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arXiv:cond-mat/0012218 · Physical review. B, Condensed matter

Numerical replica limit for the density correlation of the random Dirac fermion

Shinsei Ryu, Yasuhiro Hatsugai

Published May 31, 2001• 10 citations
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Abstract

The zero-mode wave function of a massless Dirac fermion in the presence of a random gauge field is studied. The density correlation function is calculated numerically, and found to exhibit a power law in the weak randomness with a disorder-dependent exponent. It deviates from the power law, and the disorder dependence becomes frozen in the strong randomness. A classical statistical system is employed through the replica trick to interpret the results, and the direct evaluation of the replica limit is demonstrated numerically. Analytical expressions of the correlation function and the free energy are also discussed with the replica symmetry breaking and the Liouville field theory.

DOI
10.1103/physrevb.63.233307

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The zero-mode wave function of a massless Dirac fermion in the presence of a random gauge field is studied. The density correlation function is calculated numerically, and found to exhibit a power law in the weak randomness with a disorder-dependent exponent. It deviates from the power law, and the disorder dependence becomes frozen in the strong randomness. A classical statistical system is employed through the replica trick to interpret the results, and the direct evaluation of the replica limit is demonstrated numerically. Analytical expressions of the correlation function and the free energy are also discussed with the replica symmetry breaking and the Liouville field theory.
We study a Dirac fermion model with a random vector field, especially paying attention to the strong disorder regime. Applying Bosonization techniques, we first derive an equivalent sine-Gordon model, and next average over the random vector field using the replica approach. The operator product expansion based on the replica action leads to scaling equations of the coupling constants (``fugacities'') with nonlinear terms, if we take into account the fusion of the vertex operators. These equations are converted into a nonlinear diffusion equation known as the Kolmogorov-Petrovsky-Piscounov (KPP) equation. Using the asymptotic solution of the equation, we calculate the spatial correlations of the generalized inverse participation ratios. The scaling exponent thus obtained reproduces the recent numerical calculations of the density correlation function. This implies that the freezing transition has actually revealed itself in such calculations.
Random-bond Hamiltonians of the $\ensuremath{\pi}$ flux state on the square lattice are investigated. It has a special symmetry and all states are paired except the ones with zero energy. Because of this, there are always zero modes. The states near E=0E=0E=0 are described by massless Dirac fermions. For the zero mode, we can construct a random lattice fermion without a doubling and quite large systems (up to $801\ifmmode\times\else\texttimes\fi{}801)$ are treated numerically. We clearly demonstrate that the zero mode is given by a critical wave function. Its multifractal behavior is also compared with the effective field theory.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
We obtain the disorder averaged (critical) four-point correlation functions for N species of (two-dimensional Euclidean) Dirac fermions subject to a (Gaussian) random su(N) gauge field. The replica approach and the strong disorder approach yield identical results, as one might expect in the light of recent developments in the N=2 case [1]. We establish a connection with `dressed' Z_N twist fields in the c=-2 logarithmic conformal field theory (LCFT), thereby extending the recent connection to `dressed' Z_2 twists [1,2]. We draw attention to the fact that `dressed' Z_2 twist operators have previously appeared in the description of half flux-quantum quasiholes in the bulk excitations of the Haldane--Rezayi Quantum Hall state [3].
Nonperturbative results are obtained for multipoint correlation functions of the model of (2+1)-dimensional relativistic fermions in a random non-Abelian gauge potential. At zero frequencies the model is a critical theory with power law behavior of correlation functions. The average distance between the energy levels scales as ${\mathit{L}}^{\mathrm{\ensuremath{-}}2+1/\mathit{N}2}$, where L is the system's size and N is the number of components of the fermionic field. We calculate the diffusion propagator and show that the dc conductivity for this model is finite.
We analytically study interacting Dirac fermions, described by the Thirring model, under weak local particle number measurements with monitoring rate $\ensuremath{\gamma}$. This system maps to a bosonic replica field theory, analyzed via the renormalization group. For a nonzero attractive interaction, a phase transition occurs at a critical measurement strength ${\ensuremath{\gamma}}_{c}$. When $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, the system enters a localized phase characterized by exponentially decaying density-density correlations beyond a finite correlation length; for $\ensuremath{\gamma}<{\ensuremath{\gamma}}_{c}$, the correlations decay algebraically. The transition is of Berezinskii-Kosterlitz-Thouless (BKT) type, reflected by a characteristic scaling of the correlation length. In the noninteracting limit, ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ shifts to zero, reducing the algebraic phase to a single point in parameter space. This identifies weak measurements in the free case as an implicit double fine tuning to the critical end point of the BKT phase transition. Along the noninteracting line, we compute the entanglement entropy from density-density correlation functions and find no entanglement transition at nonzero measurement strength in the thermodynamic limit.