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arXiv:0801.1646 · Physical Review B

Particle-hole symmetry and the dirty boson problem

Peter B. Weichman, Ranjan Mukhopadhyay

Published Jun 18, 2008• 55 citations
arXivHTMLPDF

Abstract

We study the role of particle-hole symmetry on the universality class of various quantum phase transitions corresponding to the onset of superfluidity at zero temperature of bosons in a quenched random medium. To obtain a model with an exact particle-hole symmetry it is necessary to use the Josephson junction array, or quantum rotor, Hamiltonian, which may include disorder in both the site energies and the Josephson couplings between wave function phase operators at different sites. The functional integral formulation of this problem in $d$ spatial dimensions yields a $(d+1)$-dimensional classical $XY$ model with extended disorder, constant along the extra imaginary time dimension---the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero site energies, which may be uniform or site dependent. We may distinguish three cases: (i) exact particle-hole symmetry, in which the site energies all vanish; (ii) statistical particle-hole symmetry, in which the site energy distribution is symmetric about zero, vanishing on average; and (iii) complete absence of particle-hole symmetry in which the distribution is generic. We explore in each case the nature of the excitations in the nonsuperfluid Mott insulating and Bose glass phases. We show, in particular, that, since the boundary of the Mott phase can be derived exactly in terms of that for the pure, nondisordered system, there can be no direct Mott-superfluid transition. Recent Monte Carlo data to the contrary can be explained in terms of rare region effects that are inaccessible to finite systems. We find also that the Bose glass compressibility, which has the interpretation of a temporal spin stiffness or superfluid density, is positive in cases (ii) and (iii), but that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then focus on the critical point and discuss the relevance of type (ii) particle-hole symmetry-breaking perturbations to the random rod critical behavior, identifying a nontrivial crossover exponent. This exponent cannot be calculated exactly but is argued to be positive and the perturbation therefore relevant. We argue next that a perturbation of type (iii) is irrelevant to the resulting type (ii) critical behavior: The statistical symmetry is restored on large scales close to the critical point, and case (ii) therefore describes the dirty boson fixed point. Using various duality transformations we verify all of these ideas in one dimension. To study higher dimensions, we attempt, with partial success, to generalize the Dorogovtsev--Cardy--Boyanovsky double-epsilon expansion technique to this problem. We find that when the dimension of time ${ϵ}_{\ensuremath{\tau}}<{ϵ}_{\ensuremath{\tau}}^{c}\ensuremath{\simeq}\frac{8}{29}$ is sufficiently small a type (ii) symmetry-breaking perturbation is irrelevant, but that for sufficiently large ${ϵ}_{\ensuremath{\tau}}>{ϵ}_{\ensuremath{\tau}}^{c}$ particle-hole asymmetry is a relevant perturbation and a new stable fixed point appears. Furthermore, for ${ϵ}_{\ensuremath{\tau}}>{ϵ}_{\ensuremath{\tau}}^{c2}\ensuremath{\approx}\frac{2}{3}$, this fixed point is stable also to perturbations of type (iii): at $ϵ={ϵ}_{\ensuremath{\tau}}^{c2}$ the generic type (iii) fixed point merges with the new fixed point. We speculate, therefore, that this new fixed point becomes the dirty boson fixed point when ${ϵ}_{\ensuremath{\tau}}=1$. We point out, however, that ${ϵ}_{\ensuremath{\tau}}=1$ may be quite special. Thus, although the qualitative renormalization group flow picture the double-epsilon expansion technique provides is quite compelling, one should remain wary of applying it quantitatively to the dirty boson problem.

