Dynamical Solitons and Boson Fractionalization in Cold-Atom Topological Insulators

D. González-Cuadra, A. Dauphin, P. R. Grzybowski, M. Lewenstein, and A. Bermudez

Phys. Rev. Lett. 125, 265301 – Published 23 December 2020

Abstract

We study the $Z_{2}$ Bose-Hubbard model, a chain of interacting bosons the tunneling of which is dressed by a dynamical $Z_{2}$ field. The interplay between spontaneous symmetry breaking (SSB) and topological symmetry protection gives rise to interesting fractional topological phenomena when the system is doped to certain incommensurate fillings. In particular, we hereby show how topological defects in the $Z_{2}$ field can appear in the ground state, connecting different SSB sectors. These defects are dynamical and can travel through the lattice carrying both a topological charge and a fractional particle number. In the hardcore limit, this phenomenon can be understood through a bulk-defect correspondence. Using a pumping argument, we show that it survives also for finite interactions, demonstrating how boson fractionalization induced by topological defects can occur in strongly correlated bosonic systems. Our results indicate the possibility of observing this phenomenon, which appears for fermionic matter in solid-state and high-energy physics, using ultracold atomic systems.

Received 5 April 2020
Accepted 24 November 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.265301

Physics Subject Headings (PhySH)

Bose gases Symmetry protected topological states Topological defects

Strongly correlated systems

Bose-Hubbard model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

D. González-Cuadra ^1,*, A. Dauphin¹, P. R. Grzybowski², M. Lewenstein ^1,3, and A. Bermudez⁴

¹ICFO—Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Avinguda Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
²Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland
³ICREA, Lluis Companys 23, 08010 Barcelona, Spain
⁴Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain

^*daniel.gonzalez@icfo.eu

$Z_{n}$ solitons in intertwined topological phases

D. González-Cuadra, A. Dauphin, P. R. Grzybowski, M. Lewenstein, and A. Bermudez

Phys. Rev. B 102, 245137 (2020)

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Vol. 125, Iss. 26 — 31 December 2020

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Images

Figure 1
Bosonic Peierls transition and topological defects: (a) Sketch of the $Z_{2}$ BHM [Eq. (1)], where bosonic particles (red spheres) tunnel between different sites with a strength that depends on the $Z_{2}$ fields (arrows) located on the bonds. (b) Qualitative phase diagram at half filling [30]. The $Z_{2}$ fields are polarized for weak Hubbard interactions, but order anti-ferromagnetically if the interactions are strong enough, according to two degenerate patterns (A and B). The SSB drives the bosons from a qSF to a BOW phase, where the bosonic tunneling is dimerized (stronger tunneling in yellow). Additionally, the B SSB sector hosts a SPT phase with localized edge states (red peaks in the figure). (c) Extra bosons create pairs of topological defects in the Néel ground state, interpolating between the different SSB sectors and hosting fractionalized particles.
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Figure 2
Ising topological defects: We show the ground state configuration for a chain of $N = 31$ sites and $N_{b} = 16$ bosons, with $U = 10 t$ and $Δ = 0.80 t$ , where a topological defect appears in the dimerized pattern of the $Z_{2}$ fields. (a) and (b) show the magnetization $⟨ σ_{i, i + 1}^{z} ⟩$ and the order parameter $φ_{j}$ for $β = 0$ , respectively, where the defect is a domain wall with topological charge $Q = 1$ (3). (c)–(d) Analogous topological defect for $β = 0.03 t$ , where quantum fluctuations broaden the defect, leading to a soliton of finite width $ξ$ , that can be accurately fitted to Eq. (2).
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Figure 3
Boson fractionalization and local Berry phase: (a) Occupation number $⟨ : n_{i} : ⟩$ for a chain with $N = 90$ sites and $N_{b} = 46$ hardcore bosons, for $Δ = 0.70 t$ and $β = 0.03 t$ . We superimpose $φ$ using dashed lines. We observe peaks in the occupation at the defects, localized on different sub-lattices (blue and red). The solid lines correspond to a fit to Eq. (4). (b) The integrated particle number $N_{i}$ shows how each peak contains half a particle. (c) The local Berry phase $γ$ is quantized to 0 or $π$ in the different SSB sectors, and interpolates between the two around the defects. (d) Softcore bosons with $U = 10 t$ , $Δ = 0.80 t$ and $β = 0.02 t$ , where we still observe peaks in the occupation. These have now support on both sublattices, since chiral symmetry is broken, but are still associated with fractionalized bosons (e).(f) The topological Berry phase is also quantized since inversion symmetry is preserved.
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Figure 4
Bosonic pumping between edges and defects: We show how the Berry phase $γ$ changes with the pumping parameter $ϕ$ , calculated at the middle of a chain with $N = 42$ and $N_{b} = 21$ . The change in this quantity (6) is associated to the Chern number of an extended 2D system, where (a) and (b) correspond to the 1D configurations A and B, respectively, at $ϕ = 0$ . (c) Bosonic occupation $⟨ : n_{i} : ⟩$ through the pumping cycle of a system with two domain walls, starting from a BAB configuration at $ϕ = 0$ . Each region transports a quantized charge in the bulk given by $Δ n = - ν$ . The arrows denote the direction of the pumped charge, flowing between edges and defects.
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