Quantum phase transitions on the noncommutative circle

Regularization aims to cure the singularities in field theory by modifying observables and making them finite; for example, lattice discretization is a traditional regularization scheme in which the lattice constant $a$ provides a hard ultraviolet cutoff. Here, we offer an alternative approach to regularize the continuous $(1+1)$-dimensional space-time, which is dubbed the ``noncommutative circle.'' This method combines two key ingredients: continuous torus geometry and Landau level projection in the quantum Hall physics. Based on this method, we study the critical phenomena of quantum phase transitions belonging to the Ising and Potts universality classes, and we systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators and organize the remaining low-energy eigenstates into conformal towers.