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arXiv:cond-mat/0507412 · Journal of Experimental and Theoretical Physics Letters

Non-fermi liquid criticality and superuniversality in the quantum hall regime

Adrianus M. M. Pruisken, I. S. Burmistrov

Published Apr 1, 2008• 23 citations
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Abstract

The results of a microscopic theory, based on the topological concept of a θ vacuum, which show that the Coulomb potential, unlike any finite-ranged interaction potential, renders the long-standing problem of the plateau transitions in the quantum Hall regime like a non-Fermi liquid are reported. These results, which are important for quantum-phase transitions in general and composite fermion ideas in particular, provide a novel understanding of the critical exponent values that have recently been (re-)taken from a series of state-of-the-art quantum Hall samples.

DOI
10.1134/s0021364008040097

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The results of a microscopic theory, based on the topological concept of a θ vacuum, which show that the Coulomb potential, unlike any finite-ranged interaction potential, renders the long-standing problem of the plateau transitions in the quantum Hall regime like a non-Fermi liquid are reported. These results, which are important for quantum-phase transitions in general and composite fermion ideas in particular, provide a novel understanding of the critical exponent values that have recently been (re-)taken from a series of state-of-the-art quantum Hall samples.
Transformations between the plateau states of the quantum Hall effect (QHE) are an archetypical example of quantum phase transitions (QPTs) between phases with non-trivial topological order. These transitions appear to be well-described by the single-particle network theories. The long-standing problem with this approach is that it does not account for Coulomb interactions. In this paper, we show that experimental data in the quantum critical regime for both integer and fractional QHEs can be quantitatively explained by the recently developed phenomenological model of QPTs in interacting systems. This model assumes that all effects of interactions are contained in the life-time of fluctuations as set by the Planckian time τP=ℏ/kBTτ_P=\hbar/k_BTτP​=ℏ/kB​T. The dephasing length is taken as the distance traveled by a non-interacting particle along the bulk edge state over this time. We show that the model also provides quantitative description of QPTs between the ground states of anomalous QHE and axion and Chern insulators. These analyzed systems are connected in that the QPTs occur via quantum percolation. Combining the presented results with the results of two companion papers, we conclude that the Planckian time is the encompassing characteristic of QPTs in interacting systems, independent of dimensionality and microscopic physics.
We introduce the concept of superuniversality in quantum Hall liquids and spin liquids. This concept has emerged from previous studies of the quantum Hall effect and states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the $\ensuremath{\theta}$ parameter in nonlinear $\ensuremath{\sigma}$ models in two dimensions. To establish superuniversality in spin liquids we revisit the mapping by Haldane who argued that the antiferromagnetic Heisenberg spin-sss chain in 1+11+11+1 space-time dimensions is effectively described by the O(3)O(3)O(3) nonlinear $\ensuremath{\sigma}$ model with a $\ensuremath{\theta}$ term. By combining the path integral representation for the dimerized spin s=1∕2s=1∕2s=1∕2 chain with renormalization-group decimation techniques we generalize the Haldane approach to include a more complicated theory, the fermionic rotor chain, involving four different renormalization-group parameters. We show how the renormalization-group calculation technique can be used to build a bridge between the fermionic rotor chain and the O(3)O(3)O(3) nonlinear $\ensuremath{\sigma}$ model with the $\ensuremath{\theta}$ term. As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain. The edge spin in spin liquids is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain such as open and closed chains, chains with an even and odd number of sides. We show that for each of the different geometries the $\ensuremath{\theta}$ term has a distinctly different physical meaning. We compare each case with a topologically equivalent quantum Hall liquid.
The long-thought charge Kondo effects have recently been experimentally realized in the quantum Hall regime. This experiment, supported by numerics, exemplifies the realization of two-channel Kondo state, a non-Fermi Liquid, and its crossover to the one-channel counterpart, a Fermi liquid. Scaling up such a platform, we find a hierarchy of non-Fermi Liquids and their tunable crossovers based on a renormalization group analysis. Utilizing results from a conformal field theory, we further examine the universal conductances of this strongly correlated system and their finite temperature scaling, which elucidate the sharp distinctions between charge and spin Kondo physics.
We study the role of electron-electron interactions near integer and abelian fractional quantum Hall (QH) transitions using composite fermion (CF) representations. Interaction effects are encapsulated in CF theories as gauge fluctuations. Without gauge fluctuations, the CF system realizes a `dual' representation of the noninteracting QH transition. With gauge fluctuations, the system is governed by a gauged nonlinear sigma model (NLSM) with a $\ensuremath{\theta}\phantom{\rule{0.28em}{0ex}}\mathrm{term}$. While the transition is described by a strong-coupling fixed point of the NLSM, we are nevertheless able to deduce two of its properties. With 1/r1/r1/r interactions, (1) the transition has a dynamical exponent z=1z=1z=1, and (2) all transitions are ``superuniversal'': fractional and integer QH transitions are in the same universality class. With short-range interactions, z=2z=2z=2 and the fate of superuniversality remains unclear.
The phase diagram of a quantum Hall bilayer at total filling ν = 1 contains an incompressible superfluid for small distances d between the layers, as well as the compressible phase corresponding to two uncoupled Fermi liquids for large d. Using exact diagonalization on the sphere and torus geometry, we investigate a long-standing question of the nature of the transition between the two regimes, and the possibility for the existence of a paired phase in the transition region. We find considerable evidence for a direct transition between the superfluid and the Fermi liquid phase, based in particular on the behavior of the ground state energy on the sphere (including appropriate finite-size corrections) as a function of d. At the critical distance dC ≈ 11.6ℓB the topological number ("shift") of the ground state changes, suggesting that tuning the layer separation d in experiment likely leads to a direct transition between the superfluid and the Fermi liquid phase.
Quantum many-particle systems in which the kinetic energy, strong correlations, and band topology are all-important pose an interesting and topical challenge. Here we introduce and study particularly simple models where all of these elements are present. We consider interacting quantum particles in two dimensions in a strong magnetic field such that the Hilbert space is restricted to the lowest Landau level (LLL). This is the familiar quantum Hall regime with rich physics determined by particle filling and statistics. A periodic potential with a unit cell enclosing one flux quantum broadens the LLL into a Chern band with a finite bandwidth. The states obtained in the quantum Hall regime evolve into conducting states in the limit of large bandwidth. We study this evolution in detail for the specific case of bosons at filling factor $\ensuremath{\nu}=1$. In the quantum Hall regime, the ground state at this filling is a gapped quantum hall state (the ``bosonic Pfaffian''), which may be viewed as descending from a (bosonic) composite Fermi liquid. At large bandwidth, the ground state is a bosonic superfluid. We show how both phases and their evolution can be described within a single theoretical framework based on a LLL composite fermion construction. Building on our previous work on the bosonic composite Fermi liquid, we show that the evolution into the superfluid can be usefully described by a noncommutative quantum field theory in a periodic potential.