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In the presence of both symmetry and interactions, a rich set of topological states, known as symmetry protected topological phases (SPTs), is known to exist. A complete classification of these systems has not yet been obtained, but rigorous results exist for static bosonic systems in one to three dimensions [18]. A natural question to ask is how this classification changes if we allow the system to vary with time. Are there new phases that are inherently dynamical in nature? Early work in this direction suggested that this is indeed the case [9]. The new driven phases of free fermionic systems [8] and some early work on interacting systems [9] suggested that this is indeed the case. 

Motivated by discussions with Curt von Keyserlingk and Shivaji Sondhi, our group was one of a number of groups [10,19–21] to provide a complete classification of time-dependent Abelian SPT phases in one dimension. Using intuition developed from the study of Floquet systems in Class D, we showed how to construct nontrivial dynamical SPT phases for an arbitrary Abelian symmetry group. This requires the construction of nontrivial unitary cycles, which may be obtained by evolving first with a trivial Hamiltonian and then with a nontrivial Hamiltonian. Uniquely, our work provides an explicit bulk-boundary correspondence for such dynamical SPT phases and describes how to calculate topological invariants in each case. 

For class D, the free fermion classification is unchanged upon the introduction of interactions, in a similar way to interacting static systems in class D [22]. A topological invariant may be constructed by inserting a flux into the system throughout the evolution, and this nontrivial topology has specific signatures in the many-body quasienergy spectrum (see figure). This is reminiscent of the topological invariant defined by Kitaev for the SPT phases in one dimension in this class [Kitaev 2001].

To generate dynamical bosonic SPT phases, we constructed nontrivial unitary cycles from the SPT models of Ref. [23]. In these systems, dynamical SPT phases correspond to irreducible representations of the symmetry group, while topological invariants may be constructed by applying twisted boundary conditions. These results generalise to all Abelian symmetry groups, providing a complete classification of such phases.

Figures a and b: Schematic splitting of a Majorana multiplet at a (0,π)-phase transition as the system is closed with (a) periodic and (b) antiperiodic boundary conditions.

References

[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).
[18]     X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013).
[19]     C. W. von Keyserlingk and S. L. Sondhi, Phys. Rev. B 93, 245145 (2016).
[20]     D. V. Else and C. Nayak, Phys. Rev. B 93, 201103(R) (2016).
[21]     A. C. Potter, T. Morimoto, and A. Vishwanath, arXiv 1602.05194v1 (2016).
[22]     L. Fidkowski, and A. Kitaev, Phys. Rev. B, 81 134509 (2010).
[23]     S. D. Geraedts and O. I. Motrunich, arXiv:1410.1580 (2014).