We investigate how topological Chern numbers can be defined when single-particle states hybridize with continua. We do so exemplarily in a bosonic Haldane model at zero temperature with an additional on-site decay of one boson into two and the conjugate fusion of two bosons into one. Restricting the Hilbert space to two bosons at maximum, the exact self-energy is accessible. We use the bilinear Hamiltonian H 0 corrected by the self-energy Σ to compute Chern numbers by two different approaches. The results are gauged against a full many-body calculation in the Hilbert space where possible. We establish numerically and analytically that the effective Hamiltonian H eff = H 0 ( k ⃗ ) + Σ ( ω , k ⃗ ) reproduces the correct many-body topology if the considered band does not overlap with the continuum. In case of overlaps, one can extend the definition of the Chern number to the non-Hermitian H eff and there is evidence that the Chern number changes at exceptional points. But the bulk-boundary correspondence appears to be no longer valid and edge modes delocalize. Published by the American Physical Society 2024
We investigate how topological Chern numbers can be defined when single-particle states hybridize with continua. We do so exemplarily in a bosonic Haldane model at zero temperature with an additional on-site decay of one boson into two and the conjugate fusion of two bosons into one. Restricting the Hilbert space to two bosons at maximum, the exact self-energy is accessible. We use the bilinear Hamiltonian H 0 corrected by the self-energy Σ to compute Chern numbers by two different approaches. The results are gauged against a full many-body calculation in the Hilbert space where possible. We establish numerically and analytically that the effective Hamiltonian H eff = H 0 ( k ⃗ ) + Σ ( ω , k ⃗ ) reproduces the correct many-body topology if the considered band does not overlap with the continuum. In case of overlaps, one can extend the definition of the Chern number to the non-Hermitian H eff and there is evidence that the Chern number changes at exceptional points. But the bulk-boundary correspondence appears to be no longer valid and edge modes delocalize. Published by the American Physical Society 2024
Understanding correlation effects in topological phases of matter is at the forefront of current research in condensed matter physics. Here we try to clarify some subtleties in studying topological behaviors of interacting Weyl semimetals. It is well known that there exist two topological invariants defined to identify their topological character. One is the many-body Chern number, which can be directly linked to the Hall conductivity and thus to the two-particle correlations. The other is the topological index constructed from the single-particle Green's functions. Because the information on Green's functions is achieved more easily than the many-body wave functions, usually only the topological index is employed in the literature. However, the approach based on the single-particle Green's function can break down in the strongly correlated phase. For illustration, an exactly solvable two-orbital model with momentum-local two-body interactions is discussed, in which both topological invariants can be calculated analytically. We find that the topological index calculated from the Green's function formalism can be nonzero even for a nontopological strongly correlated phase with a vanishing many-body Chern number. In addition, we stress that the physical surface states implied by nonzero many-body Chern numbers should be the edge modes of particle-hole collective excitations, rather than those of a quasiparticle nature derived from the Green's function formalism. Our observations thus demonstrate the limitation of the validity of the Green's function formalism in the investigations of interacting topological materials.
We investigate the properties of a few interacting bosons in a Creutz ladder, which has become a standard model for topological systems, and which can be realized in experiments with cold atoms in optical lattices. At the single-particle level, this system may exhibit a completely flat energy landscape with nontrivial topological properties. In this scenario, we identify topological two-body edge states resulting from the bonding of single-particle edge and flat-band states. We also explore the formation of two- and three-body bound states in the strongly interacting limit, and we show how these quasiparticles can be engineered to replicate the flat-band and topological features of the original single-particle model. Furthermore, we show that in this geometry perfect Aharonov-Bohm caging of two-body bound states may occur for arbitrary interaction strengths, and we provide numerical evidence that the main features of this effect are preserved in an interacting many-body scenario resulting in many-body Aharonov-Bohm caging. Published by the American Physical Society 2024
Building on the non-uniqueness of the macroscopic electric polarisation in the topological phase of a Chern insulator, we construct a many body Chern invariant. Under a smooth unitary temporal evolution, this quantity evolves with time and successfully reflects the topology of the out-of-equilibrium state of the system by assuming an integer-quantised value in the adiabatic limit. Considering a linear ramping of the staggered Semenoff mass of the paradigmatic Haldane model of graphene, we illustrate that starting from a trivial state, it is indeed possible to obtain a topologically non-trivial state even by an adiabatic unitary driving protocol. We further propose a counter-diabatic protocol that enables us to suppress the otherwise inevitable diabatic excitations generated while crossing the gapless quantum critical point. Notably, the corresponding emergence of an edge current in the final evolved state of the model under a semi-periodic boundary condition necessitates the application of a novel counter-diabatic field manifested as anisotropic hopping in the real space Hamiltonian. The generalised topology of this dynamical Chern invariant takes into account the dynamical occupation of instantaneous single-particle bands, and consequently ensconces an emergent bulk-boundary correspondence that is expected to manifest in experiments.
The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest-neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.
Large Chern-number topological phase is always an important topic in modern physics. Here we investigate the topological superfluids in a coupled-layer system, in which transitions between different topological superfluids can be realized by controlling the binding energy, interlayer tunneling, and layer asymmetry, etc. These topological transitions are characterized by energy gap closing and reopening at the critical points at zero momentum, where the Chern number and sign of Pfaffian undergo a discontinuous change. Topological protected edge modes at the boundaries are ensured by the bulk-edge correspondence. In a trapped potential the edge modes are spatially localized at the interfaces between distinct topological superfluids, where the number of edge modes is equal to the Chern-number difference between the left and right superfluids. These topological transitions can be detected by spin texture at or near zero momentum, which changes discretely across the critical points due to band inversion. The model can be generalized to a multilayer system in which the Chern number can be equal to any positive integer. These large Chern-number topological superfluids provide fertile grounds for exploring exotic quantum matters in the context of ultracold atoms.
Topological phase transitions and the phase diagram of Haldane-Hubbard model are investigated using the variational cluster approach. With increasing the interaction strength, there are two successive topological phase transitions. The first is characterized by the closing and reopening of the single particle gap with the Chern number changing its sign, while the second is featured by the divergence of the topological Hamiltonian's spectrum with the Chern number changing from a non-zero integer to zero. The second topological phase transition is not accompanied with the single particle gap closing but is associated with the zeros of the single particle Green's function. Between these two topological phase transitions, there exists a non-magnetic topological Mott insulating phase with no gapless single particle edge states which cannot be adiabatically connected to any non-interacting Chern insulator.