The goal of this diploma thesis is to develop a field theory, for a specific class of disordered bosonic systems, in order to study, whether disordered bosons generically \nhave universal statistical properties, similar to fermions. \nNow, if disordered bosons show system independent statistical features, the evident \nquestion is whether the bosonic universality classes are different from the fermionic \nones. For fermions, the classification in terms of symmetric spaces is a largely algebraical one, which happens in a complexified setting before compact or non-compact \nreal forms are specified. Hence one can rightly state that all effective bosonic non-linear \nσ-models are already classified within the ten fold way together with the fermionic ones. \nAnd indeed, in the system studied in [LSZ06], well known universal GUE statistics \nwere found in the bulk and Lück finds GOE statistics in [Lü09], both for systems of \ndisordered bosons. \nBut for bosons non-linear σ-models are not everything. A subtle point about noninteracting, i.e. quadratic, bosonic systems is stability. One single particle state \nof negative energy for a bosonic system immediately leads to an unbounded many \nparticle Hamiltonian. For fermionic systems, a finite number of such single particle \nstates is of no concern, due to the Pauli principle. This important difference leads \nto the well studied Gaussian ensembles not providing feasible distributions of bosonic \nHamiltonians. If, however, more complicated probability distributions are taken into \naccount, one has to be prepared to face a more complicated effective model than a \npure non-linear σ-model in the end. In this sense, an interesting question is whether \nbosonic systems if they show universal behaviour lead to novel universality classes. \nA hint to those might be the unusual density of states near zero energy that was \nfound in [LSZ06] or [GC02] and [GA04]. On physical grounds, one can expect the \ncommutation relations of bosons to have important effects at low frequencies, which \nfits to the observations of [GC02] and [GA05]. \nIn this thesis, we will develop and study a specific model, which is the next step \nafter the work of Lück, Sommers and Zirnbauer, [LSZ06], towards a more realistic \ndescription of a disordered system of bosonic degrees of freedom. As in [LSZ06], our \nmodel is still purely random, i.e. there is no limit of an underlying pure system and \nwe also stay with a model without global symmetries, such as time reversal or charge \nconjugation. But we turn from the homogeneous, zero-dimensional model to a spatially \nextended one and implement the physically important feature of locality. Furthermore, \nwe add an additional parameter to tune the number of modes per volume. In particular \nwe can hereby enforce a macroscopical number of zero modes. The goal is to study the \ndensity of states of the new model and therefore to develop an effective field theory. \nThroughout we will pay special attention to the region near zero frequency and watch \nout for unusual, possibly universal features.
The goal of this diploma thesis is to develop a field theory, for a specific class of disordered bosonic systems, in order to study, whether disordered bosons generically \nhave universal statistical properties, similar to fermions. \nNow, if disordered bosons show system independent statistical features, the evident \nquestion is whether the bosonic universality classes are different from the fermionic \nones. For fermions, the classification in terms of symmetric spaces is a largely algebraical one, which happens in a complexified setting before compact or non-compact \nreal forms are specified. Hence one can rightly state that all effective bosonic non-linear \nσ-models are already classified within the ten fold way together with the fermionic ones. \nAnd indeed, in the system studied in [LSZ06], well known universal GUE statistics \nwere found in the bulk and Lück finds GOE statistics in [Lü09], both for systems of \ndisordered bosons. \nBut for bosons non-linear σ-models are not everything. A subtle point about noninteracting, i.e. quadratic, bosonic systems is stability. One single particle state \nof negative energy for a bosonic system immediately leads to an unbounded many \nparticle Hamiltonian. For fermionic systems, a finite number of such single particle \nstates is of no concern, due to the Pauli principle. This important difference leads \nto the well studied Gaussian ensembles not providing feasible distributions of bosonic \nHamiltonians. If, however, more complicated probability distributions are taken into \naccount, one has to be prepared to face a more complicated effective model than a \npure non-linear σ-model in the end. In this sense, an interesting question is whether \nbosonic systems if they show universal behaviour lead to novel universality classes. \nA hint to those might be the unusual density of states near zero energy that was \nfound in [LSZ06] or [GC02] and [GA04]. On physical grounds, one can expect the \ncommutation relations of bosons to have important effects at low frequencies, which \nfits to the observations of [GC02] and [GA05]. \nIn this thesis, we will develop and study a specific model, which is the next step \nafter the work of Lück, Sommers and Zirnbauer, [LSZ06], towards a more realistic \ndescription of a disordered system of bosonic degrees of freedom. As in [LSZ06], our \nmodel is still purely random, i.e. there is no limit of an underlying pure system and \nwe also stay with a model without global symmetries, such as time reversal or charge \nconjugation. But we turn from the homogeneous, zero-dimensional model to a spatially \nextended one and implement the physically important feature of locality. Furthermore, \nwe add an additional parameter to tune the number of modes per volume. In particular \nwe can hereby enforce a macroscopical number of zero modes. The goal is to study the \ndensity of states of the new model and therefore to develop an effective field theory. \nThroughout we will pay special attention to the region near zero frequency and watch \nout for unusual, possibly universal features.
