Condensed Matter Physics in Time Crystals

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Condensed Matter Physics in Time Crystals K Giergiel1 and K Sacha1,2 1M. Smoluchowski Institute of Physics, Jagiellonian University, 11, ul. prof. Stanisława Łojasiewicza, 30-348, Kraków, Poland. Contact Phone: +48126644890 2Mark Kac Complex Systems Research Center, Jagiellonian University, 11, ul. prof. Stanisława Łojasiewicza, 30-348, Kraków, Poland. Contact Phone: +48126644890 Contact Email: [email protected] Periodically driven systems can display new many-body phases. One kind of such phases are time crystals – quantum many-body systems that, due to interactions between particles, are able to spontaneously self-organise their motion in a periodic way in time by analogy with the formation of crystalline structures in space in condensed matter Figure 1: Example of a system with exotic long-range physics. interactions: ultra-cold atoms bouncing resonantly on In solid-state physics properties of space crys- a harmonically oscillating mirror in a 1D model. The tals are often investigated with the help of exter- 20:1 resonance condition between the mirror oscillations and atomic motion is considered and the system is nal potentials that are spatially periodic and reflect described by the Bose-Hubbard Hamiltonian: Hêff = various crystalline structures. A similar approach J P y 1 P y y − 2 hi;ji aî a^j + 2 i;j Uijaî a^ja^jaî. Center panel shows can be applied for time crystals, as periodically the interaction coefficients Uij, which depend on the dis- driven systems constitute counterparts of spatially tance between sites i − j of the Bose-Hubbard Hamilto- periodic systems, but in the time domain. nian. The long-range interactions presented in the centre Employing periodically driven systems we pro- panel can be realised if the s-wave scattering length of pose quantum simulators, which promise access atoms g0(t) is modulated in time as presented in the right to programmable exotic long-range interactions in panel solid state-like many-body problems, systems with topological protection of time correlations, methods to investigate Anderson localisation physics in the time domain and quasi-crystalline structures in time. These propositions can be realised experimentally, for example, in ultra-cold atoms bouncing resonantly on an oscillating atom mirror..

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Arxiv:2005.03138V2 [Cond-Mat.Quant-Gas] 23 May 2020 Contents
Condensed Matter Physics in Time Crystals Lingzhen Guo1 and Pengfei Liang2;3 1Max Planck Institute for the Science of Light (MPL), Staudtstrasse 2, 91058 Erlangen, Germany 2Beijing Computational Science Research Center, 100193 Beijing, China 3Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy E-mail: [email protected] Abstract. Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time- translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research ﬁeld with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research. arXiv:2005.03138v2 [cond-mat.quant-gas] 23 May 2020 Contents 1 Brief introduction to time crystals3 1.1 Wilczek's time crystal . .3 1.2 No-go theorem . .3 1.3 Discrete time-translation symmetry breaking .
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