Equation of State and Thermometry of the 2D $\mathrm{SU}(N)$ Fermi-Hubbard Model

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Equation of State and Thermometry of the 2D $SU (N)$ Fermi-Hubbard Model

G. Pasqualetti, O. Bettermann, N. Darkwah Oppong, E. Ibarra-García-Padilla, S. Dasgupta, R. T. Scalettar, K. R. A. Hazzard, I. Bloch, and S. Fölling

Phys. Rev. Lett. 132, 083401 – Published 21 February 2024

See synopsis: A General Equation of State for a Quantum Simulator

Abstract

We characterize the equation of state (EoS) of the $SU (N > 2)$ Fermi-Hubbard Model (FHM) in a two-dimensional single-layer square optical lattice. We probe the density and the site occupation probabilities as functions of interaction strength and temperature for $N = 3$ , 4, and 6. Our measurements are used as a benchmark for state-of-the-art numerical methods including determinantal quantum Monte Carlo and numerical linked cluster expansion. By probing the density fluctuations, we compare temperatures determined in a model-independent way by fitting measurements to numerically calculated EoS results, making this a particularly interesting new step in the exploration and characterization of the $SU (N)$ FHM.

Received 30 May 2023
Accepted 9 January 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.083401

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. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Cold gases in optical lattices Equations of state Fermi gases

Mott insulators

Hubbard model SU(N) symmetries

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

synopsis

A General Equation of State for a Quantum Simulator

Published 23 February 2024

Researchers have characterized the thermodynamic properties of a model that uses cold atoms to simulate condensed-matter phenomena.

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Authors & Affiliations

G. Pasqualetti^1,2,3,*, O. Bettermann^1,2,3, N. Darkwah Oppong ^1,2,3, E. Ibarra-García-Padilla ^4,5,6,7, S. Dasgupta ^4,5, R. T. Scalettar ⁶, K. R. A. Hazzard ^4,5,6, I. Bloch^1,2,3, and S. Fölling ^1,2,3

¹Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany
²Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
³Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany
⁴Department of Physics and Astronomy, Rice University, Houston, Texas 77005-1892, USA
⁵Rice Center for Quantum Materials, Rice University, Houston, Texas 77005-1892, USA
⁶Department of Physics, University of California, Davis, California 95616, USA
⁷Department of Physics and Astronomy, San José State University, San José, California 95192, USA

^*giulio.pasqualetti@lmu.de

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Vol. 132, Iss. 8 — 23 February 2024

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Figure 1
Probing the 2D $SU (N)$ Fermi-Hubbard model with ultracold atoms. (a) The ${}^{1}{S_{0}}$ ground state of ${}^{173}{Yb}$ naturally features an SU(6) symmetry, which can be freely tuned to $N \leq 6$ by preparing a suitable combination of the nuclear spin states $m_{F} = - 5 / 2, - 3 / 2, \dots, + 5 / 2$ (colored circles and arrows). (b) Schematic of the experimental setup showing a gas in a single layer 2D square lattice with harmonic confinement detected with absorption imaging along gravity. The square lattice is created by superimposing two orthogonal retro-reflected in-plane lattices. (c) Spatial distribution of the density $n (x, y)$ for $N = 6$ . Each cloud shown in the horizontal frame has been prepared with the same initial entropy in the bulk and loaded into the lattice to a different $U / t$ value. (d) Singly occupied sites after parity projection. Each horizontal frame corresponds to the same state shown in the same column of (c). (e) Density profiles for the data shown in (c) and (d) along the corresponding dashed lines in the left frames. Each image was produced using the averaging of eight shots after center of mass alignment [68].
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Figure 2
Equation of state for the $SU (N)$ Fermi-Hubbard model with $N = 6$ (blue), $N = 4$ (purple), and $N = 3$ (red). (a)–(d) Density (circles) and singly occupied sites (diamonds) as a function of the chemical potential. Data for $N = 4$ and $N = 6$ have been offset by $0.5 n d^{2}$ and $1.0 n d^{2}$ along the vertical axis, respectively. Continuous lines associated to the density curves correspond to the fit of the EoS calculations to the total density as described in the text. The theory used for the fit is DQMC for $U / t = 2.3 (1)$ and NLCE for the other values of $U / t$ . The results from this fit model are also used to calculate the expected pair and single-site distribution measurement. The chemical potential is defined with respect to the reference half filling [ $n d^{2} (μ = 0) = N / 2$ ]. For each $U / t$ , we fit the average of 15 frames with similar atom number after center-of-mass alignment [68]. Error bars are the standard error of the mean (s.e.m.). (e) Temperature according to the fit of the EoS shown in (a)–(d). (f)–(h) Entropy per particle. Horizontal line: entropy in the 2D bulk before loading into the lattice; triangles, squares, hexagons: entropy in the lattice according to the fit of the EoS; small circles: entropy in the 2D bulk after a round-trip experiment. The entropy in the bulk takes into account the effect of the interactions and the 3D anisotropic density of states [68]. Error bars correspond to the s.e.m. of the fit results.
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Figure 3
(a) Measured density fluctuations (blue) for $N = 6$ as a function of the chemical potential for different interaction strengths. The data points have been obtained from the variance of 15 frames [same as Figs. 2] computed on spatially binned probe areas of size $\approx 5.1 \times 5.1 d^{2}$ ( $4 \times 4$ square camera pixels). The photon shot noise has been subtracted and a PSF correction has been taken into account [68]. The green line corresponds to the numerically differentiated compressibility $κ$ times the temperature $T_{EoS}$ obtained from the EoS fit of the averaged data, while the black dashed line corresponds to the theory-derived compressibility times $T_{EoS}$ . The vertical line indicates $μ (n d^{2} = 1)$ . The gray dashed line corresponds to the on-site density fluctuations $δ n_{0}^{2} = ⟨ {\hat{n}}^{2} ⟩ - ⟨ \hat{n} ⟩^{2}$ calculated with NLCE for $T_{EoS}$ . (b) Comparison of the temperatures $T_{FDT}$ (dark blue diamonds) and $T_{EoS}$ (light blue hexagons). Error bars are the s.e.m.
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Physical Review Letters

Equation of State and Thermometry of the 2D SU(N) Fermi-Hubbard Model

G. Pasqualetti, O. Bettermann, N. Darkwah Oppong, E. Ibarra-García-Padilla, S. Dasgupta, R. T. Scalettar, K. R. A. Hazzard, I. Bloch, and S. Fölling

Phys. Rev. Lett. 132, 083401 – Published 21 February 2024

See synopsis: A General Equation of State for a Quantum Simulator

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A General Equation of State for a Quantum Simulator

Published 23 February 2024

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Equation of State and Thermometry of the 2D $SU (N)$ Fermi-Hubbard Model