A hidden Goldstone mechanism in the Kagom\'e lattice antiferromagnet

In this paper, we study the phases of the Heisenberg model on the \kagome lattice with antiferromagnetic nearest neighbour coupling $J_1$ and ferromagnetic next neighbour coupling $J_2$. Analysing the long wavelength, low energy effective action that describes this model, we arrive at the phase diagram as a function of $\chi = \frac{J_2}{J_1}$. The interesting part of this phase diagram is that for small $\chi$, which includes $\chi =0$, there is a phase with no long range spin order and with gapless and spin zero low lying excitations. We discuss our results in the context of earlier, numerical and experimental work.Comment: 21 pages, latex file with 5 figure

A Farewell to Liouvillians

We examine the Liouvillian approach to the quantum Hall plateau transition, as introduced recently by Sinova, Meden, and Girvin [Phys. Rev. B {\bf 62}, 2008 (2000)] and developed by Moore, Sinova and Zee [Phys. Rev. Lett. {\bf 87}, 046801 (2001)]. We show that, despite appearances to the contrary, the Liouvillian approach is not specific to the quantum mechanics of particles moving in a single Landau level: we formulate it for a general disordered single-particle Hamiltonian. We next examine the relationship between Liouvillian perturbation theory and conventional calculations of disorder-averaged products of Green functions and show that each term in Liouvillian perturbation theory corresponds to a specific contribution to the two-particle Green function. As a consequence, any Liouvillian approximation scheme may be re-expressed in the language of Green functions. We illustrate these ideas by applying Liouvillian methods, including their extension to $N_L > 1$ Liouvillian flavors, to random matrix ensembles, using numerical calculations for small integer $N_L$ and an analytic analysis for large $N_L$. We find that behavior at $N_L > 1$ is different in qualitative ways from that at $N_L=1$. In particular, the $N_L = \infty$ limit expressed using Green functions generates a pathological approximation, in which two-particle correlation functions fail to factorize correctly at large separations of their energy, and exhibit spurious singularities inside the band of random matrix energy levels. We also consider the large $N_L$ treatment of the quantum Hall plateau transition, showing that the same undesirable features are present there, too

Multifractality and critical fluctuations at the Anderson transition

Critical fluctuations of wave functions and energy levels at the Anderson transition are studied for the family of the critical power-law random banded matrix ensembles. It is shown that the distribution functions of the inverse participation ratios (IPR) $P_q$ are scale-invariant at the critical point, with a power-law asymptotic tail. The IPR distribution, the multifractal spectrum and the level statistics are calculated analytically in the limits of weak and strong couplings, as well as numerically in the full range of couplings.Comment: 14 pages, 13 eps figure

Electrostatic theory for imaging experiments on local charges in quantum Hall systems

We use a simple electrostatic treatment to model recent experiments on quantum Hall systems, in which charging of localised states by addition of integer or fractionally-charged quasiparticles is observed. Treating the localised state as a compressible quantum dot or antidot embedded in an incompressible background, we calculate the electrostatic potential in its vicinity as a function of its charge, and the chemical potential values at which its charge changes. The results offer a quantitative framework for analysis of the observations.Comment: 4 pages, 3 figure

Network models for localisation problems belonging to the chiral symmetry classes

We consider localisation problems belonging to the chiral symmetry classes, in which sublattice symmetry is responsible for singular behaviour at a band centre. We formulate models which have the relevant symmetries and which are generalisations of the network model introduced previously in the context of the integer quantum Hall plateau transition. We show that the generalisations required can be re-expressed as corresponding to the introduction of absorption and amplification into either the original network model, or the variants of it that represent disordered superconductors. In addition, we demonstrate that by imposing appropriate constraints on disorder, a lattice version of the Dirac equation with a random vector potential can be obtained, as well as new types of critical behaviour. These models represent a convenient starting point for analytic discussions and computational studies, and we investigate in detail a two-dimensional example without time-reversal invariance. It exhibits both localised and critical phases, and band-centre singularities in the critical phase approach more closely in small systems the expected asymptotic form than in other known realisations of the symmetry class.Comment: 14 pages, 15 figures, Submitted to Physical Review

Critical statistics in a power-law random banded matrix ensemble

We investigate the statistical properties of the eigenvalues and eigenvectors in a random matrix ensemble with $H_{ij}\sim |i-j|^{-\mu}$. It is known that this model shows a localization-delocalization transition (LDT) as a function of the parameter $\mu$. The model is critical at $\mu=1$ and the eigenstates are multifractals. Based on numerical simulations we demonstrate that the spectral statistics at criticality differs from semi-Poisson statistics which is expected to be a general feature of systems exhibiting a LDT or `weak chaos'.Comment: 4 pages in PS including 5 figure

On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering

We describe analytical and numerical results on the statistical properties of complex eigenvalues and the corresponding non-orthogonal eigenvectors for non-Hermitian random matrices modeling one-channel quantum-chaotic scattering in systems with broken time-reversal invariance.Comment: 4 pages, 2 figure

Single electron magneto-conductivity of a nondegenerate 2D electron system in a quantizing magnetic field

We study transport properties of a non-degenerate two-dimensional system of non-interacting electrons in the presence of a quantizing magnetic field and a short-range disorder potential. We show that the low-frequency magnetoconductivity displays a strongly asymmetric peak at a nonzero frequency. The shape of the peak is restored from the calculated 14 spectral moments, the asymptotic form of its high-frequency tail, and the scaling behavior of the conductivity for omega -> 0. We also calculate 10 spectral moments of the cyclotron resonance absorption peak and restore the corresponding (non-singular) frequency dependence using the continuous fraction expansion. Both expansions converge rapidly with increasing number of included moments, and give numerically accurate results throughout the region of interest. We discuss the possibility of experimental observation of the predicted effects for electrons on helium.Comment: RevTeX 3.0, 14 pages, 8 eps figures included with eps

Quantum and classical localisation, the spin quantum Hall effect and generalisations

We consider network models for localisation problems belonging to symmetry class C. This symmetry class arises in a description of the dynamics of quasiparticles for disordered spin-singlet superconductors which have a Bogoliubov - de Gennes Hamiltonian that is invariant under spin rotations but not under time-reversal. Our models include but also generalise the one studied previously in the context of the spin quantum Hall effect. For these systems we express the disorder-averaged conductance and density of states in terms of sums over certain classical random walks, which are self-avoiding and have attractive interactions. A transition between localised and extended phases of the quantum system maps in this way to a similar transition for the classical walks. In the case of the spin quantum Hall effect, the classical walks are the hulls of percolation clusters, and our approach provides an alternative derivation of a mapping first established by Gruzberg, Read and Ludwig, Phys. Rev. Lett. 82, 4254 (1999).Comment: 11 pages, 5 figure

Layering in the Ising model

We consider the three-dimensional Ising model in a half-space with a boundary field (no bulk field). We compute the low-temperature expansion of layering transition lines