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. The functional integral formulation of this problem in d spatial dimensions yields a (d+1)-dimensional classical XY-model with extended disorder--the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero site energies. 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 non-superfluid Mott insulating and Bose glass phases. We find 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. We argue 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. To study higher dimensions we attempt, with partial success, to generalize the Dorogovtsev-Cardy-Boyanovsky double epsilon expansion technique to this problem. The qualitative renormalization group flow picture this technique provides is quite compelling.
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