Exploring phononlike interactions in one-dimensional Bose-Fermi mixtures

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Exploring phononlike interactions in one-dimensional Bose-Fermi mixtures

Axel Gagge, Th. K. Mavrogordatos, and Jonas Larson

Phys. Rev. Research 6, 013138 – Published 2 February 2024

Abstract

With the objective of simulating the physical behavior of electrons in a dynamic background, we investigate a cold atomic Bose-Fermi mixture confined in an optical lattice potential solely affecting the bosons. The bosons, residing in the deep superfluid regime, inherit the periodicity of the optical lattice, subsequently serving as a dynamic potential for the polarized fermions. Owing to the atom-phonon interaction between the fermions and the condensate, the coupled system exhibits a Berezinskii-Kosterlitz-Thouless transition from a Luttinger liquid to a Peierls phase. However, under sufficiently strong Bose-Fermi interaction, the Peierls phase loses stability, leading to either a collapsed or a separated phase. We find that the primary function of the optical lattice is to stabilize the Peierls phase. Furthermore, the presence of a confining harmonic trap induces a diverse physical behavior, surpassing what is observed for either bosons or fermions individually trapped. Notably, under attractive Bose-Fermi interaction, the insulating phase may adopt a fermionic wedding-cake-like configuration, reflecting the dynamic nature of the underlying lattice potential. Conversely, for repulsive interaction, the trap destabilizes the Peierls phase, causing the two species to separate.

Received 20 March 2023
Accepted 3 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013138

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Bose-Einstein condensates Cold atoms & matter waves Cold gases in optical lattices Fermi gases

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Axel Gagge ¹, Th. K. Mavrogordatos ^1,2, and Jonas Larson ^1,*

¹Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden
²ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

^*jonas.larson@fysik.su.se

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Vol. 6, Iss. 1 — February - April 2024

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Figure 1
Schematic description of the Peierls distortion. At a fermion filling corresponding to a wave number $k_{F}$ , opening up a gap of the dispersion at $k_{F}$ will lower the kinetic energy of the fermions, as depicted in (a). The gap opening results from a lattice distortion of the dynamical lattice, as shown in (b). Here, the period has been doubled through shifting every second atom by a distance $δ$ , and consequently, the size of the Brillouin zone has been halved. Thus, for half filling, we encounter a realization of the Su-Schrieffer–Heeger (SSH) model, while for other fillings, the lattice distortion will generate another periodicity. In the normal phase, the condensate will share the same periodicity as the lattice and predominantly populate the lattice minima (c). In the Peierls phase, for half filling, the bosonic density will alternate between every second site (d). Upon comparison of (b) and (d), we note that the periodicity has been broken in two different ways: in (b), the densities are held fixed, but the locations of the sites have been shifted, while in (d), the locations are fixed and the densities have been altered. The former can be seen as a ‘phase modulation,’ and the latter as an “amplitude modulation” of the densities. Finally, in (e), we depict the collapsed phase where all atoms, bosons, and fermions have been compressed to a small region of the lattice.
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Figure 2
Phase properties in the absence of a trap. (a) Phase diagram in the $(g_{bf}, g_{b})$ plane, marking the different phases: LL, Peierls, collapsed, and separated. The solid black lines mark the boundaries of first-order transitions, while the solid red lines represent the BKT transitions. The mean-field prediction (8) is plotted with dotted black lines. The critical coupling $g_{bf}^{c}$ for the LL-to-Peierls transition is calculated from a least square fit of the Peierls gap to the expression (7), using the hybrid mean-field method. (b) Same as for (a) but in the $(g_{bf}, V_{0})$ plane. Here it becomes clear that the presence of the optical lattice alters the phase diagram quantitatively, in particular through stabilizing the Peierls phase. (c) The Peierls gap as a function of $g_{bf}$ , once more calculated using the hybrid code, for $V_{0} = 0$ (yellow), 4 (green), and 16 (blue), fitted to the BKT formula (7) (dotted lines). (d) Boson and fermion densities in the Peierls phase for the parameters $V_{0} = 1$ , $g_{b} = 0.4$ , and $g_{bf} = - 1$ . In all plots, we consider half filling, $ν_{f} = 1 / 2$ .
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Figure 3
Density profiles in the presence of a harmonic trap. The top row shows the boson density (magenta) and the fermion density (green), while the bottom row shows the fermion density per site. Only $x > 0$ is shown since the densities are symmetric around $x = 0$ . Both plots pertain to the ground state in a BF mixture in a trap, obtained using the hybrid method. (a) “Wedding-cake” structured fermionic insulator, with parameters $V_{0} = 4$ , $g_{bf} = - 48$ , $ω_{b} = 0.03$ , and $ω_{f} = 0.3$ . (b) Peierls phase, with parameters $V_{0} = 1$ , $g_{bf} = - 4$ , $ω_{b} = 0.072$ , and $ω_{f} = 0.0072$ . The inset in the upper plot zooms on the density profile in the region $0 \leq x / a \leq 8$ , evincing density modulations in the bulk, with their origin in the Peierls instability. (c) Separated phase with parameters $V_{0} = 4$ , $g_{bf} = 4$ , $ω_{b} = 0.08$ , and $ω_{f} = 0.8$ . In all simulations, we have set the remaining parameters to $g_{b} = 0.4$ , $r = 0.04$ , $N_{f} = 32$ , and $N_{b} = 400$ .
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Figure 4
From a Mott to a Peierls insulator with extended eigenstates in a trap. One-particle density matrix, $| 〈 {\hat{ψ}}_{f} (x) {\hat{ψ}}_{f}^{†} (x^{'}) 〉 |^{2}$ , where only the top-right quadrant is plotted owing to the symmetry about $x = 0$ . Two cases are shown: (a) a trapped mixture in the wedding cake phase, for the same state as the one presented in the left column of Fig. 3; (b) Peierls phase for the same state as in the central column of Fig. 3. The nondiagonal part of the one-particle density matrix is significantly suppressed in the wedding cake phase, where the decay is Gaussian and on a length scale similar to the characteristic length of the optical lattice. In the Peierls phase, there are instead periodic oscillations signifying coherence.
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Figure 5
Scaling of the two-point density matrix with the width of the harmonic trap. The one-point density matrix $n (x, x^{'})$ is calculated for $x^{'} = - x$ and plotted with $x$ scaled with $a \sqrt{k}$ . The oscillations of the one-point density matrix are fitted to the expression $n (x, - x) = C sinc (b x)$ in line with Eq. (C2). Left column: normal phase ( $g_{bf} = 0, ω_{0, b} = 0.005, ω_{0, f} = 0.07$ ). Right column: Peierls phase ( $g_{bf} = - 1.5, ω_{0, b} = ω_{0, f} = 0.0036$ ). The harmonic confinements have been set as $ω_{b / f} = k ω_{0, b / f}$ , with $k = 1, 2, 4$ for the top, middle and bottom rows. In all simulations, we have fixed the remaining parameters to $g_{b} = 0.1$ and $V_{0} = 0.25$ .
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