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arXiv:1608.07341 · Physical Review A

Two-body physics in the Su-Schrieffer-Heeger model

Marco Fedele Di Liberto, Alessio Recati, Iacopo Carusotto, C. Menotti

Published Dec 22, 2016• 100 citations
arXivHTMLPDF

Abstract

We consider two interacting bosons in a dimerized Su-Schrieffer-Heeger (SSH) lattice. We identify a rich variety of two-body states. In particular, for open boundary conditions and moderate interactions, edge bound states (EBS) are present even for the dimerization that does not sustain single-particle edge states. Moreover, for large values of the interactions, we find a breaking of the standard bulk-boundary correspondence. Based on the mapping of two interacting particles in one dimension onto a single particle in two dimensions, we propose an experimentally realistic coupled optical fibers setup as quantum simulator of the two-body SSH model. This setup is able to highlight the localization properties of the states as well as the presence of a resonant scattering mechanism provided by a bound state that crosses the scattering continuum, revealing the closed-channel population in real time and real space.

DOI
10.1103/physreva.94.062704

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We consider two interacting bosons in a dimerized Su-Schrieffer-Heeger (SSH) lattice. We identify a rich variety of two-body states. In particular, for open boundary conditions and moderate interactions, edge bound states (EBS) are present even for the dimerization that does not sustain single-particle edge states. Moreover, for large values of the interactions, we find a breaking of the standard bulk-boundary correspondence. Based on the mapping of two interacting particles in one dimension onto a single particle in two dimensions, we propose an experimentally realistic coupled optical fibers setup as quantum simulator of the two-body SSH model. This setup is able to highlight the localization properties of the states as well as the presence of a resonant scattering mechanism provided by a bound state that crosses the scattering continuum, revealing the closed-channel population in real time and real space.
We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.
In the low-energy regime, the Su-Schrieffer-Heeger (SSH) model's key characteristics are encapsulated by a Dirac-type Hamiltonian in continuum space, i.e., localized states emerge at the boundaries. Building on this, we have developed an effective Hamiltonian to model ultracold interacting bosons on an SSH-like lattice through variational minimization under the mean-field approximation. To pinpoint the boundary states, we have developed an algorithm by generalizing the imaginary time propagator, where an initial state evolves under the squared Hamiltonian to converge to the targeted state. This algorithm has broader applicability, enabling the identification of specific eigenstates in various contexts. Furthermore, we draw a parallel to an experimentally physical setup involving a gas of ultracold bosons confined to an array of potential wells with alternating depths. By establishing the system's analogy with the SSH system, we apply our algorithm to investigate boundary states in the presence of interactions, demonstrating how these findings align with those of the continuous system.
We derive the bosonization of the interacting fermionic Su-Schrieffer-Heeger (SSH) model with open boundaries. We use the classical Euler-Lagrange equations of motion of the bosonized theory to compute the density profile of the zero-energy edge mode and observe excellent agreement with numerical results, notably the localization of the mode near the boundaries. Remarkably, we find that repulsive or attractive interactions do not systematically localize or delocalize the edge mode but their effects depend on the value of the staggering parameter. We provide quantitative predictions of these effects on the localization length of the edge mode. Our study shows that bosonization is able to quantitatively describe edge modes of interacting topological one-dimensional systems and pave the way to generalization to other models.
We present a strategy based on two-dimensional arrays of coupled linear optical resonators to investigate the two-body physics of interacting bosons in one-dimensional lattices. In particular, we want to address the bound pairs in topologically nontrivial Su-Schrieffer-Heeger arrays. Taking advantage of the driven-dissipative nature of the resonators, we propose spectroscopic protocols to detect and tomographically characterize bulk doublon bands and doublon edge states from the spatially resolved transmission spectra, and to highlight Feshbach resonance effects in two-body collision processes. We discuss the experimental feasibility using state-of-the-art devices, with a specific eye on arrays of semiconductor micropillar cavities.
We study the effects of electron-electron interactions on the charge excitation spectrum of the spinful Su-Schrieffer-Heeger (SSH) model, a prototype of a one-dimensional bulk obstructed topological insulator. In view of recent progress in the fabrication of dopant-based quantum simulators we focus on experimentally detectable signatures of interacting topology in finite lattices. To this end we use Lanczos-based exact diagonalization to calculate the single-particle spectral function in real space which generalizes the local density of states to interacting systems. Its spatial and spectral resolution allows for the direct investigation and identification of edge states. By studying the noninteracting limit, we demonstrate that the topological in-gap states on the boundary are robust against both finite-size effects as well as random bond and onsite disorder which suggests the feasibility of simulating the SSH model in engineered dopant arrays in silicon. While edge excitations become zero-energy spinlike for any finite interaction strength, our analysis of the spectral function shows that the single-particle charge excitations are gapped out on the boundary. Despite the loss of topological protection, we find that these edge excitations are quasiparticlelike as long as they remain within the bulk gap. Above a critical interaction strength of ${U}_{c}\ensuremath{\approx}5t$ these quasiparticles on the boundary lose their coherence which is explained by the merging of edge and bulk states. This is in contrast to the many-body edge excitations which survive the limit of strong coupling, as established in the literature. Our findings show that for moderate repulsive interactions the nontrivial phase of the interacting SSH model can be detected through remnant signatures of topological single-particle states using single-particle local measurement techniques such as scanning tunneling spectroscopy.
We investigate theoretically the Bose-Hubbard version of the celebrated Su-Schrieffer-Heeger topological model, which essentially describes a one-dimensional dimerized array of coupled oscillators with on-site interactions. We study the physics arising from the whole gamut of possible dimerizations of the chain, including both the weakly and the strongly dimerized limiting cases. Focusing on two-excitation subspace, we systematically uncover and characterize the different types of states which may emerge due to the competition between the inter-oscillator couplings, the intrinsic topology of the lattice, and the strength of the on-site interactions. In particular, we discuss the formation of scattering bands full of extended states, bound bands full of two-particle pairs (including so-called 'doublons', when the pair occupies the same lattice site), and different flavors of topological edge states. The features we describe may be realized in a plethora of systems, including nanoscale architectures such as photonic cavities, optical lattices and qubits, and provide perspectives for topological two-particle and many-body physics.