arXiv subjects
Publications and source records attributed to J. T. Chalker.
These lecture notes are intended to provide a simple overview of the physics of geometrically frustrated magnets. The emphasis is on classical and semiclassical treatments of the statistical mechanics and dynamics of frustrated Heisenberg models, and on the ways in which the results provide an understanding of some of the main observed properties of these systems.
We develop a theoretical description of a Mach-Zehnder interferometer built from integer quantum Hall edge states, with an emphasis on how electron-electron interactions produce decoherence. We calculate the visibility of interference fringes and noise power, as a function of bias voltage and of temperature. Interactions are treated exactly, by using bosonization and considering edge states that are only weakly coupled via tunneling at the interferometer beam-splitters. In this weak-tunneling limit, we show that the bias-dependence of Aharonov-Bohm oscillations in source-drain conductance and noise power provides a direct measure of the one-electron correlation function for an isolated quantum Hall edge state. We find the asymptotic form of this correlation function for systems with either short-range interactions or unscreened Coulomb interactions, extracting a dephasing length $\ell_ϕ$ that varies with temperature $T$ as $\ell_ϕ \propto T^{-3}$ in the first case and as $\ell_ϕ \propto T^{-1} \ln^2(T)$ in the second case.
We study the disorder-induced localisation transition in a three-dimensional network model that belongs to symmetry class C. The model represents quasiparticle dynamics in a gapless spin-singlet superconductor without time-reversal invariance. It is a special feature of network models with this symmetry that the conductance and density of states can be expressed as averages in a classical system of dense, interacting random walks. Using this mapping, we present a more precise numerical study of critical behaviour at an Anderson transition than has been possible previously in any context.
We derive a continuum theory for the phase transition in a classical dimer model on the cubic lattice, observed in recent Monte Carlo simulations. Our derivation relies on the mapping from a three-dimensional classical problem to a two-dimensional quantum problem, by which the dimer model is related to a model of hard-core bosons on the kagome lattice. The dimer-ordering transition becomes a superfluid-Mott insulator quantum phase transition at fractional filling, described by an SU(2)-invariant continuum theory.
We examine bosons hopping on a one-dimensional lattice in the presence of a random potential at zero temperature. Bogoliubov excitations of the Bose-Einstein condensate formed under such conditions are localized, with the localization length diverging at low frequency as $\ell(ω)\sim 1/ω^α$. We show that the well known result $α=2$ applies only for sufficiently weak random potential. As the random potential is increased beyond a certain strength, $α$ starts decreasing. At a critical strength of the potential, when the system of bosons is at the transition from a superfluid to an insulator, $α=1$. This result is relevant for understanding the behavior of the atomic Bose-Einstein condensates in the presence of random potential, and of the disordered Josephson junction arrays.
Certain classical statistical systems with strong local constraints are known to exhibit Coulomb phases, where long-range correlation functions have power-law forms. Continuous transitions from these into ordered phases cannot be described by a naive application of the Landau-Ginzburg-Wilson theory, since neither phase is thermally disordered. We present an alternative approach to a critical theory for such systems, based on a mapping to a quantum problem in one fewer spatial dimensions. We apply this method to spin ice, a magnetic material with geometrical frustration, which exhibits a Coulomb phase and a continuous transition to an ordered state in the presence of a magnetic field applied in the [100] direction.
We examine the statistical mechanics of spin-ice materials with a [100] magnetic field. We show that the approach to saturated magnetisation is, in the low-temperature limit, an example of a 3D Kasteleyn transition, which is topological in the sense that magnetisation is changed only by excitations that span the entire system. We study the transition analytically and using a Monte Carlo cluster algorithm, and compare our results with recent data from experiments on Dy2Ti2O7.
Many classical, geometrically frustrated antiferromagnets have macroscopically degenerate ground states. In a class of three-dimensional systems, the set of degenerate ground states has power-law correlations and is an example of a Coulomb phase. We investigate Neel ordering from such a Coulomb phase, induced by weak additional interactions that lift the degeneracy. We show that the critical point belongs to a universality class that is different from the one for the equivalent transition out of the paramagnetic phase, and that it is characterised by effective long-range interactions; alternatively, ordering may be discontinuous. We suggest that a transition of this type may be realised by applying uniaxial stress to a pyrochlore antiferromagnet.
