Simulational and theoretical studies of the Anderson transition in the chiral symmetry classes with weak topology

Technical Summary: Simulational and Theoretical Studies of the Anderson Transition in Chiral Symmetry Classes with Weak Topology

Problem Statement
The paper investigates the nature of the Anderson transition (disorder-driven metal-insulator transition) within the three chiral symmetry classes of the Altland-Zirnbauer (AZ) classification: chiral unitary (AIII), chiral orthogonal (BDI), and chiral symplectic (CII). Specifically, the study focuses on systems possessing a one-dimensional (1D) weak band topology. Previous research has established that in chiral symmetry classes, the 1D weak topology can induce an intermediate "quasi-localized" (QL) phase between the metallic and Anderson insulator phases, where the localization length diverges exclusively along the topological direction. While this phenomenon was numerically confirmed in the chiral unitary and orthogonal classes, its existence and critical properties in the chiral symplectic class (CII) remained unexplored. Furthermore, there is a theoretical discrepancy regarding the nature of the strong-coupling fixed point in these systems: previous renormalization group (RG) analyses suggested a stable QL phase, whereas the current work aims to re-evaluate this by incorporating previously neglected terms.

Methodology
The authors employ a dual approach combining large-scale lattice model simulations with field-theoretic renormalization group (RG) analysis.

1. Numerical Simulations:

   • Models: Tight-binding models for 3D and 2D lattices in the chiral symplectic class (CII) were constructed, both with and without 1D weak topology. The models utilize non-Hermitian off-diagonal blocks to enforce chiral symmetry and specific hopping parameters to introduce non-reciprocity along a specific direction, generating the weak topological index.
   • Transfer Matrix Method: The authors calculated the Lyapunov exponents (LEs) of the non-Hermitian transfer matrices to determine the localization lengths ($\xi$) along different spatial directions.
   • Finite-Size Scaling (FSS): A standard FSS analysis was performed assuming spatially isotropic scaling. The normalized localization length ($\Lambda = \xi/L$) was analyzed to extract critical disorder strengths ($W_c$) and critical exponents ($\nu$). The analysis distinguished between transitions along the topological direction and non-topological directions.

2. Theoretical Analysis:

   • Field Theory: The authors utilized nonlinear sigma models (NLSMs) for the chiral symmetry classes, mapping them to dual sine-Gordon (sG) models to describe vortex excitations.
   • Renormalization Group (RG): A one-loop RG analysis was conducted for the 2D sG models. Crucially, this analysis newly incorporates the one-loop renormalization of the weak topological term ($\chi_y$), which was omitted in previous studies (e.g., Ref. [1]). The authors derived RG flow equations for conductivities ($\sigma_x, \sigma_y$), Gade constants, vortex fugacity, and the topological term for all three chiral classes.

Key Contributions and Results

• Numerical Findings in Chiral Symplectic Class (CII):

  • Without Topology: The critical exponent for the 3D Anderson transition in the CII class without weak topology was determined to be $\nu \approx 0.878$. This value aligns with previous results for non-Hermitian class AII models but differs significantly from the chiral unitary ($\nu \approx 1.06$) and orthogonal ($\nu \approx 1.12$) classes, highlighting the distinct role of Kramers time-reversal symmetry.
  • With 1D Weak Topology: The simulations confirm the emergence of an intermediate quasi-localized (QL) phase in both 3D and 2D CII models. In this phase, the localization length diverges along the topological direction while remaining finite along non-topological directions.
  • Critical Exponents: The metal-to-QL transition in the 3D CII class exhibits a critical exponent $\nu \approx 0.66$, distinct from the $\nu \approx 0.82$ observed in the AIII and BDI classes. In 2D, the metal-to-QL transition yields $\nu \approx 1.45$, differing from the $\nu \approx 2.0$ of the standard 2D Anderson transition.

• Theoretical Revisions (RG Analysis):

  • Instability of the QL Fixed Point: The revised RG analysis, which includes the one-loop renormalization of the weak topological term, reveals that the strong-coupling fixed point previously identified as a "quasi-localized" phase (characterized by $\sigma_y \neq 0, \sigma_x = 0$) is actually unstable.
  • Conventional Localization: Instead, the strong-coupling phase is governed by a stable fixed point where conductivities vanish in both spatial directions ($\sigma_x = \sigma_y = 0$), indicating a conventional Anderson insulator rather than a QL phase.
  • Anisotropic Scaling: Despite the absence of a stable QL fixed point, the RG analysis confirms that the 1D weak topology fundamentally alters the criticality. The saddle fixed point governing the metal-insulator transition is characterized by spatially anisotropic scaling, specifically a vanishing conductivity ratio $\sigma_x/\sigma_y \to 0$. This implies that the critical point requires distinct scaling dimensions for the topological and non-topological directions.

Significance and Claims
The paper claims that the interplay between disorder and 1D weak topology in chiral symmetry classes is governed by the specific nature of time-reversal symmetry, leading to distinct universality classes for the metal-to-QL transition across the AIII, BDI, and CII classes.

A central claim of the work is the reconciliation of numerical and theoretical results. The authors posit that the "quasi-localized" regime observed numerically in 2D models is likely an artifact arising from the assumption of spatially isotropic scaling in the Finite-Size Scaling analysis. Because the true critical point exhibits spatially anisotropic scaling (as revealed by the RG analysis), the standard isotropic FSS ansatz may incorrectly identify a finite QL regime. The paper concludes that a conclusive numerical identification of the QL phase requires a finite-size scaling framework capable of accommodating generic anisotropic spatial scaling. The work thus refines the theoretical understanding of disorder-driven topological phase transitions, correcting previous interpretations of the strong-coupling fixed point while confirming the robustness of the anisotropic critical behavior induced by weak topology.