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arXiv:1411.5992 · Physical Review B

Topology versus Anderson localization: Nonperturbative solutions in one dimension

Alexander Altland, Dmitry Bagrets, Alex Kamenev

Published Feb 27, 2015• 66 citations
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Abstract

We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters $(g,χ)$ representing localization and topological properties, respectively. Certain critical values of $χ$ (half-integer for $\Bbb{Z}$ classes, or zero for $\Bbb{Z}_2$ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In $\Bbb{Z}_2$ classes we uncover a hidden supersymmetry, present at the quantum critical point.

DOI
10.1103/physrevb.91.085429

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We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters (g,χ)(g,χ)(g,χ) representing localization and topological properties, respectively. Certain critical values of χχχ (half-integer for Z\Bbb{Z}Z classes, or zero for Z2\Bbb{Z}_2Z2​ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In Z2\Bbb{Z}_2Z2​ classes we uncover a hidden supersymmetry, present at the quantum critical point.
We present an analytic theory of quantum criticality in the quasi-one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters (ggg, $\ensuremath{\chi}$) representing localization and topological properties, respectively. Surfaces of half-integer valued $\ensuremath{\chi}$ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two-parameter flow describing the class AAA quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given. We check the quantitative validity of our theory by comparison to numerical transfer matrix computations.
A one-dimensional boundary of a two-dimensional topological superconductor can host a number of topologically protected chiral modes. Combining two topological superconductors with different topological indices, it is possible to achieve a situation when only a given number of channels (mmm) are topologically protected, while others are not and therefore are subject to Anderson localization in the presence of disorder. We study transport properties of such quasi-one-dimensional quantum wires with broken time-reversal and spin-rotational symmetries (class D) and calculate the average conductance, its variance and the third cumulant, as well as the average shot noise power. The results are obtained for arbitrary wire length, tracing a crossover from the diffusive Drude regime to the regime of strong localization where only mmm protected channels conduct. Our approach is based on the nonperturbative treatment of the nonlinear supersymmetric sigma model of symmetry class D with two replicas developed in our recent publication [D. S. Antonenko et al., Phys. Rev. B 102, 195152 (2020)]. The presence of topologically protected modes results in the appearance of a topological Wess-Zumino-Witten term in the sigma-model action, which leads to an additional subsidiary series of eigenstates of the transfer-matrix Hamiltonian. The developed formalism can be applied to study the interplay of Anderson localization and topological protection in quantum wires of other symmetry classes.
The topological phase transitions and localization properties in a 1D p-wave superconductor under Fibonacci quasi-periodic potential modulation are investigated in this work. By calculating the Z2 topological invariant, the topological phase diagram of the system is determined numerically. It is found that the system can transition from a topologically trivial phase to a topological Anderson superconductor phase through the Fibonacci quasi-periodic modulation. Moreover, under certain parameters, the system undergoes multiple topological Anderson superconductor phase transitions, accompanied by the emergence of zero-energy modes. However, in the case of strong disorder, the topological Anderson superconductor phase is destroyed, indicating that the topological Anderson superconductor phase can be induced only within a finite range of parameters. Furthermore, by calculating and analyzing the fractal dimension and the mean inverse participation ratio (MIPR) order parameter, the localization properties of the system are analyzed. The results show that regardless of how the disorder intensity increases, the fractal dimension values of most eigenstates always remain within a range of 0–1. Subsequently, the variations in the fractal dimensions of all eigenstates for different system sizes are studied. The results show that the fractal dimension values of most eigenstates are away from 0 and 1. These results indicate that the wavefunction in the bulk of the topological Anderson superconductor phase induced by Fibonacci quasi-periodic potential is a critical state wavefunction, with the system overall being in a critical phase. The stability of the critical phase is confirmed by scale behavior of MIPR as shown in figure. It differs from the traditional topological Anderson superconductor phase induced by random disorder or AA-type quasi-periodic disorder. The results provide new insights into and references for studying topological phase transitions and localization transitions in 1D p-wave superconductors.
Due to the absence of local order parameters, it is a challenging task to characterize the quantum topological phase transitions between topologically ordered phases in two dimensions. In this paper, we construct a topologically ordered tensor network wave function with one parameter $\ensuremath{\lambda}$, describing both the toric code state $(\ensuremath{\lambda}=1)$ and double semion state $(\ensuremath{\lambda}=\ensuremath{-}1)$. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\ensuremath{\lambda}$, and three different quantum critical points (QCPs) at $\ensuremath{\lambda}=0,\ifmmode\pm\else\textpm\fi{}1.73$ are identified. The first one separates the toric code phase and double semion phase, while latter two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry-breaking phase, respectively. When mapping the quantum tensor network wave function to the exactly solved statistical model, the norm of the wave function is identified as the partition function of the classical eight-vertex model, and both QCPs at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}1.73$ correspond to the eight-vertex model at the critical point $\ensuremath{\lambda}=\sqrt{3}$, while the QCP at $\ensuremath{\lambda}=0$ corresponds to the critical six-vertex model. Actually such a quantum-classical mapping cannot yield the complete low-energy excitations at these three QCPs. We further demonstrate that the full eigenvalue spectra of the transfer operators without/with the flux insertions can give rise to the complete quantum criticalities, which are described by the (2+0)-dimensional free boson conformal field theories (CFTs) compactified on a circle with the radius R=6R=\sqrt{6}R=6​ at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}\sqrt{3}$ and R=8/3R=\sqrt{8/3}R=8/3​ at $\ensuremath{\lambda}=0$. From the complete transfer operator spectra, the finite-size spectra of the CFTs for the critical eight-vertex model are obtained, and the topological sectors of anyonic excitations are yielded as well. Furthermore, for the QCP at $\ensuremath{\lambda}=0$, no anyon condensation occurs, but the emerged symmetries of the matrix product operators significantly enrich the topological sectors of the CFT spectra. Finally, we provide our understanding of the (2+0)-dimensional conformal quantum criticalities and their possible connection with the generic (2+1)-dimensional CFTs for quantum topological phase transitions.
The localization problem of electronic states in a two-dimensional quantum spin Hall system (that is, a symplectic ensemble with topological term) is studied by the transfer matrix method. The phase diagram in the plane of energy and disorder strength is exposed, and demonstrates "levitation" and "pair annihilation" of the domains of extended states analogous to that of the integer quantum Hall system. The critical exponent nu for the divergence of the localization length is estimated as nu congruent with 1.6, which is distinct from both exponents pertaining to the conventional symplectic and the unitary quantum Hall systems. Our analysis strongly suggests a different universality class related to the topology of the pertinent system.
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.