We study the strong localization of hard-core bosons. Using a locator expansion we find that in the insulator, unlike for typical fermion problems, nearly all low-energy scattering paths come with positive amplitudes and hence interfere constructively. As a consequence, the localization length of bosonic excitations shrinks when the constructive interference is suppressed by a magnetic field, entailing an exponentially large positive magnetoresistance, opposite to and significantly stronger than the analogous effect in fermions. Within the forward-scattering approximation, we find that the lowest-energy excitations are the most delocalized. A similar analysis applied to random field Ising models suggests that the ordering transition is due to a delocalization initiated at zero energy rather than due to the closure of a mobility gap in the paramagnet.
We study the strong localization of hard-core bosons. Using a locator expansion we find that in the insulator, unlike for typical fermion problems, nearly all low-energy scattering paths come with positive amplitudes and hence interfere constructively. As a consequence, the localization length of bosonic excitations shrinks when the constructive interference is suppressed by a magnetic field, entailing an exponentially large positive magnetoresistance, opposite to and significantly stronger than the analogous effect in fermions. Within the forward-scattering approximation, we find that the lowest-energy excitations are the most delocalized. A similar analysis applied to random field Ising models suggests that the ordering transition is due to a delocalization initiated at zero energy rather than due to the closure of a mobility gap in the paramagnet.
We study the strong localization of hard core bosons. Using a locator expansion we find that in the insulator, unlike for typical fermion problems, nearly all low-energy scattering paths come with positive amplitudes and hence interfere constructively. As a consequence, the localization length of bosonic excitations shrinks when the constructive interference is suppressed by a magnetic field, entailing an exponentially large positive magnetoresistance, opposite to and significantly stronger than the analogous effect in fermions. Within the forward scattering approximation, we find that the lowest energy excitations are the most delocalized. A similar analysis applied to random field Ising models suggests that the ordering transition is due to a delocalization initiated at zero energy rather than due to the closure of a mobility gap in the paramagnet.
Bosonic excitations within long-range ordered, but strongly inhomogeneous phases have been studied in quite some detail. My thesis focuses instead on understanding the insulating, localized phase of disordered bosonic systems. In particular I study localization properties of strongly interacting bosons and spin systems in a disorder potential at zero temperature. I focus on simple, prototypical spin models (Ising model and XY model) in random fields on a Cayley tree with large connectivity. Regarding the nature of the quantum phase transition in strong disorder I find the following results: i) With a uniformly distributed disorder non-extensive excitations in the disordered phase are all localized. ii)Moreover, I find that the order arises due to a collective condensation, which is qualitatively distinct from a Bose Einstein condensation of single particle excitations into a delocalized state. In particular, in non-frustrated Bose glasses, I do not find evidence for a boson mobility edge in the Bose glass. These results are qualitatively different from claims in the recent literatures . Considering that (many body) localization of bosons is a kind of quantum glass transition, it is an interesting question to ask what phenomena occur, if the ingredients for more conventional (classical) glassy physics are added to a disordered bosons system, namely: random, frustrated interactions between the bosons. One can still think about such a system as bosons in a disordered potential, where the disordered potential is, at least partly, self-generated by random frustrated interactions between the bosons. This question takes us to the study of another type of disordered systems: glassy systems. Those are typically characterized by low temperature phases with an inhomogeneous density or magnetization pattern, which is extremely long-lived due to the occurrence of non-trivial ergodicity breaking. I study a solvable model of hard core bosons (pseudospins) subject to disorder and frustrating interactions. This solvable model provides insight into the possibility of coexistence of super uidity and glassy density order, as well as into the nature of the coexistence phase (the superglass). In particular, for the considered mean field model I prove the existence of a superglass phase. This complements the numerical evidence for such phases provided by quantum Monte Carlo investigations in finite dimensions and on random graphs. Those were, however, limited to finite temperature, and could thus not fully elucidate the structure of the phases at T = 0. In contrast, my analytical approach allows one to understand the quantum phase transition between glassy superfluid and insulator, and the non-trivial role played by glassy correlations. When the frustrated interactions are strong enough, the superfluid order may be destroyed. As I will show in a mean field model, this happens within the glass phase of the system, where a disorder induced superfluid-insulator phase transition takes place to give way to a frustrated Bose glass. The glassy background on top of which this happens leads to many interesting phenomena which seem not to have been noticed before. To understand the nature of the glassy superfluid-insulator quantum phase transition at zero temperature and the transport properties on the insulating, Bose glass side of the transition is the goal of the third part of my thesis. To address the above questions, I studied an exactly solvable model of a glassy superfluid-insulator quantum phase transition on a Bethe lattice geometry with high connectivity. My main results can be summarized as follows: i) I found that the superfluid-insulator transition is shifted to stronger hopping. This is a result of the pseudo gap in the density of states of the glass state, which tends to strongly disfavor the onset of superfluidity. ii) In the glassy insulator, the discrete local energy levels become broadened due to the quantum fluctuations.The level-broadening process appears as a phase transition which has strong similarities with an Anderson localization transition, and has implications on many body localization. By using the locator expansion for bosons I found that, the glassy insulator has a finite mobility edge for the bosonic excitations, which, however, does not close upon approaching the SI quantum phase transition point. This finding helps to understand the nature of the superfluid-to-frustrated Bose glass transition: the superfluid emerges as a collective phase ordering phenomenon at zero temperature, and not as a condensation in to a single particle delocalized state, in contrast to opposite predictions in the recent literatures. The existence of a mobility edge in the insulator suggests the possibility of phononless, activated transport in the bosonic insulator, which might be a candidate explanation for the experimentally seen activated transport, which has remained a mystery for a long time.
