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arXiv:2108.12133 · Physical review. B./Physical review. B

Multicriticality of two-dimensional class-D disordered topological superconductors

Tong Wang, Zhiming Pan, Tomi Ohtsuki, Ilya A. Gruzberg, Ryuichi Shindou

Published Nov 1, 2021• 15 citations
arXivHTMLPDF

Abstract

A generic two-dimensional disordered topological superconductor in symmetry class D exhibits rich phenomenology and multiple phases: diffusive thermal metal (DTM), Anderson insulator (AI), and thermal quantum Hall (TQH) phase (a topological superconductor). We numerically investigate the phase diagram of a lattice model of such class-D superconductor, specifically focusing on transitions between the phases and the associated universal critical behaviors. We confirm the existence of a tricritical point and its repulsive nature at the point on the phase diagram where the three phases meet. We characterize the critical behaviors at various critical points and the tricritical point using numerical evaluation of the localization length, the conductance (or conductivity), and the density of states. We conclude that the two metal-insulator transitions (DTM-TQH and DTM-AI) belong to the same universality class, whereas the tricritical point represents a distinct universality class.

DOI
10.1103/physrevb.104.184201

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A generic two-dimensional disordered topological superconductor in symmetry class D exhibits rich phenomenology and multiple phases: diffusive thermal metal (DTM), Anderson insulator (AI), and thermal quantum Hall (TQH) phase (a topological superconductor). We numerically investigate the phase diagram of a lattice model of such class-D superconductor, specifically focusing on transitions between the phases and the associated universal critical behaviors. We confirm the existence of a tricritical point and its repulsive nature at the point on the phase diagram where the three phases meet. We characterize the critical behaviors at various critical points and the tricritical point using numerical evaluation of the localization length, the conductance (or conductivity), and the density of states. We conclude that the two metal-insulator transitions (DTM-TQH and DTM-AI) belong to the same universality class, whereas the tricritical point represents a distinct universality class.
Two-dimensional (2D) disordered superconductor (SC) in class D exhibits a disorder-induced quantum multicritical phenomenon among diffusive thermal metal (DTM), topological superconductor (TS), and conventional localized (AI) phases. To characterize the quantum tricritical point where these three phases meet, we carry out a two-loop renormalization group (RG) analysis for 2D Dirac fermion with random mass in terms of the $\ensuremath{\epsilon}$-expansion in the spatial dimension $d=2\ensuremath{-}\ensuremath{\epsilon}$. In two dimensions $(\ensuremath{\epsilon}=0)$, the random mass is marginally irrelevant around a clean-limit fixed point of the gapless Dirac fermion, while there exists an IR unstable fixed point at finite disorder strength that corresponds to the tricritical point. The critical exponent, dynamical exponent, and scaling dimension of the (uniform) mass term are evaluated around the tricritical point by the two-loop RG analysis. Using a mapping between an effective theory for the 2D random-mass Dirac fermion and the (1+1)(1+1)(1+1)-dimensional Gross-Neveu model, we further deduce the four-loop evaluation of the critical exponent, and the scaling dimension of the uniform mass around the tricritical point. Both the two-loop and four-loop results suggest that criticalities of a AI-DTM transition line as well as TS-DTM transition line are controlled by other saddle-point fixed point(s) at finite uniform mass.
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Abstract The interplay between disorder, topology, and many-body interactions in two-dimensional superconducting systems continues to reveal unexpected critical phenomena beyond the reach of conventional universality theory. We propose and characterize a novel universality class — the Discrete Scale Invariance Multicritical Point (DSI-MCP) — emerging from Efimov-type discrete scale invariance at the tricritical point of a two-dimensional disordered Class D topological superconductor. Using a two-loop renormalization group analysis within the replica framework, we demonstrate that when disorder and three-body couplings simultaneously exceed their critical thresholds, the renormalization group flow bypasses the conventional tricritical point and spirals into a new stable fixed point with complex eigenvalues. The imaginary component of these eigenvalues — absent at any conventional critical point with real scaling exponents — encodes log-periodic scaling and generates an infinite geometric tower of bound states, establishing the condensed matter analogue of the atomic Efimov effect. The DSI-MCP is distinguished from the tricritical point of Pan et al. by three independent criteria: a localization exponent that is fixed by symmetry and independent of the dimensional regularization parameter, a two-parameter multicritical threshold requiring simultaneous supercriticality in both disorder and three-body coupling, and the presence of log-periodic oscillations in the correlation length that are strictly forbidden at any conventional critical point. We further identify a competition between the DSI mechanism and Quantum Griffiths Singularity physics, governed by the statistical character of the disorder distribution, and construct a unified phase diagram delineating three experimentally distinguishable regimes: activated scaling, log-periodic power-law modulation, and a coexistence corridor exhibiting a doubled oscillation frequency with no precedent in the existing literature on disordered topological systems. All predictions are parameter-free once the operating point in coupling space is specified, and are directly accessible to STM spectroscopy, mesoscopic transport, and cold-atom platforms simulating Class D symmetry. These findings establish the DSI-MCP as a new entry in the classification of critical phenomena in disordered topological matter, bridging Efimov physics, quantum Griffiths singularities, and multicritical disorder-driven phase transitions into a unified theoretical framework.
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.
The quantum phase transition between two clean, noninteracting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point is at the heart of the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase. In the global phase diagram, the disorder controlled fixed point where superuniversality is lost, serves as a multicritical point, where the delocalized diffusive and two topologically distinct localized phases meet and the nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features, we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. We also establish a similar structure of the phase diagram and the superuniversality for weak disorder in higher spatial dimensions. By noting that 1/r21/{r}^{2}1/r2 power-law correlated disorder acts as a marginal perturbation for massless Dirac fermions in any spatial dimension ddd, we have established a general renormalization group framework for addressing disorder driven critical phenomena for fixed spatial dimension d>2d>2d>2.
Topological phase transitions can be described by the theory of critical phenomena and identified by critical exponents that define their universality classes. This is a consequence of the existence of a diverging length at the transition that has been identified as the penetration depth of the surface modes in the nontrivial topological phase. In this paper we characterize different universality classes of topological transitions by determining their correlation length exponents directly from numerical calculations of the penetration length of the edge modes as a function of the distance to the topological transition. We consider generalizations of the topological nontrivial Su-Schriefer-Heeger (SSH) model, for the case of next next nearest neighbors hopping terms and in the presence of a synthetic potential. The latter allows the system to transit between two universality classes with different correlation length and dynamic critical exponents. It presents a line of multicritical point in its phase diagram since the behavior of the Berry connection depends on the path it is approached. We compare our results with those obtained from a scaling approach to the Berry connection.