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arXiv:0902.2822

Transport Studies of Lattice Bosons: Paradigms for Fluctuating Superconductivity

Assa Auerbach, Netanel H. Lindner

Published Feb 17, 2009• 0 citations
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Abstract

A strong periodic potential generally enhances the short wavelength fluctuations of a superfluid beyond the validity of standard continuum approaches. Here we report some recent results on hard core bosons on finite lattices. We find several interesting effects of the periodic potential on the ground state, vortex dynamics, and and Hall conductivity. For example, the Magnus field on a vortex abruptly reverses direction at half filling. A rotating Bose condensate on an optical lattice may allow an experimental test of our results. Insight may also be gained about strongly fluctuating superconductors modelled by charge 2e lattice bosons.

DOI
10.48550/arxiv.0902.2822

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A strong periodic potential generally enhances the short wavelength fluctuations of a superfluid beyond the validity of standard continuum approaches. Here we report some recent results on hard core bosons on finite lattices. We find several interesting effects of the periodic potential on the ground state, vortex dynamics, and and Hall conductivity. For example, the Magnus field on a vortex abruptly reverses direction at half filling. A rotating Bose condensate on an optical lattice may allow an experimental test of our results. Insight may also be gained about strongly fluctuating superconductors modelled by charge 2e lattice bosons.
Magnetotransport of hard-core bosons is studied using an XXZXXZXXZ quantum spin model representation, appropriately gauged on the torus to allow for an external magnetic field. We find strong lattice effects near half filling. An effective quantum mechanical description of the vortex degrees of freedom is derived. Using semiclassical and numerical analysis we compute the vortex-hopping energy tV{t}_{\mathsf{V}}tV​, which at half filling is close to magnitude of the boson hopping energy. The critical quantum melting density of the vortex lattice is estimated at $6.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ vortices per unit cell. The Hall conductance is computed from the Chern numbers of the low-energy eigenstates. At zero temperature, it reverses sign abruptly at half filling. At precisely half filling, all eigenstates are doubly degenerate for any odd number of flux quanta. We prove the exact degeneracies on the torus by constructing an SU(2) algebra of point-group symmetries, associated with the center of vorticity. This result is interpreted as if each vortex carries an internal spin-half degree of freedom, which can manifest itself as a charge density modulation in its core. Our findings suggest interesting experimental implications for vortex motion of cold atoms in optical lattices and magnet transport of short coherence length superconductors.
We study hard-core lattice bosons in a magnetic field near half filling. The bare vortex hopping rate is extracted from exact diagonalizations of square clusters. We deduce a quantum melting of the vortex lattice above vortex density of 6.5x10(-3) per lattice site. The Hall conductivity reverses sign abruptly as the density crosses half filling, where its characteristic temperature scale vanishes. We prove that at precisely half filling, each vortex carries a spin-1/2 quantum number ("v spin"). Experimental implications of these results are discussed.
Conductivities and Hall coefficients of two-dimensional hard-core bosons (HCB) are calculated using the thermodynamic expansions of Kubo formulas. At temperatures above the superfluid transition, the resistivity rises linearly and is weakly dependent on boson filling. The zeroth-order Hall coefficient diverges toward zero and unit fillings, and reverses its sign at half filling. The correction terms, which are calculated up to fourth (Krylov) orders, do not alter this behavior. The high temperature thermal Hall coefficient is reversed relative to the electric Hall coefficient. We discuss relevance of HCB transport to the metallic state of short coherence length superconductors.
Two-dimensional hard core bosons suffer strong scattering in the high-temperature resistive state at half filling. The dynamical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ is calculated using nonperturbative tools such as continued fractions, series expansions, and exact diagonalization. We find a large temperature range with linearly increasing resistivity and broad dynamical conductivity, signaling a breakdown of Boltzmann-Drude quasiparticle transport theory. At zero temperature, a high-frequency peak in $\ensuremath{\sigma}(\ensuremath{\omega})$ appears above a ``Higgs mass'' gap and corresponds to order-parameter magnitude fluctuations. We discuss the apparent similarity between conductivity of hard core bosons and phenomenological characteristics of cuprates, including the universal scaling of Homes et al. [Nature (London) 430, 539 (2004)].
Atomic clouds in a rotating optical lattice are at the intellectual intersection of several major paradigms of condensed matter physics. An optical lattice simulates the periodic potential ubiquitous in solid state physics, while rotation probes the superfluid character of these cold atomic gases by driving the formation of quantized vortices. Here we explore the theory of vortices in an optical lattice. We first provide a detailed introduction section aimed at providing the reader with the information necessary to understand and appreciate the research presented in later chapters. Next we study an infinite square lattice configuration of vortices in a rotating optical lattice near the superfluid-Mott-insulator transition. We find a series of abrupt structural phase transitions where vortices are pinned with their cores only on plaquettes or only on sites. We discuss connections between these vortex structures and the Hofstadter-butterfly spectrum of free particles on a rotating lattice. We then investigate vortex configurations within a harmonically trapped Bose-Einstein condensate in a rotating optical lattice. We find that proximity to the Mott insulating state dramatically affects the vortex structures. To illustrate we give examples in which the vortices: (i) all sit at a fixed distance from the center of the trap, forming a ring, or (ii) coalesce at the center of the trap, forming a giant vortex. We model time-of-flight expansion to demonstrate the experimental observability of our predictions. Finally for a trapped gas far from the Mott regime, the competition between vortex-vortex interactions and pinning to the optical lattice results in a complicated energy landscape, which leads to hysteretic evolution. The qualitative structure of the vortex configurations depends on the commensurability between the vortex density and the site density - with regular lattices when these are commensurate, and concentric rings when they are not. Again we model the imaging of these structures by calculating time-of-flight column densities. As in the absence of the optical lattice, the vortices are much more easily observed in a time-of-flight image than in-situ.
Quantum simulations are quickly becoming an indispensable tool for studying particle transport in correlated lattice models. One of the central topics in the study of transport is bad-metal behavior, characterized by the direct-current resistivity linear in temperature. In the fermionic Hubbard model, optical conductivity has been studied extensively, and a recent optical lattice experiment demonstrated bad-metal behavior in qualitative agreement with theory. Far less is known about transport in the bosonic Hubbard model. We investigate the conductivity in the Bose-Hubbard model and focus on the regime of strong interactions and high temperatures. We use numerically exact calculations for small lattice sizes. At weak tunneling, we find multiple peaks in the optical conductivity that stem from the Hubbard bands present in the many-body spectrum. This feature slowly washes out as the tunneling rate gets stronger. At high temperature, we identify a regime of TTT-linear resistivity, as expected. When the interactions are very strong, the leading inverse-temperature coefficient in conductivity is proportional to the tunneling amplitude. As the tunneling becomes stronger, this dependence takes quadratic form. At very strong coupling and half filling, we identify a separate linear resistivity regime at lower temperature, corresponding to the hard-core boson regime. Additionally, we unexpectedly observe that at half filling, in a big part of the phase diagram, conductivity is an increasing function of the coupling constant before it saturates at the hard-core-boson result. We explain this feature based on the analysis of the many-body energy spectrum and the contributions to conductivity of individual eigenstates of the system.