We study the ac conduction in a system of fermions or bosons strongly localized in a disordered array of sites with short-range interactions at frequencies larger than the intersite tunneling but smaller than the characteristic fluctuation of the on-site energy. While the main contribution ${\ensuremath{\sigma}}_{0}(\ensuremath{\omega})$ to the conductivity comes from local dipole-type excitations on close pairs of sites, coherent processes on three or more sites lead to an interference correction ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$, which depends on the statistics of the charge carriers and can be suppressed by a magnetic field. For bosons the correction is always positive, while for fermions it can be positive or negative depending on whether the conduction is dominated by effective single-particle or single-hole processes. We calculate the conductivity explicitly assuming a constant density of states of single-site excitations. Independently of the statistics, ${\ensuremath{\sigma}}_{0}(\ensuremath{\omega})=\text{const}$. For bosons, ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})\ensuremath{\propto}\mathrm{log}(C/\ensuremath{\omega})$. For fermions, ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})\ensuremath{\propto}\mathrm{log}[\mathrm{max}(A,\ensuremath{\omega})/\ensuremath{\omega}]\ensuremath{-}\mathrm{log}[\mathrm{max}(B,\ensuremath{\omega})/\ensuremath{\omega}]$, where the first and the second term are, respectively, the particle and hole contributions, A and B being the particle and hole energy cutoffs. The ac magnetoresistance has the same sign as ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$.
We analyze the weak-localization correction to the conductivity of a spinless two-leg ladder model in the limit of strong dephasing ${\ensuremath{\tau}}_{\ensuremath{\varphi}}\ensuremath{\ll}{\ensuremath{\tau}}_{\mathrm{tr}}$, paying particular attention to the presence of a magnetic field, which leads to an unconventional magnetoresistance behavior. We find that the magnetic field leads to three different effects: (i) negative magnetoresistance due to the regular weak-localization correction, (ii) effective decoupling of the two chains, leading to positive magnetoresistance, and (iii) oscillations in the magnetoresistance originating from the nature of the low-energy collective excitations. All three effects can be observed depending on the parameter range, but it turns out that large magnetic fields always decouple the chains and thus lead to the curious effect of magnetic-field-enhanced localization.
We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-21 models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number K⪢1 and weak dimensionless coupling g⪡1. The characteristic feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point $K\ensuremath{\ge}{K}_{c}(g)$, where the critical temperature Tc of the ordering transition vanishes. We find that the local probability distribution P(B) of the order parameter B has a long power-law tail in the region where B is much larger than its typical value B0. Near the quantum-critical point, at $K\ensuremath{\rightarrow}{K}_{c}(g)$, the typical value of the order parameter vanishes exponentially, ${B}_{0}\ensuremath{\propto}{e}^{\ensuremath{-}C/[K\ensuremath{-}{K}_{c}(g)]}$ while the spatial scale Ninh of the order parameter inhomogeneities diverges as ${[K\ensuremath{-}{K}_{c}(g)]}^{\ensuremath{-}2}$. In the disordered regime, realized at K<Kc(g) we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of ${K}^{\ensuremath{\ast}}(g)<K<{K}_{c}(g)$. It has two regimes of energies: at low excitation energies, $\ensuremath{\omega}<{\ensuremath{\omega}}_{d}(K,g)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\ensuremath{\omega}>{\ensuremath{\omega}}_{d}$ the level acquire a nonzero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller $K<{K}^{\ensuremath{\ast}}(g)$, where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.
Most of our quantitative understanding of disorder-induced metal-insulator transitions comes from numerical studies of simple noninteracting tight-binding models, like the Anderson model in three dimensions.An important outstanding problem is the fate of the Anderson transition in the presence of additional Hubbard interactions of strength U between particles.Based on large-scale numerics, we compute the position of the mobility edge for a system of two identical bosons or two fermions with opposite spin components.The resulting phase diagram in the interaction-energy-disorder space possesses a remarkably rich and counterintuitive structure, with multiple metallic and insulating phases.We show that this phenomenon originates from the molecular or scatteringlike nature of the pair states available at given energy E and disorder strength W .The disorder-averaged density of states of the effective model for the pair is also investigated.Finally, we discuss the implications of our results for ongoing research on many-body localization.