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 \chi_F can detect a QCP provided that the correlation length exponent satisfies \nu < 2. We then show that \chi_F can be used to locate a QCP even if \nu \ge 2 if we introduce boundary conditions labeled by a twist angle N\theta, where N is the system size. If the QCP lies at g = 0, we find that if N is kept constant, \chi_F has a scaling form given by \chi_F \sim \theta^{-2/\nu} f(g/\theta^{1/\nu}) if \theta \ll 2\pi/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 \nu = q, and in a XY spin-1/2 chain in which \nu = 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 \chi_F. The peak value and width of \chi_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.