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Quantum fidelity for degenerate groundstates in quantum phase transitions

Spontaneous symmetry breaking mechanism in quantum phase transitions manifests the existence of degenerate groundstates in broken symmetry phases. To detect such degenerate groundstates, we introduce a quantum fidelity as an overlap measurement between system groundstates and an arbitrary reference state. This quantum fidelity is shown a multiple bifurcation as an indicator of quantum phase transitions, without knowing any detailed broken symmetry, between a broken symmetry phase and symmetry phases as well as between a broken symmetry phase and other broken symmetry phases, when a system parameter crosses its critical value (i.e., multiple bifurcation point). Each order parameter, characterizing a broken symmetry phase, from each of degenerate groundstates is shown similar multiple bifurcation behavior. Furthermore, to complete the description of an ordered phase, it is possible to specify how each order parameter from each of degenerate groundstates transforms under a symmetry group that is possessed by the Hamiltonian because each order parameter is invariant under only a subgroup of the symmetry group although the Hamiltonian remains invariant under the full symmetry group. Examples are given in the quantum $q$-state Potts models with a transverse magnetic field by employing the tensor network algorithms based on infinite-size lattices. For any $q$, a general relation between the local order parameters is found to clearly show the subgroup of the $Z_q$ symmetry group. In addition, we systematically discuss the criticality in the $q$-state Potts model.