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.
The existence of Anderson localization, characterized by vanishing diffusion due to strong disorder, has been demonstrated in numerous ways. A systematic approach based on the Anderson quantum model of the Fermi gas in random lattices that can describe both diffusive and localized regimes has not yet been fully established. We build on a recent publication [V. Janiš, ] and present a microscopic theory of disordered electrons that covers both the metallic phase with extended Bloch waves and the localized phase, where a propagating particle forms a quantum bound state with the hole left behind at the origin. The general theory provides a framework for constructing controlled approximations to one- and two-particle Green functions that satisfy the necessary conservation laws and causality requirements across the full range of disorder strength. It is used explicitly to derive a local, mean-field-like approximation for the two-particle irreducible vertices, enabling quantitative analysis of the solution's dynamic properties in both metallic and localized phases, including critical behavior at the mobility edge. A new instability line for the dynamical electron-hole correlation function of the metallic phase is introduced.
We study particle-hole symmetry at the integer quantum Hall plateau transition using composite fermion mean-field theory. Because this theory implicitly includes some electron-electron interactions, it also has applications to certain fractional quantum Hall plateau transitions. Previous work [P. Kumar et al., Phys. Rev. B 100, 235124 (2019)] using this approach showed that the diffusive quantum criticality of this transition is described by a nonlinear sigma model with topological $\ensuremath{\theta}=\ensuremath{\pi}$ term. This result, which holds for both the Dirac and Halperin, Lee, and Read composite fermion theories, signifies an emergent particle-hole (reflection) symmetry of the integer (fractional) quantum Hall transition. Here we consider the stability of this result to various particle-hole symmetry-violating perturbations. In the presence of quenched disorder that preserves particle-hole symmetry, we find that finite longitudinal conductivity at this transition requires the vanishing of a symmetry-violating composite fermion effective mass, which if present would generally lead to $\ensuremath{\theta}\ensuremath{\ne}\ensuremath{\pi}$ and a corresponding violation of particle-hole symmetric electrical transport ${\ensuremath{\sigma}}_{xy}\ensuremath{\ne}\frac{1}{2}\frac{{e}^{2}}{h}$. When the disorder does not preserve particle-hole symmetry, we find that $\ensuremath{\theta}$ can vary continuously within the diffusive regime. Our results call for further study of the universality of the quantum Hall plateau transition.
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.
We study the excited states of interacting fermions in one dimension with particle-hole symmetric disorder (equivalently, random-bond XXZ chains) using a combination of renormalization group methods and exact diagonalization. Absent interactions, the entire many-body spectrum exhibits infinite-randomness quantum critical behavior with highly degenerate excited states. We show that though interactions are an irrelevant perturbation in the ground state, they drastically affect the structure of excited states: Even arbitrarily weak interactions split the degeneracies in favor of thermalization (weak disorder) or spontaneously broken particle-hole symmetry, driving the system into a many-body localized spin glass phase (strong disorder). In both cases, the quantum critical properties of the noninteracting model are destroyed, either by thermal decoherence or spontaneous symmetry breaking. This system then has the interesting and counterintuitive property that edges of the many-body spectrum are less localized than the center of the spectrum. We argue that our results rule out the existence of certain excited state symmetry-protected topological orders.
We study the transport properties of generic out-of-equilibrium quantum systems connected to fermionic reservoirs. We develop a perturbation scheme in the inverse system size, named
1/N expansion, to study a large class of out of equilibrium diffusive/ohmic systems. The bare theory is described by a Gaussian action corresponding to a set of independent two level systems at equilibrium. This allows a simple and compact derivation of the diffusive current as a first-order pertubative term. In addition, we obtain exact solutions for a large class of quantum stochastic Hamiltonians (QSHs) with time and space dependent noise, using a self-consistent Born diagrammatic method in the Keldysh representation. We show that these QSHs exhibit diffusive regimes, which are encoded in the Keldysh component of the single particle Green's function. The exact solution for these QSHs models confirms the validity of our system size expansion ansatz, and its efficiency in capturing the transport properties. We consider in particular three fermionic models: (i) a model with local dephasing, (ii) the quantum simple symmetric exclusion process model, and (iii) a model with long-range stochastic hopping. For (i) and (ii) we compute the full temperature and dephasing dependence of the conductance of the system, both for two- and four-points measurements. Our solution gives access to the regime of finite temperature of the reservoirs, which could not be obtained by previous approaches. For (iii), we unveil a ballistic-to-diffusive transition governed by the range and the nature (quantum or classical) of the hopping. As a byproduct, our approach equally describes the mean behavior of quantum systems under continuous measurement.