The quantum spin Hall effect shares many similarities (and some important differences) with the quantum Hall effect for the electric charge. As with the quantum (electric charge) Hall effect, there exists a correspondence between bulk and boundary physics that allows to characterize the quantum spin Hall effect in diverse and complementary ways. In this paper, we derive from the network model that encodes the quantum spin Hall effect, the so-called Z_2 network model, a Dirac Hamiltonian in two dimensions. In the clean limit of this Dirac Hamiltonian, we show that the bulk Kane-Mele Z_2 invariant is nothing but the SU(2) Wilson loop constructed from the SU(2) Berry connection of the occupied Dirac-Bloch single-particle states. In the presence of disorder, the non-linear sigma model (NLSM) that is derived from this Dirac Hamiltonian describes a metal-insulator transition in the standard two-dimensional symplectic universality class. In particular, we show that the fermion doubling prevents the presence of a topological term in the NLSM that would change the universality class of the ordinary two-dimensional symplectic metal-insulator transition. This analytical result is fully consistent with our previous numerical studies of the bulk critical exponents at the metal-insulator transition encoded by the Z_2 network model. Finally, we improve the quality and extend the numerical study of boundary multifractality in the Z_2 topological insulator. We show that the hypothesis of two-dimensional conformal invariance at the metal-insulator transition is verified within the accuracy of our numerical results.
We analyze the critical behavior of the dephasing rate induced by short-range electron-electron interaction near an Anderson transition of metal-insulator or quantum Hall type. The corresponding exponent characterizes the scaling of the transition width with temperature. Assuming no spin degeneracy, the critical behavior can be studied by performing the scaling analysis in the vicinity of the non-interacting fixed point, since the latter is stable with respect to the interaction. We combine an analytical treatment (that includes the identification of operators responsible for dephasing in the …
Scaling of various local observables with a system size at Anderson transition criticality is characterized by a generalized multifractality. We study the generalized multifractality in the spin quantum Hall symmetry class (class C) in the presence of interaction. We employ Finkel'stein nonlinear sigma model and construct the pure scaling derivativeless operators for class C in the presence of interaction. Within the two-loop renormalization group analysis we compute the anomalous dimensions of the pure scaling operators and demonstrate that they are affected by the interaction. We find that t…
Entanglement entropy and multifractality at localization transitions
von Neumann纠缠熵是表征量子相变的一个有用的度量。我们研究了非相互作用电子系统中无序主导的量子相变中熵的非解析性。在这些临界点上，冯诺伊曼熵由单个粒子的波函数强度决定，它表现出复杂的标度不变波动。我们发现，多重分形的概念自然适合于研究临界波函数的冯诺伊曼熵。我们对三维安德森局域化跃迁和整数量子Hall平台跃迁的数值模拟表明，多重分形分析可以很好地描述这些跃迁处的纠缠.
We present the first experimental evidence for the multifractality of a transport property at a topological phase transition. In particular, we show that conductance fluctuations display multifractality at the integer-quantum-Hall $ν=1 \longleftrightarrow ν=2$ plateau-to-plateau transition in a high-mobility mesoscopic graphene device. We establish that to observe this multifractality, it is crucial to work with very high-mobility devices with a well-defined critical point. This multifractality gets rapidly suppressed as the chemical potential moves away from these critical points. Our combina…
We offer a new perspective to the problem of characterizing mesoscopic fluctuations in the inter-plateau region of the integer quantum Hall transition. We found that longitudinal and transverse conductance fluctuations, generated by varying the external magnetic field within a microscopic model, are multifractal and lead to distributions of conductance increments (magnetoconductance) with heavy tails (intermittency) and signatures of a hierarchical structure (a cascade) in the corresponding stochastic process, akin to Kolmogorov's theory of fluid turbulence. We confirm this picture by interpre…
We generalize universal relations between the multifractal exponent α_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude Λ_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude Λ_c^o is universally related to the boundary multifractal exponent α_0^s for the typical wav…
Higher-order topological superconductors and superfluids host lower-dimensional Majorana corner and hinge states since novel topology exhibitions on boundaries. While such topological nontrivial phases have been explored extensively, more possible schemes are necessary for engineering Majorana states. In this paper we propose Majorana corner states could be realized in a two-dimensional attractive quantum spin-Hall insulator with opposite in-plane Zeeman energy at two sublattice sites. The appropriate Zeeman field leads to the opposite Dirac mass for adjacent edges of a square sample, and natu…
The electronic states in a corner-overgrown bent GaAs/AlGaAs quantum well heterostructure are studied with numerical Hartree simulations. Transmission electron microscope pictures of the junction justify the sharp-corner assumption. In a tilted magnetic field both facets of the bent quantum well are brought to a quantum Hall (QH) state, and the corner hosts an unconventional hybrid system of two coupled counter-propagating quantum Hall edges and an additional one-dimensional accumulation wire. A subsystems model is introduced, whereby the total hybrid dispersion and wavefunctions are explained…
