Edge modes in a frustrated quantum Ising chain

The ground state properties of an Ising chain with nearest ($J_1$) and next-nearest neighbor ($J_2$) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered region), one of which has simple ferromagnetic or Néel order depending upon the sign of $J_1$, and the other is the double-staggered antiferromagnetic phase. The presence of the edge modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and Néel phases support two edge modes, except very near $J_1=0$, where it seems they have four edge modes like the exact case at $J_1=0$ itself.

Figure 1

The $J_1-J_2$ quantum Ising chain of $L$ spin-1/2's. The solid lines denote $J_1$ interaction, and the dashed lines denote $J_2$.

Figure 2

The frustrated and unfrustrated regions of $\widehat{H}$ in the $J_1-J_2$ plane. The dotted line, $2J_2=\left|J_1\right|$, is the line of maximum frustration, which together with $J_1=0$ line for negative $J_2$ form the phase boundaries between the FM (ferromagnetic), N-AFM (Néel antiferromagnetic), and DS-AFM (double-staggered antiferromagnetic) ground states in the corresponding classical $J_1-J_2$ Ising chain (with $h=0$).

Figure 3

The spin-spin correlation function for $J_1=-1.5$ and different values of $J_2$, from a DMRG calculation ($L=206$, and the states kept after truncation $=32$). For $J_2=2.0$, it shows double-staggered AFM order. In the middle panel, note the slow and rapid decays of the spin-spin correlations for $J_2=1.2$ and $0.8$, respectively. It reveals two qualitatively different quantum paramagnetic (Q-PM) phases. For $J_2=0$, the spins are ferromagnetically correlated. The transverse field, $h$, is equal to 1 here and in the rest of the paper.

Figure 4

The FM ($p$) and DS-AFM ($p_{\mathrm{ds}}^{}$) order parameters vs. $J_2$ for fixed values of $J_1$, from a DMRG calculation on a chain of length $L=206$ (and 64 states kept in truncation).

Figure 5

Energy-gap vs. $J_2$ for different values of $J_1$, from a DMRG calculation ($L=406$, states kept $=64$). The gap on the left side closes at the FM to Q-PM phase transition point, and on the right, it vanishes for the transition from DS-AFM to a critical Q-PM (cQ-PM) phase. In between, the gap is zero in the cQ-PM phase and nonzero in the Q-PM phase.

Figure 6

Quantum phase diagram of the $J_1-J_2$ quantum Ising chain (in units of the transverse field) from DMRG and CMFT calculations.

Figure 7

The $log$ of the spin-spin correlations vs. $logp$ in the FM phase for $J_2=-2.0$. This data is generated by varying $J_1$ from 0 (the left-most points) to $-4$ using DMRG ($L=206$, and states kept=32). Here, 1-L in the plot legends denotes $\langle {\sigma }_1^x{\sigma }_L^x\rangle$, and likewise for the other correlations. The “bulk” is a correlation between two faraway spins that are also far away from the ends. The inset zooms in the data near $J_1=-4$.

Figure 8

The $log{p}_i$ vs. $logp$ in the FM phase. Here, $p_i=\langle {\sigma }_i^x\rangle$. This data is from a CMFT calculation, wherein the cluster is built using DMRG ($L=206$, states kept $=32$). It shows that $p_1$ goes as $p^4$, while $p_2$ and $p_3$ approach the bulk ($\sim p$), as $J_1$ goes deep into the FM phase (toward the upper-right corner).

Figure 9

Free-ends DMRG. While the shaded parts of the left and right blocks are iteratively renormalized, the two boundary spins on each side are kept unrenormalized.

Figure 10

The $log{\rho }_{2,L-1}^x$ vs. $log{p}_{\mathrm{ds}}^{}$ from a DMRG calculation ($L=106$, and states kept = 64) in the DS-AFM phase for two different values of $J_1$. This data is generated by varying $J_2$ from the upper critical line (in Fig. 6) to $J_2=8$. (Inset) The $log$ of ${\rho }_{1,L}^x$, ${\rho }_{1,L-1}^x$, and ${\rho }_{2,L}^x$ vs. $log{p}_{\mathrm{ds}}^{}$. Here, and in the following figures, $J_2$ increase toward the upper-right corner.

Figure 11

The $log{p}_i$ vs. $log{p}_{\mathrm{ds}}$ from CMFT (combined with DMRG; $L=206$, states kept $=16$) calculations in the DS-AFM phase. It shows that $p_1$ and $p_2$ always follow $p_{\mathrm{ds}}^4$ behavior, while the rest of the $p_i$'s are always bulk.

Figure 12

Exact numerical diagonalization results for the $log$ of various end-to-end correlations in the ground state of $\widehat{H}$ for 16 spins.

Figure 13

The quantum phase diagram of the $J_1-J_2$ Ising chain in a transverse field ($h=1$), characterized in terms of the number of edge modes in each phase.