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 nontrivial powers of $q$ at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.

Figure 1

Plot of ${\chi }_F$ versus $h$ for $q=1,\varphi =0$ (red solid), $q=2,\varphi =\pi /4$ (black dashed), $q=3,\varphi =0$ (blue dotted), and $q=4,\varphi =\pi /8$ (magenta plus), with $N=240$ and $\theta =\pi /N$.

Figure 2

Plot of ${\chi }_F$ versus $h$ for $q=2,\varphi =\pi /4$ (black dashed), $q=3,\varphi =0$ (blue dotted), and $q=4,\varphi =\pi /8$ (magenta plus), with $N=240$ and $\theta =0.001$.

Figure 3

Plot of ${\chi }_F\theta$ versus $h/{\theta }^{1/2}$ for $\theta =0.0004$ (black dashed), $\theta =0.0002$ (blue dotted), and $\theta =0.0001$ (magenta plus), with $q=2$, $N=240$, and $\varphi =\pi /4$.

Figure 4

Plot of ${\chi }_F{\theta }^{2/3}$ versus $h/{\theta }^{1/3}$ for $\theta =0.0001$ (black dashed), $\theta =0.00003$ (blue dotted), and $\theta =0.00001$ (magenta plus), with $q=3$, $N=240$, and $\varphi =0$.

Figure 5

Plot of ${\chi }_F\theta$ versus $h/{\theta }^{1/2}$ for $\varphi =0$ (black dashed), $\varphi =\pi /8$ (blue dotted), and $\varphi =\pi /4$ (magenta plus), with $q=2$, $N=240$, and $\theta =0.0004$.

Figure 6

Plot of ${\chi }_F\theta$ versus ${h\left|\sin \left(2\varphi \right)\right|}^{1/2}/{\theta }^{1/2}$ for $\varphi =\pi /16$ (black dashed), $\varphi =\pi /8$ (blue dotted), and $\varphi =\pi /4$ (magenta plus), with $q=2$, $N=240$, and $\theta =0.0001$.

Figure 7

Plot of ${\chi }_F\theta$ versus $\delta /{\theta }^{1/2}$ for $\theta =0.0004$ (black dashed), $\theta =0.0002$ (blue dotted), and $\theta =0.0001$ (magenta plus) for the anisotropic spin-1/2 chain with $N=240$.

Figure 8

Plot of ${\chi }_F$ versus $h$ for $q=120$ (black dashed), $q=240$ (blue dotted), and $q=480$ (magenta solid), with $N=2q$, $\theta =\pi /N$, and $\varphi =\pi /N$ in each case.

Figure 9

Log-log plot of the peak value (blue solid, $y$ axis on left) and the full width at half maximum (red dashed, $y$ axis on right) versus $q$ for the peak in ${\chi }_F$ at $h=2$.

Figure 10

Plot of ${\chi }_F/q^{2.25}$ versus $(h-2)q^{0.70}$ for $q=120$ (black dashed), $q=240$ (blue dotted), and $q=480$ (magenta solid), with $N=2q$, $\theta =\pi /N$, and $\varphi =\pi /N$ in each case.