Effective models for the lowest-energy subspaces

We can obtain effective models for the subspaces of two-boson states with lowest self-energy by projecting the Hamiltonian (6.1) into the two lowest flat bands. In order to do so, we remove from the expansions (6.12) all the contributions from higher bands and define a new set of operators

ˆ¯

By plugging in Eq. (6.1) the projected OAM operators (6.20) instead of the original ones, we obtain a Hamiltonian expressed in the basis of the single-particle eigenstates

and restricted to the two degenerate flat bands of lowest energy. As expected, the kinetic part of the projected Hamiltonian is diagonal and reads

ˆ¯ The interaction part of the projected Hamiltonian contains products of the type ˆ¯

j_α^i†ˆ¯j_α^i†ˆ¯j_αⁱˆ¯j_αⁱ and ˆ¯j_α^i†ˆ¯j_α^i†ˆ¯j_−αⁱ ˆ¯j_−αⁱ , with j = {a, b, c} and α = ±. Therefore, it yields terms that involve products of operators that belong to the H2, H3 and H4 subspaces. How-ever, after a long but straightforward calculation it can be shown that all the coupling terms cancel out and separate projected Hamiltonians can be written for each of these subspaces. In the next subsections, we present the effective models that are obtained for the two lowest-energy subspaces, i.e.,H4 and H3.

6.3.1 H

subspace

Introducing the projected OAM operators (6.20) into Eq. (6.3), we find that the projected interaction Hamiltonian contains the following terms consisting of products of operators associated to the H4 subspace

ˆ¯

The first term of Eq. (6.22) is associated with the self-energy of the states of the H4

subspace, Eq. (6.19), whereas the other two terms correspond to hoppings between two-boson states belonging to neighbouring plaquettes induced by the overlap between the localized modes, as shown in Fig. 6.2 (a). By mapping the two-body states of H4 into single particle states according to the definition |ii ≡ Wˆ_−,i^1†Wˆ_−,i^2† |0i, we can compute all the matrix elements of Eqs. (6.21) and (6.22) over the two-boson states ofH4 and write an effective single-particle tight-binding model for this subspace

H_eff^H4 =V_H4 illustrated in Fig. 6.2 (b), Eq. (6.23) describes a Su-Schrieffer-Heeger (SSH) chain with

128 Chapter 6 – Interaction-induced two-boson topological states a unit cell formed by two sites that correspond to neighbouring plaquettes of the original diamond chain. The intra- and inter-cell hoppings of this chain are given by t₂ and t₁ respectively, and the energy of all sites is shifted by a uniform potential V_H4.

Figure 6.2: Effective model for the H4 subspace. (a) Original states of H4 in the diamond chain. The blue squares framed in dashed lines indicate the plaquettes where the states are localized. In the sites with a darker blue shade, neighbouring states overlap and interact with a strength U₁ or U₂. (b) Effective SSH chain describing the H4 subspace. The blue shaded area indicates the unit cell, which is formed by two neighbouring plaquettes of the original diamond chain. (c) Band structure of the SSH chain for values of the interaction parameters {U_A/J = −0.2, U₁/J = −0.2, U₂/J =

−0.2} (left plot), {U_A/J =−0.2, U₁/J = −0.2(0), U₂/J = 0(−0.2)} (middle plot), and {U_A/J = −0.2, U₁/J = −0.2(−0.1), U₂/J = −0.1(−0.2)} (right plot). In the middle and right plots, we also indicate the value of the Zak’s phases of the bands for each case.

By Fourier-transforming the Hamiltonian (6.23), we find that the two energy bands

of this model are given by The shape of the band structure (6.24) is shown in Fig. 6.2 (c) for different values of the hopping parameters. As shown in the left plot, for the particular case t1 = t2 the gap closes atk =±π. As can be seen in the middle plot, fort₁ = 0 or t₂ = 0 the bands are flat and separated by a gap. In the general case t₁ 6=t₂, which is illustrated in the right plot, both bands are dispersive and the gap remains open. In the situations where there is an energy gap, the relative value of the hopping parameters determines the topological properties of the system. If |t₁|<|t₂|, the Zak’s phases of the bands are γ_H¹₄ =γ_H²₄ = 0 and the system is in a topologically trivial phase. On the other hand, if the values of the hoppings are such that|t₁|>|t₂|, the Zak’s phases are γ_H¹

4 =γ_H²

4 =π and the system is in a topological phase. Thus, by introducing the different interaction strengths U₁ and U₂ at B and C sites of odd and even unit cells respectively, we are able to control the shape of the band structure and to render the H4 subspace topologically non-trivial.

