Abstract
We formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality at k = 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory when k > 2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.
Highlights
Many efforts have been done on the classification of gapped parafermionic topological orders without symmetry [12, 14,15,16,17,18] and with symmetries [19]
The construction of the general critical theory of the Zk-parafermionic systems still remains an open problem for k > 2, and the significant distinctions of parafermion statistics from boson and fermion statistics imply that these parafermionic critical theories with k > 2 may not be described by any existing conformal field theories (CFTs), e.g., bosonic or fermionic CFTs
The parafermionization method enables us to study the general properties of partition functions of critical parafermionic chains, which obey unconventional modular transformations, distinct from any existing bosonic or fermionic CFT
Summary
Where 1k is the k × k unit matrix and the matrices other than 1k appear only in the k-dimensional local Hilbert space at the site j Such a generalized Zk-spin picture in a finite chain is, roughly speaking, equivalent to a parafermionic system by the following Fradkin-Kadanoff transformation [20] generalizing the Jordan-Wigner transformation: γ2j−1 ≡ σj τi; γ2j ≡ ω(k−1)/2σj τi, i
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