2013/10/31 by Paul Fendley · 2 citations
Mathematics · Physics and Astronomy · #Condensed matter physics #Fermion #Generalization #Hamiltonian (control theory) #Hermitian matrix #Ising model #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum many-body systems #Quantum mechanics #Spectrum (functional analysis) #Spins #Theoretical physics #Topological Materials and Phenomena #cond-mat.stat-mech #cond-mat.str-el #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/47/7/075001
published as J. Phys. A 47 (2014) 075001 · 44 pages. v2: minor rewriting, added several references
arxiv created 2013/11/18 · openalex publication_date 2014/01/30 · arxiv updated 2015/06/17 · openalex created_date 2020/07/02 · openalex updated_date 2026/08/06
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 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 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.