Interaction induced topological states remain one of the fascinating phases in condensed matter physics. The exciton condensate has recently sparked renewed interest due to the discovery of new candidate materials and its driving force to realize exotic topological states. In this work, we explore the exciton orders induced high-order topology in the bilayer quantum spin Hall insulators and find that the topological excitonic corner states can be realized by tuning the gate and magnetic field. When an in-plane Zeeman field is applied to the system, two or four excitonic boundary-obstructed cor…
We study spin conductance in a ballistic and quasi-ballistic two dimensional electron system with Rasbha spin-orbit coupling. The system has a four-terminal geometry with round corners at the connection to the leads. It is found that by going from sharp corners to more round corners in the ballistic system the energy depended spin conductance goes from being relatively flat to a curve showing a series of minima and maxima. It is also found that when changing the size of the terminal area by modifying the roundness of the terminal corners the maxima and minima in the transverse spin conductance…
We study the quantum entanglement structure of integer quantum Hall states via the reduced density matrix of spatial subregions. In particular, we examine the eigenstates, spectrum and entanglement entropy (EE) of the density matrix for various ground and excited states, with or without mass anisotropy. We focus on an important class of regions that contain sharp corners or cusps, leading to a geometric angle-dependent contribution to the EE. We unravel surprising relations by comparing this corner term at different fillings. We further find that the corner term, when properly normalized, has …
In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT). In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales. Normally QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field. In this paper we show that QHPT states can be realized throughout an energy continuum, i.e. as an "energy sta…
Instanton analysis for the spin quantum Hall symmetry class: Non-perturbative corrections to physical observables and generalized multifractal spectrum
The unusual electronic properties of the quantum spin Hall or Chern insulator become manifest in the form of robust edge states when samples with boundaries are studied. In this work, we ask if and how the topologically non-trivial electronic structure of these two-dimensional systems can be passed on to their zero-dimensional relatives, namely fullerenes or other closed-cage molecules. To address this question, we study Haldane's honeycomb lattice model on polyhedral nano-surfaces. We find that for sufficiently large surfaces characteristic corner states appear for parameters for which the pl…
We study the critical behavior near the integer quantum Hall plateau transition by focusing on the multifractal (MF) exponents $X_q$ describing the scaling of the disorder-average moments of the point contact conductance $T$ between two points of the sample, within the Chalker-Coddington network model. Past analytical work has related the exponents $X_q$ to the MF exponents $Δ_q$ of the local density of states (LDOS). To verify this relation, we numerically determine the exponents $X_q$ with high accuracy. We thereby provide, at the same time, independent numerical results for the MF exponents…
Topological corner states are exotic topological boundary states that are bounded to zero-dimensional geometry even the dimension of systems is large than one. As an elegant physical correspondence, their numbers are dictated by the bulk topological invariants. So far, all previous realizations of HOTIs are hallmarked by $\mathbb{Z}_2$ topological invariants and therefore have only one corner state at each corner. Here we report an experimental demonstration of $\mathbb{Z}$-class HOTI phases in electric circuits, hosting $N$ corner modes at each single corner structure. By measuring the impeda…
We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibit an unusual mechanism of topological protection against Anderson localization. These describe surface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological superconductors (TSCs), subject to the effects of quenched disorder. Numerics reveal a surprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall effects in classes A, C, and D. Conventional arguments derived from the non-linear $σ$-model picture imply that most TSC surface states should A…
We realize an elastic second-order topological insulator hosting both one-dimensional gapped edge states and zero-dimensional in-gap corner modes in the double-sided pillared phononic crystal plates with square lattice. Changing the width of two neighbor pillars breaks the inversion symmetry and induces the band inversion to emulate the quantum spin Hall effect where the gapless edge states are obtained. Further breaking the space-symmetry at interface, the gapless edge states are gapped and inducing the edge topological transitions and then giving rise to the zero-dimensional in-gap corner mo…
Boundaries constitute a rich playground for quantum many-body systems because they can lead to novel degrees of freedom such as protected boundary states in topological phases. Here, we study the groundstate of integer quantum Hall systems in the presence of boundaries through the reduced density matrix of a spatial region. We work in the lowest Landau level and choose our region to intersect the boundary at arbitrary angles. The entanglement entropy (EE) contains a logarithmic contribution coming from the chiral edge modes, and matches the corresponding conformal field theory prediction. We u…
Second-order topological superconductors host Majorana corner and hingemodes in contrast to conventional edge and surface modes in two and three dimensions. However, the realization of such second-order corner modes usually demands unconventional superconducting pairing or complicated junctions or layered structures. Here we show that Majorana corner modes could be realized using a 2D quantum spin Hall insulator in proximity contact with an $s$-wave superconductor and subject to an in-plane Zeeman field. Beyond a critical value, the in-plane Zeeman field induces opposite effective Dirac masses…
