Fidelity susceptibility of one-dimensional models with twisted boundary conditions

Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility χ_F can detect a QCP provided that the correlation length exponent satisfies ν< 2. We then show that χ_F can be used to locate a QCP even if ν\ge 2 if we introduce boundary conditions labeled by a twist angle Nθ, where N is the system size. If the QCP lies at g = 0, we find that if N is kept constant, χ_F has a scaling form given by χ_F \sim θ^{-2/ν} f(g/θ^{1/ν}) if θ\ll 2π/N. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude h and period 2q in which ν= q, and in a XY spin-1/2 chain in which ν= 2. Finally we show that when q is very large, the model has two additional QCPs at h = \pm 2 which cannot be detected by studying the energy spectrum but are clearly detected by χ_F. The peak value and width of χ_F seem to scale as non-trivial powers of q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.

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