Rice-Mele model with topological solitons in an optical lattice

Attractive ultra-cold fermions trapped in a one-dimensional periodically shaken opticla lattices are considered. For an appropriate resonant shaking the system realizes paradigmatic dimes physics described by Rice-Mele model. The important feature of our system is the possible presence of controlled defects. They result in the creation of topologically protected loclaized modes carrying fractional particle number. Their possible experimental signatures are discussed.


I. INTRODUCTION
Conducting polymers [1] are particularly interesting one-dimensional systems due to their unusual topological properties characterized by the non-trivial Zak phase [2,3], degenerate ground states, topological solitons [4] and the fractional charge [5]. Such polymers can be described in a simple manner by fermions moving in a lattice with dimerized tunneling amplitudes [4,6]. Advances in quantum matter engineering have raised the possibility of realizing such models with ultracold atoms with unprecedented tunability and control. Recently, ultracold bosons in optical superlattice were used to simulate experimentally [7] a model of conducting polymers (namely Rice-Mele (RM) model [6], one of the simplest 1D models of nontrivial topology) and the corresponding Zak phase was measured. In a parallel work topological edge states in a similar potential were studied theoretically [8].
Models based on superlattice potentials are relatively easy to realize in experiment, however, they have also some limitations. Namely, the fact that an optical lattice is perfect in structure results in a dificulty of realizing topological solitons. Such solitons typically emerge on defects that are the domain walls between topologicaly distinct phases. Forming the signature of nontrivial topology, they are the essence of RM model.
In the present paper, we show how to realize RM model with controlled defects using a system of attractive ultracold fermions [9][10][11] in a simple shaken one-dimensional optical lattice. We show that there exist a regime of parameters where atoms self-organize into a dimerized structure. The ground state is then two-fold degenerate with the states representing two topologically distinct dimerized configurations. Due to the emergent nature of the dimerized state, both configurations, separated by a domain wall, can be simultaneously present in the lattice. Moreover, by controlling the filling fraction, impurities may be added to the configurations. Such defects -domain walls and impurities -naturally give rise to topologically protected solitons or bound states with a fractionalized particle number.