According to the bulk-boundary correspondence, in the topological phase we expect a chain with open boundaries to display two edge states of energy V_H4 in its spectrum.

Remarkably, these edge states consist of bound pairs of bosons and are induced by the interplay between the strength of different on-site interactions in a system where the kinetic energy plays no role due to the fact that all the bands are flat.

6.3.2 H

subspace

Doing the same procedure as in the case of theH4 subspace, we find that the part of the interaction Hamiltonian projected to the two lowest flat bands containing products of operators associated to the H3 subspace reads

¯ˆ

130 Chapter 6 – Interaction-induced two-boson topological states The first term of Eq. (6.25) corresponds to the self-energy of the states belonging to H3, Eq. (6.17), and the other ones can be regarded as pair-tunneling terms induced by the interactions at the sites where the single-particle localized modes interact. More specifically, the second term corresponds to a coupling between two-boson states local-ized at the same site but belonging to different bands, and the third and fourth terms to hoppings between states localized in neighbouring plaquettes belonging to the same and different bands, respectively. By mapping the two-body states ofH3 into single-particle states according to|i,1i ≡ ^√1₂Wˆ_−,i^1†Wˆ_−,i^1† |0i and |i,2i ≡ ^√1₂Wˆ_−,i^2†Wˆ_−,i^2† |0i, we can combine all the non-zero matrix elements of (6.25) and (6.21) over two-boson states of H3 and write an effective single-particle model for this subspace

Hˆ_eff^H3 =V_H3 Eq. (6.23). As illustrated in Fig. 6.3 (a), Eq. (6.26) describes a Creutz ladder [309] with a unit cell formed by two legs, each of which corresponds to the two states ofH3 localized in each plaquette, |i,1i and |i,2i. The intra-leg coupling is given by ^3(t1₂^+t2), and the inter-leg hoppings are−^t₂¹ or−^t₂² depending on the parity of the unit cell. Additionally, the energy of all sites is shifted by a uniform potential V_H3. By Fourier-transforming the Hamiltonian (6.26), we find the following energy bands

E_H¹

In Fig. 6.3 (c) we plot the band structure (6.27) for different values of the hopping parameters. As long as t₁ 6= t₂ there is an energy gap between the dispersive bands E_H¹₃(k) and E_H²₃(k), which only become flat in the cases t₁ = 0 or t₂ = 0, as shown in the middle plot of Fig. 6.3 (c). Whenever the gap remains open, if |t₁| < |t₂| the Zak’s phases of the bands are γ_H¹₃ = γ_H²₃ = γ_H³₃ = γ_H⁴₃ = 0 and the system is in a topologically trivial phase. On the other hand, if the relation between the values of the

hopping parameters is |t₁| > |t₂|, the Zak’s phases are γ_H¹₃ = γ_H²₃ = π, γ_H³₃ =γ_H⁴₃ = 0 and therefore H3 is in a topologically non-trivial phase.

Thus, the H3 and H4 subspaces share the same topological phase diagram as a function of the values of U1 and U2. In the non-trivial regime |U1| > |U2|, we expect that in a lattice with open boundaries the spectrum ofH3 displays two edge states with energy ^3(t1₂^+t2)+V_H3, which are of the same nature as those thatH4 exhibits in the same situation.

Figure 6.3: (a) Effective Creutz ladder describing the H3 subspace. The blue shaded area framed in dashed lines indicates the unit cell, which is formed by two neigh-bouring legs that are in turn composed of the two states of the original diamond chain loacalized in each plaquette. (b) Band structure of the Creutz ladder for values of the interaction parameters {UA/J = −0.2, U1/J = −0.2, U2/J = −0.2}

(left plot), {U_A/J = −0.2, U₁/J = −0.2(0), U₂/J = 0(−0.2)} (middle plot), and {U_A/J = −0.2, U₁/J = −0.2(−0.1), U₂/J = −0.1(−0.2)} (right plot). In the mid-dle and right plots, we also indicate the value of the Zak’s phases of the bands for each case.

132 Chapter 6 – Interaction-induced two-boson topological states