Quantum fidelity for degenerate ground states in quantum phase transitions

Abstract

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.

Geometry of quantum phase transitions

“…Since its early development this approach has been applied successfully to a wide variety of systems, ranging from quasi-free Fermions [198,199], the Dicke model [158,161,193,200], matrix product state systems [201], Bose-Hubbard models [202,203], Stoner-Hubbard and BCS models [51]. Various methods have been developed to evaluate the fidelity, such as exact diagonalisation, density matrix renormalisation group [160,204,205], quantum Monte Carlo methods [195,206,207], tensor network algorithms [208,209]. This provides the means to venture into the study of models where analytical solutions are not available [160,204,205,207,210,211].…”

Degenerate ground states and multiple bifurcations in a two-dimensionalq-state quantum Potts model

“…As discussed [23], spontaneous symmetry breaking leads to a degenerate groundstate for the broken-symmetry phase. Consequently, the relations between the local order parameters calculated from degenerate groundstates are determined by a symmetry group of the system Hamiltonian.…”

“…For one-dimensional spin lattice systems, doubly degenerate ground states for broken-symmetry phases have been detected by means of the quantum fidelity bifurcations with the tensor network algorithm in various spin lattice models such as the quantum Ising model, spin-1/2 XYX model with transverse magnetic field, among others [20][21][22]. Very recently, Su et al [23] have further demonstrated that the quantum fidelity measured by an arbitrary reference state can detect and identify explicitly all degenerate ground states (N-fold degenerate groundstates) due to spontaneous symmetry breaking in broken-symmetry phases for the infinite matrix product state (iMPS) representation in the one-dimensional (1D) q-state quantum Potts model. It has been also discussed how each order parameter calculated from degenerate ground states transforms under a subgroup of a symmetry group of the Hamiltonian.…”

“…The infinite time-evolving block decimation (iTEBD) method [5] will be used to calculate iPEPS groundstate wavefunctions with randomly chosen 2D initial states. In order to distinguish degenerate groundstate wavefunctions of broken-symmetry phases and to determine phase transition points, the quantum fidelity [23], defined as an overlap measurement between an arbitrary 2D reference state and the iPEPS groundstates of the system, will be employed. The defined quantum fidelity corresponds to a projection of each 2D iPEPS groundstate onto a chosen 2D reference state.…”

Universal Order Parameters and Quantum Phase Transitions: A Finite-Size Approach

“…For q = 2, 3 and 4 there is a continuous phase transitions at λ = 1, whilst for q = 5 the phase transition is first-order (discontinuous) at λ = 1. Here we remark that the fidelity per site has been demonstrated to be capable of detecting the discontinuous phase transitions in this model through the so-called multiple bifurcation points 31 .…”