A Haldane conjecture is revealed for spin-singlet charge modes in
2N-component fermionic cold atoms loaded into a one-dimensional optical lattice. By means of a low-energy approach and density matrix renormalization group calculations, we show the emergence of gapless and gapped phases depending on the parity of
N for attractive interactions at half-filling. The analog of the Haldane phase of the spin-1 Heisenberg chain is stabilized for
N=2 with nonlocal string charge correlation and pseudo-spin-1/2 edge states. At the heart of this even-odd behavior is the existence of a spin-singlet pseudo-spin-
N/2 operator which governs the low-energy properties of the model for attractive interactions and gives rise to the Haldane physics.
We develop a framework to analyze the topological properties of one-dimensional systems with charge conservation and tendency towards topological superconducting order. In particular, we consider models with
N flavors of fermions and
(Z2)N symmetry, associated with the conservation of the fermionic parity of each flavor. For
N=1, and with no other symmetry other than charge conservation, we recover the result that there is no distinct topological phase with exponentially localized zero modes. For
N>1, however, we show that the ends of the system can host low-energy, exponentially-localized modes. To illustrate these ideas, we focus on lattice models with
SO(N) symmetric interactions and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of
N=3. In this case, the topologically nontrivial superconducting phase corresponds to a gapless analog of the Haldane phase in spin-1 chains. In this phase, although the bulk hosts gapless modes, corresponding to composite fermionic excitations with an enlarged Fermi surface, the ends host spin-
1/2 degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B 231, 445 (1984)] as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-
1/2 kinks in the Andrei-Destri model as the topological end modes at the domain walls between the two phases.
Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an odd-Haldane phase of a dimerized
S=21 Heisenberg chain and a Haldane phase of an
S=1 chain realized when Hund coupling locks two
S=21 spins per monomer into
S=1. Density-functional theory places the active manifold deep in the Mott regime (
U/t≈126), justifying a spin-only Heisenberg description; a compact
(t,U)→J mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon/Haldane features in
S+−(q,ω). Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.
Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an even-Haldane phase of a dimerized S = 1 2 Heisenberg chain and a Haldane phase of an S = 1 chain realized when Hund coupling locks two S = 1 2 spins per monomer into S = 1. Density-functional theory places the active manifold deep in the Mott regime ( U / t > 100), justifying a spin-only Heisenberg description; a compact ( t, U ) → J mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon-Haldane features in S +– ( q,ω). Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.
A large number of symmetry-protected topological (SPT) phases have been hypothesized for strongly interacting spin-1/2 systems in one dimension. Realizing these SPT phases, however, often demands fine-tunings hard to reach experimentally. And the lack of analytical solutions hinders the understanding of their many-body wave functions. Here we show that two kinds of SPT phases naturally arise for ultracold polar molecules confined in a zigzag optical lattice. This system, motivated by recent experiments, is described by a spin model whose exchange couplings can be tuned by an external field to reach parameter regions not studied before for spin chains or ladders. Within the enlarged parameter space, we find the ground state wave function can be obtained exactly along a line and at a special point, for these two phases, respectively. These exact solutions provide a clear physical picture for the SPT phases and their edge excitations. We further obtain the phase diagram by using infinite time-evolving block decimation and discuss the phase transitions between the two SPT phases and their experimental signatures.
We discuss a platform for the synthetic realization of key physical properties of helical Tomonaga Luttinger liquids (HTLLs) with ultracold fermionic atoms in one-dimensional optical lattices. The HTLL is a strongly correlated metallic state where spin polarization and propagation direction of the itinerant particles are locked to each other. We propose an unconventional one-dimensional Fermi-Hubbard model which, at quarter filling, resembles the HTLL in the long wavelength limit, as we demonstrate with a combination of analytical (bosonization) and numerical (density matrix renormalization group) methods. An experimentally feasible scheme is provided for the realization of this model with ultracold fermionic atoms in optical lattices. Finally, we discuss how the robustness of the HTLL against backscattering and imperfections, well known from its realization at the edge of two-dimensional topological insulators, is reflected in the synthetic one-dimensional scenario proposed here.