Frustrated magnets without geometrical frustration in bosonic flux ladders

Letter
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Frustrated magnets without geometrical frustration in bosonic flux ladders

Luca Barbiero, Josep Cabedo, Maciej Lewenstein, Leticia Tarruell, and Alessio Celi

Phys. Rev. Research 5, L042008 – Published 6 October 2023

Abstract

We propose a scheme to realize a frustrated Bose-Hubbard model with ultracold atoms in an optical lattice that comprises the frustrated spin-1/2 quantum $XX$ model. Our approach is based on a square ladder of magnetic flux $\sim π$ with one real and one synthetic spin dimension. Although this system does not have geometrical frustration, we show that at low energies it maps into an effective triangular ladder with staggered fluxes for specific values of the synthetic tunneling. We numerically investigate its rich phase diagram and show that it contains bond-ordered-wave and chiral superfluid phases. Our scheme gives access to minimal instances of frustrated magnets without the need for real geometrical frustration, in a setup of minimal experimental complexity.

Received 22 December 2022
Revised 7 August 2023
Accepted 14 September 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.L042008

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)

Bose gases Cold gases in optical lattices Frustrated magnetism Synthetic gauge fields

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Luca Barbiero ^1,*, Josep Cabedo ^2,*,†, Maciej Lewenstein ^3,4, Leticia Tarruell ^3,4, and Alessio Celi ²

¹Institute for Condensed Matter Physics and Complex Systems, DISAT, Politecnico di Torino, I-10129 Torino, Italy
²Departament de Física, Universitat Autónoma de Barcelona, E-08193 Bellaterra, Spain
³ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Avenida Carl Friedrich Gauss 3, E-08860 Castelldefels (Barcelona), Spain
⁴ICREA, Passeig de Lluís Companys 23, E-08010 Barcelona, Spain

^*These authors contributed equally to this work.
^†josep.cabedo@uab.cat

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Issue

Vol. 5, Iss. 4 — October - December 2023

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Images

Figure 1
Mapping semisynthetic square ladders into frustrated triangular ladders. (a) Left: Original square flux-ladder model (1), with intra- and interleg tunnelings $t$ and $Ω / 2$ , interleg offset $δ$ , on- and off-site interactions $U$ , and magnetic flux $γ$ . Right: Truncated lower-band Hamiltonian (2), describing an effective triangular ladder with complex tunnelings $t_{1}$ and $t_{2}$ , on-site interactions $U$ , and staggered flux $Φ$ . Inset: Single-particle dispersion relation of the lower-energy band of the square flux ladder for strong $Ω$ and $γ \sim π$ . (b)–(d) Current (arrow) and density (circle) patterns of the flux-ladder phases at average density $ρ = 1 / 4$ (left), and corresponding phases of the effective triangular model at half filling (right). (b) Meissner superfluid (M-SF): Superfluid (SF) of the triangular model. (c) Vortex lattice insulator ( ${VL}_{1 / 2}$ -MI): Bond-ordered wave (BOW). (d) Biased-ladder superfluid phase (BLP): Chiral superfluid (CSF). Circles and arrows in the square ladder sketches are scaled according to their numerical values at $Ω = 10 t$ and $δ = 0$ , with $γ$ and $U$ adjusted so that $t_{2} / | t_{1} | = 0.2$ in (b), $t_{2} / | t_{1} | = 0.5$ in (c), and $t_{2} / | t_{1} | = 1.0$ in (d), and $U = 10 | t_{1} |$ . The BLP is signaled by the difference of circle size between both legs. Color scale: Spin composition $m_{z}$ of the ground states in the semisynthetic ladder.
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Figure 2
Phase diagram. (a) DMRG-computed values of chiral correlation function $k^{2} (L / 2)$ (top panel), charge gap $Δ_{c}$ (central panel), and two-point bond-order operator $O_{BO}$ (bottom panel) as a function of $| t_{2} / t_{1} |$ extrapolated to $L \to \infty$ . Triangles: Effective triangular staggered-flux ladder (2) at half filling, with $U = 10 | t_{1} |$ and $Φ = π$ . Squares: Square flux ladder (1) at $ρ = 1 / 4$ and $δ = 0$ , where we have fixed $Ω = 20 t$ and adjusted $γ$ and $U$ to match the corresponding values of $t_{2} / t_{1}$ and $U / | t_{1} |$ . All quantities are extracted via finite-size extrapolation using system lengths up to $L = 120$ ( $L = 80$ ) for the triangular (square) ladder model. Only the central half of the sites is used to compute the expected values. (b) Phase diagram of the semisynthetic square flux ladder (1) at $ρ = 1 / 4$ for $Ω = 20 t$ and $δ = 0$ . Black circles: Phase boundary from DMRG simulations. Blue triangles: Set of points where $U = 10 t cos (γ / 2) = 10 | t_{1} |$ and the effective triangular ladder (2) tunneling ratio $| t_{2} / t_{1} |$ takes values from 0.9 to 0.1, corresponding to the curves of (a).
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Figure 3
(a) Magnetization $m_{z}$ as a function of the Raman detuning $δ$ , for the ground state of Hamiltonian (1) at $ρ = 1 / 4$ , with $Ω = 10 t$ and adjusting $γ$ and $U$ so that $U = 10 | t_{1} |$ and $| t_{2} / t_{1} | = 0.2$ (yellow triangles), 0.5 (teal squares), 0.8 (dark red circles), with $t_{1} < 0$ . (b) Staggered leg current $j_{s l}$ defined in (5) as a function of the lattice dimerization $Δ$ . All quantities are extrapolated to the thermodynamic limit by considering system sizes up to $L = 80$ .
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