site occupation by magnetic ${\mathrm{Cu}}^{2+}$ and nonmagnetic ${\mathrm{Ga}}^{3+}$ or ${\mathrm{Al}}^{3+}$ ions. A microscopic analysis based on density-functional-theory calculations for ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ reveals a frustrated quasi-two-dimensional spin model featuring five inequivalent antiferromagnetic exchanges. A broad low-temperature ${}^{11}\mathrm{B}$ nuclear magnetic resonance points to a considerable spin disorder in the system. In zero magnetic field, antiferromagnetic order sets in below $T_{\text{N}}\approx 4.1$ K and $\sim 2.4$ K for the Ga and Al compounds, respectively. From neutron diffraction, we find that the magnetic propagation vector in ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ is commensurate and lies on the Brillouin-zone boundary in the $\left(H0L\right)$ plane, ${\mathbf{q}}_{\text{m}}=(0.45,0,-0.7)$, corresponding to a complex noncollinear long-range ordered structure with a large magnetic unit cell. Muon spin relaxation is monotonic, consisting of a fast static component typical for complex noncollinear spin systems and a slow dynamic component originating from the relaxation on low-energy spin fluctuations. Gapless spin dynamics in the form of a diffuse quasielastic peak is also evidenced by inelastic neutron scattering. Most remarkably, application of a magnetic field above 1 T destroys the static long-range order, which is manifested in the gradual broadening of the magnetic Bragg peaks. We argue that such a crossover from a magnetically long-range ordered state to a spin-glass regime may result from orphan spins on the structurally disordered magnetic sublattice, which are polarized in magnetic field and thus act as a tuning knob for field-controlled magnetic disorder.

The significantly enhanced superconducting transition temperature ($T_c$) of an FeSe monolayer on ${\mathrm{SrTiO}}_3$(001) substrate has attracted extensive attention in recent years. Here, based on first-principles electronic structure calculations, we propose another candidate substrate, LaO(001), for the epitaxial growth of an FeSe monolayer to realize superconductivity. Our calculations show that for the optimal adsorption structure of the FeSe monolayer on LaO(001), the stripe antiferromagnetic state and the dimer antiferromagnetic state are almost energetically degenerate, indicating the existence of strong magnetic fluctuation that is beneficial to the appearance of superconductivity. According to the Bader charge analysis, the calculated electron doping from the LaO substrate to the FeSe monolayer is about 0.18 electrons per Fe atom, even larger than that in the case of FeSe/${\mathrm{SrTiO}}_3$(001). Since LaO was also reported to be a superconductor with $T_c\sim 5$ K, it may have a superconducting proximity effect on the epitaxial FeSe film and vice versa. These results suggest that LaO would be an interesting substrate to study interface-related superconductivity.