Long-range doublon transfer in a dimer chain induced by topology and ac fields

The controlled transfer of particles from one site of a spatial lattice to another is essential for many tasks in quantum information processing and quantum communication. In this work we study how to induce long-range transfer between the two ends of a dimer chain, by coupling states that are localized just on the chain’s end-points. This has the appealing feature that the transfer occurs only between the end-points – the particle does not pass through the intermediate sites–making the transfer less susceptible to decoherence. We first show how a repulsively bound-pair of fermions, known as a doublon, can be transferred from one end of the chain to the other via topological edge states. We then show how non-topological surface states of the familiar Shockley or Tamm type can be used to produce a similar form of transfer under the action of a periodic driving potential. Finally we show that combining these effects can produce transfer by means of more exotic topological effects, in which the driving field can be used to switch the topological character of the edge states, as measured by the Zak phase. Our results demonstrate how to induce long range transfer of strongly correlated particles by tuning both topology and driving.

The dimer chain consists of two sub-lattices connected by hopping, and so has sub-lattice symmetry. Consequently for any positive eigenvalue of the Hamiltonian (1) there exists a negative eigenvalue with the same absolute value, as we can see in Fig. 2. However, if the chain contains an odd number of sites (a half-integer number of dimers) there is only one edge state, with energy equal to 0. For λ < λ c this edge state is localized on the first site of the lattice, whereas for λ > λ c it is localized on the last site, as shown in Fig.4.

