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 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 investigate the topological properties of one-dimensional weakly interacting topological insulators using bosonization. To do that we study the topological edge states that emerge at the edges of a model realized by a strong impurity or at the boundary between topologically distinct phases. In the bosonic model, the edge states are manifested as degenerate bosonic kinks at the boundaries. We first illustrate this idea on the example of the interacting Su-Schrieffer-Heeger (SSH) chain. We compute the localization length of the edge states as the width of an edge soliton that occurs in the SSH model in the presence of a strong impurity. Next, we examine models of two capacitively coupled SSH chains that can be either identical or in distinct topological phases. We find that weak Hubbard interaction reduces the ground-state degeneracy in the topological phase of identical chains. We then prove that, similarly to the noninteracting model, the degeneracy of the edge states in the interacting case is protected by chiral symmetry. We then study topological insulators built from two SSH chains with interchain hopping that represent models of different chiral symmetric universality classes. We demonstrate in bosonic language that the topological index of a weakly coupled model is determined by the type of interchain coupling, invariant under one of two possible chiral symmetry operators. Finally, we show that a general one-dimensional model in a phase with topological index $\ensuremath{\nu}$ is equivalent at low energies to a theory of at least $\ensuremath{\nu}$ SSH chains. We illustrate this idea on the example of an SSH model with longer-range hopping.
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.
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 investigate interacting Su-Schrieffer-Heeger (SSH) chains with two- and three-site unit cells using density matrix renormalization group (DMRG) simulations. By selecting appropriate filling fractions and sweeping across interaction strength Jz and dimerization $\ensuremath{\delta}$, we map out their phase diagrams and identify transition lines via entanglement entropy and magnetization measurements. In the two-site model, we observe the emergence of an interaction-induced antiferromagnetic intermediate phase between the topologically trivial and nontrivial regimes, as well as a critical region at negative Jz with suppressed magnetization and finite-size scaling of entanglement entropy. In contrast, the three-site model lacks an intermediate phase and exhibits asymmetric edge localization and antiferromagnetic ordering in both positive and negative Jz regimes. We further examine the response of edge states to Ising perturbations. In the two-site model, zero-energy edge modes are topologically protected and remain robust up to a finite interaction strength. However, in the three-site model, where the edge states reside at finite energy, this protection breaks down. Despite this, the edge-localized nature of these states survives in the form of polarized modes whose spatial profiles reflect the noninteracting limit.
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.
We analyze interacting ultracold bosonic atoms in a one-dimensional superlattice potential with alternating tunneling rates t1 and t2 and inversion symmetry, which is the bosonic analogue of the Su-Schrieffer-Heeger model. A Z2 topological order parameter is introduced which is quantized for the Mott insulating (MI) phases. Depending on the ratio t1/t2 the n=1/2 MI phase is topologically nontrivial, which results in many-body edge states at open boundaries. In contrast to the Su-Schrieffer-Heeger model the bosonic counterpart lacks chiral symmetry and the edge states are no longer midgap. This leads to a generalization of the bulk-edge correspondence, which we discuss in detail. The edge states can be observed in cold atom experiments by creating a step in the effective confining potential, e.g., by a second heavy atom species, which leads to an interface between two MI regions with filling n=1 and n=1/2. The shape and energy of the edge states as well as the conditions for their occupation are determined analytically in the strong coupling limit and in general by density-matrix renormalization group simulations.