We investigate numerically the quasiparticle density of states $\varrho(E)$ for a two-dimensional, disordered superconductor in which both time-reversal and spin-rotation symmetry are broken. As a generic single-particle description of this class of systems (symmetry class D), we use the Cho-Fisher version of the network model. This has three phases: a thermal insulator, a thermal metal, and a quantized thermal Hall conductor. In the thermal metal we find a logarithmic divergence in $\varrho(E)$ as $E\to 0$, as predicted from sigma model calculations. Finite size effects lead to superimposed oscillations, as expected from random matrix theory. In the thermal insulator and quantized thermal Hall conductor, we find that $\varrho(E)$ is finite at E=0. At the plateau transition between these phases, $\varrho(E)$ decreases towards zero as $|E|$ is reduced, in line with the result $\varrho(E) \sim |E|\ln(1/|E|)$ derived from calculations for Dirac fermions with random mass.
We investigate the consequences for geometrically frustrated antiferromagnets of weak disorder in the strength of exchange interactions. Taking as a model the classical Heisenberg antiferromagnet with nearest neighbour exchange on the pyrochlore lattice, we examine low-temperature behaviour. We show that random exchange generates long-range effective interactions within the extensively degenerate ground states of the clean system. Using Monte Carlo simulations, we find a spin glass transition at a temperature set by the disorder strength. Disorder of this type, which is generated by random strains in the presence of magnetoelastic coupling, may account for the spin freezing observed in many geometrically frustrated magnets.
We present calculations of the magnetic ground state of Cs_2CuCl_4 in an applied magnetic field, with the aim of understanding the commensurately ordered state that has been discovered in recent experiments. This layered material is a realization of a Heisenberg antiferromagnet on an anisotropic triangular lattice. Its behavior in a magnetic field depends on field orientation, because of weak Dzyaloshinskii-Moriya interactions.We study the system by mapping the spin-1/2 Heisenberg Hamiltonian onto a Bose gas with hard core repulsion. This Bose gas is dilute, and calculations are controlled, close to the saturation field. We find a zero-temperature transition between incommensurate and commensurate phases as longitudinal field strength is varied, but only incommensurate order in a transverse field. Results for both field orientations are consistent with experiment.
A set of stacked two-dimensional electron systems in a perpendicular magnetic field exhibits a three-dimensional version of the quantum Hall effect if interlayer tunneling is not too strong. When such a sample is in a quantum Hall plateau, the edge states of each layer combine to form a chiral metal at the sample surface. We study the interplay of interactions and disorder in transport properties of the chiral metal, in the regime of weak interlayer tunneling. Our starting point is a system without interlayer tunneling, in which the only excitations are harmonic collective modes: surface magnetoplasmons. Using bosonization and working perturbatively in the interlayer tunneling amplitude, we express transport properties in terms of the spectrum for these collective modes, treating electron-electron interactions and impurity scattering exactly. We calculte the conductivity as a function of temperature, finding that it increases with increasing temperature as observed in recent experiments. We also calculate the autocorrelation function of mesoscopic conductance fluctuations induced by changes in a magnetic field component perpendicular to the sample surface, and its dependence on temperature. We show that conductance fluctuations are characterised by a dephasing length that varies inversely with temperature.
We examine the effects of electron-electron interactions on transport between edge states in a multilayer integer quantum Hall system. The edge states of such a system, coupled by interlayer tunneling, form a two-dimensional, chiral metal at the sample surface. We calculate the temperature-dependent conductivity and the amplitude of conductance fluctuations in this chiral metal, treating Coulomb interactions and disorder exactly in the weak-tunneling limit. We find that the conductivity increases with increasing temperature, as observed in recent experiments, and we show that the correlation length characterising conductance fluctuations varies inversely with temperature.
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.
We present detailed calculations of the magnetic ground state properties of Cs$_2$CuCl$_4$ in an applied magnetic field, and compare our results with recent experiments. The material is described by a spin Hamiltonian, determined with precision in high field measurements, in which the main interaction is antiferromagnetic Heisenberg exchange between neighboring spins on an anisotropic triangular lattice. An additional, weak Dzyaloshinkii-Moriya interaction introduces easy-plane anisotropy, so that behavior is different for transverse and longitudinal field directions. We determine the phase diagram as a function of field strength for both field directions at zero temperature, using a classical approximation as a first step. Building on this, we calculate the effect of quantum fluctuations on the ordering wavevector and components of the ordered moments, using both linear spinwave theory and a mapping to a Bose gas which gives exact results when the magnetization is almost saturated. Many aspects of the experimental data are well accounted for by this approach.
We consider classical normal modes and non-interacting bosonic excitations in disordered systems. We emphasise generic aspects of such problems and parallels with disordered, non-interacting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of non-interacting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low frequency limit, above a critical dimension $d_c$, which in different circumstances takes the values $d_c=2$ and $d_c=0$. For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $ρ(ω) \propto ω^4$ is a universal feature in systems with ground states that depend on the disorder realisation. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension.
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.
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.