Abstract
The supersymmetric reformulation of physical observables in the Chalker-Coddington model (CC) for the plateau transition in the integer quantum Hall effect leads to a reformulation of its critical properties in terms of a 2D non-compact loop model or a 1D non-compact gl(2|2) spin chain. Following a proposal by Ikhlef, Fendley and Cardy [11], we define and study a series of truncations of these loop models and spin chains, involving a finite and growing number of degrees of freedom per site. The case of the first truncation is solved analytically using the Bethe-ansatz. It is shown to exhibit many of the qualitative features expected for the untruncated theory, including a quadratic spectrum of exponents with a continuous component, and a normalizable ground state below that continuum. Quantitative properties are however at odds with the results of simulations on the CC model. Higher truncations are studied only numerically. While their properties are found to get closer to those of the CC model, it is not clear whether this is a genuine effect, or the result of strong finite-size corrections.
Highlights
The transition between plateaux of the integer quantum Hall effect remains one of the most fascinating open problems in the field of quantum criticality: it is simple enough to admit very accessible formulations, is well documented experimentally—and yet, analytical predictions or any kind of “exact solution” remain essentially out of reach.The universality class of this problem is believed to be well captured by the Chalker-Coddington network model [1]
It can be expected that crossovers might occur slowly enough to allow higher truncations to approach the Brownian exponents at intermediate lengths, any naive model of “truncated Brownian motion” can probably be expected to fail at providing good insights on the Brownian motion itself. It is not clear whether the situation is so bad for the Chalker-Coddington model itself—in part, because we know so little about the universality class we are after
With the common notation q = eiγ, we find that the continuum limit of the b(21) and a(32) model are essentially identical
Summary
The transition between plateaux of the integer quantum Hall effect remains one of the most fascinating open problems in the field of quantum criticality: it is simple enough to admit very accessible formulations, is well documented experimentally—and yet, analytical predictions or any kind of “exact solution” remain essentially out of reach (despite a lot of recent progress, see below). It can be expected that crossovers might occur slowly enough to allow higher truncations to approach the Brownian exponents at intermediate lengths, any naive model of “truncated Brownian motion” can probably be expected to fail at providing good insights on the Brownian motion itself It is not clear whether the situation is so bad for the Chalker-Coddington model itself—in part, because we know so little about the universality class we are after (see [14] for a discussion of this problem). Original model is expected to exhibit a continuous spectrum of critical exponents, this would require the target of the continuum theory for the truncated model to be non-compact. In some parts of the literature, x happens to be denoted by “∆”: in this case, extra care has to be exercised in comparing results
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