II. SYSTEM
Our system consists of two-species (denoted as ↓, ↑) fermionic mixture trapped in an optical lattice potential V latt = V sin 2 (πx/a) + V ⊥ (sin 2 (πy/a) + sin 2 (πz/a)), where a is the lattice constant. For V ⊥ V the system is effectively one dimensional. To control the system we use a familiar lateral (horizontal) lattice shaking [12]. Importantly, however, we introduce periodic changes in the potential depth which we call here vertical shaking: V = V 0 + δV 0 cos ωt. δV 0 is an amplitude of the lattice depth shaking and ω -the frequency, common to the lateral and vertical shaking. We assume fermionic species of equal mass, M , with different fillings: n ↓ ≈ 1 and n ↑ ≈ 1/2. The interaction between atoms of different species is attractive and it results in creation of composites [13] with density given by ↑-fermion density n ↑ . We include p-bands in the model and effectively have the composites that occupy s-bands and excess ↓-fermions that occupy p and s-bands.
To write the effective Hamiltonian of the model, we construct the time dependent Hamiltonian, H(t) and average it in time [12]. The minimal Hamiltonian of our system contains tunnelings, density induced tunnelings, renormalized interactions and shaking:Ĥ = H tun +Ĥ dit +Ĥ int +Ĥ sh (t), where: Here,ŝ † i ,ŝ i ,p † i ,p i are the creation and annihilation operators of the ↓-fermions in the s-and p-bands respectively, whileŝ † ↑i ,ŝ ↑i are s-band creation and annihilation oper-arXiv:1407.6533v1 [cond-mat.quant-gas] 24 Jul 2014 ators for ↑-fermion. Accordingly,n i ,n p i , andn ↑ i are the corresponding number operators. In the on-site interaction Hamiltonian,Ĥ int , the renormalized self-energy of the composites is denoted by U 0 , the on-site renormalized interaction between the composite and an excess ↓-fermion is characterized by U 1 , and E 1 is the energy of the p-band.Ĥ tun corresponds to standard tunnelings with amplitudes J 0 and J 1 for the s and p bands,Ĥ dit describes the often neglected density induced tunnelings ( [14][15][16][17], for details see Appendix A) with amplitudes T i . H sh (t) is a time-periodic Hamiltonian with K denoting the amplitude of the lateral lattice shaking while δE 1 denotes the strength of time-variation of single-particle energy in the p-band which is induced by periodic driving of the lattice depth. Additional effects due to the vertical shaking that are negligible for moderate δV 0 are discussed in Appendices.
Next, we describe the averaging process (see Appendix B for more details). First, we apply the unitary transfor- In comparison withĤ the shaking and the on-site interaction parts are removed byÛ while the tunneling part is dressed in oscillating terms. To find the effective Hamiltonian, we assume the resonant condition E 1 +U 1 = N ω+2∆, where N is integer and ∆ ω is the detuning and we time average the Hamiltonian. In the process fast oscillating terms ∼ 1/ω are neglected. Necessarily the shaking frequency ω is chosen large compared to all the tunneling amplitudes. We obtain H eff =H tun +H dit +H ons , with The intra-band tunneling parts above are modified in the standard manner [12]: is the ordinary Bessel function of order zero. In the case of the inter-band part, time averaging brings us, however, a new effect. The inter-band hopping is modified by the Bessel function of order N with different amplitudes depending on the direction of this process (+ or −): T 01 . The detuning, ∆, leads to residual on-site potentialH ons . Now we can tune the hopping parameters. The intraband amplitudes may be made very small by choosing K/ω such that J 0 (K/ω) ≈ 0. For slightly different K/ω ss-hopping remains negligible (so the composites may be still considered as immobile) while the typically much larger pp-hopping start to plays a role and has to be taken Dependence of different hopping amplitudes on shaking parameter K/ω for the exemplary system of V0 = 8, V ⊥ = 25, α = −0.1. Gray area marks the interval of K/ω in which we obtain density wave (DW) structure of ↑-fermionsoutside this region phase separation (PS) occurs.
into account. At the same time the inter-band hopping is large since it depends on Bessel functions of order N = 0 .
From now on we set the recoil energy, E R = h 2 /(8M a 2 ), as an energy unit and consider the ground state structure of H eff on an exemplary case of lattice depths V 0 = 8, V ⊥ = 25 and interaction strength α = a s /a = −0.1 (with a s being the (negative) scattering length). In the region with dominant interorbital tunneling, we expect that the ground state is given by the density wave configuration (DW) with every second side occupied by composites. To confirm that prediction, we first assume the composites to be immobilized due to the negligible value of J 0 in the whole regime under consideration. Then finding the ground state configuration boils down to solving the single particle Hamiltonian for a group of all possible configurations ofn ↑ i (that determines composites distribution). Since the search space grows exponentially with the number of sites, it quickly becomes too large for the exact diagonalization. Therefore we apply Simmulated Annealing [18] to find the dependence of lowest energy configuration on K/ω for the lattice of 24 sites with periodic boundary conditions. Results are shown in Fig. 1. We find indeed two possible configurations of composites: 1) phase separation (PS) where all the composites cluster together; 2)DW phase (shadowed region) where we have alternate occupations. We see that DW structures occur for the shaking parameter, K/ω, for which |T − 01 | + |T + 01 | |J 1 |J 0 (K/ω).

