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arXiv:1904.03660 · Physical review. B./Physical review. B

Fermion parity gap and exponential ground state degeneracy of the one-dimensional Fermi gas with intrinsic attractive interaction

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

Published Sep 21, 2020• 1 citation
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Abstract

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.

DOI
10.1103/physrevb.102.125135

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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 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 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.
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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 revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.