We unravel a fundamental connection between supersymmetry (SUSY) and a wide class of two-dimensional (2D) second-order topological insulators (SOTI). This particular supersymmetry is induced by applying a half-integer Aharonov-Bohm flux $f=\mathrm{\Phi }/{\mathrm{\Phi }}_0=1/2$ through a hole in the system. Here, three symmetries are essential to establish this fundamental link: chiral symmetry, inversion symmetry, and mirror symmetry. At such a flux of half-integer value, the mirror symmetry anticommutes with the inversion symmetry leading to a nontrivial $n=1$ SUSY representation for the absolute value of the Hamiltonian in each chiral sector, separately. This implies that a unique zero-energy state and an exact twofold degeneracy of all eigenstates with nonzero energy is found even at a finite system size. For arbitrary smooth surfaces, the link between 2D-SOTI and SUSY can be described within a universal low-energy theory in terms of an effective surface Hamiltonian which encompasses the whole class of supersymmetric periodic Witten models. Applying this general link to the prototypical example of a Bernevig-Hughes-Zhang(BHZ) model with an in-plane Zeeman field, we analyze the entire phase diagram and identify a gapless Weyl phase separating the topological from the nontopological gapped phase. Surprisingly, we find that topological states localized at the outer surface remain in the Weyl phase, whereas topological hole states move to the outer surface and change their spatial symmetry upon approaching the Weyl phase. Therefore the topological hole states can be tuned in a versatile manner opening up a route toward magnetic-field-induced topological engineering in multihole systems. Finally, we demonstrate the stability of localized states against deviation from half-integer flux, flux penetration into the sample, surface distortions, and random impurities for impurity strengths up to the order of the surface gap.

We develop a dynamical mean-field theory approach for the spiral magnetic order, changing to a local coordinate frame with preferable spin alignment along the $z$ axis, which can be considered with the impurity solvers treating the spin diagonal local Green's function. We furthermore solve the Bethe-Salpeter equations for nonuniform dynamic magnetic susceptibilities in the local coordinate frame. We apply this approach to describe the evolution of magnetic order with doping in the $t\text{−}{t}^{\prime }$ Hubbard model with $t^{\prime }=0.15$, which is appropriate for the description of the doped ${\mathrm{La}}_2{\mathrm{CuO}}_4$ high-temperature superconductor. We find that with doping the antiferromagnetic order changes to the $(Q,\pi )$ incommensurate one and then to the paramagnetic phase. The spectral weight at the Fermi level is suppressed near half filling and continuously increases with doping. The dispersion of holes in the antiferromagnetic phase shows qualitative agreement with the results of the $t\text{−}J$ model consideration. In the incommensurate phase we find two branches of hole dispersions, one of which crosses the Fermi level. The resulting Fermi surface forms hole pockets. We also consider the dispersion of the magnetic excitations, obtained from the nonlocal dynamic magnetic susceptibilities. The transverse spin excitations are gapless, fulfilling the Goldstone theorem; in contrast to the mean-field approach the obtained magnetic state is found to be stable. The longitudinal excitations are characterized by a small gap, showing the rigidity of the spin excitations. For realistic hopping and interaction parameters we reproduce the experimentally measured spin-wave dispersion of ${\mathrm{La}}_2{\mathrm{CuO}}_4$.

Hall conductivity for the intrinsic anomalous quantum Hall effect in homogeneous systems is given by the topological invariant composed of the Green function depending on momentum of quasiparticle. This expression reveals correspondence with the mathematical notion of the degree of mapping. A more involved situation takes place for the quantum Hall effect in the presence of external magnetic field. In this case, the mentioned expression remains valid if the Green function is taken in a specific representation, where it becomes the infinite-dimensional matrix [N. Imai et al., Phys. Rev. B 42, 10610 (1990)] or if it is replaced by its Wigner transformation while ordinary products are replaced by the Moyal products [M. A. Zubkov and X. Wu, Ann. Phys. 418, 168179 (2020)]. Both these expressions, unfortunately, are much more complicated and might be useless for the practical calculations. Here we represent the alternative representation for the Hall conductivity of a uniform system in the presence of constant magnetic field. The Hall conductivity is expressed through the Green function taken in Harper representation, when its nonhomogeneity is attributed to the matrix structure while functional dependence is on one momentum that belongs to magnetic Brillouin zone. Our consideration for the interacting systems is nonperturbative and is based on the Schwinger-Dyson equations truncated in a reasonable way. We demonstrate that in this approximation the expression for the Hall conductivity in Harper representation remains valid, where the interacting Green function is to be used instead of the noninteracting one. We, therefore, propose that the obtained expression may be used for the topological description of fractional quantum Hall effect.

