We study spontaneous particle-hole symmetry breaking in a disordered two-site quantum breakdown model. The model is solvable in the limit of a large number of fermionic modes on each site. For a small ratio of the numbers of modes on the two sites, we derive the free energy analytically and find a finite-temperature symmetry-breaking transition. This transition changes from continuous to first order as the number of fermions produced in each conversion increases. Numerical results show that both types of transition persist beyond this regime, with an order-parameter exponent close to
1/2 for continuous transitions.
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.
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.