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arXiv:cond-mat/9907502

Disorder Induced Phase Transition for Random 2D Dirac Fermions

Victor Gurarie

Published Jul 30, 1999• 0 citations
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Abstract

We present evidence that two dimensional Dirac fermions in the presence of random Abelian gauge potential exhibit a phase transition when the disorder strength exceeds a certain critical value. We argue that this phase transition has novel properties unique to disordered systems. It resembles in many ways the transition from dilute to dense polymers phase in two dimension. In particular, we argue that the central charge of the disordered Dirac fermions, being $c=0$ before the transition, changes to $c=-2$. We discuss possible implications to quantum Hall transitions, in view of recently proposed model for quantum Hall transitions with $c=-2$.

DOI
10.48550/arxiv.cond-mat/9907502

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We present evidence that two dimensional Dirac fermions in the presence of random Abelian gauge potential exhibit a phase transition when the disorder strength exceeds a certain critical value. We argue that this phase transition has novel properties unique to disordered systems. It resembles in many ways the transition from dilute to dense polymers phase in two dimension. In particular, we argue that the central charge of the disordered Dirac fermions, being c=0c=0c=0 before the transition, changes to c=−2c=-2c=−2. We discuss possible implications to quantum Hall transitions, in view of recently proposed model for quantum Hall transitions with c=−2c=-2c=−2.
Recently, in an attempt to study disordered criticality in quantum Hall systems and ddd-wave superconductivity, it was found that two-dimensional random Dirac-fermion systems contain a line of critical points that is connected to the pure system. We use bosonization and current algebra to study properties of the critical line and calculate the exact scaling dimensions of all local operators. We find that the critical line contains an infinite number of relevant operators with negative scaling dimensions.
Metallic phases have been observed in several disordered two-dimensional (2D) systems, including thin films near superconductor-insulator transitions and quantum Hall systems near plateau transitions. The existence of 2D metallic phases at zero temperature generally requires an interplay of disorder and interaction effects. Consequently, experimental observations of 2D metallic behavior have largely defied explanation. We formulate a general stability criterion for strongly interacting, massless Dirac fermions against disorder, which describe metallic ground states with vanishing density of states. We show that (2+1)-dimensional quantum electrodynamics (QED3{\mathrm{QED}}_{3}QED3​) with a large, even number of fermion flavors remains metallic in the presence of weak scalar potential disorder due to the dynamic screening of disorder by gauge fluctuations. We also show that QED3{\mathrm{QED}}_{3}QED3​ with weak mass disorder exhibits a stable, dirty metallic phase in which both interactions and disorder play important roles.
We calculate nonperturbatively the multifractal scaling exponents of the critical wave function for two dimensional Dirac fermions in the presence of a random magnetic field. We do so by arguing that the multifractal scaling exponents can be expressed in terms of the free energy of random directed polymers on a Cayley tree. We find a weak-strong disorder transition for the multifractal scaling exponents of the wave function that is parallel to the freezing or glassy transition of the random polymer model.
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the random Ising phase transition. The values are consistent with recent estimates obtained from high-temperature series.
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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