By using the constrained-phase quantum Monte Carlo method, we performed a systematic study of the ground state of the half filled Hubbard model for a trilayer honeycomb lattice. We analyze the effect of the perpendicular electric field on the electronic structure, magnetic property, and pairing correlations. It is found that the antiferromagnetism is suppressed by the perpendicular electric field, especially the long-range parts, and the dominant magnetic fluctuations are still antiferromagnetic. The electronic correlation drives a $d+id$ superconducting pairing to be dominant over other pairing patterns among various electric fields and interaction strengths. We also found that the $d+id$ pairing correlation is greatly enhanced as the on-site Coulomb interaction is increased. Our intensive numerical results may unveil the nature of the recently observed superconductivity in rhombohedral trilayer graphene under an electric field.

Topology plays an important role in non-Hermitian systems. How to characterize a non-Hermitian topological system under open-boundary conditions (OBCs) is a challenging problem. A one-dimensional (1D) topological invariant defined on a generalized Brillion zone (GBZ) was recently found to successfully describe the topological property of the two-band Su-Schrieffer-Heeger model. But for a 1D multiband chiral symmetric system under OBCs, it is still controversial how to define the topological invariant. We show in this paper by exact proof and detailed demonstration that to acquire the topological invariant for multiband non-Hermitian models with chiral symmetry, the GBZ as the integral domain should be replaced by a more generalized closed loop. Our work thus establishes the non-Bloch bulk-boundary correspondence for 1D multiband chiral symmetric non-Hermitian systems.

Valleytronics, which uses the inequivalent electronic states at the band extrema in semiconductors, has been considered to play a vital role in the future information read/write technology. In this paper, we show that a sizable valley contrasting bulk photovoltaic (BPV) effect could exist even when the total BPV photocurrent component is symmetrically forbidden. We illustrate this argument by using a prototypical two-dimensional (2D) antiferromagnetic (AFM) semiconductor, $\mathrm{MnP}{\mathrm{Se}}_3$ monolayer, that is $\mathcal{P}\mathcal{T}$ symmetric ($\mathcal{P}$ and $\mathcal{T}$ refer to spatial inversion and time-reversal operators, respectively). We show that the Néel vector controls the magnetic point group well, so that the BPV current direction can be altered. In this material, the mirror reflection constrains some BPV components to be symmetrically forbidden. Nonetheless, we show that the local symmetry at the valleys allows finite BPV photocurrent components correspondingly. This would lead to hidden valley-polarized photoconductivity, reaching a magnitude of $\sim 300\mu \mathrm{A}/{\mathrm{V}}^2$, observable experimentally. We further predict that the $\mathrm{MnP}{\mathrm{Se}}_3$ monolayer is an example of a 2D ferrotoroidic system, also depending on the Néel vector direction, which can be characterized via magnetoelectric response measurements. Hence, we provide an exemplary platform for paving the route to future optospintronic and optovalleytronic devices in a single AFM material.

Certain fractional quantum Hall edges have been predicted to undergo quantum phase transitions which reduce the number of edge channels and at the same time bind electrons together. However, detailed studies of experimental signatures of such a “binding transition” remain lacking. Here, we propose quantum transport signatures with focus on the edge at filling $\nu =9/5$. We demonstrate theoretically that in the regime of nonequilibrated edge transport, the bound and unbound edge phases have distinct conductance and noise characteristics. We also show that for a quantum point contact in the strong back-scattering (SBS) regime, the bound phase produces a minimum Fano factor $F_{\mathrm{SBS}}=3$ corresponding to three-electron tunneling, whereas single-electron tunneling is strongly suppressed at low energies. Together with recent experimental developments, our results will be useful for detecting binding transitions in the fractional quantum Hall regime.

We study the scattering processes and the associated waiting time distributions (WTDs) in heterostructures based on one-dimensional helical edge states of a two-dimensional topological insulator. In combination with a proximitized $s$-wave superconductor and an applied magnetic field a topological transition occurs. Along this transition the WTD reveals specific features related to the presence of Andreev bound states at finite energy or Majorana bound states at zero energy under ideal conditions. The effects of several imperfections such as disorder and finite temperature are discussed.