We consider classical normal modes and noninteracting bosonic excitations in disordered systems. We emphasize generic aspects of such problems and parallels with disordered, noninteracting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground-state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of noninteracting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low-frequency limit, above a critical dimension dc, which in different circumstances takes the values dc=2 and dc=0. For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ is a universal feature in systems with ground states that depend on the disorder realization. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension.
We consider noninteracting bosonic excitations in disordered systems, emphasizing generic features of quadratic Hamiltonians in the absence of Goldstone modes. We discuss relationships between such Hamiltonians and the symmetry classes established for fermionic systems. We examine the density $\ensuremath{\rho}(\ensuremath{\omega})$ of excitation frequencies $\ensuremath{\omega}$, showing how the universal behavior $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{4}$ for small $\ensuremath{\omega}$ can be obtained both from general arguments and by detailed calculations for one-dimensional models.
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 have seen in the previous chapter how the existence of an order-disorder duality structure allows the obtainment of a full quantum theory of topological excitations. In the present chapter, conversely, we will see how the same structure is at the very roots of a method by which one can generate, out of bosonic fields, new composite fields with different statistics, either fermionic or generalized. In the first case, the method is usually called bosonization and allows a full description of fermions within the bosonic theory, whereas in the second, the method provides a complete description of anyons, as the particles with generalized statistics have been called [59], also in the framework of the bosonic theory. From their inception, both bosonization and the construction of fields with generalized statistics are deeply related to the order-disorder duality structure, since both fermion and anyon fields can be expressed as products of order and disorder operators, which respectively carry charge and topological charge [31, 32, 45]. The statistics of the resulting composite field, then, is proportional to the product of the charge and topological charge borne by this field. In this chapter we will expose the basic features of bosonization, as well as its relation with generalized statistics, and apply it to different quantum field theory systems in D = 2, 3 and 4, thereby verifying it is an extremely powerful tool for solving interacting field theories. Indeed, in some cases, this method can lead to the exact solution of highly nonlinear systems. The Symmetrization Postulate and Its Violation: Bosons, Fermions and Anyons Physical systems most frequently possess identical objects. Such is the case, for instance, of electrons, photons, phonons, etc. Also billiard balls, perhaps. The description of identical objects, however, changes dramatically according to whether we use a classical or quantum-mechanical approach. In the first case, one can always distinguish identical objects because we can follow their trajectories, whereas in the second case, this is no longer possible. Indeed, within a quantummechanical framework the wave-functions of different identical particles may overlap, thus causing us at the end of a physical process, to completely loose track of which particle corresponds to each of the initial ones.
Disorder is everywhere in nature and it has a fundamental impact on the behavior of many quantum systems. The presence of a small amount of disorder, in fact, can dramatically change the coherence and transport properties of a system. Despite the growing interest in this topic, a complete understanding of the issue is still missing. An open question, for example, is the description of the interplay of disorder and interactions, which has been predicted to give rise to exotic states of matter such as quantum glasses or many-body localization. In this review, we will present an overview of experimental observations with disordered quantum gases, focused on one-dimensional bosons, and we will connect them with theoretical predictions.
The topological classification of insulators and superconductors, the "ten-fold way", is grounded on fermionic many-body symmetries and has had a dramatic impact on many fields of physics. Therefore, it seems equally important to investigate a similar approach for bosons as tightly analogous to the fermionic prototype as possible. There are, however, several obstacles coming from the fundamental physical differences between fermions and bosons. Here, we propose a theory of free boson topology (topological classification and bulk-boundary correspondence) protected by bosonic many-body symmetry operations, namely, squeezing transformations, particle number, and bosonic time reversal. We identify two symmetry classes that are topologically non-trivial in one dimension. They include key models like the bosonic Kitaev chain, protected by a squeezing symmetry within our framework, and the celebrated bosonic SSH model, protected by a squeezing symmetry and particle number. To provide a robust experimental platform for testing our theory, we introduce a new quantum meta-material: photo-magnonic crystals. They are remarkable for their experimental flexibility and natural affinity for displaying band topological physics at microwave frequencies. We engineer a many-body symmetry-protected topological photo-magnonic chain with boundary modes mandated by a Pfaffian invariant. Using an electromagnetic finite-element modelling, we simulate its reflection and transmission and identify experimental signatures of its boundary modes. The experimental tuning of the crystal to its symmetry-protected topological phase is also addressed. Our modelling of the photo-magnonic chain provides a thorough blueprint for its experimental realisation and the unambiguous observation of its exotic physics.