Fidelity susceptibility of one-dimensional models with twisted boundary conditions

It is well-known that the ground state fidelity 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 a 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 in a tight-binding model of fermions with a spatially varying chemical potential with amplitude $h$ and period $2q$ in which $\nu = q$. Finally we show that when $q$ is very large, the model has two 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 $\chi_F$ 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.