Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

Castillo, Horacio E.; Chamon, Claudio; Fradkin, Eduardo; Goldbart, Paul M.; Mudry, Christopher

doi:10.1103/physrevb.56.10668

Cited by 116 publications

(204 citation statements)

References 30 publications

Supporting

Mentioning

189

Contrasting

Order By: Relevance

“…In fact, the multifractal property of this wave function has been revealed quantitatively by a close analogy to a generalized random energy model [11,12]. As the disorder strength g varied, the multifractal spectrum exhibits a sharp transition which is similar to the freezing phenomenon in spin glasses.…”

Section: 15rn 7510nrmentioning

confidence: 87%

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

Ryu

Hatsugai

2001

Phys. Rev. B

View full text Add to dashboard Cite

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 power law in the weak randomness with the 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. The analytic expression of the correlation function and the free energy are also discussed with the replica symmetry breaking and the Liouville field theory.72.15. Rn, 75.10.Nr Although the scaling theory of localization for twodimensional disordered systems generally predicts the absence of extended states, we have some examples of nonlocalized states in two dimensions which are marginally allowed to appear. Among them, systems with chiral symmetry such as the Anderson bond-disordered model [1], the random flux model [2][3][4] or the π-flux model with link disorders [5,6] have attracted lots of attention. Density of states of the models have singularities at the zero energy and the corresponding wave functions exhibit multifractal behavior. Actually, one generally expects the existence of zero energy states for these models. Consider a Hamiltonian of the above models with chiral symmetry H = c † i t ij c j + h.c. on a bipartite lattice Λ which can be decomposed into two sublattices Λ A and Λ B . After performing a unitary transformation (redefinition of indices), the Hamiltonian is expressed asThe off-diagonal block structure of H implements the fact that hopping is restricted between the interpenetrating sublattices Λ A and Λ B . The zero mode wave functions ψ = t (ψ A , ψ B ) satisfy the Schrödinger equations

show abstract

Section: 15rn 7510nrmentioning

confidence: 87%

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

Ryu

Hatsugai

2001

Phys. Rev. B

View full text Add to dashboard Cite

show abstract

“…Subsequent work 41,42 on the random vector potential model elucidated the mechanisms of termination and freezing, transitions that occur in the spectral statistics for large moments q > q c (∆ A ) or strong disorder ∆ A ≥ 2π. For this broken-T class, we can also examine the spin LDOS fluctuations, utilizing the same nonperturbative bosonization treatment employed in Ref.…”

Section: B Resultsmentioning

confidence: 99%

“…Outside of this range, the τ (q) associated to a fixed disorder realization is linear, a phenomenon known as spectral termination. [41][42][43] [This assumes that higher order corrections can be ignored for q ≥ q c . Regardless, the τ (q) spectrum is always linear for sufficiently large q].…”

Section: A Definitionsmentioning

confidence: 99%

Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder

Foster

2012

Phys. Rev. B

View full text Add to dashboard Cite

We compute the multifractal spectra associated to local density of states (LDOS) fluctuations due to weak quenched disorder, for a single Dirac fermion in two spatial dimensions. Our results are relevant to the surfaces of Z2 topological insulators such as Bi2Se3 and Bi2Te3, where LDOS modulations can be directly probed via scanning tunneling microscopy. We find a qualitative difference in spectra obtained for magnetic versus non-magnetic disorder. Randomly polarized magnetic impurities induce quadratic multifractality at first order in the impurity density; by contrast, no operator exhibits multifractal scaling at this order for a non-magnetic impurity profile. For the time-reversal invariant case, we compute the first non-trivial multifractal correction, which appears at two loops (impurity density squared). We discuss spectral enhancement approaching the Dirac point due to renormalization, and we survey known results for the opposite limit of strong disorder.

show abstract

“…Furthermore, we propose a new method for investigating analytical properties of thermodynamical quantities of finite-dimensional systems. There is another, perhaps more fundamental motivation for the present investigation: it was obtained resently that REM at complex temperatures is closely connected with strings [4] [5].In disordered systems there are averaging with the canonical Gibbs distribution at given temperature, and averaging by frozen disorder. In REM and related models energy levels are considered as random quantities, and the distribution is derived from a concrete "microscopic" Hamiltonian [1] [2].…”

mentioning

confidence: 99%

Random energy model at complex temperatures

Saakian

2000

Phys. Rev. E

View full text Add to dashboard Cite

The complete phase diagram of Random Energy Model (REM) is obtained for complex temperatures using the method proposed by Derrida. We find the density of zeroes for statistical sum. Then the method is applied to Generalized Random Energy Model (GREM). This allows us to propose new analytical method for investigating zeroes of statistical sum for finite-dimensional systems.Random Energy Model (REM) [1] is one of the most popular contemporary models of statistical mechanics. Besides its direct applications in spin-glasses the model has a wide range of employments in very different areas of the modern theoretical physics and biophysics. Recently, several new, and more intimate applications have been founded in the theories of well-developed turbulence [2], and strings [3][4][5]. On the other hand, it is well-known that the model allows to consider the fundamental problems of information theory in the language of statistical mechanics. In particular, problems connected with transmission of information through a noisy channel can be formulated and effectively solved in the language of REM [6,7]. This astonishing range of different applications cannot be accidental one: It seems, that REM is a paradigm for a complex system physics, or at least contains all essential ingredients of that physics. In other words, the nature sings his sacral songs using REM's language.The thermodynamical structure of REM has simple structure but at the same contains all essential ingradients of phase transitions [1,11,12]: There is well defined critical temperature T c which can be obtained from clear and semi-intuitive physical reasons. For T < T c the system is frozen in the spin-glass phase with one level of replica symmetry breaking. Besides this rough thermodynamical phase transition there are some "mild", mesoscopic phase transitions in the domain (T , T c ),T > T c [1,12]. The temperatureT strongly depends from the concrete form of energy's distribution; it can be infinite for the gaussian distribution [1] but it has some finite value for REM generated by more physical dilute hamiltonians [12]. This rich and interesting spectra of phase transitions waits its complete and exhaustive investigation, and we hope that the present work will shed some light to this question also.It is well-known that structure and properties of phase transitions can be investigated through analytical properties of thermodynamical quantities in the complex plane of temperature or/and magnetic field [18] [17] [16]. Indeed, if a phase transition is assosiated with singular behavior of thermodynamic potential in the thermodynamic limit (in our case it is free energy or statistical sum), then by consideration of its analytical properties in the complex plane the important physical information can be obtained. The method was proposed by Yang and Lee [14] (for the case of complex magnetic fields), and Fisher [15] (for the case of complex temperatures). It has large variety of different applications in statistical physics; furthermore it is one one of the most ex...

show abstract

Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

Cited by 116 publications

References 30 publications

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

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

Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder

Random energy model at complex temperatures

Contact Info

Product

Resources

About