Long-range particle transfer.
Let us consider a finite dimer chain with an even number of sites. In the topological regime, where there are two edge states |e + , |e − , with energies E + , E − , one can consider FIG. 2: a) Energy spectrum of the SSH Hamiltonian for a finite chain containing 10 dimers. The energies corresponding to edge states (in red) enter the bulk bands for λ > 1 − 1/11 0.91 and the system no longer supports them. b) The energies for a chain consisting on 10 and a half dimers. As can be seen, now there is always an edge state. This is a consequence of the sub-lattice symmetry.  Fig. 2b), for λ = 0.2 (left) and λ = 5 (right). In each case there is just one edge state with weight just at one edge, which changes from left to right as a function of λ If the initial condition is |ψ(t = 0) = |1 , the state evolves in time as The mean occupation on the edges of the chain is therefore given by We plot this behavior in Fig. 5, and see that this approximation gives an excellent description of the transfer dynamics. This approximation improves as the edge states becomes more localized at the edges of the chain. In the non-topological regime, however, for the same initial condition, the state will in general be a superposition of more eigenstates and the particle will spread over the entire lattice, see Fig. 6 (center).
In the following we will consider two interacting fermions in the singlet subspace. Usually, the interaction destroys the charge oscillation between edges described in the previous section, as shown in Fig 8. When this interaction is large enough, however, the two particles bind together forming what is called a doublon, whose dynamics can be modeled by an effective Hamiltonian (Eq. 2 in the main article). In Fig. 7, we compare the energy spectrum obtained from the original two-particle Hubbard Hamiltonian, and that obtained from the effective doublon Hamiltonian, as a function of the interaction strength, U . The agreement is clearly excellent for U ≥ 4J.
If one compares the spectral density of a non-interacting dimer array in the topological regime ( Fig. 9 left) with that of the effective Hamiltonian for interacting particles ( Fig. 9 right), one can see that the inhomogeneity in the system due to the effective chemical potential induced by the interaction removes the edge states. This is why the long-range transfer of doublons does not occur naturally in the dimer chain unless a perturbation is added to the system, such as a gate potential, in order to recover the symmetry. The initial condition corresponds to a particle being localized on site 1, i.e. |ψ(0) = |1 . On the left, the system in the topological regime, the particle oscillates between both ends of the chain barely occupying the middle sites (long-range transfer). On the center, the system is in the trivial regime. The particle spreads over the entire lattice and the occupation looks practically homogeneous over the lattice. On the right, time evolution of the occupation for a chain containing 10 and a half dimers in the topological regime. Now the particle remains at the initial position at all times; in this case |1 is approximately an eigenstate of the system and is therefore stationary. In this section, we derive the effective Hamiltonian for doublons in a dimer chain coupled to an external periodic driving. The Hamiltonian reads: where J ij is the hopping rate between neighboring sites i and j, and U is the interaction strength between particles occupying the same lattice site. c † iσ (c iσ ) is the usual fermionic creation (annihilation) operator of one particle with spin σ on site i and n iσ = c † iσ c iσ is the number operator. The periodic potential, V i (t + T ) = V i (t), has frequency For small U there is still long-range transfer of particles but they are less localized at the edges than in the non interacting case (Fig. 6  left). As U increases, the two particles spread over the entire lattice. ω = 2π/T and is the same for all spin species. Going to the rotating frame with respect to both interaction and driving amounts to: Where we have defined: Hopping processes described by the operators h + ijσ and h − ijσ raise and lower by one the total double occupancy of the system respectively, whereas those described by h 0 ijσ leave it invariant. A(t) is a vector potential from which the ac field comes, and d ij is the distance between neighboring sites i and j (note that d ji = −d ij ). In order to apply the HFE we need to find a common frequency. We will consider first the resonant regime, U = lω, and then, by means of analytical continuation, obtain the limits U ω and U ω [5]. We will approximate H eff only up to first order in 1/ω. The different terms in the HFE can be found following several perturbative approaches [6] [7] and are given by the following expressions: with H Now assuming a particular shape of the periodic driving: V i (t) = E cos(ωt)x i ⇒ A(t) = E ω sin(ωt), the Fourier components of H int (t) are found to be: We make some remarks on the different terms of the HFE. First, in the zeroth-order effective Hamiltonian, terms already appear that raise and lower the double occupancy. These are proportional to the Bessel functions of the first kind of order l (the order of resonance) and correspond to the doublon association and dissociation processes assisted by the ac field. However, they can be neglected for small driving amplitudes. As can be seen in Fig. 10, for higher driving amplitudes, the effective model we derive does not give accurate results. Second, in the first-order correction to the effective Hamiltonian, three different kind of terms appear. Those which are a product of h 0 ijσ and h ± mnσ do not conserve the total double occupancy. Terms proportional to the product h + ijσ h − mnσ do preserve the total double occupancy as well as terms proportional to the reversed product of the same operators, but the former act on the doublon subspace whereas the latter act on the single-occupancy subspace. Since we want to describe the dynamics of doublons alone, there are no single-occupancy states in the system, and we have to keep only terms proportional to h + ijσ h − mnσ . Of these, the only non-vanishing terms are: Since J ij = J ji and d ij = −d ji , the first terms can be written as: And the second term can be written as: Strong interacting regime: U ω > J eff , J eff In the limit U ω we can ignore pω/U in the denominators of the above expressions. We note that in the analytical continuation of these expressions, the restriction p = −l has no meaning. Now using the identities: we arrive at: which has been written as a function of the doublon operators d † i = c † i↑ c † i↓ and n d i = d † i d i . Here J ij is either J or J , and d ij is what we have called b 0 or (a 0 − b 0 ). The sum in the definition of µ eff i is carried over the neighbours of site i. The first term corresponds to the hopping processes of doublons. The second one corresponds to a chemical potential that depends on the specific lattice characteristics, and the last term corresponds to an attractive interaction between neighboring doublons. As we consider just one doublon in our system, we can neglect this term. In order to back up our reasoning, we present in Fig. 10 a comparison between the derived effective Hamiltonian and exact numerical results.
High-frequency regime: ω U > J eff , J eff In the limit ω U all terms in the series are very small except those for p = 0 and we can approximate: In Fig. 11 we again compare the quasienergy spectrum given by this effective theory with the exact results. It is important to note that now the chemical potential, µ i , also depends on the parameters of the driving and we do not expect Shockley transfer to occur. In addition, the dependence on the field amplitude is equivalent to that of a single particle.
Here we have performed an energy shift to bring the chemical potential in every site equal to zero, except for those at the edges of the chain where µ = −J eff . Now, the subspace of states spanned by |1 and |N is degenerate, with energy E D = µ. The perturbation does not couple those states at first order, so one needs to go to second order to find the linear combination of states that approximates the eigenstate of the whole Hamiltonian. First we diagonalise the Hamiltonian in the degenerate subspace with energy equal to zero. We are going to consider the case where λ = 1, which can be analytically treated. In the base of localized orbitals, {|i } i=2..N −1 , the matrix representation of H is a special case of a tridiagonal Toeplitz symmetric matrix, whose eigenvalues and eigenstates have the following analytical expressions Up to second order, the perturbation in the subspace spanned by {|1 , |N } is Here |α , |β ∈ {|1 , |N } and β|H |α = 0. After some algebra we find which has eigenvalues and eigenvectors In the limit where J 0 (x) 1, for b = 0 and λ = 1, we can approximate the transfer time as: T 0 = π/2|b|. To calculate G (Eq. 5 main text) to first order, we must first compute the next-order correction for the lowest energy eigenstates Ea0 ω |x n .
We then have which must be normalized, yielding If we approximate J0 Ea0 ω µ− n J0 Ea0 ω

J0
Ea0 ω µ , noting that: finally we find Clearly in the limit of J 0 Ea0 ω → 0, G[|ψ ± n ] → 1/2. This implies that the localization of the eigenstates at the edges increases, and therefore the long-range dynamics is particularly clean, although slower. As the Bessel function approaches a zero, the atoms become disconnected because the tunneling rates are renormalized to zero.