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arXiv:2307.10012 · Physical review. B./Physical review. B

Drude weight and the many-body quantum metric in one-dimensional Bose systems

G. Salerno, Tomoki Ozawa, Päivi Törmä

Published Oct 11, 2023• 16 citations
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Abstract

We study the effect of quantum geometry on the many-body ground state of one-dimensional interacting bosonic systems. We find that the Drude weight is given by the sum of the kinetic energy and a term proportional to the many-body quantum metric of the ground state. Notably, the many-body quantum metric determines the upper bound of the Drude weight. We validate our results on the Creutz ladder, a flat-band model, using exact diagonalization at half and unit densities. Our work sheds light on the importance of the many-body quantum geometry in one-dimensional interacting bosonic systems.

DOI
10.1103/physrevb.108.l140503

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We study the effect of quantum geometry on the many-body ground state of one-dimensional interacting bosonic systems. We find that the Drude weight is given by the sum of the kinetic energy and a term proportional to the many-body quantum metric of the ground state. Notably, the many-body quantum metric determines the upper bound of the Drude weight. We validate our results on the Creutz ladder, a flat-band model, using exact diagonalization at half and unit densities. Our work sheds light on the importance of the many-body quantum geometry in one-dimensional interacting bosonic systems.
We introduce the concept of quantum weight as a ground-state property of quantum many-body systems that is encoded in the static structure factor and characterizes density fluctuation at long wavelengths. The quantum weight carries a wealth of information about dielectric responses and optical properties of the system and is closely related to its quantum geometry. For systems with short-range interactions or low-dimensional Coulomb systems, we show that the many-body quantum metric (which measures the change of the ground state under twisted boundary conditions) can be determined directly from the quantum weight. Notably, the quantum weight is a property of a single ground state and independent of boundary conditions in the thermodynamic limit. Our finding thus enables direct experimental measurement and numerical calculation of the many-body quantum metric. On the other hand, for three-dimensional Coulomb systems, we show that the quantum weight is distinct from the many-body quantum metric due to dielectric screening in three dimensions. We further use dielectric sum rules to derive upper and lower bounds on the quantum weight of real materials in terms of their electron density, static dielectric constant, and plasmon energy. Our work highlights quantum weight as a fundamental material parameter, which can be experimentally determined by x-ray scattering or electron-loss spectroscopy.
Quantum geometry provides important information about the structure and topology of quantum states in various forms of quantum matter. The information contained therein has profound effects on observable quantities such as superconducting weight, Drude weight, and optical responses. Motivated by the recent advances in flat-band interacting systems, we investigate the role of interaction effects on the quantum metric. By using the fermionic Creutz ladder as a representative system, we show that the repulsive Hubbard interaction monotonically suppresses the quantum metric. While the eigenstates and their overlap quantifying the quantum metric can be obtained exactly in the presence of interactions through exact diagonalization, this method is limited to small system sizes. Alternatively, two theoretical proposals, the generalized quantum metric and the dressed quantum metric, suggest using renormalized Green's functions to define the interacting quantum metric. By comparing these analytical approaches with results from exact diagonalization, we show that the dressed quantum metric provides a better fit to the exact diagonalization results. Our conclusion holds for both flat-band and dispersive systems.
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.
The quantum geometry of Bloch states fundamentally affects a wide range of physical phenomena. The quantum Hall effect, for example, is governed by the Chern number, and flat-band superconductivity by the distance between the Bloch states: the quantum metric. While understanding quantum geometry phenomena in the context of fermions is well established, less is known about the role of quantum geometry in bosonic systems where particles can undergo Bose-Einstein condensation (BEC). In conventional single-band or continuum systems, excitations of a weakly interacting BEC are determined by the condensate density and the interparticle interaction energy. In contrast to this, we discover here fundamental connections between the properties of a weakly interacting BEC and the underlying quantum geometry of a multiband lattice system. We show that, in the flat-band limit, the defining physical quantities of BEC, namely, the speed of sound and the quantum depletion, are dictated solely by the quantum geometry. We find that the speed of sound becomes proportional to the quantum metric of the condensed state. Furthermore, the quantum distance between the Bloch functions forces the quantum depletion and the quantum fluctuations of the density-density correlation to obtain finite values for infinitesimally small interactions. This is in striking contrast to dispersive bands where these quantities vanish with the interaction strength. Additionally, we show how in the flat-band limit the supercurrent is carried by the quantum fluctuations and is determined by the Berry connections of the Bloch states. Our results reveal how nontrivial quantum geometry allows reaching strong quantum correlation regime of condensed bosons even with weak interactions. This is highly relevant, for example, for polariton and photon BECs where interparticle interactions are inherently small. Our predictions can be experimentally tested with flat-band lattices already implemented in ultracold gases and various photonic platforms.
Flatband systems form a new class of materials that challenge the conventional wisdom of transport. The intrinsically strong electronic correlations combined with the vanishing kinetic energy scale suggest a sensitive dependence of transport properties on the flat band states and make interacting flat bands promising candidates for exotic quantum transport. Utilizing the Drude weight, we investigate the low-frequency spectral properties of the electrical conductivity within a controlled analytic treatment of the many-body response at temperatures above the bandwidth and the interaction strength and below the band gap. Focusing on this new transport regime, we demonstrate the potential of a quantum geometric approach for interacting systems and intermediate temperatures. The derived spectral weight yields unexplored four-point geometric contributions unrelated to the quantum metric, which questions the previously proposed projection methods. For long-ranged interactions, we show that the low-frequency spectral weight reduces to the variance of the Berry curvature.
We apply well-established finite-temperature quantum Monte Carlo techniques to one-dimensional Bose systems with soft- and hard-core constraints, as well as to spinless fermion systems. We give clear and robust numerical evidence that, as expected, no superfluid density for bosons or Meissner fraction for fermions is possible at any nonzero temperature in one-dimensional interacting Bose or Fermi lattice models, whereas a finite Drude weight is generally observed in gapless systems, in partial disagreement with previous expectations.