Parafermionization, bosonization, and critical parafermionic theories

TL;DR

The paper develops a comprehensive Z_k parafermionization/bosonization framework in (1+1)D, built from a generalized Jordan-Wigner transformation, to study parafermionic critical chains whose fundamental degrees of freedom obey fractional statistics. It demonstrates that these parafermionic critical theories generally evade description by existing bosonic/fermionic CFTs and exhibit unconventional modular transformations on the torus, providing explicit partition functions across broad classes and using the construction to exhaust all Z_k parafermionic minimal models. By treating parafermionization as an attachment to a gapped parafermionic phase and proving invertibility with a bosonization inverse, the work connects parafermionic and bosonic descriptions and reveals a rich spectrum with fractional spins and nontrivial locality. Extending to Z_k-clock models, the authors derive a large family of parafermionic critical theories, yielding insights into fractional statistics and generalized locality with potential implications for anomalies and topological phases.

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.

Figures (3)

• Figure 1: The visualization of the partition function $\mathcal{Z}_{s_1,s_2}$: the dash arrow points along the vector $(\tau_1,\tau_2)$ and the state $|\Psi\rangle$ is summed over. The solid arrow denotes the $s_1$-twisted link between $\gamma_{2L}$ and $\gamma_1$ which is effectively moved by $V_\text{transl}$ during the time evolution. The temporal twisting $Q_f^{s_2+1}$ is not explicitly shown in this figure. Here we have included $\gamma_{2j-1}$ and $\gamma_{2j}$ into one unit cell thereby denoted by $\gamma_{2j-1,2j}$ for the sake of clarity.
• Figure 2: (a) $\mathcal{T}$ transformation of spacetime torus: the transformation $\tau\rightarrow\tau+1$ crosses an additional branch cut during the continuum limit $a_0,\beta_0\rightarrow0$. (b) $\mathcal{S}$ transformation: the space-time plane is effectively rotated by 90 degrees, after which the branch-cutting link along the spatial direction is reversed.
• Figure 3: The visualization of the partition function $Z^\text{latt}_{r,s}$: the dashed arrows point along the vector $(\tau_1,\tau_2)$ and the state $|\Psi\rangle$ is summed over. The solid arrows denote the $r$-twisted link between the sites $L$ and $1$ and the correspondng links moved by $V_\text{transl}$ during the time evolution. The temporal $s$-twisting is not shown explicitly in this figure.