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We investigate the ground-state Riemannian metric and the cyclic quantumdistance of an inhomogeneous quantum Ising spin-1/2 chain in a transversefield. This model can be diagonalized by using a general canonicaltransformation to the fermionic Hamiltonian mapped from the spin system. Theground-state Riemannian metric is derived exactly on a parameter manifold ring$S^1$, which is introduced by performing a gauge transformation to the spinHamiltonian through a twist operator. The ground-state cyclic quantum distanceand the second derivative of the ground-state energy are studied in differentinhomogeneous exchange coupling parameter region. Particularly, we show thatthe quantum ferromagnetic phase in the uniform Ising chain can be characterizedby an invariant cyclic quantum distance with a constant ground-state Riemannianmetric, and this metric will rapidly decay to zero in the paramagnetic phase.

Ground-state Riemannian metric, cyclic quantum distance, and the quantum criticality in an inhomogeneous Ising spin chain