III. EMERGENT RICE-MELE MODEL
When the DW-configuration of the composites minimizes the energy of the system, the dominant hopping process is the intra-band sp-one and the effective Hamiltonian for the excess ↓-fermions corresponds to the Rice-Mele model [6], From now we shall drop the˜sign over tunneling amplitudes as we shall consider effective tunnelings only restricting to (3).
To write the Hamiltonian in the momentum space we specify a unit cell to contain two neighboring sites of which only one is always occupied by a composite. Such unit cell can be chosen in two different ways depending whether the composite resides in the first or the second site of the open chain (see Fig. 2). As expected for RM model [6] these two choices of the unit cell give rise to topologically distinct states. When written in the momentum space, the Hamiltonian (3) reads: where the H + DW (k) corresponds to "PS" configuration, H − DW (k) corresponds to the "SP"-one, and σ x,y,x are Pauli matrices. The dispersion relations are the same for both configurations: Topologically distinct configurations are characterized by different Zak phases [2] (i.e. Berry phases acquired across the Brillouin zone). Zak phases of particular states depend on the choice of the unit cell, their difference forms an invariant of the system. The Zak phase is given by [2,3]: where |u k are Bloch functions of the system i.e. eigenfunctions of the Hamiltonian (4). For ∆ = 0, when the Hamiltonian is equivalent to the SSH model [1], we obtain φ SP Zak −φ P S Zak = π. This indicates that "SP" and "PS" phases are topologically distinct -one of them must be nontrivial. For nonzero ∆ we obtain fractional (in units of π) Zak phase differences changing from π to 1.67π for ∆ ∈ [0, 0.002ω]. To determine which configuration has a nontrivial topology, we investigate the existence of edge modes.

IV. LOCALIZED MODES
In our model, defects arise naturally due to the emergent nature of the DW structure. As discussed in [13], the time scale required to reach a particular DW lattice configuration is set by the minority component tunneling rate. Subsequently the timescale to form the entire DW configuration is governed by the corresponding Lieb-Robinson bound [19]. When the time of creation is not sufficiently long, smaller regions of different DW configurations, separated by domain walls, will be created. Moreover, due to number fluctuations present for trapped atoms, the composites will not be exactly at half-filling. Any deviation from this filling will result in a defect in the form of a vacancy or a filled site.
Both kinds of impurities -domain walls and lattice defects -give rise to topologically protected localized modes (see Fig. 3) [20]. If we tune the shaking such that both ss and pp tunnelings are negligibly small, then on domain walls we effectively create open boundary conditions. This will result in an appearance of edge modes in SP configuration that vanish sharply on the edges (compare Fig. 3a and Fig. 3c ). They certify that SP is the configuration of nontrivial topology. These edge modes have energies ±∆ and their eigenvectors are given by the spinor: (ψ s (x), ψ p (x)). Depending on which edge we are, setting x = 0 on the boundary, we get: edge mode on the left end (the one ending with S site) with energy −∆ where ψ s (x) = A(e −λ + x − e −λ − x ), ψ p (x) = 0; edge mode on the right end (the one ending with P site) with energy and A being the normalization constant. When we now tune K/ω further from the zero point of the Bessel function, ss hopping is still negligible, but pp hopping becomes significant. Therefore, on those boundaries that are separated by P-sites ( Fig. 3b and Fig. 3d), particles can tunnel through the boundary and the mode vanishes exponentially on the both sites giving topological solitons with energy ∆ . Defects occurring inside the "SP" configuration ( Fig. 3e) give rise to two localized modes on both sides of the impurity. Depending on ss and pp tunneling rates they may end sharply on the boundary or smoothly vanish inside the defect. The width of the edge states depends on the hopping amplitudes and can be changed by tuning the value of K/ω. For K/ω = 2.3 the edge state is about 15 lattice sites long and it becomes narrower with higher values of K/ω.
Defects present in the lattice are associated with local changes of fermion number by fraction f 0 = tan −1 |T + 01 − T − 01 |/2∆ [6] at zero temperature. At finite temperatures, T , the corresponding fractional fermion number for the localized mode is given by [21], which shows the presence of fractionalized charge as long as δ ≡ ∆/πT 1.

V. EXPERIMENTAL REALIZATION AND PROBING
Having the system prepared we may probe its topological properties. The Zak phase can be measured experimentally in a way it has been proposed in Ref. [7] -with application of coherent Bloch oscillations combined with Ramsey interferometry. Localized states can be observed with photo-emission spectroscopy [22]. As the number of edge modes can be controlled by the filling fraction, this results in an increased peak intensity near zero momentum making it less susceptible to noise. At half filling, the only localized states that can be occupied are those of the negative energy. Fermion number fractionalization can be probed on defects with use of the single site imaging.

