We elaborate that for topological insulators and superconductors described by Dirac models in any dimension symmetry class, the order can be mapped to lattice sites a universal marker. Deriving from recently discovered momentum-space invariant, we introduce operator consists of alternating projectors filled empty eigenstates position operators, multiplied matrices are omitted Hamiltonian. The projected yields marker, whose form is explicitly constructed every topologically nontrivial class...

A universal topological marker has been proposed recently to map the invariants of Dirac models in any dimension and symmetry class lattice sites. Using this marker, we examine conditions under which global order, represented by spatially averaged remains unchanged presence disorder for 1D 2D systems. We find that if an impurity corresponds varying a nonzero-matrix element Hamiltonian, regardless represents hopping, chemical potential, pairing, etc., then average is conserved. However, there...

We introduce and analyze a lattice model of anyons in periodic potential an external magnetic field, which exhibits transition from Mott insulator to quantum Hall fluid. The is characterized by the anyon statistics \ensuremath{\alpha}, can vary between fermions, \ensuremath{\alpha}=0, bosons, \ensuremath{\alpha}=1. For bosons universality class classical three-dimensional XY model. Near fermion limit, described massless 2+1 Dirac theory coupled Chern-Simons gauge field. Analytic calculations...

The correlation functions related to topological phase transitions in inversion-symmetric lattice models described by $2\ifmmode\times\else\texttimes\fi{}2$ Dirac Hamiltonians are discussed. In one dimension, the function measures charge-polarization between Wannier states at different positions, while two dimensions it itinerant-circulation states. is nonzero both topologically trivial and nontrivial states, allows us extract a length that diverges transitions. curvature defines invariants...

A unified expression for topological invariants has been proposed recently to describe the order in Dirac models belonging any dimension and symmetry class. We uncover a correspondence between curvature function that integrates this invariant quantum metric measures distance properly defined many-body Bloch states momentum space. Based on metric-curvature correspondence, time-resolved angle-resolved photoemission spectroscopy experiment is measure violation of spectral sum rule caused by...

The quantum geometry in the momentum space of semiconductors and insulators, described by metric valence band Bloch state, has been an intriguing issue owing to its connection various material properties. Because Brillouin zone is periodic, integration over represents average distance between neighboring states, which we call fidelity number. We show that this number can further be expressed real as a marker, local quantity calculated directly from diagonalizing lattice Hamiltonian. A linear...

Through $s-d$ coupling, a superconducting thin film interfaced to noncollinear magnetic insulator inherits its order, which may induce unconventional superconductivity that hosts Majorana edge states. From the cycloidal, helical, or (tilted) conical order of multiferroics, Bloch and Neel domain walls ferromagnetic insulators, induced pairing is ($p_{x}+p_{y}$)-wave, state supports modes without adjusting chemical potential. In this setup, states can be separated over distance long range...

Graphene is a promising material for designing next-generation electronic and valleytronic devices, which often demand the opening of bandgap in otherwise gapless pristine graphene. To date, several conceptually different mechanisms have been extensively exploited to induce bandgaps graphene, including spin–orbit coupling inversion symmetry breaking monolayer quantum confinement graphene nanoribbons (GNRs). Here, we present multiscale study competing gap overlayer GNRs proximity-coupled...

We propose that the magnetoelectric effect in multiferroic insulators with a coplanar antiferromagnetic spiral order, such as BiFeO_{3}, enables electrically controlled magnonics without need of magnetic field. Applying an oscillating electric field these materials frequency low household can activate Goldstone modes manifest fast planar rotations spin, whose motion is essentially unaffected by crystalline anisotropy. Combining spin ejection mechanisms, rotation deliver electricity at room...

The quantum geometric properties of a Bloch state in momentum space are usually described by the Berry curvature and metric. In realistic gapped materials where interactions disorder render not viable starting point, we generalize these concepts introducing dressed metric at finite temperature, which effect many-body can be included perturbatively. These quantities extracted from charge polarization susceptibility caused linearly or circularly polarized electric fields, whose spectral...

In 2D semiconductors and insulators, the Chern number of valence band Bloch state is an important quantity that has been linked to various material properties, such as topological order. We elaborate opacity materials circularly polarized light over a wide range frequencies, measured in units fine structure constant, can be used extract spectral function frequency-integrates number, offering simple optical experiment measure it. This method subsequently generalized finite temperature locally...