It has recently been realized that the gap nodes of multiband superconductors that break time-reversal symmetry generically take the form of Fermi surfaces of Bogoliubov quasiparticles. However, these Fermi surfaces lead to a nonzero density of states (DOS) at the Fermi energy, which typically disfavors such superconducting states. It has thus not been clear whether they can be stable for reasonable pairing interactions or are in practice preempted by time-reversal-symmetric states with vanishing DOS. In this Letter, we show within BCS theory applied to a paradigmatic model that the time-reversal-symmetry-breaking states are indeed stabilized over broad parameter ranges at weak coupling. Moreover, we introduce a fast method that involves solving the inverse BCS gap equation, does not require iteration, does not suffer from convergence problems, and can handle metastable solutions.

Attempts to unravel the nature of magnetic ordering in ${\mathrm{LaSrCoO}}_4$ (${\mathrm{Co}}^{3+}$), a compound intermediate between antiferromagnetic (AFM) ${\mathrm{La}}_2{\mathrm{CoO}}_4$ (${\mathrm{Co}}^{2+}$) and ferromagnetic (FM) ${\mathrm{Sr}}_2{\mathrm{CoO}}_4$ (${\mathrm{Co}}^{4+}$), have met with limited success so far. In this paper, the results of a thorough investigation of dc magnetization and ac susceptibility in single-phase ${\mathrm{LaSrCoO}}_4$ provide clinching evidence for a thermodynamic paramagnetic (PM)-ferromagnetic (FM) phase transition at $T_c=220.5$ K, followed at lower temperature ($T_g=7.7$ K) by a transition to the cluster spin glass state (CSG). Analysis of the low-field Arrott plot isotherms, in the critical region near $T_c$, in terms of the Aharony-Pytte scaling equation of state clearly establishes that the PM-FM transition is basically driven by random magnetic anisotropy (RMA). For temperatures below $\approx 30$ K, large enough RMA destroys long-range FM order by breaking up the infinite FM network into FM clusters of finite size and leads to the formation of a CSG state at temperatures $T \lesssim$ 8 K by promoting freezing of finite FM clusters in random orientations. Increasing strength of the single-ion magnetocrystalline anisotropy (and hence RMA) with decreasing temperature is taken to reflect an increase in the number of low-spin ${\mathrm{Co}}^{3+}$ ions at the expense of that of high-spin ${\mathrm{Co}}^{3+}$ ions. At intermediate temperatures ($30 \mathrm{K}\lesssim T\lesssim 180 \mathrm{K}$), spin dynamics has contributions from the infinite FM network (fast relaxation governed by a single anisotropy energy barrier) and finite FM clusters (extremely slow stretched exponential relaxation due to hierarchical energy barriers).

The structural, magnetic, and thermodynamic properties of a quasi-one-dimensional (1D) $S=1$ alternating spin chain compound ${\mathrm{Sr}}_2\mathrm{Ni}{\left(\mathrm{Se}{\mathrm{O}}_3\right)}_3$ are investigated by using synchrotron x-ray powder diffraction, magnetic susceptibility $\chi (H,T)$, and heat capacity $C_{\mathrm{P}}(H,T)$ measurements together with density functional theory (DFT) calculations. The $\chi (H,T)$ and $C_{\mathrm{P}}(H,T)$ data reveal long-range antiferromagnetic order at $T_{\mathrm{N}}=3.4\left(3\right)$ K and short-range order at $T_{\mathrm{m}}\approx 7.8\mathrm{K}$. The short-range magnetic order together with 95% of spin entropy release above $T_{\mathrm{N}}$ signifies the importance of 1D spin correlations persisting to $\sim 8T_{\mathrm{N}}$. Theoretical DFT calculations with generalized gradient approximation determine leading exchange interactions, suggesting that interchain interactions are responsible for the observed long-range magnetic ordering. In addition, the temperature-field phase diagram of ${\mathrm{Sr}}_2\mathrm{Ni}{\left(\mathrm{Se}{\mathrm{O}}_3\right)}_3$ is determined based on the $\chi (T,H)$ and $C_{\mathrm{P}}(T,H)$ data. Interestingly, a nonmonotonic phase boundary of $T_{\mathrm{m}}$ is found for an external field applied along a hard axis. Our results suggest that the ground state and magnetic behavior of ${\mathrm{Sr}}_2\mathrm{Ni}{\left(\mathrm{Se}{\mathrm{O}}_3\right)}_3$ rely on the interplay of single-ion anisotropy, bond alternation, and interchain interactions.

