Free parafermions

Abstract

The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with $Z_n$ symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-hermitian (but PT-symmetric) clock Hamiltonian is "free", in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are $n$ possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a $Z_n$ generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter's and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a "free parafermion". A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.

44 pages. v2: minor rewriting, added several references

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Cited by in corpus (93)

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• Topological phases of parafermions: a model with exactly-solvable ground states
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• Tunable Splitting of the Ground-State Degeneracy in 1D Parafermionic Wires
• Parafermions in the tau-2 model
• Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT
• Reflection Positivity for Parafermions
• Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains
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• Spin Structures and Exact Dualities in Low Dimensions
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• Families of Gapped Interfaces Between Fractional Quantum Hall States
• Onsager algebra and algebraic generalization of Jordan-Wigner transformation
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• A Brief History of Free Parafermions
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• Exact real time dynamics with free fermions in disguise
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• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• Probing Majorana Modes via Local Spin Dynamics
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• Conformal Field Theories as Scaling Limit of Anyonic Chains
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• Dynamically Induced Topology and Quantum Monodromies in a Proximity Quenched Gapless Wire
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