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Parafermions are emergent excitations which generalize Majorana fermions and are potentially relevant to topological quantum computation. Using the concept of Fock parafermions, we present a mapping between lattice $mathbb{Z}_4$ parafermions and lattice spin-$1/2$ fermions which preserves the locality of operators with $mathbb{Z}_4$ symmetry. Based on this mapping, we construct an exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian which hosts zero-energy modes obeying parafermionic algebra. We numerically show that this parafermionic phase remains stable in a wide range of parameters, and discuss its signatures in the fermionic spectral function.
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We discuss a one-dimensional fermionic model with a generalized $mathbb{Z}_{N}$ even multiplet pairing extending Kitaev $mathbb{Z}_{2}$ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a $mathbb{Z}_{N}$ symmetry enforcing an $N$-fold degenerate ground state robust to certain disorder. Interestingly, we show that the system supports a pair of parafermions but they are non-local instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy. Surrounded by two band insulators, the model exhibits for $N=4$ the dual of an $8 pi$ fractional Josephson effect highlighting the presence of parafermions.
We study interaction-induced Mott insulators, and their topological properties in a 1D non-Hermitian strongly-correlated spinful fermionic superlattice system with either nonreciprocal hopping or complex-valued interaction. For the nonreciprocal hopping case, the low-energy neutral excitation spectrum is sensitive to boundary conditions, which is a manifestation of the non-Hermitian skin effect. However, unlike the single-particle case, particle density of strongly correlated system does not suffer from the non-Hermitian skin effect due to the Pauli exclusion principle and repulsive interactions. Moreover, the anomalous boundary effect occurs due to the interplay of nonreciprocal hopping, superlattice potential, and strong correlations, where some in-gap modes, for both the neutral and charge excitation spectra, show no edge excitations defined via only the right eigenvectors. We show that these edge excitations of the in-gap states can be correctly characterized by only biorthogonal eigenvectors. Furthermore, the topological Mott phase, with gapless particle excitations around boundaries, exists even for the purely imaginary-valued interaction, where the continuous quantum Zeno effect leads to the effective on-site repulsion between two-component fermions.
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 DMRG calculations, we show the emergence of gapless and gapped phases depending on the parity of $N$ for attractive interactions at half-filling. The analogue of the Haldane phase of the spin-1 Heisenberg chain is stabilized for N=2 with non-local 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.
The physical properties of arbitrary half-integer spins $F = N - 1/2$ fermionic cold atoms trapped in a one-dimensional optical lattice are investigated by means of a low-energy approach. Two different superfluid phases are found for $F ge 3/2$ depending on whether a discrete symmetry is spontaneously broken or not: an unconfined BCS pairing phase and a confined molecular superfluid instability made of $2N$ fermions. We propose an experimental distinction between these phases for a gas trapped in an annular geometry. The confined-unconfined transition is shown to belong to the $Z_N$ generalized Ising universality class. We discuss on the possible Mott phases at $1/2N$ filling.
Using quantum Monte Carlo simulations, we show that density-density and pairing correlation functions of the one-dimensional attractive fermionic Hubbard model in a harmonic confinement potential are characterized by the anomalous dimension $K_rho$ of a corresponding periodic system, and hence display quantum critical behavior. The corresponding fluctuations render the SU(2) symmetry breaking by the confining potential irrelevant, leading to structure form factors for both correlation functions that scale with the same exponent upon increasing the system size, thus giving rise to a (quasi)supersolid.
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