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arXiv:2307.08748

Unconventional superfluidity and quantum geometry of topological bosons

Lukin, Ilya, Andrii Sotnikov, Alexander Kruchkov

Published Jul 17, 2023• 0 citations
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Abstract

We investigate superfluidity of bosons in gapped topological bands and discover a new phase that has no counterparts in the previous literature. This phase is characterized by a highly unconventional modulation of the order parameter, breaking the crystallographic symmetry, and for which the condensation momentum is neither zero nor any other high-symmetry vector of the Brillouin zone. This unconventional structure impacts the spectrum of Bogoliubov excitations and, consequently, the speed of sound in the system. Even in the case of perfectly flat bands, the speed of sound and Bogoliubov excitations remain nonvanishing, provided that the underlying topology and quantum geometry are nontrivial. Furthermore, we derive detailed expressions for the superfluid weight using the Popov hydrodynamic formalism for superfluidity and provide estimates for the Berezinskii-Kosterlitz-Thouless temperature, which is significantly enhanced by the nontriviality of the underlying quantum metric. These results are applicable to generic topological bosonic bands, with or without dispersion. To illustrate our findings, we employ the Haldane model with a tunable bandwidth, including the narrow lowest-band case. Within this model, we also observe a re-entrant superfluid behavior: As the Haldane's magnetic flux is varied, the Berezinskii-Kosterlitz-Thouless transition temperature initially decreases to almost zero, only to resurface with renewed vigor.

