Topological Time Crystals

By analogy with the formation of space crystals, crystalline structures can also appear in the time domain. While in the case of space crystals we often ask about periodic arrangements of atoms in space at a moment of a detection, in time crystals the role of space and time is exchanged. That is, we fix a space point and ask if the probability density for detection of a system at this point behaves periodically in time. Here, we show that in periodically driven systems it is possible to realize topological insulators, which can be observed in time. The bulk-edge correspondence is related to the edge in time, where edge states localize. We focus on two examples: Su-Schrieffer-Heeger (SSH) model in time and Bose Haldane insulator which emerges in the dynamics of a periodically driven many-body system.

Introduction. Spontaneous formation of space crystals is a well known phenomenon. Two famous examples are the generation of a Wigner crystal [1] for a gas of interacting electrons and the Peierls instability leading to the celebrated SSH model [2] for a system of electrons interacting with phonons. It turns out that quantum many-body systems can spontaneously self-organize their motion in a periodic way in time as well in a full analogy to the spontaneous formation of space crystals [3][4][5][6][7] (for classical version of time crystals, including topologically protected crystals, see [8][9][10][11][12][13]). Spontaneous formation of the so-called discrete time crystals has been predicted and experimentally realized [14][15][16][17][18] (see also [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]). While time crystals base often on spatially periodic structures which are driven periodically in time, crystalline structures in the time domain can also emerge in periodically driven systems which have nothing to do with any space crystallization [7,37] (for the so-called phase-space crystals see [38][39][40][41]). We would like to stress that not every periodically driven system can be identified with a time crystal. It has been shown that in certain Hilbert subspaces, resonantly driven systems can behave as a space crystal with a crystalline structure which is observed in the time domain even if no spatially periodic potential is present [37,[42][43][44][45][46].
In this letter we show that it is possible to drive periodically a system so that the emerging crystalline structure in time is a symmetry protected topological (SPT) phase [47]. The topological time crystals we consider should not be confused with the so-called Floquet topological systems. In the latter, a crystalline structure (usually an optical lattice) is present in space and it is periodically driven so that its effective parameters can be changed and the system can reveal topological properties in space but no crystalline structure can be observed versus time. In the models considered in the present letter, resonant dynamics leads to an emergence of a time crystalline structure that may possess a SPT phase. SPT phases constitute a new paradigm: they are characterized by a global topological invariant and, therefore, can not be described by the Landau theory of phase transitions. Remarkable examples of these phases are, among others, the Haldane phase [48,49] and the topological insulators [50]. The study of the latter with and without interactions has attracted much interest in condensed matter and in quantum simulators [50][51][52]. Quantum simulators constitute very versatile platforms with an unprecedented degree of control of the parameters of the system such as the hopping or the interactions [53]. Quantum simulators have successfully simulated and detected topological insulators in 1D [54][55][56][57][58][59], 2D [60][61][62] and 4D [63,64]. All these realizations of the topological insulators rely on an initial underlying lattice. In the present letter we do not assume any underlining spatially periodic structure with a non-trivial topology [28]. We show that time crystals with topological properties can emerge due to an appropriate resonant driving and discuss possible implementation and detection schemes in quantum simulators.
The model. We focus on ultra-cold atoms bouncing on an oscillating mirror in the presence of the gravitational field [65] (for the stationary mirror experiments see [66][67][68][69][70][71][72][73]) but the phenomena we investigate can be realized in any periodically driven system which can reveal non-linear resonances in the classical description [74]. The single-particle Hamiltonian, in the gravitational units and in the frame oscillating with the mirror [75], reads H = H 0 + H 1 where H 0 = p 2 /2 + x and with f (t) = f (t+2π/ω) = k f k e ikωt and integer-valued s 1. In order to describe a resonant driving of the system it is convenient to perform canonical transformation to the so-called action angle variables, where the unperturbed Hamiltonian depends only on the new momentum (action), H 0 (I) = (3πI) 2/3 /2 [74,75]. In the absence of the perturbation (H 1 = 0) the action I is a constant of motion, and the conjugate position variable (angle) changes linearly in time θ(t) = Ωt + θ(0), where Ω(I) = dH 0 (I)/dI is the frequency of periodic evolution of a particle. We assume the resonant driving Ω(I 0 ) = ω, where I 0 is the resonant value of the action. Then, by means of the secular approximation [74,75], in the frame moving along the classical resonant orbit Θ = θ − ωt, we obtain the effective Hamiltonian This result is valid in the regime close to resonance, i.e. when P = I − I 0 ≈ 0. Equation (2) indicates that for s 1 and V 2 (Θ) = 0, a resonantly driven particle behaves like an electron in a one-dimensional (1D) space crystal.
