A \(q\)-Virasoro algebra at roots of unity, free fermions, and Temperley-Lieb Hamiltonians (Q2804961)

The author considers the \(q\)-deformation of the Virasoro algebra [\textit{M. Chaichian} and \textit{P. Prešnajder}, ``Sugawara construction and the \(q\)-deformation of Virasoro (super) algebra'', Phys. Lett. B 277, No. 1--2, 109--118 (1992; \url{doi:10.1016/0370-2693(92)90965-7})] expressed in terms of free fermions and checks that in the limit \(q\rightarrow 1\), one can get to the first nontrivial order of the expansion of the Virasoro algebra. This algebra is realized, when the deformation parameter is a root of unity, it can be realized on the lattice in a truncated form in terms of the Clifford algebra of \(\Gamma\) matrices, albeit without the central extension term.NEWLINENEWLINEThis lattice truncation enjoys several useful properties, such as the existence of null vectors on the lattice, and especially, it is related to the Temperley-Lieb Hamiltonians of \textit{A. Nigro} [``Lattice integrals of motion of the Ising model on the cylinder'', \url{arXiv:1010.4426}]. The eigenvalues of the Temperley-Lieb Hamiltonians are found, and their simple form can be easily guessed from their simple form in terms of lattice Fermi modes.