Figure 1

Decorated square lattice in a near-diagonal tilt: Panel (a) shows the physical lattice for the bosons. The spins reside on the links, which form the lattice shown in (b). The effective resonant subspace of the boson model of the decorated square lattice (a) maps to an antiferromagnet on the octagon-square-cross lattice (b) in a strong longitudinal and in a transverse magnetic field. If the tilt is not exactly diagonal, then spins on horizontal lines experience a different longitudinal field than spins on vertical lines. The spin lattice is not bipartite, and in this sense the antiferromagnet is frustrated.Reuse & Permissions

Figure 2

Phase diagram of the near-diagonally tilted decorated square lattice, determined by quantum Monte Carlo calculations. The tuning parameters are ${\lambda }_x$ and ${\lambda }_y$ which parametrize the cost for having a dipole in $x$ and $y$ directions, respectively. (a) For realistic system sizes there appears to be only one-dimensional order. This is a finite-size artifact, arising from the fact that for these system sizes the finite-size gap dominates over the (antiferromagnetic) interchain coupling. (b) In the thermodynamic limit we predict two-dimensional (2D) quantum Ising transitions to a phase where neighboring chains are aligned antiferromagnetically.Reuse & Permissions

Figure 3

Effective five-state model: each unit cell of the decorated square lattice can be in one out of five possible states: having no dipole, or having a dipole on one of the four links. There is also a constraint that dipoles may not overlap: two neighboring unit cells may not have dipoles directed toward each other. Thus the effective model is a constraint five-state model on a simple square lattice.Reuse & Permissions

Figure 4

Phase diagram obtained from QMC study. We find only one-dimensional order. The two-dimensional order parameters are zero everywhere. The phase boundary was obtained from crossing of the Binder (Ref. 30) cumulant of the one-dimensional magnetization, Eq. (3.3), see Appendix 6a. Square lattice of size $L_x=L_y=(4,8,16,32,64)$ the imaginary time slice thickness was $a=0.04$, imaginary time direction was scaled with the linear system size, $M_{\tau }=(40,80,160,320,640)$, corresponding to temperatures $T=(0.625,0.3125,0.1562,0.0781,0.0391).$ Essentially the same phase diagram is obtained from order-parameter scaling assuming the Ising exponent $\eta =1/4.$Reuse & Permissions

Figure 5

Absence of two-dimensional order in the QMC study. Here we show the scaling of the total magnetization with system size, $\langle {\vec{M}}^2\rangle \times L$. For randomly aligned magnetized chains we expect $\langle {\vec{M}}^2\rangle \propto 1/L$, while for perfectly aligned chains $\langle {\vec{M}}^2\rangle =\mathrm{const}$ and for antiferromagnetic order in transverse direction $\langle {\vec{M}}^2\rangle =0$. In the disordered phase $\langle {\vec{M}}^2\rangle \propto 1/\left(L_x{L}_y\right)$. This shows that in the ordered phase the system behaves as a collection of independent chains with one-dimensional order within each chain. Here ${\lambda }_x=-5$ was kept fixed; similar results are obtained for different cuts through the phase diagram.Reuse & Permissions

Figure 6

Two-chain model used for the exact-diagonalization study. Each central site (marked with a square) can be in one out of three states. A chain the length of four unit cells is shown. This model should capture the interchain behavior of our model qualitatively.Reuse & Permissions

Figure 7

Strong finite-size effects: Logarithmic plot for energy of the first three excited states relative to the ground state as a function of system length, for (a) ${\Delta }_a=-5,{\Delta }_b=-4$, (b) ${\Delta }_a=-7,{\Delta }_b=-1$. The lines $E_1-E_0$ and $E_2-E_0$ appear on top of each other. At first sight this suggests that there are four ground states in the thermodynamic limit and that thus the order parameters of the chains are not coupled. However, there is a small splitting between the first two levels above the ground state. As can be seen in Fig. 8 this splitting grows linearly with system size and eventually dominates over the finite-size gap for long enough chains.Reuse & Permissions

Figure 8

Splitting between first and second excited state grows linearly with system size. This suggests that there is indeed a coupling between the order parameters of these two chains.Reuse & Permissions

Figure 9

The interchain coupling is antiferromagnetic, which we show by fixing the boundary conditions so that the chains are forced to be either aligned or antialigned. The energy difference grows linearly with system size and antiferromagnetic boundary conditions are energetically favorable.Reuse & Permissions

Figure 10

The antiferromagnetic coupling appears in 10th order in perturbation theory. This logarithmic plot shows that the coupling per unit length scales as ${\left|\Delta \right|}^{-9}$, where $\Delta ={\Delta }_b-{\Delta }_a$.Reuse & Permissions

Figure 11

A kink (a) and antikink (b) configuration. Kinks in different chains cannot reside on the same rung, because of the hard-core constraint; no such restriction exists for the antikinks. (c) Schematic space-time trajectories for kink-antikink pairs. The kinks (antikinks) are represented by solid (dashed) lines, respectively. The arrows represent the direction of the Ising order parameter. The average order parameter is assumed to be pointing to the left. In this configuration, the kinks are typically to the right of the antikinks, and their world lines typically curve left. Since the two chains couple only through the kinks, this provides a mechanism for coupling the order parameters of the two chains. (d), (e) A simplified model showing the source of the interaction between the order parameters of the two chains. The space-time trajectories of kinks on the upper chain are approximated, for simplicity, by the rectangular U-shaped curves shown in blue. We define the “position” of a world line as the center of the rectangle enclosing it. The hatched region shows the space-time volume blocked for the position of kinks in the lower chain, assuming that the order parameters of the two chains are aligned (d) or antialigned (e), due to the hard-core repulsion between kinks in the upper and lower chains. The space-time trajectories of kinks in the lower chain in the two cases are shown by the dashed blue curves.Reuse & Permissions

Figure 12

Binder cumulant for order parameters $\langle \langle M_{\mathrm{LR}}^2\rangle \rangle$ and $\langle \langle M_{\mathrm{UD}}^2\rangle \rangle$ for a horizontal cut through the phase diagram, keeping ${\lambda }_x=-5$ fixed, plotted for different system sizes, $L_x=L_y=8,16,32,64.$ Similar plots are obtained for different cuts through the phase diagram.Reuse & Permissions

Figure 13

Order-parameter scaling, assuming the correlation length exponent $\eta =1/4$ of the classical two-dimensional Ising model. There is good agreement of the crossing points, also with the ones found from the Binder cumulant.Reuse & Permissions

Figure 14

Data collapse for the critical exponent $\nu =1$ of the classical two-dimensional Ising model. The $x$ axis is centered around the critical point and rescaled by $L^{1/\nu }$.Reuse & Permissions