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arXiv:cond-mat/9301037 · Physical review. B, Condensed matter

Mott transition in an anyon gas

Wei Chen, Matthew P. A. Fisher, Yong-Shi Wu

Published Nov 1, 1993• 142 citations
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Abstract

We introduce and analyze a lattice model of anyons in a periodic potential and an external magnetic field, which exhibits a transition from a Mott insulator to a quantum Hall fluid. The transition is characterized by the anyon statistics \ensuremath{\alpha}, which can vary between fermions, \ensuremath{\alpha}=0, and bosons, \ensuremath{\alpha}=1. For bosons the transition is in the universality class of the classical three-dimensional XY model. Near the fermion limit, the transition is described by a massless 2+1 Dirac theory coupled to a Chern-Simons gauge field. Analytic calculations perturbative in \ensuremath{\alpha}, and also a large-N expansion, show that due to gauge fluctuations, the critical properties of the transition are dependent on the anyon statistics. Comparison with previous calculations at and near the boson limit, strongly suggest that our lattice model exhibits a fixed line of critical points, with universal critical properties that vary continuously and monotonically as one passes from fermions to bosons. Possible relevance to experiments on the transitions between plateaus in the fractional quantum Hall effect and the magnetic-field-tuned superconductor-insulator transition are briefly discussed.

DOI
10.1103/physrevb.48.13749

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We introduce and analyze a lattice model of anyons in a periodic potential and an external magnetic field, which exhibits a transition from a Mott insulator to a quantum Hall fluid. The transition is characterized by the anyon statistics \ensuremath{\alpha}, which can vary between fermions, \ensuremath{\alpha}=0, and bosons, \ensuremath{\alpha}=1. For bosons the transition is in the universality class of the classical three-dimensional XY model. Near the fermion limit, the transition is described by a massless 2+1 Dirac theory coupled to a Chern-Simons gauge field. Analytic calculations perturbative in \ensuremath{\alpha}, and also a large-N expansion, show that due to gauge fluctuations, the critical properties of the transition are dependent on the anyon statistics. Comparison with previous calculations at and near the boson limit, strongly suggest that our lattice model exhibits a fixed line of critical points, with universal critical properties that vary continuously and monotonically as one passes from fermions to bosons. Possible relevance to experiments on the transitions between plateaus in the fractional quantum Hall effect and the magnetic-field-tuned superconductor-insulator transition are briefly discussed.
Critical properties of transitions between a variety of quantum fluids and the Mott insulator are examined in terms of the transition to an incompressible state. In one dimension, they show universally common properties each other irrespective of the statistics, components and the interaction of particles. In two dimensions, they show qualitative differences originated from the differences in the nature of the quantum fluid. It is useful to categorize the singularity at the Mott transition into two types, namely, one characterized by the charge mass divergence and the other by the vanishment of the carrier number. This difference originates from the number of components. Its variety is further explored by examining various cases such as strongly correlated fermions, hard core bosons and boson t - J models. Critical properties to other incompressible states such as spin gap states and the fractional quantum Hall state are also discussed from the same viewpoint. Relevance of our analyses to the understanding of normal state properties in high- T c cuprates and other transition metal oxides is discussed.
We study one-dimensional lattice anyons with extended Hubbard interactions at unit filling using bosonization and numerical simulations. The behavior can be continuously tuned from bosonic to fermionic statistics by adjusting the topological exchange angle θ, which leads to a competition of different instabilities. We present the bosonization theory in the presence of dynamic gauge fields, which predicts a phase diagram of four different gapped phases with distinct dominant correlations. Advanced numerical simulations determine and analyze the exact phase transitions between Mott insulator, charge density wave, dimerized state, and Haldane insulator, all of which meet at a multicritical line in the parameter space of anyonic angle θ, on-site interaction U, and nearest neighbor repulsion V. Superfluid and pair-superfluid phases are stable in a region of small V.
We study the consequences of long-range Coulomb interactions at the critical points between integer/fractional quantum Hall states and an insulator. We use low energy theories for such transitions in anyon gases in the presence of an external periodic potential. We find that Coulomb interactions are marginally irrelevant for the integer quantum Hall case. For the fractional case, depending upon the anyon statistics parameter, we find behavior similar to the integer case, or flow to a novel line of fixed points with exponents z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1, $\ensuremath{\nu}>1$ stable against weak disorder in the position of the critical point, or runaway flow to strong coupling.
We study Mott phases and superfluid-insulator (SI) transitions of ultracold bosonic atoms in a two-dimensional square optical lattice at commensurate filling and in the presence of a synthetic periodic vector potential characterized by a strength ppp and a period l=qal=qal=qa, where qqq is an integer and aaa is the lattice spacing. We show that the Schr\"odinger equation for the noninteracting bosons in the presence of such a periodic vector potential can be reduced to an one-dimensional Harper-like equation which yields qqq energy bands. The lowest of these bands have either single or double minima whose position within the magnetic Brillouin zone can be tuned by varying ppp for a given qqq. Using these energies and a strong-coupling expansion technique, we compute the phase diagram of these bosons in the presence of a deep optical lattice. We chart out the ppp and qqq dependence of the momentum distribution of the bosons in the Mott phases near the SI transitions and demonstrate that the bosons exhibit several reentrant field-induced SI transitions for any fixed period qqq. We also predict that the superfluid density of the resultant superfluid state near such a SI transition has a periodicity qqq (q/2)(q/2)(q/2) in real space for odd (even) qqq and suggest experiments to test our theory.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
We develop a theory of a direct, continuous quantum phase transition between a bosonic Laughlin fractional quantum Hall state and a superfluid, generalizing the Mott insulator to superfluid phase diagram of bosons to allow for the breaking of time-reversal symmetry. The direct transition can be protected by a spatial symmetry, and the critical theory is a pair of Dirac fermion fields coupled to an emergent Chern-Simons gauge field. The transition may be achieved in optical traps of ultracold atoms by starting with a $\ensuremath{\nu}=1/2$ bosonic Laughlin state and tuning an appropriate periodic potential to change the topology of the composite fermion band structure.