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arXiv:cond-mat/0201305 · Physical review. B, Condensed matter

Two-dimensional Dirac fermions with a random axial vector potential

Takahiro Fukui, 勇人 江村, Hiroki Yamada

Published Mar 20, 2002• 0 citations
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Abstract

A Dirac fermion model with random axial vector potential is proposed. At a special strength of randomness, the symmetry of the action is enhanced, which is due to the gauge symmetry in the manner of Nishimori. Some exact scaling exponents of single-particle Green functions are computed. The relationship with the $\mathrm{XY}$ gauge glass model is discussed.

DOI
10.1103/physrevb.65.153301

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A Dirac fermion model with random axial vector potential is proposed. At a special strength of randomness, the symmetry of the action is enhanced, which is due to the gauge symmetry in the manner of Nishimori. Some exact scaling exponents of single-particle Green functions are computed. The relationship with the XY\mathrm{XY}XY gauge glass model is discussed.
Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function, which has only poles on the lower half plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods. \textcopyright{} 1996 The American Physical Society.
We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wave functions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavors, are obtained by deriving from conformal field theory an effective Gaussian model for the wave function amplitudes and mapping it to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wave functions remain delocalized in the presence of strong disorder.
Two-dimensional random Dirac fermions are studied numerically. They are realized on a square lattice by the $\ensuremath{\pi}$-flux model with random hopping. It preserves a symmetry denoted by ${H,\ensuremath{\gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ in an effective field theory. Although it belongs to the orthogonal ensemble, the zero-energy states do not localize but become critical. The density of states vanishes as $\ensuremath{\sim}{E}^{\ensuremath{\alpha}}$ and the exponent $\ensuremath{\alpha}$ changes with strength of the randomness (the critical line). Rapid enhancement of the Thouless number is observed near the zero energy. The level-spacing distribution is also investigated, which is consistent with the existence of the critical states at E=0E\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0E=0.
We construct an exactly solvable Hamiltonian lattice model realizing a 1+1d Dirac fermion, with exact microscopic vector and axial vector U(1)U(1)U(1) symmetries. The mixed anomaly between these is accommodated by the not-on-site action of the symmetries. Our Hilbert space is a Z2Z_2Z2​-graded tensor product of local Z2Z_2Z2​-graded Hilbert spaces which include infinite dimensional rotor degrees of freedom. The Hamiltonian becomes manifestly exactly solvable after a locality-preserving unitary mapping to an equivalent fermionic Villain Hilbert space. Our construction also allows an exactly solvable realization of interacting fermionic Luttinger liquids. At the free Dirac fixed point, our Hamiltonian contains irrelevant interactions, which we compute to leading order.
Three-dimensional quantum electrodynamics exhibits a number of interesting properties, such as dynamical chiral symmetry breaking, weak confinement, and non-Fermi-liquid behavior, and also has wide applications in condensed-matter physics. We study the effects of random potentials, which exist in almost all realistic condensed-matter systems, on the low-energy behaviors of massless Dirac fermions by means of renormalization-group method, and show that the role of random mass is significantly enhanced by the gauge interaction, whereas random scalar and vector potentials are insusceptible to the gauge interaction at the one-loop order. The static random potential breaks the Lorentz invariance, and as such induces unusual renormalization of fermion velocity. We then consider the case in which three types of random potentials coexist in the system. The random scalar potential is found to play a dominant role in the low-energy region, and drives the system to undergo a quantum phase transition.
We argue that massless Dirac particles in two spatial dimensions with 1∕r1∕r1∕r Coulomb repulsion and quenched random gauge field are described by a manifold of fixed points which can be accessed perturbatively in disorder and interaction strength, thereby confirming and extending the results of Herbut, Juricic, and Vafek [arXiv:0707.4171 (unpublished)]. At small interaction and small randomness, there is an infrared stable fixed curve which merges with the strongly interacting infrared unstable line at a critical endpoint, along which the dynamical critical exponent z=1z=1z=1.