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arXiv:hep-th/9912143 · Modern Physics Letters A

LOGARITHMIC OPERATORS IN THE THEORY OF PLATEAU TRANSITION

Ian I. Kogan, Alexei M. Tsvelik

Published May 10, 2000• 28 citations
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Abstract

We show that SL (2; C)/ SU (2) model which had been recently proposed to describe the behavior of the local densities of states at the plateau transition in integer quantum Hall effect, has logarithmic operators. Their unusual properties are studied in this letter.
DOI
10.1142/s0217732300000931

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In this paper, I revise arguments in favor of the PSL(2\ensuremath{\mid}2) Wess-Zumino-Novikov-Witten (WZNW) model as a theory of the plateau transition in integer quantum Hall effect. I show that all available numerical data (including the correlation length exponent $\ensuremath{\nu}$) are consistent with the predictions of such WZNW model with the level k=8k=8k=8.
We study the effects of the electron-electron interaction on the critical properties of the plateau transitions in the integer quantum Hall effect. We find the renormalization group dimension associated with short-range interactions to be $\ensuremath{-}0.66\ifmmode\pm\else\textpm\fi{}0.04$. Thus the noninteracting fixed point (characterized z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2 and $\ensuremath{\nu}\ensuremath{\approx}2.3$) is stable. For the Coulomb interaction, we find the correlation effect is a marginal perturbation at a Hartree-Fock fixed point ( z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1, $\ensuremath{\nu}\ensuremath{\approx}2.3$) by dimension counting. Further calculations are needed to determine its stability upon loop corrections.
In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on the correspondence between the Chalker–Coddington network model and a supersymmetric vertex model.
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.