A time crystal topological insulator. Let us focus on the first energy band of the quantum version of the effective Hamiltonian of Eq. (2) with V 2 (Θ) = 0. The Wannier states w i (Θ) of the first energy band [89] [77], which are localized in single sites of the periodic potential in Eq. (2), correspond to localized wavepackets w i (x, t) moving along the classical resonant orbit with the period 2π/ω. For both non-vanishing V 0 and V 1 , the effective periodic potential describes a Bravais lattice with a twopoint basis. The restriction to the first energy band and the resulting tight-binding Hamiltonian, where J 2i = J and J 2i−1 = J, lead to the SSH model [2] which describes spinless fermions hopping on a 1D-lattice with staggered hopping amplitudes. Changing the ratio λ 1 /λ in (1), allow one to control the ratio J /J. This model belongs to the BDI class of the periodic table of the topological insulators and superconductors [78] and is characterized by a Z topological invariant, the winding number ν. For a periodic system with J > J (J < J), the system is in a topological (trivial) phase with winding number ν = 1 (ν = 0). For a finite system, the topological phase exhibits zero energy edge states protected by the topology of the bulk. The SSH model has been experimentally realized in quantum simulators and both the presence of edge states and the winding number have been measured [56][57][58]. While these implementations required an initial underlying superlattice the SSH model in space may also be realized in an emergent superlattice structure due to periodic driving of a basic lattice [55,[79][80][81]. Detection of the topology. In the laboratory frame, the crystalline behavior described by Eq. (2) is reproduced in the time domain due to the linearity of the transformation Θ = θ−ωt [37,42]. In other words, the clicking probability of a detector in the laboratory frame close to the classical resonant orbit, reflects periodic behavior of the Bloch waves ψ(Θ) of the Hamiltonian (2) in the moving frame. The experimental detection of the edge states and the corresponding bulk winding number ν can be realized in ultra-cold atoms which form a Bose-Einstein condensate (BEC). The Hamiltonian of Eq. (3) fulfills periodic boundary conditions, i.e. a s+1 = a 1 . However, it is possible to introduce an edge in our system by means of a proper modulation of the mirror motion. Indeed, if f k 's in the definition of f (t) are suitably chosen, the resulting effective potential V 2 (θ) can have a shape of a barrier localized on two adjacent sites of the periodic potential in Eq. (2). Then, the edge is created and two eigenstates can localize exponentially close to it in the topological phase. The corresponding quasi-energy spectrum of the full Floquet Hamiltonian H(t) = H − i∂ t is shown in Fig. 1(a). For J /J > 1 two degenerate zero-energy levels form which correspond to two eigenstates localized close to the barrier created by the potential V 2 (Θ). In the laboratory frame these two eigenstates are related to Floquet states that evolve periodically along the classical resonant orbit. For a detector located close to the orbit, the probability of a detector clicking reveals an edge in time and these two Floquet states localize close to it, as shown in Fig. 1(b) for increasing values of J /J. In Ref. [35] it is shown how a BEC prepared in a trap can be loaded on a classical resonant orbit: a localized atomic cloud has to be released from a trap at the position of the turning point of the classical resonant orbit. Then, the initial state of the system corresponds to a Wannier state w i (x, t) evolving along the orbit. An atomic cloud loaded when the edge state is passing close to the turning point remains there as the edge states do not penetrate much the bulk [56]. The probability density of one of the zero-energy eigenstates in the configuration space is depicted in Fig. 2 for sωt = 39π/2. Measurements of atomic density at different times therefore allow one to confirm the localization properties of the state loaded on the edge.
On the other hand, if an atomic cloud is released in the bulk, then the subsequent time evolution leads to nonzero populations |a i (t)| 2 of many Wannier wavepackets. Measurement of the atomic density allows one to obtain |a i (t)| 2 and consequently the winding number which is determined by the mean chiral displacement, i.e. ν ≈ where i 0 is a number of a cell of the Bravais lattice where the atomic cloud is initially loaded [55,59,81]. The relation between ν and the mean chiral displacement is valid after a long-time evolution when time averaged [59].
A time crystal Haldane phase. We now switch from single-particle to many-body systems which are periodically driven and which can be characterized by nontrivial topology. It is known that a Bose gas in a timeindependent spatially periodic lattice with repulsive onsite and nearest-neighbor interactions, described by the following Bose-Hubbard Hamiltonian can reveal a topological behavior. For large J the Hamiltonian (4) describes superfluid phase of bosons, for large U the Mott insulator (MI) emerges and for large V the density wave (DW) phase is present where translation symmetry of the Hamiltonian (4) is spontaneously broken. However, between the MI phase and the DW phase there is the topological Haldane insulator (HI) phase [82][83][84][85] -or more precisely a bosonic analog of HI in spin-1 chain [48,49]. As discussed in Ref. [82] this phase breaks a hidden Z 2 symmetry related to a highly nonlocal string order parameter [86].
