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arXiv:1311.5233 · Physical Review Letters

Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder

Ian Mondragon-Shem, Taylor L. Hughes, Juntao Song, Emil V. Prodan

Published Jul 23, 2014• 279 citations
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Abstract

The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.

DOI
10.1103/physrevlett.113.046802

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The chiral AIII symmetry class in the classification table of topological insulators contains topological phases classified by a winding number ν for each odd space dimension. An open problem for this class is the characterization of the phases and phase boundaries in the presence of strong disorder. In this work, we derive a covariant real-space formula for ν and, using an explicit one-dimensional disordered topological model, we show that ν remains quantized and nonfluctuating when disorder is turned on, even though the bulk energy spectrum is completely localized. Furthermore, ν remains robust even after the insulating gap is filled with localized states, but when the disorder is increased even further, an abrupt change of ν to a trivial value is observed. Using exact analytic calculations, we show that this marks a critical point where the localization length diverges. As such, in the presence of disorder, the AIII class displays markedly different physics from everything known to date, with robust invariants being carried entirely by localized states and bulk extended states emerging from an absolutely localized spectrum. Detailed maps and a clear physical description of the phases and phase boundaries are presented based on numerical and exact analytic calculations.
The winding number has been widely used as an invariant for diagnosing topological phases in one-dimensional chiral-symmetric systems. We put forward a real-space representation for the winding number. Remarkably, our method reproduces an exactly quantized winding number even in the presence of disorders that break translation symmetry but preserve chiral symmetry. We prove that our real-space representation of the winding number, the winding number defined through the twisted boundary condition, and the real-space winding number derived previously [Phys. Rev. Lett. 113, 046802 (2014)], are equivalent in the thermodynamic limit at half-filling. Our method also works for the case of filling less than one half, where the winding number is not necessarily quantized. Around the disorder-induced topological phase transition, the real-space winding number has large fluctuations for different disordered samples, however, its average over an ensemble of disorder samples may well identify the topological phase transition. Besides, we show that our real-space winding number can be expressed as a Bott index, which has been used to represent the Chern number for two-dimensional systems.
Anderson localization is studied for two flavors of massless Dirac fermions in two-dimensional space perturbed by static disorder that is invariant under a chiral symmetry (chS) and a time-reversal symmetry (TRS) operation which, when squared, is equal either to plus or minus the identity. The former TRS (symmetry class BDI) can, for example, be realized when the Dirac fermions emerge from spinless fermions hopping on a two-dimensional lattice with a linear energy dispersion such as the honeycomb lattice (graphene) or the square lattice with $\ensuremath{\pi}$ flux per plaquette. The latter TRS is realized by the surface states of three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological band insulators in symmetry class CII. In the phase diagram parametrized by the disorder strengths, there is an infrared stable line of critical points for both symmetry classes BDI and CII. Here we discuss a ``global phase diagram'' in which disordered Dirac fermion systems in all three chiral symmetry classes, AIII, CII, and BDI, occur in four quadrants, sharing one corner which represents the clean Dirac fermion limit. This phase diagram also includes symmetry classes AII [e.g., appearing at the surface of a disordered three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological band insulator in the spin-orbit (symplectic) symmetry class] and D (e.g., the random bond Ising model in two dimensions) as boundaries separating regions of the phase diagram belonging to the three chS classes AIII, BDI, and CII. Moreover, we argue that physics of Anderson localization in the CII phase can be presented in terms of a non-linear-$\ensuremath{\sigma}$ model (NL$\ensuremath{\sigma}$M) with a Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NL$\ensuremath{\sigma}$M description of disordered fermionic models in all ten symmetry classes relevant to Anderson localization in two spatial dimensions.
The imposition of crystalline symmetries is known to lead to a rich variety of insulating and superconducting topological phases. These include higher-order topological phases and obstructed atomic limits with and without filling anomalies. We here comprehensively classify such topological crystalline phases (TCPs) with mirror, twofold rotation, and inversion symmetries in the presence of disorder that preserves the crystalline symmetry on average. We find that the inclusion of disorder leads to a simplification of the classification in comparison to the clean case. We also find that, while clean TCPs evade a general bulk-boundary principle, disordered TCPs admit a complete bulk-boundary correspondence, according to which (bulk) topological phases are topologically equivalent if and only if they have the same anomalous boundary states and filling anomaly. We corroborate the stability of disordered TCPs by way of field-theoretic, numerical, and symmetry-based analyses in various case studies. While the boundary signatures of most disordered TCPs are similar to their clean counterparts, the addition of disorder to certain mirror-symmetric TCPs results in novel higher-order statistical topological phases, in which zero-energy hinge states have critical wave-function statistics, while remaining protected from Anderson localization.