To understand the 2D triangular Heisenberg antiferromagnetic system, we investigated the magnetic structures and the dynamics of ${\mathrm{Lu}}_y{\mathrm{Y}}_{1-y}{\mathrm{MnO}}_3$ in detail. The substitutions are adjusted to the Mn atomic position close to $x_{\text{Mn}}=\frac{1}{3}$. The neutron powder diffraction data claims that the magnetic structure of ${\mathrm{Lu}}_y{\mathrm{Y}}_{1-y}{\mathrm{MnO}}_3$ is described as a mixture of ${\mathrm{\Gamma }}_3$ ($P{6}_3^{\prime }c{m}^{\prime }$) and ${\mathrm{\Gamma }}_4$ ($P{6}_3^{\prime }{c}^{\prime }m$) at the $x_{\text{Mn}}$ position for $y=0.15$, 0.30, and 0.45. The ratio of ${\mathrm{\Gamma }}_3$ and ${\mathrm{\Gamma }}_4$ depends on temperature and composition and the fraction of ${\mathrm{\Gamma }}_3$ increases upon cooling, while no clear trimerization was observed at the $x_{\text{Mn}}$ position. We estimated exchange parameters from the analysis of the low-energy part of the spin waves. The results showed a weak trimerization effect on cooling because the nearest-neighbor exchange interaction is slightly enhanced. The temperature dependence of the spin-wave dispersion around the $\mathrm{\Gamma }$ point shows that the spin gap closes with increasing temperature because the exchange interactions in the nearest Mn-Mn neighbor become smaller. Gapless diffusive magnetic excitation from a Mn triangular lattice has been observed in a wide range in $Q$ and $E$ space of ${\mathrm{Lu}}_y{\mathrm{Y}}_{1-y}{\mathrm{MnO}}_3$. We found that ${\mathrm{Lu}}_{0.3}{\mathrm{Y}}_{0.7}{\mathrm{MnO}}_3$ could be an ideal case to investigate the trimerization, frustrated magnetism, and magnetoelastic coupling often observed in two-dimensional triangular lattice Heisenberg antiferromagnet systems.

Interatomic potential development using machine learning (ML) approaches has attracted a lot of attention in recent years because these potentials can effectively describe the structural and dynamical properties of complex materials at the atomistic level. In this work, we present the development of a neural network (NN) deep ML interatomic potential for Fe-Si alloys, and we demonstrate the effectiveness of the NN-ML potential in predicting the structures and energies of liquid and crystalline phases of Fe-Si alloys in comparison with the results from ab initio molecular dynamics simulations or experimental data. The developed NN-ML potential is also used to perform molecular dynamics simulations to study the structures of Fe-Si alloys with various compositions under rapid solidification conditions. The short-ranged orders in the rapidly solidified Fe-Si alloys are also analyzed by a cluster alignment method.

Antiferromagnets (AFMs) with anomalous quantum responses have led to new progress for the understanding of their magnetic and electronic structures from symmetrical and topological points of view. Two typical topological states are the collinear antiferromagnetic Weyl semimetal (WSM) and Weyl nodal loop semimetal (WNLSM). In comparison with their counterparts in ferromagnets and noncollinear AFMs, WSMs and WNLSMs in collinear AFMs are still awaiting experimental verification. In this paper, we theoretically predict the coexistence of Weyl points (WPs) and Weyl nodal loops (WNLs) in transition metal oxide ${\mathrm{RuO}}_2$. Owing to the small magnetocrystalline anisotropy energy, the WPs and WNLs can transform into each other via tuning the Néel vector. Moreover, since the WPs are very close to the Fermi level and the WNLs are even crossing the Fermi level, the topological states in ${\mathrm{RuO}}_2$ can be easily probed by photoemission and scanning tunneling microscopy methods. Our result provides a promising material platform for the study of WSM and WNLSM states in collinear AFMs.

Moiré systems offer an exciting playground to study the many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on ${\text{WSe}}_2/{\text{WS}}_2$ moiré bilayers [Y. Tang et al., Nature (London) 579, 353 (2020)], which realized a triangular superlattice with a small hopping (of approximately 10 Kelvin), with tunable density of holes, we explore the Hubbard model on the triangular lattice for intermediate temperatures $t\lesssim T<U$. Employing finite-temperature Lanczos calculations and closely following the fitting protocols used in the experiment, we recover the observed trends in the reported Curie-Weiss temperature $\mathrm{\Theta }$ with filling, using the reported interaction strength $U/t=20$. We focus on the large increase of $\left|\mathrm{\Theta }\right|$ on decreasing the density below half filling and the sign change of $\mathrm{\Theta }$ at higher fillings, which signals the onset of ferromagnetism. The increase in $\left|\mathrm{\Theta }\right|$ is also seen in the $t-J$ model (the low-energy limit of the Hubbard model) in the intermediate temperature range, which we clarify is opposite to the trend in its high temperature limit. Differences between the low, intermediate, and high temperature regimes are discussed. Our numerical calculations also capture the crossover between short-range antiferromagnetic to ferromagnetic order in the intermediate temperature regime, a result broadly consistent with the experimental findings. We find that this behavior is a finite-temperature remnant of the underlying zero-temperature phase transition, which we explore with ground-state density matrix renormalization group calculations. We provide evidence of ferromagnetism characterized by weak (but robust) correlations that explain the small $\mathrm{\Theta }$ seen in the experiment.

