Plaquette Ordered Phase and Quantum Phase Diagram in the Spin-$\frac{1}{2}$ $J_1\text{−}{J}_2$ Square Heisenberg Model

We study the spin-$1/2$ Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions $J_1$ and $J_2$, which possesses a nonmagnetic region that has been debated for many years and might realize the interesting $Z_2$ spin liquid. We use the density matrix renormalization group approach with explicit implementation of $SU(2)$ spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing $J_2$, we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at $J_2/J_1\simeq 0.44$. For $0.5<J_2/J_1<0.61$, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the $s$-wave character of the PVB order. For $0.44<J_2/J_1<0.5$, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies.

Figure 1

Phase diagram of spin-$1/2$ $J_1\text{−}{J}_2$ SHM obtained by our $SU(2)$ DMRG studies. With growing $J_2$, the model has a Néel phase for $J_2<0.44$ and a PVB phase for $0.5<J_2<0.61$. Between these two phases, the finite-size magnetization and spin gap appear small in our calculations, consistent with a near-critical behavior. The main panel shows Néel order parameter $m_s$ and spin gap ${\mathrm{\Delta }}_T$ in the thermodynamic limit. The inset is a sketch of a RC4-6 cylinder; $J_{\text{pin}}$ shows the modified odd vertical bonds providing the boundary pinning for dimer orders.

Figure 2

(a) $m_s^2$ plotted versus $1/L$ for $\mathrm{RC}L\text{−}2L$ cylinder with $L=4,6,8,10,12,14$; lines are polynomial fits up to fourth order. The inset is $J_2$ dependence of the obtained magnetic order in the 2D limit $m_{s,\infty }^2$. (b) Same data as (a) shown as log-log plot of $m_s^2$ versus width $L$.

Figure 3

(a) Log-linear plot of vDOP for $J_2=0.5$ and $J_{\text{pin}}=2.0$ on the RC cylinder. The inset is the comparison of width dependence of the vertical dimer decay length ${\xi }_y$ with Ref. [40]. (b),(c) ${\xi }_y$ and ${\xi }_x$ versus $W_y$ on RC cylinders with $J_{\text{pin}}=2.0$ for a range of $J_2$ shown with the same symbols in both panels.

Figure 4

(a) The NN $J_1$ bond energy for $J_2=0.55$ on the left half of the tilted TC8-25 cylinder, where we have subtracted a constant $-0.2948$ from all the bond energies. Note that the TC cylinders have the square lattice rotated by 45 deg compared to Fig. 1. We trim every other site on both boundaries to make lattice select unique PVB order. $E_s$ ($E_w$) denotes the sum of four NN bond energies of the red (blue) plaquette with negative (positive) numbers. (b) Dependence of the pDOP decay length ${\xi }_P$ on the cylinder width $W_y$.

Figure 5

(a),(b) Ground-state energies for RC10-20 cylinder at $J_2=0.5$ in the $S=0$ ($E_{S=0,M}$) and $S=1$ ($E_{S=1,M}$) sectors versus the DMRG truncation error $\epsilon$. All the energies have subtracted the ground-state energy $E_{S=0,\infty }=-99.022(1)$. $M$ is the number of kept $U(1)$-equivalent DMRG states and is indicated next to the symbols. (c) Finite-size extrapolations of spin gap ${\mathrm{\Delta }}_T$ on $\mathrm{RC}L\text{−}2L$ cylinders ($L=4,6,8,10$). For $J_2<0.5$, the data are fitted using the formula ${\mathrm{\Delta }}_T(L)={\mathrm{\Delta }}_T(\infty )+\alpha /L^2+\beta /L^3+\gamma /L^4$, while for $J_2\ge 0.5$, we fit the data using ${\mathrm{\Delta }}_T(L)={\mathrm{\Delta }}_T(\infty )+a/L+b/L^2+c/L^4$. We estimate ${\mathrm{\Delta }}_T(\infty )=0.018\pm 0.01$ and $0.04\pm 0.01$ for $J_2=0.5$ and 0.55, respectively.