DOI
10.1103/physrevb.77.214516

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We study the role of particle-hole symmetry on the universality class of various quantum phase transitions corresponding to the onset of superfluidity at zero temperature of bosons in a quenched random medium. To obtain a model with an exact particle-hole symmetry it is necessary to use the Josephson junction array, or quantum rotor, Hamiltonian, which may include disorder in both the site energies and the Josephson couplings between wave function phase operators at different sites. The functional integral formulation of this problem in ddd spatial dimensions yields a (d+1)(d+1)(d+1)-dimensional classical XYXYXY model with extended disorder, constant along the extra imaginary time dimension---the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero site energies, which may be uniform or site dependent. We may distinguish three cases: (i) exact particle-hole symmetry, in which the site energies all vanish; (ii) statistical particle-hole symmetry, in which the site energy distribution is symmetric about zero, vanishing on average; and (iii) complete absence of particle-hole symmetry in which the distribution is generic. We explore in each case the nature of the excitations in the nonsuperfluid Mott insulating and Bose glass phases. We show, in particular, that, since the boundary of the Mott phase can be derived exactly in terms of that for the pure, nondisordered system, there can be no direct Mott-superfluid transition. Recent Monte Carlo data to the contrary can be explained in terms of rare region effects that are inaccessible to finite systems. We find also that the Bose glass compressibility, which has the interpretation of a temporal spin stiffness or superfluid density, is positive in cases (ii) and (iii), but that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then focus on the critical point and discuss the relevance of type (ii) particle-hole symmetry-breaking perturbations to the random rod critical behavior, identifying a nontrivial crossover exponent. This exponent cannot be calculated exactly but is argued to be positive and the perturbation therefore relevant. We argue next that a perturbation of type (iii) is irrelevant to the resulting type (ii) critical behavior: The statistical symmetry is restored on large scales close to the critical point, and case (ii) therefore describes the dirty boson fixed point. Using various duality transformations we verify all of these ideas in one dimension. To study higher dimensions, we attempt, with partial success, to generalize the Dorogovtsev--Cardy--Boyanovsky double-epsilon expansion technique to this problem. We find that when the dimension of time ϵ\ensuremathτ<ϵ\ensuremathτc\ensuremath≃829{ϵ}_{\ensuremath{\tau}}<{ϵ}_{\ensuremath{\tau}}^{c}\ensuremath{\simeq}\frac{8}{29}ϵ\ensuremathτ​<ϵ\ensuremathτc​\ensuremath≃298​ is sufficiently small a type (ii) symmetry-breaking perturbation is irrelevant, but that for sufficiently large ϵ\ensuremathτ>ϵ\ensuremathτc{ϵ}_{\ensuremath{\tau}}>{ϵ}_{\ensuremath{\tau}}^{c}ϵ\ensuremathτ​>ϵ\ensuremathτc​ particle-hole asymmetry is a relevant perturbation and a new stable fixed point appears. Furthermore, for ϵ\ensuremathτ>ϵ\ensuremathτc2\ensuremath≈23{ϵ}_{\ensuremath{\tau}}>{ϵ}_{\ensuremath{\tau}}^{c2}\ensuremath{\approx}\frac{2}{3}ϵ\ensuremathτ​>ϵ\ensuremathτc2​\ensuremath≈32​, this fixed point is stable also to perturbations of type (iii): at ϵ=ϵ\ensuremathτc2ϵ={ϵ}_{\ensuremath{\tau}}^{c2}ϵ=ϵ\ensuremathτc2​ the generic type (iii) fixed point merges with the new fixed point. We speculate, therefore, that this new fixed point becomes the dirty boson fixed point when ϵ\ensuremathτ=1{ϵ}_{\ensuremath{\tau}}=1ϵ\ensuremathτ​=1. We point out, however, that ϵ\ensuremathτ=1{ϵ}_{\ensuremath{\tau}}=1ϵ\ensuremathτ​=1 may be quite special. Thus, although the qualitative renormalization group flow picture the double-epsilon expansion technique provides is quite compelling, one should remain wary of applying it quantitatively to the dirty boson problem.
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We study the nature of various quantum phase transitions corresponding to the onset of superfluidity, at zero temperature, of bosons in a quenched medium. Particle-hole symmetry plays an essential role in determining the universality class of the transitions. To obtain a model with an exact particle-hole symmetry it is necessary to use the Josephson junction array Hamiltonian, which may include disorder in the Josephson couplings between phases at different sates. The functional integral formulation of this problem in d spatial dimensions yields a (d + 1)-dimensional classical XY-model with extended disorder, constant along the extra imaginary time dimension -- the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero chemical potentials or site energies, which may also be site dependent and random. We may then distinguish three cases: (i) exact particle-hole symmetry, in which the site energies all vanish, (ii) statistical particle-hole symmetry in which the site energy distribution is symmetric about zero and hence vanishes on average, and (iii) complete absence of particle-hole symmetry in which the distribution is generic. We explore in each case the nature of the excitations in the nonsuperfluid Pose glass phase. We find, for example, that the compressibility, which has the interpretation of a temporal spin stiffness or superfluid density, is positive in cases (ii) and (iii), but that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then focus on the critical point and discuss the validity of various scaling arguments. In particular, we argue that the dynamical exponent z could be different from d, and the arguments leading to their equality are incorrect. We then discuss the relevance of