Phase transitions in the frustrated Ising ladder with stoquastic and nonstoquastic catalysts

The role of nonstoquasticity in the field of quantum annealing and adiabatic quantum computing is an actively debated topic. We study a strongly-frustrated quasi-one-dimensional quantum Ising model on a two-leg ladder to elucidate how a first-order phase transition with a topological origin is affected by interactions of the $\pm XX$-type. Such interactions are sometimes known as stoquastic (negative sign) and nonstoquastic (positive sign) “catalysts”. Carrying out a symmetry-preserving real-space renormalization group analysis and extensive density-matrix renormalization group computations, we show that the phase diagrams obtained by these two methods are in qualitative agreement with each other and reveal that the first-order quantum phase transition of a topological nature remains stable against the introduction of both $XX$-type catalysts. This is the first study of the effects of nonstoquasticity on a first-order phase transition between topologically distinct phases. Our results indicate that nonstoquastic catalysts are generally insufficient for removing topological obstacles in quantum annealing and adiabatic quantum computing.

Figure 1

Frustrated Ising ladder with transverse fields and $XX$ interactions. Spin-$1/2$ particles are located at the black circles on the top row and white circles on the bottom row. The black solid lines are ferromagnetic interactions of magnitude $K$ and the black dashed lines are antiferromagnetic interactions of magnitude $K$. Local longitudinal fields $K$ and $U/2$ are applied at each black circle and each white circle, respectively, and have opposite directions. A transverse field ${\mathrm{\Gamma }}_{\mathrm{t}}$ is appended at every site on the top row and ${\mathrm{\Gamma }}_{\mathrm{b}}$ on the bottom row. An $XX$ interaction ${\mathrm{\Xi }}_{\mathrm{t}\mathrm{t}}$ is applied on each horizontal bond on the top row, ${\mathrm{\Xi }}_{\mathrm{b}\mathrm{b}}$ on each horizontal bond on the bottom row, and ${\mathrm{\Xi }}_{\mathrm{t}\mathrm{b}}$ on each vertical bond between the two rows. The interactions and fields are indicated by operators in the figure, where the subscripts of the Pauli operators ${\widehat{X}}_{a,i}$, ${\widehat{Y}}_{a,i}$, and ${\widehat{Z}}_{a,i}$ are omitted. Periodic boundary conditions are imposed by identifying the sites at the horizontal position $i=L+1$ with those at $i=1$.

Figure 2

Phase diagram of the Ising ladder with uniform transverse field $\mathrm{\Gamma }$ and no $XX$ terms, where the staggered magnetization of the top row $m_{\mathrm{t}}^{\prime }$ is color coded in (a) and the magnetization of the bottom row $m_{\mathrm{b}}$ in (b). The black squares in (a) and (b) are the locations of the minimum energy gap for fixed values of $K/U$. Here, we set the system size to $L=10$. In both (a) and (b), the first- and second-order phase boundaries predicted by perturbation theory are shown as solid and dotted lines, respectively. The first-order transition line for $K/U\gg 1$ is $\mathrm{\Gamma }/U\approx 1/c+U/\left(4K{c}^3\right)$ ($c\approx 0.6$) and the second-order transition line for $K/U\ll 1$ is $\mathrm{\Gamma }/U\approx K/U$ [11].

Figure 3

Examples of the correspondence between states in the frustrated Ising model and the dimer model on two-leg ladders. The lattice for the former model is depicted in a similar way to Fig. 1 (namely, the black circles, the black-solid lines, and the black-dashed lines are longitudinal fields, ferromagnetic interactions, and antiferromagnetic interactions, respectively, of equal magnitude $K$), while the dual lattice for the latter model is indicated by the dotted lines. The rightmost points are identified with the leftmost points for each lattice, which is subject to periodic boundary conditions. The red arrows are spins and the blue thick line segments are dimers. The topological number is $w=0,+1,-1$ from top to bottom, the first being a columnar configuration and the second and the third being staggered ones.

Figure 4

Coupling constants $U\left(l\right)$, $\overline{U}\left(l\right)$, $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$ as functions of $l$ for the bare couplings $(\mathrm{\Gamma }\left(0\right),V\left(0\right),\mathrm{\Xi }\left(0\right))=(1,0,-1)$ in (a), (1,0,0) in (b), and (1,0,1) in (c). In each of the cases, $U\left(0\right)=0.3,{\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)},0.7$. Note that $\overline{U}\left(l\right)$, $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$ do not depend on $U\left(0\right)$.

Figure 5

RG flow diagrams of the frustrated Ising ladder in the dimer limit $K\to \infty$. We plot $(\mathrm{\Gamma }\left(l\right),U\left(l\right))$, $(\mathrm{\Gamma }\left(l\right),V\left(l\right))$, and $(\mathrm{\Gamma }\left(l\right),\mathrm{\Xi }\left(l\right))$ ($l=0,1,\cdots ,10$) in (a), (b), and (c), respectively. The bare couplings are set to $\mathrm{\Gamma }\left(0\right)=1$, $V\left(0\right)=0$, $\mathrm{\Xi }\left(0\right)=0,\pm 0.5,\pm 1$, and $U\left(0\right)=0.3,{\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)},0.7$. Note that $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$ do not depend on $U\left(0\right)$. The inset in (a) is a magnification of $(\mathrm{\Gamma }\left(l\right),U\left(l\right))$ for $U\left(0\right)={\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}$. The arrows on each line indicate the direction of the RG flow. It can be seen that $l=10$ is sufficient for convergence of $U\left(l\right)$ $[U\left(0\right)={\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}]$, $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$.