The modern theory of electric polarization in crystals associates the dipole moment of an insulator with a Berry phase of its electronic ground state [1, 2]. This concept constituted a breakthrough that not only solved the long-standing puzzle of how to calculate dipole moments in crystals, but also lies at the core of the theory of topological band structures in insulators and superconductors, including the quantum anomalous Hall insulator [3, 4] and the quantum spin Hall insulator [5-7], as well as quantized adiabatic pumping processes [8-10]. A recent theoretical proposal extended the Berry…
We propose theoretically a reconfigurable two-dimensional (2D) hexagonal sonic crystal with higher-order topology protected by the six-fold, $C_6$, rotation symmetry. The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit-cell. In the nontrivial phase, the sonic crystal realizes the topological spin Hall effect in a higher-order fashion: (i) The edge states emerging in the bulk band gap exhibits partial spin-momentum locking and are gapped due to the reduced spatial symmetry at the edges. (ii) The gapped edge states, on the other hand, stab…
Higher order topological superconductors hosting Majorana-Kramers pairs (MKPs) as corner modes have recently been proposed in a two-dimensional (2D) quantum spin Hall insulator (QSHI) proximity-coupled to unconventional cuprate or iron-based superconductors. Here, we show that such MKPs can be realized using a conventional s-wave superfluid with a soliton in cold atom systems governed by the Hubbard-Hofstadter model. The MKPs emerge in the presence of interaction at the "corners" defined by the intersections of line solitons and the one-dimensional edges of the system. Our scheme is based on t…
We consider a quantum Hall (QH) region in contact with a superconductor (SC), i.e., a QH-SC junction. Due to successive Andreev reflections, the QH-SC interface hosts hybridized electron and hole edge states called chiral Andreev edge states (CAES). We theoretically study the transport properties of these CAES by using a microscopic, tight-binding model. We find that the transport properties strongly depend on the contact geometry and the value of the filling factor. We notice that it is necessary to add local barriers at the corners of the junction in order to reproduce such properties, when …
As platforms of Majorana modes, topological insulator (quantum anomalous Hall insulator)/superconductor (SC) heterostructures have attracted tremendous attention over the past decade. Here we substitute the topological insulator by its higher-order counterparts. Concretely, we consider second-order topological insulators (SOTIs) without time-reversal symmetry and investigate SOTI/SC heterostructures in both two and three dimensions. Remarkably, we find that such novel heterostructures provide natural realizations of second-order topological superconductors (SOTSCs) which host Majorana corner m…
Photonic topological systems, the electromagnetic analog of the topological materials in condensed matter physics, create many opportunities to design optical devices with novel properties. We present an experimental realization of the bi-anisotropic meta waveguide photonic system replicating both quantum Hall (QH) and quantum spin-Hall (QSH) topological insulating phases. With careful design, a composite QH-QSH photonic topological material is created and experimentally shown to support reflection-free edgemodes, a heterogeneous topological structure that is unprecedented in condensed matter …
A Quantum Spin-Hall Insulator (QSHI) is implemented into a simple mass-spring Kagome lattice. The transition from the trivial state to the topological one is described by an invariant Chern number function of a contrast parameter. The band diagram and helical edge states characteristic of QSHI are obtained by a combination of numerical and analytical methods. In particular, these states are shown to be Stoneley wave solutions to a set of asymptotic continuous motion equations. Last, scatterless propagation of polarized topological edge waves around sharp corners is demonstrated and robustness …
Electrical control and detection of spin precession are experimentally demonstrated by using spin-resolved edge states in the integer quantum Hall regime. Spin precession is triggered at a corner of a biased metal gate, where electron orbital motion makes a sharp turn leading to a nonadiabatic change in the effective magnetic field via spin-orbit interaction. The phase of precession is controlled by the group velocity of edge-state electrons tuned by gate bias voltage: A spin-FET device is thus realized by all-electrical means, without invoking ferromagnetic material. The effect is also interp…
Generally, the topological corner state in two-dimensional second-order topological insulator (2D SOTI) is equivalent to the well-known domain wall state, originated from the mass-inversion between two adjacent edges with phase shift of pi. In this work, go beyond this conventional physical picture, we report a fractional mass-kink induced 2D SOTI in monolayer FeSe with canted checkerboard antiferromagnetic (AFM) order by analytic model and first-principles calculations. The canted spin associated in-plane Zeeman field can gap out the quantum spin Hall edge state of FeSe, forming a fractional …
We develop the theoretical framework of nonequilibrium ultrafast photonics in monolayer quantum spin-Hall insulators supporting a multitude of topological states. In these materials, ubiquitous strong light-matter interactions in the femtosecond scale lead to non-adiabatic quantum dynamics, resulting in topology-dependent nonlinear optoelectronic transport phenomena. We investigate the mechanism driving topological Dirac fermions interacting with strong ultrashort light pulses and uncover various experimentally accessible physical quantities that encode fingerprints of the quantum material's t…