We study quantum phase transitions of three-dimensional disordered systems in the chiral classes (AIII and BDI) with and without weak topological indices. We show that the systems with a nontrivial weak topological index universally exhibit an emergent thermodynamic phase where wave functions are delocalized along one spatial direction but exponentially localized in the other two spatial directions, which we call the quasilocalized phase. Our extensive numerical study clarifies that the critical exponent of the Anderson transition between the metallic and quasilocalized phases, as well as that between the quasilocalized and localized phases, are different from that with no weak topological index, signaling the new universality classes induced by topology. The quasilocalized phase and concomitant topological Anderson transition manifest themselves in the anisotropic transport phenomena of disordered weak topological insulators and nodal-line semimetals, which exhibit the metallic behavior in one direction but the insulating behavior in the other directions.
We discuss the effects of disorder in time-reversal-invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in the symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field-theory analysis (the large-Nf{N}_{f}Nf​ expansion, where Nf{N}_{f}Nf​ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a nontrivial critical point separating a metallic phase and a Dirac semimetal. A similar-field theory approach is developed for disorder-driven transitions in symmetry classes AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the nonlinear sigma model supplemented with a topological term, which is a three-dimensional analog of the Pruisken term in the integer quantum Hall effect.
Topological phases of matter describe systems beyond the paradigm of spontaneous symmetry breaking, giving rise to unconventional phenomena with possible applications in topological quantum computing. Almost 40 years after their discovery, topological phases remain among the most active research fields of condensed matter physics. As evidenced by the recent discovery of higher order topological insulators (HOTIs), novel physics can emerge even in non-interacting systems. Theoretical investigations of strongly correlated systems are extremely difficult due to the exponential growth of the quantum-mechanical Hilbert space and frequently require the application of approximative methods such as model wave functions. Tensor network states (TNS) are a class of variational wave functions which allow an efficient encoding of relevant quantum many body states and which form the basis for very successful numerical algorithms. Projected entangled pair states (PEPS), a class of TNS in two and higher dimensions, permit the exact representation of many interacting topological phases with time reversal symmetry. However, the description of experimentally relevant chiral topological phases like the integer and fractional quantum Hall effects is much more subtle using the TNS framework. The first part of this thesis focuses on the description of chiral topological phases using PEPS. Firstly, we study a previously proposed chiral PEPS that was conjectured to possess anyonic excitations. By a careful analysis of its symmetries we are able to align important entanglement observables more closely with the expected universal behavior. This highlights that efficient PEPS can possess characteristic properties of chiral topological phases. Secondly, we focus on the issue that all known chiral PEPS have algebraic bulk correlation functions and therefore cannot be the ground states of gapped local Hamiltonians. Since this problem arises already for non-interacting chiral topological phases, we focus on two examples of such systems and show that their ground states can be represented exactly by efficient PEPS in a hybrid lattice with one momentum-space direction. After an inverse Fourier transform, the PEPS with only real-space coordinates requires an exponentially growing number of parameters for an exact representation. This provides a concrete illustration of the impossibility to encode gapped chiral phases exactly with efficient PEPS. Finally, we analyze a model wave function for a three-dimensional (3D) HOTI with strong intrinsic correlations using large-scale variational Monte Carlo simulations. This wave function is obtained by projection of two copies of a non-interacting HOTI with chiral hinge states. We characterize the gapless hinge states of the interacting system and show that they are of the same nature as the edge states of the 1/21/21/2 Laughlin state. Surprisingly, the gapped surfaces host a two-dimensional (2D) phase whose topological entanglement entropy is half of that of the 1/21/21/2 Laughlin state. Such a value cannot be obtained by any of the known 2D topological orders, showing a clear departure not only from the Laughlin 1/21/21/2 physics but from conventional 2D topological order. This demonstrates that 3D topological phases can host rich phenomena that are not yet fully understood.