VI. CONCLUSIONS
We have shown that a combination of shaking and attractive interactions in 1D optical lattice can give rise to topologically nontrivial system. We have used standard lateral shaking but also introduced an additional vertical shaking. Together, they result in a dimerized tunneling structure. Moreover, by tuning the onsite energy slightly out of the resonance we can induce the staggered potential. By controlling the filling, we have shown further the presence of topologically protected localized modes. We believe that such modes can be experimentally verified at accessible temperatures.

APPENDIX A
We derive the minimal model in a standard manner starting from many body Hamiltonian of dilute gas of atoms in a second quantization representation [17,23]. We consider two species (denote by ↑-fermions and ↓fermions) of equal masses which can occupy the lowest band. The ↓-fermions have occupation close to unity, for them we consider also the excited, p orbital. Different species undergo contact interactions. The parameters in the Hamiltonian (1) in the main text are given by integrals of Wannier functions W 0(1) i (x, y) on s(p)-bands, where i is a site index.
Specifically the single particle ss and pp hoppings do not depend on the type of species and read where H latt = − ∂ 2 ∂x 2 + V 0 sin 2 (πx/a) is a single particle Hamiltonian for a static lattice. Observe the lack of two in the kinetic energy as we work in recoil units. The contact interactions between different species lead to density induced tunnelings [14][15][16][17]. The corresponding part of Visualisation of different density dependent tunneling processes present in the system. Blue and pink circles denote ↑-fermion and ↓-fermion, respectively.
the Hamiltonian may be expressed aŝ where, let us recall,ŝ † i ,ŝ i ,p † i ,p i are the creation and annihilation operators of the ↓-fermions in the s-and p-bands respectively, whileŝ † ↑i ,ŝ ↑i are s-band creation and annihilation operators for ↑-fermion.n i ,n p i , andn ↑ i denote the corresponding number operators. Throughout the paper we assume that minority ↑-fermions appear in pairs only due to strong attractive interactions. Thus some of the processes included above vanish. In particular, the term proportional to T 0 should be excluded as occupation of isite by ↑-fermion means that there is a ↓-fermion occupying this site already, so Pauli principle inhibits tunneling into this site. Similarly T 1 may be omitted as the occupation of i-site by p fermion is possible energetically only if there is a composite there. This presence prohibits tunneling into this site of s-fermion. The remaining terms formĤ dit included in the Hamiltonian (1) of the paper.
The amplitudes, T 's, are given by integrals over four Wannier functions and take the form is a renormalized 1D coupling constant [24] and α = a s /a is the ratio of the interaction strength to the lattice spacing.
Pictorial representation of different tunneling processes is shown in Fig. 4. Note that, since ↑ -fermions are minority fermions, they are always paired and probability of their tunneling to the p-band is negligibly small. That is why in case of these fermions we consider only ss tunneling. On the other hand, the presence of ↑ -fermions (and therefore composites) stimulates pp tunneling of ↓fermions. Observe that the first term in (9) is the interband sp hopping which has a staggered nature [reflected by (j − i) sign] and may happen when composite-empty site adjoins composite-occupied one.
Let us now discuss the on-site energies present in the Hamiltonian. The corresponding term reads: U 0 , U 1 are given by: U 0 is by far the biggest (on the modulus) energy scale and is responsible for pairing. Single particle energy of occupying p-band E 1 reads: with the origin of the energy axis corresponding to the s-fermion single particle energy. Consider now the effects due to latteral and vertical shaking. The former is quite standard [12] and leads a familar term K cos ωt j j(n ↑ i +ŝ † jŝ j +p † jp j ), where K is the shaking amplitude. The vertical shaking of the lattice depth (assumed to be not too large) causes periodic changes of single particle hoppings J z (t) = J z + δJ z cos ωt for z = 0, 1, with amplitudes: On time averaging we will see that these periodic changes have negligibly small influence on the system and can be omitted. That is the reason, why they do not appear in the Hamiltonian (1) of the main text.
Next we have periodic changes in the onsite energy with amplitudes: On the contrary to changes in the tunneling, this onsite effect is very important for the model and allow us to realize tunneling dimerization of the RM model.