Magnetic order-disorder transitions on a one-third-depleted square lattice

Quantum Monte Carlo simulations are used to study the magnetic and transport properties of the Hubbard model, and its strong coupling Heisenberg limit, on a one-third-depleted square lattice. This is the geometry occupied, after charge ordering, by the spin-$\frac{1}{2}{\mathrm{Ni}}^{1+}$ atoms in a single layer of the nickelate materials ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ and (predicted) ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$. Our model is also a description of strained graphene, where a honeycomb lattice has bond strengths which are inequivalent. For the Heisenberg case, we determine the location of the quantum critical point (QCP) where there is an onset of long range antiferromagnetic order (LRAFO), and the magnitude of the order parameter, and then compare with results of spin wave theory. An ordered phase also exists when electrons are itinerant. In this case, the growth in the antiferromagnetic structure factor coincides with the transition from band insulator to metal in the absence of interactions.

Figure 1

The one-third-depleted square lattice. A regular diagonal stripe array of black crosses is removed, leaving the red site structure. We will assume two types of bonds exist corresponding to connections between NN (black) and NNN (green) sites of the original square geometry. (See text.)

Figure 2

Finite size scaling of the square of the AF order parameter $m^2$ for the spin-1/2 Heisenberg model. For a ratio $g>1.75=g_c$ of exchange couplings, a transition to a disordered spin liquid state occurs. LRAFO persists to small values of the interchain exchange. Data were obtained with the SSE algorithm.

Figure 3

Extrapolated values of the SSE results for the AF order parameter from Fig. 2 and the results of LSWT analysis, Eq. (5). With LSWT (SSE), LRAFO disappears above $g_c=6.20\pm 0.02$ $(1.75\pm 0.01)$. The QMC data are shown in the Supplemental Material [32].

Figure 4

(a) Band gap $\mathrm{\Delta }$ as a function of the ratio of hopping. $\mathrm{\Delta }$ vanishes for $t^{\prime }/t<2$. The noninteracting limit is a band insulator ($\mathrm{\Delta }>0$) for $t^{\prime }/t>2$. (b) Semimetallic band structure at $t^{\prime }/t=0.5$. (c) Insulating band structure at $t/t^{\prime }=0.25$.

Figure 5

The AF structure factor $S_{\mathrm{AF}}$ is shown as a function of hopping anisotropy for different $U$. The linear lattice size $L=8$ so that the number of sites $N=128$. (There are 64 unit cells each with two sites.) The inverse temperature discretization $\mathrm{\Delta }\tau =\beta /L=1/2U$ except for $U=1$ where $\mathrm{\Delta }\tau =1/4$. Data were acquired from 25 simulations of 1000 equilibration and 4000 measurement sweeps for each $t^{\prime }/t$. The QMC data is shown in the Supplemental Material [32].

Figure 6

Finite size scaling of the AF structure factor $S_{\mathrm{AF}}$ for $t/t^{\prime }=0.95$ (a) and $t^{\prime }/t=0.90$ (b). In both cases $U_c>4.5$ is well above the critical interaction strength $U_c=3.869$ for isotropic hopping [42].

Figure 7

Near-neighbor (singlet) spin correlation function across intra- and interchain bonds $m_t$ and $m_{t^{\prime }}$, respectively. $\langle S_i·S_j\rangle$ is large and independent of $t^{\prime }/t$ for $t^{\prime }/t\gtrsim 2$. This value matches the point at which a nonzero gap $\mathrm{\Delta }$ opens in the spectrum, Fig. 4. The limiting value at $t^{\prime }=0$ ($t=0$) is 0.4515 [42] (0.75). The QMC data are shown in the Supplemental Material [32].

Figure 8

Phase diagram. The $U=\infty$ Heisenberg limit is along the top of the figure, $U/(4+U)=1$, and is extracted from the data of Fig. 3. The critical interaction strength diverges even prior to entry into the band insulator phase at $t/t^{\prime }=0.5$.