a type (ii) particle-hole symmetry breaking perturbation to the random rod critical behavior, identifying a nontrivial crossover exponent. This exponent cannot be calculated exactly but is argued to be positive and the symmetry breaking perturbation therefore relevant. We argue next that a perturbation of type (iii) is irrelevant to the resulting type (ii) critical behavior: the statistical symmetry is restored on large scales close to the critical point, and case (ii) therefore describes the dirty boson fixed point. Using various duality transformations we verify all of these ideas in one dimension. To study higher dimensions we attempt, with partial success, to generalize the Dorogovtsev-Cardy-Boyonovsky double epsilon expansion technique to this problem. We find that when the dimension of time [...] is sufficiently small the symmetry breaking perturbation of type (ii) is irrelevant, but that for sufficiently large [...] this is a relevant perturbation and a new stable commensurate fixed point appears. We speculate that this new fixed point becomes the dirty boson fixed point when [...] = 1. We also show that for [...], there exists a particle-hole asymmetric fixed point of type (iii), but we provide evidence that it merges with the commensurate fixed point for some finite [...]. This tends to confirm symmetry restoration at the physical [...] = 1.
The generic Hamiltonian describing the zero temperature transition between the insulating Bose glass phase and the superfluid phase lacks particle-hole symmetry, but a statistical version of this symmetry is believed to be restored at the critical point. We show that the renormalization group relevance of particle-hole asymmetry may be explored in a controlled fashion only for small time dimensions, ${\ensuremath{\epsilon}}_{\ensuremath{\tau}}\ensuremath{\ll}1$, where we find a stable particle-hole asymmetric and an unstable particle-hole symmetric fixed point, but we provide evidence that the two merge for some finite ${\ensuremath{\epsilon}}_{\ensuremath{\tau}}\ensuremath{\approx}\frac{2}{3}$, which tends to confirm symmetry restoration at the physical ${\ensuremath{\epsilon}}_{\ensuremath{\tau}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.
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 study the zero-temperature superfluid-insulator transition for a two-dimensional model of interacting, lattice bosons in the presence of quenched disorder and particle-hole symmetry. We follow the approach of a recent series of papers by Altman, Kafri, Polkovnikov, and Refael, in which the strong disorder renormalization group is used to study disordered bosons in one dimension. Adapting this method to two dimensions, we study several different species of disorder and uncover universal features of the superfluid-insulator transition. In particular, we locate an unstable finite disorder fixed point that governs the transition between the superfluid and a gapless, glassy insulator. We present numerical evidence that this glassy phase is the incompressible Mott glass and that the transition from this phase to the superfluid is driven by a percolation-type process. Finally, we provide estimates of the critical exponents governing this transition.
The superfluid to insulator quantum phase transition of a three-dimensional particle-hole symmetric system of disordered bosons is studied. To this end, a site-diluted quantum rotor Hamiltonian is mapped onto a classical (3+1)-dimensional XYXYXY model with columnar disorder and analyzed by means of large-scale Monte Carlo simulations. The superfluid-Mott insulator transition of the clean, undiluted system is in the four-dimensional XYXYXY universality class and shows mean-field critical behavior with logarithmic corrections. The clean correlation length exponent $\ensuremath{\nu}=1/2$ violates the Harris criterion, indicating that disorder must be a relevant perturbation. For nonzero dilutions below the lattice percolation threshold of pc=0.688392{p}_{c}=0.688\phantom{\rule{0.16em}{0ex}}392pc​=0.688392, our simulations yield conventional power-law critical behavior with dilution-independent critical exponents z=1.67(6)z=1.67(6)z=1.67(6), $\ensuremath{\nu}=0.90(5)$, $\ensuremath{\beta}/\ensuremath{\nu}=1.09(3)$, and $\ensuremath{\gamma}/\ensuremath{\nu}=2.50(3)$. The critical behavior of the transition across the lattice percolation threshold is controlled by the classical percolation exponents. Our results are discussed in the context of a classification of disordered quantum phase transitions, as well as experiments in superfluids, superconductors, and magnetic systems.
We use a large-N renormalization-group (RG) method to study a model of interacting boson system with a quenched random potential. In the absence of impurities, the pure boson system has a critical point that describes the superfluid--Mott-insulator (SF--MI) transition. The SF--MI transition of d-dimensional bosons belongs to the (d+1)-dimensional XY model universality class. In this paper, we study the dirty-boson critical points in the neighborhood of this pure SF--MI critical point. In general, the on-site random potential in the original lattice model gives two types of randomness in the effective-field-theoretic action. One is the randomness in the effective on-site repulsion w(x) and the other is the randomness of the chemical potential u(x). It turns out that d=2 is the critical dimension for both types of disorder but the roles of these two types of disorder are reversed as d=2 is crossed. Applying \ensuremath{\epsilon}=d-2 expansion, we found coupled RG equations for both kinds of randomness which reveal several nontrivial critical points. All the weak random fixed points we found have three or more relevant directions. We conclude that the direct SF--MI transition is unlikely to occur near two dimensions.