We propose a general construction scheme of topological pump for correlated systems with local gauge symmetry. The bulk-edge correspondence is one of its natural consequences. As a concrete example, an $SU\left(Q\right)$ quantum chain associated with a local ${\left[U\left(1\right)\right]}^{\otimes Q}$ gauge invariance of colored fermions is discussed in detail. The $SU\left(Q\right)$ invariant dimer phases are characterized by the $Z_Q$ Berry phases as a topological order parameter with a $d$-dimensional twist space ($d=Q-1$) as a synthetic Brillouin zone. By inclusion of the symmetry breaking perturbation specified by a rational parameter $\mathrm{\Phi }=P/Q$, the pump that encloses around the phase boundary, is characterized by the $Q$ Chern numbers associated with the currents due to uniform infinitesimal twists. The analysis of the systems under the open/periodic/twisted boundary conditions clarifies the bulk-edge correspondence of the pump where the large gauge transformation generated by the center-of-mass (CoM) plays a central role. An explicit formula for the Chern number is given by using the Diophantine equation. Numerical demonstrations by the exact diagonalization and the DMRG for finite systems ($Q=3,4,$ and 5) have been presented to confirm the general discussions for low-energy spectra, edge states, CoM's, Chern numbers, and the bulk-edge correspondence. A modified Lieb-Schultz-Mattis type argument for the general $SU\left(Q\right)$ quantum chain is also mentioned.

The electromagnetic coupling between heterostructures made of different materials is of great interest, both from the perspective of discovering new phenomena, as well as for its potential applications in novel devices. In this work, we study the electromagnetic coupling of a heterostructure made of a topological insulator (TI) slab and a single graphene layer, where the later presents a diluted concentration of ionized impurities. We explore the topological effects of the magnetoelectric polarizability of the TI, as well as its relative dielectric permittivity on the electrical conductivity in graphene at low but finite temperatures.

Identifying the topological order of the fractional quantum Hall state at filling $\nu =5/2$ is an important step towards realizing non-Abelian Majorana modes in condensed matter physics. However, to unambiguously distinguish between various proposals for this order is a formidable challenge. Here, we present a detailed study of transport along interfaced edge segments of fractional quantum Hall states hosting non-Abelian Majorana modes. With an incoherent model approach, we compute, for edge segments based on Pfaffian, anti-Pfaffian, and particle-hole-Pfaffian topological orders, thermal conductances, voltage biased charge current noise, and delta-$T$ noise. We determine how the thermal equilibration of edge modes impacts these observables and identify the temperature scalings of transitions between regimes of differently quantized thermal conductances. In combination with recent experimental data, we use our results to estimate thermal and charge equilibration lengths in real devices. We also propose an experimental setup, which permits measuring several transport observables for interfaced fractional quantum Hall edges in a single device. It can, e.g., be used to rule out edge reconstruction effects. In this context, we further point out some subtleties in two-terminal thermal conductance measurements and how to remedy them. Our findings are consistent with recent experimental results pointing towards a particle-hole-Pfaffian topological order at filling $\nu =5/2$ in GaAs/AlGaAs, and provide further means to pinpoint the edge structure at this filling and possibly also other exotic fractional quantum Hall states.

Frustrated triangular molecular magnets are a very important class of magnetic molecules since the absence of inversion symmetry allows an external electric field to couple directly with the spin chirality that characterizes their ground state. The spin-electric coupling in these molecular magnets leads to an efficient and fast method of manipulating spin states, making them an exciting candidate for quantum information processing. The efficiency of the spin-electric coupling depends on the spin-induced electric-dipole moment of the frustrated spin configurations contributing to the chiral ground state. In this paper, we report on first-principles calculations of spin-electric coupling in a $\left\{{\text{V}}_3\right\}$ triangular magnetic molecule. We have explicitly calculated the spin-induced charge redistribution within the magnetic centers that is responsible for the spin-electric coupling. Furthermore, we have generalized the method of calculating the strength of the spin-electric coupling to calculate any triangular spin-1/2 molecule with $C_3$ symmetry and have applied it to calculate the coupling strength in $\left\{{\text{V}}_{15}\right\}$ molecular magnets.