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Fermion parity gap and topological degeneracy of one dimensional chain of fermions with spin-orbit coupling, Zeeman field, and intrinsic attractive interaction

Monalisa Singh Roy, Manoranjan Kumar, Jay Deep Sau, Sumanta Tewari

Published Apr 7, 2019• 0 citations
OpenAlex

Abstract

We examine the topological properties of a one-dimensional (1D) chain of fermions with spin-orbit coupling, Zeeman field, and attractive Hubbard interaction by numerically computing the pair binding energy, excitation gap, and susceptibility to local perturbations by density matrix renormalization group. Such a system can in principle be realized in a 1D optical lattice. We find that, in the presence of spatial interfaces introduced by a smooth parabolic potential, the variation of the pair binding energy and the excitation gap with the system size indicate an exponentially vanishing fermion parity gap and topological ground state degeneracy in the thermodynamic limit, consistent with recent works. However, the susceptibility of the ground state degeneracy to local perturbations indicate that the vanishing of the fermion parity gap in this number conserving system scales as a power-law in system size. We compare the present system with the more familiar system of an Ising antiferromagnet in the presence of a transverse field realized with Rydberg atoms, and argue that the degeneracy of this conventional symmetry-breaking system is identical to the 1D chain of fermions considered. Therefore, the degeneracy properties of the 1D chain of fermions cannot be attributed to topology.

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We examine the topological properties of a one-dimensional (1D) chain of fermions with spin-orbit coupling, Zeeman field, and attractive Hubbard interaction by numerically computing the pair binding energy, excitation gap, and susceptibility to local perturbations by density matrix renormalization group. Such a system can in principle be realized in a 1D optical lattice. We find that, in the presence of spatial interfaces introduced by a smooth parabolic potential, the variation of the pair binding energy and the excitation gap with the system size indicate an exponentially vanishing fermion parity gap and topological ground state degeneracy in the thermodynamic limit, consistent with recent works. However, the susceptibility of the ground state degeneracy to local perturbations indicate that the vanishing of the fermion parity gap in this number conserving system scales as a power-law in system size. We compare the present system with the more familiar system of an Ising antiferromagnet in the presence of a transverse field realized with Rydberg atoms, and argue that the degeneracy of this conventional symmetry-breaking system is identical to the 1D chain of fermions considered. Therefore, the degeneracy properties of the 1D chain of fermions cannot be attributed to topology.
We examine the properties of a one-dimensional (1D) Fermi gas with attractive intrinsic (Hubbard) interactions in the presence of spin-orbit coupling and Zeeman field by numerically computing the pair binding energy, excitation gap, and susceptibility to local perturbations using the density matrix renormalization group. Such a system can, in principle, be realized in a system of ultracold atoms confined in a 1D optical lattice. We note that, in the presence of spatial interfaces introduced by a smooth parabolic potential, the pair binding and excitation energy of the system decays exponentially with the system size, pointing to the existence of an exponential ground state degeneracy, and is consistent with recent works. However, the susceptibility of the ground state degeneracy of this number-conserving system to local impurities indicates that the energy gap vanishes as a power law with the system size in the presence of local perturbations. We compare this system with the more familiar system of an Ising antiferromagnet in the presence of a transverse field realized with Rydberg atoms and argue that the exponential splitting in the clean number-conserving 1D Fermi system is similar to a phase with only conventional order.
We describe a novel topological superfluid state, which forms in a one-dimensional Fermi gas with Rashba-like spin-orbit coupling, a Zeeman field, and intrinsic attractive interactions. In spite of total number conservation and the presence of gapless excitations, Majorana-like zero modes appear in this system and can be linked with interfaces between two distinct phases that naturally form at different regions of the harmonic trap. As a result, the low lying collective excitations of the system, including the dipole oscillations and the long-wavelength phonons are all doubly degenerate. While backscattering from point impurities can lead to a splitting of the degeneracies that scales algebraically with the system size, the smooth confining potential can only cause an exponentially small splitting. We show that the topological state can be uniquely probed by a pumping effect induced by a slow sweep of the Zeeman field from a high initial value down to zero. The effect is expected to be robust to introducing a finite temperature as long as it is much smaller than the interaction induced single particle gap in the final state of the sweep.
We examine quantum phases emerging from the double degeneracy of ppp-orbital bands in attractive atomic Fermi gases loaded on a one-dimensional (1D) optical lattice. Our numerical simulations via the density-matrix renormalization group predict the emergence of a state with a charge excitation gap, namely the Haldane insulator phase. A mapping onto an effective spin-1 model reveals its physical origin. Moreover, we show that population imbalance leads to richer diversity of the quantum phases, including a phase-separated polarized state. Finally, we study the effects of the harmonic trap potential in this 1D chain.
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.
We revisit the low-energy physics of one-dimensional spinless fermion liquids, showing that with sufficiently strong interactions the conventional Luttinger liquid can give way to a strong-pairing phase. While the density fluctuations in both phases are described by a gapless Luttinger liquid, single fermion excitations are gapped only in the strong-pairing phase. Smooth spatial Interfaces between the two phases lead to topological degeneracies in the ground-state and low-energy spectrum of the density fluctuations. Using a concrete microscopic model, with both single-particle and pair hopping, we show that the strong-pairing state is established through emergence of a new low-energy fermionic mode. We characterize the two phases with numerical calculations using the density matrix renormalization group. In particular we find enhancement of the central charge from c=1c=1c=1 in the two Luttinger liquid phases to c=3/2c=3/2c=3/2 at the critical point, which gives direct evidence for an emergent critical Majorana mode. Finally, we confirm the existence of topological degeneracies in the low-energy phonon spectrum, associated with spatial interfaces between the two phases.
We investigate the effects of attractive Hubbard interaction on two-component fermionic atoms in narrow two-dimensional (2D) energy bands that exhibit Rashba spin-orbit coupling (SOC) in the presence of an applied Zeeman field. This narrow-band 2D spin-orbit-coupled attractive Fermi-Hubbard model can potentially be realized in cold-atom systems in optical lattices with artificially engineered Rashba SOC and Zeeman field. Employing a self-consistent mean-field approximation for the pairing potential, we uncover a complex phase diagram featuring various topological superfluid (TS) phases, dependent on the chemical potential and the Zeeman field. We focus on the pairing potential and the corresponding quasiparticle gap characterizing the TS phases, which are notably small for a wide-band model with quadratic dispersion near the $\mathrm{\ensuremath{\Gamma}}$ point, as found in earlier work. We identify the parameter regimes that maximize the gap. We find that, while generally the value of the pairing potential increases with the reduction of the fermionic bandwidth, as expected for narrow- or flat-band systems, the magnitude of the topological gap characterizing the TS phases reaches a significantly improved maximum of about 10−−12.510--12.5%10−−12.5 of the interaction strength at finite values of the hopping amplitude, Rashba coupling, and Zeeman field. Our work establishes a robust operating regime for experimenting with topological superfluids in cold-atom optical lattices.