Classifying parafermionic gapped phases using matrix product states

Abstract

In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for Zp parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of Zp parafermions without extra symmetry other than Zp charge symmetry, including topological phases, spontaneous symmetry breaking phases and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the non-trivial simple Zp graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. But the spontaneous symmetry breaking phases correspond to the trivial semisimple Z p/n graded algebras, which can be further reduced to the trivial simple Z p/n graded algebras, where n is the divisor of p. So the present results provide the complete classification of the parafermionic gapped phases and deepen our understanding of topological phases in one dimension.

Decoding quantum criticality from fermionic/parafermionic topological states

“…where |Ψ m is a many-body state defined in the H m with the charge m = 0, 1, 2, .., N − 1. Recently, the MPS for the one-dimensional parafermionic topological phase has been constructed 10,11 . It can be expressed in terms of a series of local parafermionic graded tensor:…”

Stability of Majorana Edge Zero Modes against Interactions

“…For those zero modes, the dynamics [8,9] and the thermal effect [10] have been studied as well. See also [11][12][13][14][15][16][17][18] for other several aspects of parafermions. However, mathematical treatments of Majorana edge zero modes for interacting systems are still rare.…”

Fragmentation of C2H4 2+ produced by 300 eV electron impact

“…Recently, numerous studies have been conducted on the dissociation of molecular ions produced by various ionizing projectile or radiation [6][7][8][9][10][11][12][13][14][15][16] and KERs for some specific dissociation channels have been measured, which advance insight into the proton/hydrogen migration [12], isomerization [13,14], fragmentation mechanism [15,16], etc.…”