General quantum fidelity susceptibilities for the $J_1$-$J_2$ chain

We study slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-$1/2$ $J_1$-$J_2$ antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\chi }_{\rho }$, ${\chi }_D$, and ${\chi }_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the $J_1$-$J_2$ model. We show that they all can accurately identify a quantum critical point in this model occurring at $J_2^c\sim 0.241J_1$ between a gapless Heisenberg phase for $J_2<J_2^c$ and a dimerized phase for $J_2>J_2^c$. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.

Figure 1

$\frac{{\chi }_{\rho }}{L}$ vs $\Delta$: The spin stiffness fidelity susceptibility [${\chi }_{\rho }\left(\Delta \right)/L$] as a function of the $z$ anistropy $\Delta$. At the $\Delta =0$ point the spin-current operator $\mathcal{J}$ and kinetic energy $\mathcal{T}$ commute with the XXZ Hamiltonian and thus such a perturbation does not change the ground state, and the fidelity is one. Thus, ${\chi }_{\rho }$ is zero at this point.

Figure 2

${\chi }_{\rho }$ vs $L$ (the XXZ model at different values of $\Delta$): This graph shows the scaling of ${\chi }_{\rho }$ with system size for different values of the $z$ anisotropy $\Delta$. The points represent numerical data and the lines represent fits to the scaling form predicted for the spin stiffness susceptibility ${\chi }_{\rho }/L=A_{1}L+A_2+A_3{L}^{-1}+A_4{L}^{3-8K}$. It can be seen that there is good agreement.

Figure 3

$\frac{{\chi }_{\rho }}{L}$ vs $J_2$ and inset: The generalized spin stiffness susceptibility ${\chi }_{\rho }$ as a function of the second-nearest-neighbor exchange coupling $J_2$. The system acquires a clearly size-invariant form in the vicinity of the critical point $J_2\sim 0.24$ (as well as tending to a global minima). Inset shows the minima for system sizes $L=16$, 20, 24, 28, 32 with $J_2^c$ indicated as the vertical dashed line. A clear dependence of the $J_2$ value of ${\chi }_{\rho }/L$ minima on the system size can be seen.

Figure 4

(a) The $J_2$ value of ${\chi }_{\rho }$ minima as a function of system size, as well as a (power-law) line of best fit. As the system size tends toward infinity the power-law best fit predicts a minima at $J_2=0.24077$ in good agreement with previously published results. (b) Scaling of ${\chi }_{\rho }$ at $J_2=0.23$ (the highest, linear curve), $J_2=0.25$ (the second highest, linear curve), and the critical point $J_2=0.241167$ (flat curve). The near constant scaling of $\frac{{\chi }_{\rho }}{L}$ at the critical point as well as nonconstant scaling on either side of the critical point can clearly be seen.

Figure 5

$\frac{{\chi }_D}{L^3}$ vs $J_2$: The generalized dimer fidelity susceptibility ${\chi }_D/L^3$ as a function of the second-nearest-neighbor exchange parameter $J_2$. A clear intersection of all curves can be seen in the vicinity of the proposed critical point at $J_2\sim 0.2–0.25$. The inset explicitly shows the crossing of $L=12$ and $L=24$. The dashed vertical lines indicate $J_2^c$.

Figure 6

(a) Scaling of ${\chi }_D$ vs $L^3$ at the points $J_2=0.0,0.1,0.2$, and $0.241167$. For $J_2<0.241167$ the scaling exponent is fitted to be less than 3. (b) The $J_2$ value of the intersection of $\frac{{\chi }_D}{L^3}$ between systems of size $L$ and $L+2$ plotted as a function of $L$. The curve can be fitted with a power-law line of best fit. The line of best fit is found to converge to $J_2=0.241$.

Figure 7

${\chi }_{\mathrm{AF}}/L^3$ vs $J_2$: ${\chi }_{\mathrm{AF}}$ is expected to approach zero exponentially with the system size for $J_2>J_2^c$, to scale as $L^3$ at $J_2^c$, and to scale as $L^{{\alpha }_{\mathrm{eff}}}$ with ${\alpha }_{\mathrm{eff}}>3$ for $J_2<J_2^c$. A crossing close to the critical point $J_2^c$ (dashed vertical line) is then visible.