Quantum fidelity approach to the ground-state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field

In this work we analyze the ground-state properties of the $s=1/2$ one-dimensional axial next-nearest-neighbor Ising model in a transverse field using the quantum fidelity approach. We numerically determined the fidelity susceptibility as a function of the transverse field $B_x$ and the strength of the next-nearest-neighbor interaction $J_2$, for systems of up to 24 spins. We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic, floating, and $〈2,2〉$ phases, and we predict an infinite number of modulated phases in the thermodynamic limit ($L\to \infty$). Paramagnetism only occurs for larger magnetic fields. The transition lines separating the modulated phases seem to be of second order, whereas the line between the floating and the $〈2,2〉$ phases is possibly of first order.

Figure 1

Fidelity susceptibility as a function of the next-nearest-neighbor coupling $J_2$ for the transverse ANNNI model with $B_x=0.2$, for the case of a chain with $L=8$ spins. Here, and also in the next figures, $J_1=1$ is set as the energy unit. The locations of the peaks give the transition points.

Figure 2

Phase diagram in the $(J_2,B_x)$ plane for a system of size $L=8$. The system displays three phase regions, ferromagnetic $F$, $P_1$, and $〈2,2〉$. No additional phases are present here. The dashed boundary is the exact Peschel-Emery line.

Figure 3

Fidelity susceptibility as a function of next-nearest-neighbor coupling $J_2$, with $B_x=0.2$, for the case $L=12$ spins. The locations of the peaks give the transition points.

Figure 4

Phase diagram in the $(J_2,B_x)$ plane. The system displays four phase regions, ferromagnetic $F$, $P_1$ and $P_2$, and $〈2,2〉$. Again, the dashed boundary is the Peschel-Emery line.

Figure 5

Fidelity susceptibility as a function of the transverse field $B_x$ for the transverse ANNNI model with $J_2=0.30$, for the case of a chain with $L=12$ spins. The location of the peak gives the transition point.

Figure 6

Fidelity susceptibility versus the transverse field $B_x$, for $J_2=0.46$, in the case $L=12$. The locations of the peaks give the transition points.

Figure 7

Fidelity susceptibility as a function of the transverse field $B_x$, with $J_2=0.70$, for the case $L=12$. The location of the peak gives the transition point.

Figure 8

Ground-state amplitude versus the basis-state index $n$ for $(J_2,B_x)=(0.345,0.200)$, within the phase $F$ for $L=12$. The two largest amplitudes correspond to the ferromagnetic phase. The smaller amplitudes are induced by the transverse magnetic field.

Figure 9

Ground-state amplitude for each basis-state index $n$, with $(J_2,B_x)=(0.438,0.200)$, located inside the phase region $P_2$, for $L=12$. The two largest amplitudes correspond to a ferromagnetic phase. The next largest amplitudes are from states with a single kink separating ferromagnetic domains.

Figure 10

Amplitude of the ground state against the basis-state index $n$ for the case $L=12$, $J_2=0.565$, and different values of $B_x$. Top: $B_x=0.200$, which lies in the phase region $P_1$ in Fig. 4. The six highest amplitudes correspond to a $〈3,3〉$ phase, while the next-highest amplitudes belong to states without sequential order for the spins. Middle: $B_x=2.000$; here there are no noticeable prominent amplitudes, since the system is already in an induced paramagnetic state, where the spins are mostly aligned to the transverse field. Bottom: Case $B_x=20.00$; now nearly all the spins are aligned with the transverse field.

Figure 11

Amplitude of the ground state for each of the basis-state indexes $n$ when $(J_2,B_x)=(0.675,0.200)$, within the phase $〈2,2〉$ for $L=12$. The four largest amplitudes correspond to the $〈2,2〉$ phase. The transverse magnetic field is responsible for the appearance of the smaller amplitudes.

Figure 12

Fidelity susceptibility as a function of the next-nearest-neighbor coupling for the case $L=16$. The four peaks shown are centered at the transition points. Here $B_x=0.2$.

Figure 13

Phase diagram in the $(J_2,B_x)$ plane for the case $L=16$. The figure shows the phase regions F, $P_1$, $P_2$, $P_3$, and $〈2,2〉$. The dashed boundary is the exact Peschel-Emery line.

Figure 14

Fidelity susceptibility as a function of next-nearest-neighbor coupling $J_2$ for $B_x=0.2$ in the case $L=20$. The five peaks are centered at the transition points.

Figure 15

Phase diagram in the $(J_2,B_x)$ plane when the system size is $L=20$. In addition to the phases $F$ and $〈2,2〉$, at the left and right of the diagram, respectively, there are four phases in between them, namely $P_1$, $P_2$, $P_3$, and $P_4$. The dashed boundary is the exact Peschel-Emery line.

Figure 16

Fidelity susceptibility as a function of next-nearest-neighbor coupling $J_2$ for the case $L=20$. The six peaks are centered at the transition points. Here $B_x=0.2$.

Figure 17

Phase diagram in the $(J_2,B_x)$ plane when the system size is $L=24$. In addition to the phases $F$ and $〈2,2〉$, at the left and right of the diagram, there are five phases in between them, namely $P_1$, $P_2$, $P_3$, $P_4$, and $P_5$. The dashed boundary is the exact Peschel-Emery line, which lies very close to transition line between $F$ and $P_5$.

Figure 18

Fidelity susceptibility at criticality as a function of the lattice size $L$ for two different transition lines. Squares are for the transition line bordering the ferromagnetic phase (Peschel-Emery line) while circles are for the antiphase.