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The introduction of time dependence into a quantum system can dramatically increase the complexity of the problem. In such dynamical systems, traditional concepts from equilibrium physics, such has the notion of a ground state, are no longer useful. Instead, an understanding of these systems—and even the notion of a 'phase' away from equilibrium— requires fundamentally different theoretical approaches. On the other hand, dynamical systems offer the chance to generate exotic new phases that cannot exist in static systems. Many such phases are directly accessible in the laboratory—where lasers and modulated electromagnetic fields may be used to drive a system out of equilibrium.

There has recently been much theoretical and experimental interest in quantum systems that are driven periodically in time, known as Floquet systems [1-2]. In these cases, fast oscillations can be used to alter the underlying band structure of the static (or time-averaged) system, generating exotic topological phases through a form of band structure engineering [1-2]. This has recently been demonstrated in the laboratory using both photonic systems and ultracold atomic gases [3–6]. Most interestingly, dynamical analogues of topological insulators and superconductors have been predicted that would be impossible to realise in a static system (see [7–8] for a complete list of references).

Our group is interested in a range of open questions that arise in driven systems, with a particular focus on the exotic topological phases that can exist only in dynamical systems. We recently uncovered a number of new examples of such driven topological phases. We also unified previously discovered driven topological phases in a periodic table of dynamical topological insulators and superconductors, which rigorously classifies all topological phases of free fermions in any number of dimensions [8].

In the presence of interactions and symmetry, dynamical analogues of symmetry protected topological phases (SPTs) had also been predicted to arise [9], with many-body localization (MBL) a crucial ingredient. We were one of a number of groups who provided a complete classification of a set of such phases in one dimension [10]. We are actively pursuing further ideas at this interface between symmetry, topology, interactions and time dependence, where many unsolved problems remain. 

Figure: Schematic representation of a Thouless pump. The numer of electrons that cross any section of the wire per cycle is quantized.

References

[1]     J. Cayssol, B. Do ́ra, F. Simon, and R. Moessner, Phys. Status Solidi RRL 7, 101 (2013).
[2]    M. Bukov, L. D’Alessio, and A. Polkovnikov, Adv. Phys. 64, 139 (2015).
[3]    M. C. Rechtsman et al., Nature 496, 196 (2013).
[4]     T. Kitagawa, M. A. Broome, A. Fedrizzi, and M. S. Rudner, Nat. Commun. 3, 882 (2012).
[5]     G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2015).
[6]     K. Jiménez-García et al., Phys. Rev. Lett. 114, 125301 (2015).
[7]      F. Nathan and M. S. Rudner, New Journal of Physics 17, 125014 (2015).
[8]     R. Roy and F. Harper, arXiv:1603.06944 (2016).
[9]     V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 116, 250401.
[10]    R. Roy and F. Harper, arXiv:1602.08089 (2016).