Orbital ordering of ultracold alkaline-earth atoms in optical lattices

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Orbital ordering of ultracold alkaline-earth atoms in optical lattices

Andrii Sotnikov, Nelson Darkwah Oppong, Yeimer Zambrano, and Agnieszka Cichy

Phys. Rev. Research 2, 023188 – Published 19 May 2020

Abstract

We report on a dynamical mean-field theoretical analysis of emerging low-temperature phases in multicomponent gases of fermionic alkaline-earth(-like) atoms in state-dependent optical lattices. Using the example of ${}^{173}{Yb}$ atoms, we show that a two-orbital mixture with two nuclear spin components is a promising candidate for studies of not only magnetic but also staggered orbital ordering peculiar to certain solid-state materials. We calculate and study the phase diagram of the full Hamiltonian with parameters similar to existing experiments and reveal an antiferro-orbital phase. This long-range-ordered phase is inherently stable, and we analyze the change of local and global observables across the corresponding transition lines, paving the way for experimental observations. Furthermore, we suggest a realistic extension of the system to include and probe a Jahn-Teller source field playing one of the key roles in real crystals.

Received 26 February 2020
Accepted 30 April 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023188

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.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Cold gases in optical lattices Fermi gases Orbital order Phase transitions

Ultracold gases

Dynamical mean field theory

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Andrii Sotnikov ^1,2,*, Nelson Darkwah Oppong ^3,4,5, Yeimer Zambrano ⁶, and Agnieszka Cichy⁶

¹Akhiezer Institute for Theoretical Physics, NSC KIPT, Akademichna 1, 61108 Kharkiv, Ukraine
²Karazin Kharkiv National University, Svobody Sq. 4, 61022 Kharkiv, Ukraine
³Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
⁴Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany
⁵Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany
⁶Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznan, Poland

^*a_sotnikov@kipt.kharkov.ua

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Vol. 2, Iss. 2 — May - July 2020

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Images

Figure 1
(a) Schematic representation of an exemplary state from the Hilbert space of Eq. (1) illustrating the interaction parameters. The lattice bonds are shown as gray lines and the blue (yellow) circles refer to $g$ ( $e$ ) atoms in different spin states as indicated by the arrows. (b) Illustration of the state-dependent lattice potential (solid lines) which introduces distinct hopping amplitudes $t_{g} > t_{e}$ for $g$ and $e$ atoms (blue and yellow circles).
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Figure 2
(a) Phase diagram obtained from DMFT (see Appendix pp3) for different densities $n_{g}$ and $n_{e}$ of ${}^{173}{Yb}$ in a SDL with the polarizability ratio $p = 2.1$ . The filled contours indicate the critical temperatures $T_{c}$ of the ordered phases, which are antiferro-orbital (AFO), antiferromagnetic (AFM), ferromagnetic (FM), and antiferromagnetic in $g$ ( $g$ -AFM). The gray-shaded regions at $T_{c} = 0$ correspond either to a normal phase without long-range order or a regime with phase-separation and the dotted lines indicate constant total filling. (b) Illustration of the local orbital and magnetic order in the different phases. Blue and yellow circles correspond to $g$ and $e$ atoms, arrows indicate the spin state, and lattice bonds are shown as gray lines. In the AFM and FM phases, the $g$ and $e$ spins are parallel on doubly-occupied lattice sites.
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Figure 3
(a) Critical temperature and (b) critical entropy per particle of the AFO phase for variable polarizability ratio $p$ at fixed densities $n_{g} = 1$ and $n_{e} = 0.5$ . The yellow star indicates $p = 2.1$ , which is the central value of our study. Green circles refer to values obtained from DMFT and lines serve as a guide to the eye.
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Figure 4
Site-averaged double occupancy $D_{g g}$ (a) as a function of temperature at $n_{g} = 1$ , $n_{e} = 0.5$ and as a function of the orbital density (b) $n_{g}$ and (c) $n_{e}$ at fixed $T / t_{g} = 0.2$ . We show the result for the AFO phase in green and for an artificially-restricted normal phase in dark gray. Green circles refer to values obtained from DMFT and lines serve as a guide to the eye.
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Figure 5
(a)–(c) Radial density profiles in a harmonic trap with the potential $V_{ho} / t_{g} = 2.3 \times 10^{- 2} {(r / a_{lat})}^{2}$ , which corresponds to the trapping frequency $ω = 2 π \times 60 Hz$ . Here, $a_{lat} = λ / 2$ is the lattice constant. We show the profiles for fixed atom number $N \approx 1.7 \times 10^{3}$ and temperatures (a) $T / t_{g} = 0.1$ , (b) 0.3 (slightly below $T_{c}$ ), as well as (c) 0.6. Solid lines refer to the total density $n = n_{e} + n_{g}$ while dotted lines show $n_{e}$ . The amplitude of the charge-density wave, $(n_{i} - n_{i + 1}) / (n_{i} + n_{i + 1})$ , indicates the orbitally-ordered region of the trap and is shown as thick green line in the background. (d),(e) Local density of the two orbitals on neighboring lattice sites, $j = i$ (red squares) and $j = i + 1$ (blue circles), as a function of temperature at mean density $n_{g} = 1$ and $n_{e} = 0.5$ . The solid lines serve as a guide to the eye.
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Figure 6
(a) Schematic representation of the lattice potentials for $e$ (yellow lines) and $g$ atoms (blue lines) in the presence (solid) and absence (dotted) of the additional superlattice potential producing an offset $Δ_{JT}^{(af)}$ in analogy to the antiferrodistortive JTE. (b) Illustration of changing the orbital densities at constant $n = 1.5$ in the AFO phase and the resulting ferrodistortive JTE analog quantified by $Δ_{JT}^{(f)}$ . We show the limit of strongly-bound $g$ atoms (blue circles) and single $e$ (yellow circles) atoms, and their renormalized chemical potentials ${\tilde{μ}}_{g}$ and ${\tilde{μ}}_{e}$ . (c),(d) Local (normalized) density of $g$ atoms on neighboring lattice sites, $j = 2 i$ (red squares) and $j = 2 i + 1$ (blue circles), for (c) probing the antiferrodistortive JTE with a variable superlattice potential at $n_{g} = 1$ , $n_{e} = 0.5$ , and $T / t_{g} = 0.36$ and (d) probing the ferrodistortive JTE at constant $n = n_{e} + n_{g} = 1.5$ and variable $n_{g} / n_{e}$ at $T / t_{g} = 0.2$ . The solid lines serve as a guide to the eye.
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