Impurity-induced quantum phase transition in finite Heisenberg spin chains: Criteria for existence and stability

A quantum phase transition may occur in a system at zero temperature when a controlling parameter is tuned towards a critical point. An important question is whether such a critical point exists in a particular system and how stable it is. Here, we identify the critical point of a quantum phase transition as a singular point in the affine algebraic variety of the characteristic equation for the Hamiltonian describing the system, with an unstable critical point being associated with an isolated singular point which has a finite Tjurina number. The theory is illustrated by studying a model system of zero-dimensional (finite) Heisenberg spin chain with an impurity, which exhibits a nontrivial first-order quantum phase transition. Both analytical and numerical calculations show that the quantum phase transition always exists when the impurity has a $Z_2$ symmetry but only remains in systems with an even number of spin sites when the $Z_2$ symmetry is broken.