Six-dimensional time-space crystalline structures

Letter

Six-dimensional time-space crystalline structures

Giedrius Žlabys, Chu-hui Fan, Egidijus Anisimovas, and Krzysztof Sacha

Phys. Rev. B 103, L100301 – Published 8 March 2021

Abstract

Time crystalline structures are characterized by regularity that single-particle or many-body systems manifest in the time domain, closely resembling the spatial regularity of ordinary space crystals. Here we show that time and space crystalline structures can be combined together and even six-dimensional time-space lattices can be realized. As an example, we demonstrate that such time-space crystalline structures can reveal the six-dimensional quantum Hall effect quantified by the third Chern number.

Received 9 December 2020
Revised 22 February 2021
Accepted 23 February 2021

DOI:https://doi.org/10.1103/PhysRevB.103.L100301

Physics Subject Headings (PhySH)

Cold atoms & matter waves Geometric & topological phases Integer quantum Hall effect Symmetry protected topological states Synthetic gauge fields

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Giedrius Žlabys ¹, Chu-hui Fan², Egidijus Anisimovas ¹, and Krzysztof Sacha ²

¹Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, LT-10257 Vilnius, Lithuania
²Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland

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Issue

Vol. 103, Iss. 10 — 1 March 2021

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Images

Figure 1
Probability densities of the $k = 0$ Floquet-Bloch state $| u_{0, α} {(x, t) |}^{2}$ (black line) and the individual Wannier states $| w_{i, β} {(x, t) |}^{2}$ (dashed colored lines) for a particle in the resonantly vibrating optical lattice potential (1), at different moments of time, i.e., at (a) $ω t = π / 5$ , (b) $ω t = π / 5 + π$ , and (c) $ω t = π / 5 + 2 π$ . The parameters of the system are the following: $s = 3, V_{0} = 4320, ω = 240$ , and $λ = 0.01$ . Three sites of the optical lattice potential are shown only and in each site only one Wannier state (out of three for $s = 3$ ) is presented. Note that the Floquet-Bloch states are periodic with the period $ω T = 2 π$ , whereas the period of the Wannier states is three times longer.
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Figure 2
Loading ultracold atoms to the resonant manifold: (a) The atoms are first loaded into the ground state of an auxiliary superlattice that consists of a periodic sequence of narrow potential wells. Their widths are chosen such that the lowest-band Wannier functions (shown by the blue, red, and green solid lines) approximate the targeted Wannier wave packets $w_{i, α} (x)$ . (b) Next, the auxiliary lattice (dashed black line) is turned off and the targeted shaken lattice (solid black line) is turned on to place the prepared Wannier states (red lines) at the classical turning points; the profiles shown here correspond to $ω t = 0$ . (c) Wannier states undergo periodic oscillations; the profiles shown here correspond to $ω t = π / 5 + π$ [cf. Fig. 1].
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Figure 3
Energy dispersion obtained from a 2D tight-binding model of (a) 3-leg and (b) 11-leg ribbons with hopping parameters $J = | J_{i, α}^{i + 1, α} |$ and $| J_{i, α}^{i, α + 1} | = 0.4 J$ in the long and short directions, respectively, and $π / 2$ flux per plaquette encoded by Peierls phases [65]. Formation of a topological subband structure with edge modes crossing the gaps is apparent. Combining three such 2D models into a 6D model, its edge mode is built as a product of states at quasimomentum $k_{x} = k_{y} = k_{z} = π / 4$ marked by the red dot in (a) and (b). The probability density $ρ_{\vec{α}}$ of this mode in (c) 3-leg and (d) 11-leg ribbons is plotted as a function of the time-lattice indices $\vec{α} = (α_{x}, α_{y}, α_{z})$ and illustrates its localized nature with respect to all three directions, i.e., on the edge of the 6D ribbon. Parameters that can realize such a ribbon are $ω = 236.2, V_{0} = 4320, V_{1} = 0.1 V_{0}, ϕ = π / 8$ , and $U_{x} = 1$ . The first ten harmonics for the tilted potential are included [cf. Eq. (6)]. This gives the hopping parameter $J \sim 10^{- 3}$ .
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