DOI
10.48550/arxiv.2307.08748

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We investigate superfluidity of bosons in gapped topological bands and discover a new phase that has no counterparts in the previous literature. This phase is characterized by a highly unconventional modulation of the order parameter, breaking the crystallographic symmetry, and for which the condensation momentum is neither zero nor any other high-symmetry vector of the Brillouin zone. This unconventional structure impacts the spectrum of Bogoliubov excitations and, consequently, the speed of sound in the system. Even in the case of perfectly flat bands, the speed of sound and Bogoliubov excitations remain nonvanishing, provided that the underlying topology and quantum geometry are nontrivial. Furthermore, we derive detailed expressions for the superfluid weight using the Popov hydrodynamic formalism for superfluidity and provide estimates for the Berezinskii-Kosterlitz-Thouless temperature, which is significantly enhanced by the nontriviality of the underlying quantum metric. These results are applicable to generic topological bosonic bands, with or without dispersion. To illustrate our findings, we employ the Haldane model with a tunable bandwidth, including the narrow lowest-band case. Within this model, we also observe a re-entrant superfluid behavior: As the Haldane's magnetic flux is varied, the Berezinskii-Kosterlitz-Thouless transition temperature initially decreases to almost zero, only to resurface with renewed vigor.
We consider the superfluid weight, speed of sound and excitation fraction of a flat band Bose–Einstein condensate (BEC) within multiband Bogoliubov theory. The superfluid weight is calculated by introducing a phase winding and minimizing the free energy with respect to it. We find that the superfluid weight has a contribution arising from the change in the condensate density and chemical potential upon the phase twist that has been neglected in the previous literature. We also point out that the speed of sound and the excitation fraction are proportional to orbital-position-independent generalizations of the quantum metric and the quantum distance, and reduce to the usual quantum metric (Fubini–Study metric) and the Hilbert–Schmidt quantum distance only in special cases. We derive a second-order perturbation correction to the dependence of the speed of sound on the generalized quantum metric, and show that it compares well with numerical calculations. Our results provide a consistent connection between flat band BEC and quantum geometry, with physical observables being independent of the orbital positions for fixed hopping amplitudes, as they should, and complete formulas for the evaluation of the superfluid weight within the Bogoliubov theory. We discuss the limitations of the Bogoliubov theory in evaluating the superfluid weight.
Quantum geometry strongly impacts physical properties in flat-band systems. We consider its role in bosonic condensation and superfluidity on flat bands, and show that the superfluid weight has an important contribution proportional to the condensate quantum metric. Based on this result, we uncover conditions under which flat-band superfluidity is unlikely. For instance, we find that stable flat-band superfluidity in a two-dimensional system requires at least three bands within Bogoliubov theory. Because the quantum geometry at the condensation momentum plays a disproportionately large role, a large integrated quantum metric is not sufficient for flat-band superfluidity, but how the quantum metric is distributed in the Brillouin zone is crucial.
In this thesis, superconductivity of fermions in two-dimensional Hubbard lattice models is theoretically studied. Each of the studied lattice structures features intriguing properties that provide valuable insight to understand superconductivity in general. While real condensed matter systems are too complicated to tackle with theoretical tools, the Hubbard model is one of the simplest quantum many-body models to possess a wide spectrum of physically relevant phenomena, from superconductivity to magnetism and topological order. Hubbard models are also interesting as they can be experimentally realized and accurately controlled by utilizing ultracold gases. The first part of the thesis explains the required theoretical background of superconductivity in the level of mean-field and linear response theories. Specifically, we primarily focus on the superfluid weight which implies the existence of the Meissner effect and dissipationless current, defining properties of superconductors. We further discuss how the Berezinskii-Kosterlitz-Thouless (BKT) transition, related to the loss of superconductivity due to the thermal fluctuations of the superconducting order parameter in high enough temperatures, can be accessed via the superfluid weight. In the second part, the main results are discussed. We show how the superconductivity of lattice models featuring dispersionless Bloch bands, so-called flat bands, can be large and originating from the quantum geometric properties of the Bloch states of the flat bands. This is in contrast to conventional dispersive bands where superconductivity is determined by the effective mass of the particles, i.e. by the dispersion relation of the Bloch bands alone and not by the Bloch states. This is well demonstrated in the Lieb lattice geometry, studied in Publication I, which features both a strictly flat and two dispersive Bloch bands, allowing direct comparison of superconductivity in dispersive and flat bands. We show how the superconductivity of the flat band is more robust than that of the dispersive bands and how the superfluid weight behaves non-monotonically as a function of the interaction strength. This non-monotonic behavior should be experimentally accessible. Flat band superconductivity is also the topic of Publication III, where the experimentally realized twisted bilayer graphene (TBG) is studied with local and non-local attractive interactions. We predict qualitative differences between local and non-local interaction schemes which could be distinguished experimentally and reveal the quantum geometric origin of superconductivity in TBG systems in case of both the assumed interaction schemes. Exotic Fulde-Ferrell (FF) superfluid phases, characterized by the non-zero momentum of Cooper pairs, were explored in Publication II. We show how the FF phases can be stabilized against thermal fluctuations in the presence of spin-orbit coupling (SOC) and that topologically non-trivial FF states, created with SOC and Zeeman fields, can be found at finite temperatures.; Tassa…
Ultracold quantum gases are highly controllable and thus capable of simulating difficult quantum many-body problems ranging from condensed matter physics to astrophysics. Although experimental realizations have so far been restricted to flat geometries, recently also curved quantum systems, with the prospect of exploring tunable geometries, have been produced in microgravity facilities as ground-based experiments are technically limited. Here, we analyze bubble-trapped condensates, in which the atoms are confined on the surface of a thin spherically symmetric shell by means of external magnetic fields. A thermally induced proliferation of vorticity yields a vanishing of superfluidity. We describe the occurrence of this topological transition by conceptually extending the theory of Berezinskii, Kosterlitz, and Thouless for infinite uniform systems to such finite-size systems. Unexpectedly, we find universal scaling relations for the mean critical temperature and the finite width of the superfluid transition. Furthermore, we elucidate how they could be experimentally observed in finite-temperature hydrodynamic excitations.
Topological phase transitions represent a new paradigm beyond conventional Landau-Ginzburg symmetry breaking. Even in the absence of symmetry breaking, the symmetries of a Hamiltonian play an important role in stabilizing topological signatures. In this thesis, we will study the interplay between symmetry and topology in many-body systems, with three distinct physical setups in mind. First, we investigate a two-dimensional system of ultracold bosons which is condensed in the p-band of a triangular lattice. Certain smooth interaction profiles result in non-Abelian symmetry generators of the order parameter manifold, leading to anomalous physical consequences. These include a lack of Berezinskii-Kosterlitz-Thouless transition due to unstable vortex configurations, additional Goldstone modes, and a marginally divergent normal fluid density which implies lack of superfluidity in the thermodynamic limit. Our results suggest possible connections to solid state experiments on 4He films. We then show that disorder which keeps a Hamiltonian in the same Altland-Zirnbauer symmetry class is capable of inducing a topological transition. This is achieved by considering the normalizability of Majorana edge modes in a topological superconductor, and corroborated with entanglement metrics. While degeneracies of the many-body spectrum are robust with respect to weak perturbations, strong disorder can both promote and destroy a topological index. Lastly, we study topological edge modes and degeneracies in atomic, molecular, and optical systems which evolve non-unitarily in time or space due to non-equilibrium effects, including: condensate instabilities after a quench, decoherence channels from an external environment, and dissipation into a medium. Mathematically, the non-unitary evolution is generated by effective Hamiltonians which break Hermiticity. These systems can possess uniquely non-Hermitian symmetries which are absent from equilibrium counterparts. We present a symmetry-based classification, and investigate several systems which can be solved exactly.
Using a nonperturbative renormalization-group technique, we compute the momentum and frequency dependence of the anomalous self-energy and the one-particle spectral function of two-dimensional interacting bosons at zero temperature. Below a characteristic momentum scale k_{G}, where the Bogoliubov approximation breaks down, the anomalous self-energy develops a square-root singularity and the Goldstone mode of the superfluid phase (Bogoliubov sound mode) coexists with a continuum of excitations, in agreement with the predictions of Popov's hydrodynamic theory. Thus our results provide a unified picture of superfluidity in interacting boson systems and connect Bogoliubov's theory (valid for momenta larger than k_{G}) to Popov's hydrodynamic approach.