[Paper Review] Wegner model in high dimension: U(1) symmetry breaking and a non-standard phase of disordered electronic matter, I. One-replica theory

Martin R. Zirnbauer|arXiv (Cornell University)|Sep 29, 2023

Physics of Superconductivity and MagnetismPhysics and Astronomy3 citations

TL;DR

This paper proposes a novel field theory for the Anderson transition in high dimensions ($d \geq 3$) by identifying spontaneous breaking of U(1) symmetry in the Wegner $N$-orbital model, which splits the nonlinear $\sigma$-model coupling into two independent parameters. This leads to a two-parameter scaling scenario and reveals a new, third phase of disordered electronic matter—distinct from metal and insulator—characterized by fractal eigenstates and singular continuous spectra, with a non-perturbative fixed point stabilized by anisotropic stiffnesses $\lambda_\sigma \gg \lambda_\tau$. The theory is derived via bosonization at strong coupling, replacing the standard weak-coupling $\sigma$-model approach.

ABSTRACT

The Anderson transition between localized and metallic states is traditionally analyzed by assuming a one-parameter scaling hypothesis. Although that hypothesis has been confirmed near two dimensions by epsilon = d-2 expansion of the Wegner-Efetov nonlinear sigma model, there exists mounting evidence that the transition in d=3 or higher may have a second branch and that two relevant parameters are needed in order to describe the universal behavior at criticality. Doubt of the standard hypothesis also comes from field theory. Indeed, increasing the space dimension pushes the Anderson transition towards strong disorder, where a strong-coupling approach very different from the usual weak-coupling analysis of the sigma model is called for. In the present work, we develop a novel field theory of Anderson transitions at strong coupling based on the key observation that the U(1) symmetry which distinguishes retarded from advanced fields may undergo spontaneous symmetry breaking. That symmetry breakdown splits the sigma model coupling into two, thus leading to a natural scenario of two-parameter scaling. While we develop the field theory from the concrete starting point of the Wegner N-orbital model, we believe our results to be of much wider applicability. The first of a series, the present paper offers a pedagogical introduction to the main ideas in the setting of the one-replica theory. Subsequent papers will employ the self-consistent approximation of Abou-Chacra et al. and develop the full supersymmetric theory. The latter establishes the existence of a new renormalization-group fixed point, whose basin of attraction constitutes a third phase of disordered electronic matter.

Motivation & Objective

• To address the limitations of the one-parameter scaling hypothesis in high-dimensional Anderson transitions, especially in the strong disorder regime.
• To develop a field-theoretic framework beyond the weak-coupling $\sigma$-model for systems where the standard Hubbard-Stratonovich approach fails.
• To establish a mechanism for two-parameter scaling through spontaneous U(1) symmetry breaking in the nonlinear $\sigma$-model.
• To identify a new, third phase of disordered electronic matter with non-ergodic extended states and fractal eigenfunctions.
• To lay the foundation for a supersymmetric extension that reveals a non-perturbative fixed point with distinct critical behavior.

Proposed method

• Introduce a bosonized field theory dual to the standard Hubbard-Stratonovich field, enabling control in the strong-coupling regime.
• Apply the one-replica theory as a pedagogical framework to derive the field-theoretic structure before extending to the supersymmetric limit.
• Identify two distinct stiffness parameters: $\lambda_\sigma$ (transverse) and $\lambda_\tau$ (longitudinal), arising from anisotropic fluctuations on a degenerate target manifold.
• Use renormalization group (RG) analysis to track flow of $\lambda_\sigma$ and $\lambda_\tau$, identifying fixed points corresponding to metal, insulator, and a new critical phase.
• Construct a reduced theory by integrating out the $\lambda_\sigma \to \infty$ limit, yielding a principal chiral model dual to the integer quantum Hall CFT.
• Leverage superbosonization techniques for $N \geq 2$ orbitals to generalize the one-replica result to the full supersymmetric theory in subsequent work.

Experimental results

Research questions

• RQ1 Can the one-parameter scaling hypothesis for the Anderson transition in $d \geq 3$ be invalidated by evidence of two relevant parameters?
• RQ2 Does spontaneous U(1) symmetry breaking in the nonlinear $\sigma$-model lead to a natural two-parameter scaling scenario?
• RQ3 Can a new, third phase of disordered electronic matter exist with fractal eigenstates and singular continuous spectra?
• RQ4 Is there a non-perturbative RG fixed point in the strong-coupling regime that differs from the standard metal and insulator fixed points?
• RQ5 Can the principal chiral model emerge as the effective theory in the $\lambda_\sigma \to \infty$ limit, and does it support flow to $\lambda_\tau = 0$ via topological excitations?

Key findings

• The U(1) symmetry distinguishing retarded and advanced fields undergoes spontaneous breaking in the strong-coupling regime, leading to two independent couplings $\lambda_\sigma$ and $\lambda_\tau$.
• The theory exhibits a third RG-fixed point where $\lambda_\sigma \to \infty$ and $\lambda_\tau \to 0$, distinct from the metal and insulator fixed points.
• This new fixed point is totally attractive and corresponds to a phase with fractal eigenstates and singular continuous energy spectrum.
• The critical length $\xi_2$ diverges with exponent $\nu_2 = \nu_D + \nu_E$, e.g., $\nu_2 = 1$ if $\nu_D = \nu_E = 1/2$, indicating power-law criticality.
• In the limit $\lambda_\sigma \to \infty$, the transverse degrees of freedom integrate out exactly, yielding a principal chiral model, which matches the CFT proposed for the integer quantum Hall effect.
• The theory resolves the conflict between the nonlinear $\sigma$-model and conformal symmetry by correcting the field-theory framework rather than violating conformal invariance.