Parafermionization, bosonization, and critical parafermionic theories

Abstract

We formulate a ${\mathbb{Z}}_k$-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 ${\mathbb{Z}}_k$-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 ${\mathbb{Z}}_k$-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 ${\mathbb{Z}}_k$-parafermionic minimal models, complementing earlier works on fermionic cases.

arXiv comment27 pages, 3 figures

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• Geometric approach to Lieb-Schultz-Mattis theorem without translation symmetry under inversion or rotation symmetry
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• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Identification of gapless phases by a twisting operator
• Extending the planar theory of anyons to quantum wire networks