Fermionized parafermions and symmetry-enriched Majorana modes

Abstract

Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abeliananyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in twodimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter-yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z 4 parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.

Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire

“…Namely, they have exactly energy E = 0, and they can be expressed as in Eqs. (19), (20) and (25). For finite even L, the left and right zero modes discussed in the previous section hybridize and the resulting subgap eigenstates have finite energy ±E, which decreases exponentially with L. For even L, the first correction to the solution presented in the previous section is of order E ∼ |z 1 | L .…”

“…The corresponding excitations are described by γ σ ,γ σ given by Eqs. (19) and (25). In the case of the finite chain we are analyzing here, the exact subgap eigenstate with energy E > 0 given by Eq.…”

“…[11][12][13][14][15][16][17][18][19][20] This property has interesting implications which can be relevant for their detection and manipulation. [21][22][23][24][25][26] For a recent review on proposals to realize the TRITOPS phase, see Ref. 27.…”

Time-Crystalline Topological Superconductors

“…Discussion. The admixture of symmetry breaking and topology is known to generate new physics in static systems; examples include 8π-periodic Josephson effects [29,30] and enrichment of Majorana braiding and fusion [31]. Our work establishes that driven systems can be similarly enriched by 'decorating' topological phases with spontaneous time-translation symmetry breaking.…”

Parafermion braiding in fractional quantum Hall edge states with a finite chemical potential

“…Therefore, non-Abelian anyons with a richer exchange statistics are needed. Recently, much attention has focused on realizing so-called Z p parafermions in solid-state devices [26][27][28][29][30][31][32][33][34][35][36][37]. An ordered set of parafermionic operators obeys the following operator algebra, χ k χ l = e 2πi/p χ l χ k , (for k > l)…”