We consider the integer quantum Hall effect on a square lattice in uniform rational magnetic field. The relation between two different interpretations of conductance as topological invariants is clarified. One Thouless--Kohmoto--Nightingale--den Nijs (TKNN) infinite system and other winding number edge state. In TKNN form conductance, phase Bloch wave function defines U(1) vortices Brillouin zone total vorticity gives ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$. find that these...

A criterion to determine the existence of zero-energy edge states is discussed for a class particle-hole symmetric Hamiltonians. "loop" in parameter space assigned each one-dimensional bulk Hamiltonian, and its topological properties, combined with chiral symmetry, play an essential role. It provides unified framework discuss modes several systems such as fully gapped superconductors, two-dimensional d-wave graphite ribbons. variance Peierls instability caused by presence edges also discussed.

We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on discretized Brillouin zone. It provides an efficient method computing (spin) Hall conductances without specifying gauge-fixing conditions. demonstrate that it correctly reproduces quantized even coarsely A gauge-dependent integer-valued field, which plays key role in formulation, is evaluated several gauges. An extension to non-Abelian also given.

We study edge states in the integral quantum Hall effect on a square lattice rational magnetic field \ensuremath{\varphi}=p/q. The system is periodic y direction but has two edges x direction. have found that energies of are given by zero points Bloch function some Riemann surface (RS) (complex energy surface) when size commensurate with flux. genus RS, g=q-1, number gaps. move around holes RS as momentum conductance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$ winding holes,...

Emergence of exceptional points in two dimensions is one the remarkable phenomena non-Hermitian systems. We here elucidate impacts symmetry on physics. Specifically, we analyze chiral symmetric correlated systems equilibrium where are induced by finite lifetime quasi-particles. Intriguingly, our analysis reveals that combination and non-Hermiticity results novel topological degeneracies energy bands which call symmetry-protected rings (SPERs). observe emergence SPERs analyzing a Dirac...

We investigate the occurrence of n-fold exceptional points (EPs) in non-Hermitian systems, and show that they are stable n-1 dimensions presence antiunitary symmetries local parameter space, such as, e.g., parity-time (PT) or charge-conjugation parity (CP) symmetries. This implies particular threefold fourfold symmetry-protected EPs stable, respectively, two three dimensions. The stability multofold (i.e., beyond usual twofold EPs) is expressed terms homotopy properties a resultant vector we...

We derive an efficient formula for Z$_2$ topological invariants characterizing the quantum spin Hall effect. It is defined in a lattice Brillouin zone, which enables us to implement numerical calculations realistic models even three dimensions. Based on this, we study effect Bi and Sb quasi-two dimensions using tight-binding model.

Numerical results obtained from Monte Carlo simulation in the ground state and at finite temperatures as well exact diagonalization transfer matrix method are reported. Efficiency of is examined. Spin, charge superconducting correlations investigated for Hubbard t - J model one two dimensions. The momentum distribution one-dimensional shows fermi-liquid-like behavior least orders magnitude smaller energy scale than band width. short-range incommensurate spin correlation observed analyzed...

Inspired by a recent discovery of peculiar integer quantum Hall effect (QHE) in graphene, we study QHE on honeycomb lattice terms the topological number, with two interests. First, how zero-mass Dirac around center tight-binding band crosses over to ordinary finite-mass fermion edges. Second, bulk is related edge for entire spectrum including and behaviors. We find following. (i) The [with ${\ensuremath{\sigma}}_{xy}=\ensuremath{\mp}(2N+1){e}^{2}∕h,N$: integer] persists, surprisingly, up van...

We study the energy spectrum and Hall effect of electrons on square lattice with next-nearest-neighbor (NNN) hopping as well nearest-neighbor hopping. This includes triangular a special case. system under general rational values magnetic flux per unit cell \ensuremath{\varphi}=p/q. The structure secular equation is studied in detail, k dependence analytically obtained. In absence NNN hopping, two bands at center touch for q even; thus conductance not defined half-filling. An gap opens there...

The entanglement entropy (von Neumann entropy) has been used to characterize the complexity of many-body ground states in strongly correlated systems. In this paper, we try establish a connection between lower bound von and Berry phase defined for quantum states. As an example, family translational invariant lattice free fermion systems with two bands separated by finite gap is investigated. We argue that, one dimensional (1D) cases, when (Zak's phase) occupied band equal $\pi \times ({odd...

The authors discover the mirror skin effect, which highlights non-Hermitian topological properties protected by crystalline symmetry. Due to system shows a significant dependence of energy spectrum on boundary condition only for invariant line in two-dimensional Brillouin zone. They also propose how experimentally observe effect making use high controllability electric circuits.

