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Recent explorations of topology in physical systems have led to a new paradigm of condensed matters characterized by topologically protected states and phase transition, for example, topologically protected photonic crystals enabled by magneto-optical effects. However, in other wave systems such as acoustics, topological states cannot be simply reproduced due to the absence of similar magnetics-related sound-matter interactions in naturally available materials. Here, we propose an acoustic topological structure by creating an effective gauge magnetic field for sound using circularly flowing air in the designed acoustic ring resonators. The created gauge magnetic field breaks the time-reversal symmetry, and therefore topological properties can be designed to be nontrivial with non-zero Chern numbers and thus to enable a topological sonic crystal, in which the topologically protected acoustic edge-state transport is observed, featuring robust one-way propagation characteristics against a variety of topological defects and impurities. Our results open a new venue to non-magnetic topological structures and promise a unique approach to effective manipulation of acoustic interfacial transport at will.

The Haldane model is a paradigmatic 2 d lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a ‘topological’ phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current–current correlations. Our method provides a full construction of the critical curves, which are modified (‘dressed’) by the electron–electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods.

We demonstrate that in a correlated 2D systems of electrons in the presence of perpendicular magnetic field the magnetic flux quantum may not achieve its value determined for a single or a noncorrelated electron. Correlations induced by the repulsion of electrons at strong magnetic field presence impose topological-type limits on planar cyclotron orbits which cause specific homotopy of trajectories resulting in constraints of the magnetic field flux quantum value. These restrictions occur at discrete series of magnetic field values corresponding to hierarchy of 2D correlated Hall states observed experimentally in GaAs thin films and in graphene. The similar homotopy property is observed in 2D Chern topological insulators when the magnetic field is substituted by the Berry field.

We discuss the tight-binding models of solid state physics with the Z 2 sublattice symmetry in the presence of elastic deformations in an important particular case—the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles, the Wigner–Weyl formalism is explored. It allows the calculation of the two-point Green’s function in the presence of two slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters that result from elastic deformations. The developed formalism allows us to consider the influence of elastic deformations and the variations of magnetic field on the quantum Hall effect.

The quantum Hall effect under the influence of gravity and inertia is studied in a unified way. We make use of an algebraic approach, as opposed to an analytic approach. We examine how both the integer and the fractional quantum Hall effects behave under a combined influence of gravity and inertia using a unified Hamiltonian. For that purpose, we first re-derive, using the purely algebraic method, the energy spectrum of charged particles moving in a plane perpendicular to a constant and uniform magnetic field either (i) under the influence of a nonlinear gravitational potential or (ii) under the influence of a constant rotation. The general Hamiltonian for describing the combined effect of gravity, rotation and inertia on the electrons of a Hall sample is then built and the eigenstates are obtained. The electrons mutual Coulomb interaction that gives rise to the familiar fractional quantum Hall effect is also discussed within such a combination.

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number σH is given as the winding number of an eigenvector of a 2×2 transfer matrix, as a function of the quasi-momentum k∈(0,2π) . This method is computationally efficient (of order O(n4) in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for σH for flux p/q in the r -th gap conforms with the Diophantine equation r=σH·p+s·q , which determines σHmodq . A window such as σH∈(-q/2,q/2) , or possibly shifted, provides a natural further condition for σH , which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition σH∈(-q,q) .

We discuss quantum Hall effect in the presence of arbitrary pair interactions between electrons. It is shown that, irrespective of the interaction strength, the Hall conductivity is given by the filling fraction of Landau levels averaged over the ground state of the system. This conclusion remains valid for both the integer and fractional quantum Hall effect.

We propose to create an effective magnetic field for photons in a two-dimensional waveguide network with strong scattering at waveguide junctions. The effective magnetic field is realized by imposing a direction-dependent phase along each waveguide link. Such a direction-dependent phase can be produced by dynamic modulation or by the magneto-optical effect. Compared to previous proposals for creating an effective magnetic field for photons, this scheme is resonator-free, thus potentially reduces the experimental complexity. We also show that such a waveguide network can be used to explore photonic analogue of integer quantum Hall effect for massless particles.

In this work we studied the properties of a two-dimensional electronic gas subjected to a strong magnetic field and cooled at a low temperature. We reported exact analytical results of energies at the ground state. The results are for systems up to Ne = 10 electrons calculated in the integer quantum Hall effect (IQHE) regime at the filling factor υ = 1. To accomplish the calculation we used the complex polar coordinates method. Note that the system of electrons in the quantum Hall regime relied heavily on the disk geometry for finite systems of electrons with arbitrary values of Ne = 2 to 10 particles. The results that we obtained by analytical calculations are in good agreement with those reported by Ciftja [Ciftja O., J. Math. Phys., 2011, 52, 122105], where the representation for certain integrals of products of Bessel functions is obtained. In the end, we have studied the composite fermions energies for the excited states for several systems at υ = 1/3 and the correspondence between the fractional quantum Hall effect (FQHE) and the IQHE.

In this work, we study how the combination of rotation and a topological defect can influence the energy spectrum of a two dimensional electron gas in a strong perpendicular magnetic field. A deviation from the linear behavior of the energy as a function of magnetic field, caused by a tripartite term of the Hamiltonian, involving magnetic field, the topological charge of the defect and the rotation frequency, leads to novel features which include a range of magnetic field without corresponding Landau levels and changes in the Hall quantization steps.