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.
We consider the model of the directed polymer in a random medium of dimension
1+3, and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights ${w}_{L}(\stackrel{P\vec}{r})$ of the endpoint of the polymer of length
L via the moments ${Y}_{q}(L)={\ensuremath{\sum}}_{\stackrel{P\vec}{r}}\phantom{\rule{0.2em}{0ex}}{[{w}_{L}(\stackrel{P\vec}{r})]}^{q}$. We measure the generalized exponents $\ensuremath{\tau}(q)$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ governing the decay of the typical values ${Y}_{q}^{\mathrm{typ}}(L)={e}^{\overline{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{Y}_{q}(L)}}\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\tau}(q)}$ and disorder-averaged values $\overline{{Y}_{q}(L)}\ensuremath{\sim}{L}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)}$, respectively. To understand the difference between these exponents, $\ensuremath{\tau}(q)\ensuremath{\ne}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ above some threshold $q>{q}_{c}\ensuremath{\sim}2$, we compute the probability distributions of
y=Yq(L)∕Yqtyp(L) over the samples: We find that these distributions becomes scale invariant with a power-law tail
1∕y1+xq. These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent $\ensuremath{\nu}\ensuremath{\sim}2$.
We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to
L3=1503, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.
Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).
For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions
d=2,3, we study numerically the statistical properties of the renormalized on-site energies
ϵ and of the renormalized hoppings
V as a function of the linear size
L. We find that the renormalized on-site energies
ϵ remain finite in the localized phase in
d=2,3 and at criticality
(d=3), with a finite density at
ϵ=0 and a power-law decay
1/ϵ2 at large
∣ϵ∣. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and
u a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality
(d=3), the statistics of renormalized hoppings
V is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.
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=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.