Logarithmic scaling of Lyapunov exponents in disordered chiral two-dimensional lattices

Abstract

We analyze the scaling behavior of the two smallest Lyapunov exponents for electrons propagating on two-dimensional lattices with energies within a very narrow interval around the chiral critical point at E = 0 in the presence of a perpendicular random magnetic flux. By a numerical analysis of the energy and size dependence we confirm that the two smallest Lyapunov exponents are functions of a single parameter. The latter is given by ln L/ ln ξ(E), which is the ratio of the logarithm of the system width L to the logarithm of the correlation length ξ(E). Close to the chiral critical point and energy |E| ≪ E0, we find a logarithmically divergent energy dependence ln ξ(E) ∝ | ln(E0/|E|)| 1/2 , where E0 is a characteristic energy scale. Our data are in agreement with the theoretical prediction of M. Fabrizio and C. Castelliani [Nucl. Phys. B 583, 542 (2000)] and resolve an inconsistency of previous numerical work.

Scaling at chiral quantum critical points in two dimensions

“…where p/q (p and q are mutually prime integers) is the rational number of magnetic flux quanta h/e per plaquette 2a 2 , and B = ph/(2qea 2 ) is the magnetic flux density perpendicular to the two-dimensional lattice. This differs from the random flux model of our previous work 25 , where the constant magnetic field part was absent. The random part originates from the magnetic fluxes φ x,y , which are uniformly distributed −f /2 ≤ φ x,y ≤ f /2 with zero mean and disorder strength 0 ≤ f /(h/e) ≤ 1.…”

“…The same exponent was found also for a similar model defined on the square lattice. 25 This result confirms theoretical predictions that for |E| ≪ E 0 the localization length is ξ(E) = ξ 0 exp[A(ln(E 0 /|E|)) κ ] with a universal exponent κ = 1/2. 15,21 However, the recent suggestion 24 that nonperturbative effects have to be taken into account applies also to these investigations.…”

“…To compare the new results with our previous data published in Ref. 25 for the chiral unitary system without a constant magnetic field, we use the present fitting method also for the analysis of our previous data. In Table II we see that the critical exponent for the model chU[0, 0.5] is consistent with the value 1/2 as reported previously, 25 but differs significantly from the value 2/3 obtained here for all other ensembles.…”

“…23 Very recently, it has been pointed out that non-perturbative effects related to topologically non-trivial excitations have to be taken into account in field-theoretical investigations of quantum phase transitions in disordered systems belonging to the chiral symmetry classes. 24 Besides the divergence of the density of states discussed above, a corresponding divergence of the localization length with κ = 1/2, also predicted by theory 21 to occur in a very narrow energy interval close to E = 0, has been successfully observed for a chU model in a numerical study of the smallest Lyapunov exponent only recently, 25 despite several previous attempts. [26][27][28][29][30] Scaling properties of 2D chiral systems can be studied numerically by the analysis of the length and disorder or energy dependence of Lyapunov exponents defined for quasi-one-dimensional (q1D) systems.…”

“…25 for the chiral unitary system without a constant magnetic field, we use the present fitting method also for the analysis of our previous data. In Table II we see that the critical exponent for the model chU[0, 0.5] is consistent with the value 1/2 as reported previously, 25 but differs significantly from the value 2/3 obtained here for all other ensembles. Also, in order to check a possible disorder dependence of κ, again a bricklayer model with zero constant magnetic field but with a stronger RMF disorder strength f = 0.7 h/e (la- beled chU[0, 0.7]) was calculated.…”