21 research outputs found
The two-dimensional random-bond Ising model, free fermions and the network model
We develop a recently-proposed mapping of the two-dimensional Ising model
with random exchange (RBIM), via the transfer matrix, to a network model for a
disordered system of non-interacting fermions. The RBIM transforms in this way
to a localisation problem belonging to one of a set of non-standard symmetry
classes, known as class D; the transition between paramagnet and ferromagnet is
equivalent to a delocalisation transition between an insulator and a quantum
Hall conductor. We establish the mapping as an exact and efficient tool for
numerical analysis: using it, the computational effort required to study a
system of width is proportional to , and not exponential in as
with conventional algorithms. We show how the approach may be used to calculate
for the RBIM: the free energy; typical correlation lengths in quasi-one
dimension for both the spin and the disorder operators; even powers of
spin-spin correlation functions and their disorder-averages. We examine in
detail the square-lattice, nearest-neighbour RBIM, in which bonds are
independently antiferromagnetic with probability , and ferromagnetic with
probability . Studying temperatures , we obtain precise
coordinates in the plane for points on the phase boundary between
ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We
demonstrate scaling flow towards the pure Ising fixed point at small , and
determine critical exponents at the multicritical point.Comment: 20 pages, 25 figures, figures correcte
Dimensional Crossover of Localisation and Delocalisation in a Quantum Hall Bar
The 2-- to 1--dimensional crossover of the localisation length of electrons
confined to a disordered quantum wire of finite width is studied in a
model of electrons moving in the potential of uncorrelated impurities. An
analytical formula for the localisation length is derived, describing the
dimensional crossover as function of width , conductance and
perpendicular magnetic field . On the basis of these results, the scaling
analysis of the quantum Hall effect in high Landau levels, and the
delocalisation transition in a quantum Hall wire are reconsidered.Comment: 12 pages, 7 figure