Topological phases and criticality in low-dimensional systems with disorder and interactions

In this work criticality arsing due to topological phase transitions in systems with disorder and interaction is investigated

Extended critical phase in quasiperiodic quantum Hall systems

We consider the effects of quasiperiodic spatial modulation on the quantum Hall plateau transition, by analyzing the Chalker-Coddington network model for the integer quantum Hall transition with quasiperiodically modulated link phases. In the conventional case (uncorrelated random phases), there is a critical point separating topologically distinct integer quantum Hall insulators. Surprisingly, the quasiperiodic version of the model supports an extended critical phase for some angles of modulation. We characterize this critical phase and the transitions between critical and insulating phases. For quasiperiodic potentials with two incommensurate wavelengths, the transitions we find are in a different universality class from the random transition. Upon adding more wavelengths they undergo a crossover to the uncorrelated random case. We expect our results to be relevant to the quantum Hall phases of twisted bilayer graphene or other Moir\'e systems with large unit cells.Comment: 11 pages, 9 figure

Spectrum-Wide Quantum Criticality at the Surface of Class AIII Topological Phases: An “Energy Stack” of Integer Quantum Hall Plateau Transitions

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 stack” of critical states wherein each state in the stack exhibits QHPT phenomenology. The stacking occurs without fine-tuning at the surface of a class AIII topological phase, where it is protected by U(1) and (anomalous) chiral or time-reversal symmetries. Spectrum-wide criticality is diagnosed by comparing numerics to universal results for the longitudinal Landauer conductance and wave function multifractality at the QHPT. Results are obtained from an effective 2D surface field theory and from a bulk 3D lattice model. We demonstrate that the stacking of quantum-critical QHPT states is a robust phenomenon that occurs for AIII topological phases with both odd and even winding numbers. The latter conclusion may have important implications for the still poorly understood logarithmic conformal field theory believed to describe the QHPT

Stability of topologically protected slow light against disorder

Slowing down light in on-chip photonic devices strongly enhances light-matter interaction, but typically also leads to increased backscattering and small-bandwidth operation. It was shown re- cently that, if one modifies the edge termination of a photonic Chern insulator such that the edge mode wraps many times around the Brillouin zone, light can be slowed to arbitrarily low group velocity over a large bandwidth, without being subject to backscattering. Here we study the robust- ness of these in-gap slow light modes against fabrication disorder, finding that disorder on scales significantly larger than the minigaps between edge bands is tolerable. We identify the mechanism for wavepacket breakup as disorder-induced velocity renormalization and calculate the associated breakup time.Comment: 5 pages, 5 figure

Generalized multifractality at metal-insulator transitions and in metallic phases of 2D disordered systems

We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Andersonlocalization transitions are considered. By using the non-linear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator). At the same time, the generalized parabolicity is strongly violated at critical points of metal-insulator transitions, signalling violation of local conformal invariance. Moreover, in classes D and DIII, even the Weyl symmetry breaks down at critical points of metal-insulator transitions. This last feature is related with a peculiarity of the sigma-model manifolds in these symmetry classes: they consist of two disjoint components. Domain walls associated with these additional degrees of freedom are crucial for ensuring Anderson localization and, at the same time, lead to the violation of the Weyl symmetry.Comment: 36 pages, 14 figure

Metal-insulator transition in a 2D system of chiral unitary class

We perform a numerical investigation of Anderson metal-insulator transition (MIT) in a twodimensional system of chiral symmetry class AIII by combining finite-size scaling, transport, density of states, and multifractality studies. The results are in agreement with the sigma-model renormalization-group theory, where MIT is driven by proliferation of vortices. We determine the phase diagram and find an apparent non-universality of several parameters on the critical line of MIT, which is consistent with the analytically predicted slow renormalization towards the ultimate fixed point of the MIT. The localization-length exponent $\nu$ is estimated as $\nu = 1.55 \pm 0.10$.Comment: 11 pages, 10 figure

Generalized surface multifractality in 2D disordered systems

Recently, a concept of generalized multifractality, which characterizes fluctuations and correlations of critical eigenstates, was introduced and explored for all ten symmetry classes of disordered systems. Here, by using the non-linear sigma-model field theory, we extend the theory of generalized multifractality to boundaries of systems at criticality. Our numerical simulations on two-dimensional (2D) systems of symmetry classes A, C, and AII fully confirm the analytical predictions of pure-scaling observables and Weyl symmetry relations between critical exponents of surface generalized multifractality. This demonstrates validity of the non-linear sigma-model field theory for description of Anderson-localization critical phenomena not only in the bulk but also on the boundary. The critical exponents strongly violate generalized parabolicity, in analogy with earlier results for the bulk, corroborating the conclusion that the considered Anderson-localization critical points are not described by conformal field theories. We further derive relations between generalized surface multifractal spectra and linear combinations of Lyapunov exponents of a strip in quasi-one-dimensional geometry, which hold under assumption of invariance with respect to a logarithmic conformal map. Our numerics demonstrate that these relations hold with an excellent accuracy. Taken together, our results indicate an intriguing situation: the conformal invariance is broken but holds partially at critical points of Anderson localization.Comment: 19 pages, 21 figure