We propose mechanical systems, described by Newton's equation of motion, as suited platforms for symmetry protection non-Hermitian topological degeneracies. point out that systems possess emergent symmetry, which is a unique properties systems. Because the in contrast to other fine-tuning parameters (e.g., gain and loss) not required preserve protecting exceptional rings two dimensions. The presence symmetry-protected (SPERs) dimensions numerically demonstrated graphene with friction....

We demonstrate the emergence of a topological ordered phase for non-Hermitian systems. Specifically, we elucidate that systems with two-body interactions show fractional quantum Hall (FQH) state. The Hamiltonian is considered to be relevant cold atoms dissipation. conclude FQH state by presence degeneracy and many-body Chern number ground multiplet showing $C_{\mathrm{tot}}=1$. robust against non-Hermiticity arises from translational symmetry. Furthermore, discover emerges without any...

We propose a new correlated topological state which we call higher-order Mott insulator (HOTMI). This exhibits striking bulk-boundary correspondence due to electron correlations. Namely, the properties in bulk, characterized by Z3 spin-Berry phase, result gapless corner modes emerging only spin excitations (i.e., single-particle remain gapped around corner). demonstrate emergence of HOTMI Hubbard model on kagome lattice, and elucidate how strong correlations change at noninteracting case.

We show that a quantum scar state, an atypical eigenstate breaking thermalization hypothesis embedded in many-body energy spectrum, can be constructed flat band systems. The key idea of our construction is to make use orthogonal compact localized states. concretely discuss scheme, taking sawtooth lattice system as example, and numerically demonstrate the presence state. Examples higher-dimensional systems are also addressed. Our method has broad applications various

Although topological phenomena attract growing interest not only in linear systems but also nonlinear systems, the bulk-edge correspondence under nonlinearity of eigenvalues has been established so far. We address this issue by introducing auxiliary eigenvalues. reveal that edge states eigenstates are topologically inherited as physical when is weak finite (i.e., monotonic for one). This result leads to with

By finite-size analysis of exact diagonalization the existence a finite spin gap is confirmed for both antiferromagnetic and ferromagnetic coupling in half-filled Kondo lattice model. For all values system belongs to class coherent singlet, whereas any it scales S=1 chain with Haldane gap.

We propose to use quantized Berry phases as local order parameters of gapped quantum liquids, which are invariant under some anti-unitary operation. After presenting a general prescription, the scheme is applied for Heisenberg models with frustrations and two dimensional extended Su-Schrieffer-Heeger associated random dimer covering. In each phases, 0 or $\pi$, describe ground state texture pattern singlet bonds bonds. Also possible applications large classes correlated electron systems discussed.

We reveal from numerical study that the optical Hall conductivity sigma(xy)(omega) has a characteristic feature even in ac ( approximately THz) regime plateaus are retained both ordinary two-dimensional electron gas and graphene quantum regime, although plateau height is no longer quantized ac. In reflects unusual Landau level structure. The effect remains unexpectantly robust against significant strength of disorder, which we attribute to an localization. predict measurements feasible...

For generic time-reversal-invariant systems with spin-orbit couplings, we clarify a close relationship between the ${\mathrm{Z}}_{2}$ topological order and spin Chern number (SChN) in quantum spin-Hall effect. It turns out that global gauge transformation connects sectors different SChNs (even integers) modulo 4, which implies SChN orders yield same classification. We present method of computing demonstrate it single double planes graphene.

Abstract Recently, quantum Hall state analogs in classical mechanics attract much attention from topological points of view. Topology is not only for mathematicians but also quite useful a world. Further it even governs the Newton’s law motion. One advantages systems over solid materials its clear controllability. Here we investigate mechanical graphene, which spring-mass model with honeycomb structure as typical nontrivial phenomena. The vibration spectrum graphene characterized by Dirac...

Adiabatic ZQ invariants by quantized Berry phases are defined for gapped electronic systems in d-dimensions (Q=d+1). This series includes Polyacetylene, Kagome and Pyrochlore lattice, respectively d=1, 2 3. The quantum Q-multimer order parameters to characterize the topological phase transitions multimerization. fractional quantization is protected global equivalence even with particle-particle interaction. As chiral symmetric case, a form of Z2 invariant explicitly given as well.

A higher-order topological insulator is a new concept of states matter, which characterized by the emergent boundary whose dimensionality lower more than two compared with that bulk, and draws considerable interest. Yet, its robustness against disorders still unclear. In this work, we investigate phase diagram phases in breathing kagome model presence using state-of-the-art machine learning technique. We find corner survive finite strength disorder potential as long energy gap not closed,...

Square-root topological insulators are recently-proposed intriguing insulators, where the topologically nontrivial nature of Bloch wave functions is inherited from square Hamiltonian. In this paper, we propose that higher-order can also have their square-root descendants, which term insulators. There, emergence in-gap corner states squared Hamiltonian hosts topology. As an example such systems, investigate tight-binding model on a decorated honeycomb lattice, whose includes breathing...