To realize a periodically driven system that is effectively described by the Hamiltonian of (4), we again consider ultra-cold bosons bouncing on an oscillating mirror. We assume that the single-particle Hamiltonian corresponds to Eq. (1) with λ 1 = 0 and f (t) = 0. For the resonant driving described above, we may restrict to the first energy band of the effective Hamiltonian (2) that leads to the tight-binding model (3). In the many-body case when we restrict to the Hilbert subspace spanned by Fock states |n 1 , . . . , n s , where n i 's denote numbers of bosons occupying Wannier wavepackets w i (x, t), we obtain the effective many-body Hamiltonian which resembles (4) but with the interaction terms 1/2 where U ij = 2 2π/ω 0 dt ∞ 0 dxg 0 |w i | 2 |w j | 2 for i = j and similar U ii but twice smaller [37]. The effective interaction coefficients U ij depend on the atomic s-wave scattering length g 0 , shapes of |w i (x, t)| 2 and how densities of different Wannier wavepackets overlap in the course of time evolution on the classical resonant orbit. Despite  [84]. To realize MI or DW phases, the appropriate g0(x, t) is similar with the amplitude of its oscillations around zero being twice smaller or greater by at least factor 1.5, respectively. The other parameters of the system are the following: ω = 0.056, s = 64 (i.e. 64 : 1 resonance), λ = 0.14 and M = 2 in the expansion g0 = M m=0 α(t, m)x m . The mirror is located at x = 0 and the turning point of the classical resonant orbit at x ≈ 1600. The maximal temporal interaction energy per particle is about 1.6 × 10 4 J and thus much smaller than the gap between the first and second quasi-energy bands of the corresponding single-particle problem which is 1.1 × 10 5 J where J = 5.5 × 10 −8 . The results are obtained within the quantum secular approach [76].
the fact that the original interactions between ultra-cold atoms are contact, the effective interactions can be longrange [37,39,87]. Moreover, they can be controlled by changing the s-wave scattering length in space and periodically in time by means of a Feshbach resonance, i.e. g 0 = g 0 (x, t). Indeed, if the applied magnetic field results in appropriate oscillations of the scattering length around zero, nearly arbitrary effective long-range interactions can be created. In order to perform a systematic analysis of the control of U ij , we assume that g 0 (x, t) = M m=0 α(t, m)x m and write the interaction coefficients in the form U ij = M m=0 2π/ω 0 dtα(t, m)u ij (t, m). To find a suitable α(t, m) one can apply the singular value decomposition of the matrix u ij (t, m) where (i, j) and (t, m) are treated as indices of rows and columns, respectively [46]. Left singular vectors tell us which sets of interaction coefficients U ij can be realized, while the corresponding right singular vectors give the recipes for α(t, m) and consequently for g 0 (x, t). In Fig. 3 we show an example of g 0 (x, t) corresponding to U ii = U , U ij = V for |i − j| = 1 and U ij = 0 when |i − j| > 1 where U/J = 3 and V /J = 2.5. These values are related to the Haldane insulator phase if the unit mean boson filling factor is assumed [84]. In order to realize topologically trivial MI or DW phases, the corresponding g 0 (x, t)/J may take a similar form with the amplitude of its oscillations around zero being about twice smaller for MI or at least a factor 1.5 greater for the DW phase.
Possible experimental detection of the temporal HI phase may proceed along the lines suggested already for the spatial HI [82] i.e. via observation of the sharp peaks predicted in the absorption spectrum under an additional weak periodic modulation. The latter has to be sufficiently slow in order not to couple the system to the Hilbert subspace complementary to the resonant subspace. In fact bosonic HI or DW has not been yet observed experimentally, the time domain may provide a promising solution in that direction. Experimental detection of DW seems to be particularly simple as the system breaks spontaneously discrete time translation symmetry of the time-periodic many-body Hamiltonian and a discrete time crystal forms similarly as in [34]. Indeed, if we locate a detector in the laboratory frame close to the classical resonant orbit, it will click with the period twice longer than the driving period 2π/(sω).
Conclusions We have shown how one can construct topological insulators in the time domain. To this end we consider bosons bouncing on the periodically oscillating mirror, topological phases are obtained by an appropriate tuning the shape of the oscillations. We present explicit schemes leading to effective SSH or extended Bose-Hubbard Hamiltonians providing explicit, realistic examples of the parameters corresponding to the realization of topological phases in the time domain. We suggest also the possible experimental detection of these phases in the time domain.
Note added: After the submission of the present manuscript we learned about the work on a similar topic but in photonic materials [88].