Figure 6

Phase diagram of the frustrated Ising ladder in the dimer limit $K\to \infty$ predicted by the real-space RG method. The first-order transition points (blue plus signs) indicate that adding $XX$ catalysts (stoquastic or nonstoquastic) does not remove the first-order transition.

Figure 7

Phase diagram of the Ising ladder in the limit $U\to \infty$ (the Ising chain) predicted by the real-space RG method. The blue line indicates second-order transition points.

Figure 8

Ground-state phase diagram of the frustrated Ising ladder obtained by the DMRG calculations with $K/U=5$ and $L=20$ under periodic boundary conditions. We set ${\mathrm{\Gamma }}_{\mathrm{t}}={\mathrm{\Gamma }}_{\mathrm{b}}=\mathrm{\Gamma }$, ${\mathrm{\Xi }}_{\mathrm{t}\mathrm{t}}=\mathrm{\Xi }$, and ${\mathrm{\Xi }}_{\mathrm{b}\mathrm{b}}={\mathrm{\Xi }}_{\mathrm{t}\mathrm{b}}=0$ in Eq. (1). The staggered and symmetric phases are separated by a line of first-order transitions.

Figure 9

Staggered order parameter on the top row $S$ (a) as a function of $\mathrm{\Gamma }/U$ with several values of $\mathrm{\Xi }/U$ and (b) as a function of $\mathrm{\Xi }/U$ with $\mathrm{\Gamma }=0$. The results are obtained by the DMRG method for $K/U=5$ and $L=20$ under periodic boundary conditions.

Figure 10

First derivatives of the ground-state energy $E_0$ (a) with respect to $\mathrm{\Gamma }$ for several values of $\mathrm{\Xi }/U$ and (b) with respect to $\mathrm{\Xi }$ for $\mathrm{\Gamma }=0$. The results are obtained by the DMRG method for $K/U=5$ and $L=20$ under periodic boundary conditions. The inset in (a) is a magnification around $\mathrm{\Gamma }/U=1$.

Figure 11

The finite-size scaling of the minimum gap through the first-order transition driven by varying $\mathrm{\Gamma }/U$ at a fixed value of $K/U=5$. The results are obtained by the DMRG method for three values of $\mathrm{\Xi }/U$ under periodic boundary conditions.

Figure 12

Locations of the minimum energy gap of the Ising ladder with uniform transverse field $\mathrm{\Gamma }$ and no $XX$ terms for several system sizes $L$, at fixed values of $K/U$. The first- and second-order phase boundaries predicted by perturbation theory are indicated by black-solid and dashed lines, respectively.

Figure 13

Energy gap $\mathrm{\Delta }E$ as a function of $\mathrm{\Gamma }/U$ for $L=10$ at different values of $K/U$. The inset is a magnification of the lines for $K/U=1.49,1.5$, which shows that the global minimum of $\mathrm{\Delta }E$ moves from the left “well” to the right “well” as $K/U$ changes.

Figure 14

Coupling constants $U\left(l\right)$, $\overline{U}\left(l\right)$, $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$ and variational parameters ${\alpha }_1\left(l\right)$, ${\alpha }_2\left(l\right)$, $z\left(l\right)$, ${\beta }_1\left(l\right)$, and ${\beta }_2\left(l\right)={\beta }_3\left(l\right)$ as functions of $l$ for the bare couplings $U\left(0\right)={\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}=0.51773501$, $\mathrm{\Gamma }\left(0\right)=1$, $V\left(0\right)=0$, and $\mathrm{\Xi }\left(0\right)=0$. The left graph is identical to Fig. 4 except that the present figure does not contain $U\left(l\right)$ for $U\left(0\right)\ne {\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}$.

Figure 15

Coupling constants $U\left(l\right)$, $\overline{U}\left(l\right)$, $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$ and variational parameters ${\alpha }_1\left(l\right)$, ${\alpha }_2\left(l\right)$, $z\left(l\right)$, ${\beta }_1\left(l\right)$, and ${\beta }_2\left(l\right)={\beta }_3\left(l\right)$ as functions of $l$ for $\mathrm{\Xi }\left(0\right)<0$. We set $\left(U\right(0),\mathrm{\Gamma }(0),V(0),\mathrm{\Xi }(0\left)\right)=(0.51144374,1,0,-0.1)$ in (a), $(0.52956828,1,0,-1)$ in (b), $(2.47583092,1,0,-10)$ in (c), and $(0.23570226,0,0,-1)$ in (d), all of which satisfy $U\left(0\right)={\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}$. The left graph of (b) is identical to Fig. 4 except that the present figure does not contain $U\left(l\right)$ for $U\left(0\right)\ne {\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}$.

Figure 16

Coupling constants $U\left(l\right)$, $\overline{U}\left(l\right)$, $\mathrm{\Gamma }\left(l\right)$, $V\left(l\right)$, and $\mathrm{\Xi }\left(l\right)$ and variational parameters ${\alpha }_1\left(l\right)$, ${\alpha }_2\left(l\right)$, $z\left(l\right)$, ${\beta }_1\left(l\right)$, and ${\beta }_2\left(l\right)={\beta }_3\left(l\right)$ as functions of $l$ for $\mathrm{\Xi }\left(0\right)>0$. We set $\left(U\right(0),\mathrm{\Gamma }(0),V(0),\mathrm{\Xi }(0\left)\right)=(0.52489031,1,0,0.1)$ in (a), (0.61474501,1,0,1) in (b), (2.46771358,1,0,10) in (c), and (0.23570226,0,0,1) in (d), all of which satisfy $U\left(0\right)={\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}$. The left graph of (b) is identical to Fig. 4 except that the present figure does not contain $U\left(l\right)$ for $U\left(0\right)\ne {\overline{U}}_{\mathrm{\Gamma }\left(0\right),\mathrm{\Xi }\left(0\right)}$.