The topological properties of the Su-Schrieffer-Heeger (SSH) model in presence nearest-neighbor interaction are investigated by means a marker, generalized from noninteracting one utilizing single-particle Green's function many-body ground state. We find that despite marker not being perfectly quantized interactions, it always remains finite topologically nontrivial phase while converging to zero trivial when approaching thermodynamic limit, and hence correctly judges phases interactions....

For three-dimensional semiconductors and insulators, the authors elaborate here that imaginary part of dielectric function integrated over frequency gives spread valence-band Wannier functions, which characterizes how insulating is material. two-dimensional materials, their opacity divided by then does same job.

The topological phase transitions in static and periodically driven Kitaev chains are investigated by means of a renormalization group (RG) approach. These transitions, across which the numbers or Floquet Majorana edge modes change, accompanied divergences relevant Berry connections. at certain high-symmetry points momentum space form basis RG approach, through boundaries identified as function system parameters. We also introduce several aspects to characterize quantum criticality both...

For $D$-dimensional weakly interacting topological insulators in certain symmetry classes, the invariant can be calculated from a $D$- or $(D+1)$-dimensional integration over curvature function that is expressed terms of single-particle Green's functions. Based on divergence at phase transition, we demonstrate how renormalization group approach circumvents these integrations and reduces necessary calculation to for alone, rendering numerically efficient tool identify transitions large...

For inversion-symmetric topological insulators and superconductors characterized by [Formula: see text] invariants, two scaling schemes are proposed to judge phase transitions driven an energy parameter. The renormalize either the gradient or second derivative of Pfaffian time-reversal operator, through which renormalization group flow driving parameter can be obtained. near invariant momentum is revealed display a universal critical behavior for great variety models examined.

In topological insulators and superconductors, the discrete jump of invariant upon tuning a certain system parameter defines phase transition. A unified framework is employed to address quantum criticality transitions in one three spatial dimensions, which simultaneously incorporates symmetry classification, order band crossing, $m$-fold rotational symmetry, correlation functions, critical exponents, scaling laws, renormalization group approach. We first classify higher-order Dirac models...

We demonstrate that the prototypical two-dimensional Chern insulator hosts exotic quantum multicriticality in presence of an appropriate periodic driving: a linear Dirac-like transition coexists with quadratic nodal looplike transition. This loop gap closure is characterized by enhanced chiral-mirror symmetry induced driving procedure. The existence multiple universality classes can be unambiguously captured extracting critical exponents and scaling laws single renormalization group approach...

The notion of fidelity susceptibility, introduced within the context quantum metric tensor, has been an important quantity to characterize criticality near phase transitions. We demonstrate that for topological transitions in Dirac models, provided momentum space is treated as manifold metric, susceptibility coincides with curvature function whose integration gives invariant. Thus a transition also describes and correlation length extracted from scale over which decays. To map out profile we...

A bipartite lattice with chiral symmetry is known to host zero-energy flat bands if the numbers of two sublattices are different. We demonstrate that this mechanism producing can be realized on graphene by introducing periodic vacancies. Using first-principles calculations, we elaborate even though pristine does not exactly preserve symmetry, applied holey still produces single or multiple as narrow $\ensuremath{\sim}0.5\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ near Fermi surface throughout...

In two-dimensional time-reversal symmetric topological insulators described by Dirac models, theZ2topological invariant can be the spin Chern number. We present a linear response theory for Berry curvature that integrates to number, and introduce its spectral function measured at finite temperature momentum- spin-resolved circular dichroism, which may achieved pump-probe type of experiments using spin- time-resolved ARPES. As result, sign Pfaffian theZ2invariant directly measured. A number...

We propose a promising electron entanglement detector consisting of two quantum spin Hall systems weakly coupled to superconductor. The detection spins along various polarization directions, which is prerequisite for testing Bell's inequality on solid state spins, can be achieved in an all-electrical-controlled manner utilizing the helical edge states. It found that violation exists large range tunneling parameters, realized mercury telluride wells.

Topologically ordered systems are characterized by topological invariants that often calculated from the momentum space integration of a certain function represents curvature many-body state. The may be Berry curvature, connection, or other quantities depending on system. Akin to stretching messy string reveal number knots it contains, scaling procedure is proposed for in inversion symmetric systems, which phase transition can identified flow driving energy parameters control topology...

Topological phases of materials are characterized by topological invariants that conventionally calculated different means according to the dimension and symmetry class system. For described Dirac models, we introduce a wrapping number as unified approach obtain in arbitrary dimensions classes. Given unit vector parametrizes momentum-dependence model, describes degree map from Brillouin zone torus sphere formed call sphere. This method is gauge-invariant originates intrinsic features...