Global phase diagram of competing ordered and quantum spin-liquid phases on the kagome lattice

We study the quantum phase diagram of the spin-$1/2$ Heisenberg model on the kagome lattice with first-, second-, and third-neighbor interactions $J_1,J_2$, and $J_3$ by means of density matrix renormalization group. For small $J_2$ and $J_3$, this model sustains a time-reversal invariant quantum spin-liquid phase. With increasing $J_2$ and $J_3$, we find in addition a $q=(0,0)$ Néel phase, a chiral spin-liquid phase, an apparent valence-bond crystal phase, and a complex noncoplanar magnetically ordered state with spins forming the vertices of a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid and cuboc1 phase break time-reversal symmetry in the sense of spontaneous scalar spin chirality. We show that the chiralities in the chiral spin liquid and cuboc1 are distinct, and that these two states are separated by a strong first-order phase transition. The transitions from the chiral spin liquid to both the $q=(0,0)$ phase and to time-reversal symmetric spin liquid, however, are consistent with continuous quantum phase transitions.

Figure 1

(a) Quantum phase diagram of the spin-$1/2$ $J_1-J_2-J_3$ kagome Heisenberg model for $0.0\le J_2\le 0.25$ and $0.0\le J_3\le 0.5$. The phases shown are a time-reversal invariant quantum spin-liquid (QSL) phase, a coplanar magnetically ordered $q=(0,0)$ Néel phase, a time-reversal broken chiral spin-liquid (CSL) phase, a noncoplanar magnetically and chiral ordered cuboc1 phase, and a valence bond crystal (VBC) phase. The cuboc1 phase remains stable for larger $J_3$ beyond the range shown here (we have checked up to $J_3\le 1.0$). The dashed region indicates the uncertainty in locating the phase boundary between the QSL and $q=(0,0)$ Néel phases. The purple dashed line shows the line of classical degeneracy between the $q=(0,0)$ Néel and cuboc1 phases [63]. (b) The configurations of spins (arrows indicate the direction of static moments) of the cuboc1 state on the kagome lattice. On each small triangle, the spins are coplanar and sum to zero. In each hexagon, sets of three consecutive spins are noncoplanar, as are the sets obtained by taking every second spin around the hexagon. This breaks time-reversal symmetry in the sense that the scalar spin chirality is nonzero and the wave function is intrinsically complex.

Figure 2

(a)–(c) Spin-spin correlations for different phases on the YC8 and XC8 cylinders. The green site is the reference spin; the blue and red colors denote positive and negative correlations, respectively, of the site in question with the reference spin. The area of circle is proportional to the magnitude of the spin correlation. The large dashed hexagon in (b) shows the short-range spin correlations in CSL phase. The arrows in (c) show the reference spin (the red solid arrow) and the direction of other spins (the blue dashed arrow) whose correlations are plotted in Fig. 3. Panel (d) plots the nearest-neighbor bond energy on the XC8 cylinder in the VBC phase.

Figure 3

Panels (a) and (c) are log-linear plots of the spin correlations between sites of one sublattice vs site distance $d$ on the XC8 and XC12 cylinders. Panel (b) illustrates the dependence upon the number of states kept, $M$, in the DMRG, for the case $J_2=0.2$ on the XC12 cylinder. The geometry of the sites in the first three panels is shown in Fig. 2. Panel (d) shows the finite size scaling of the (squared) magnetic order parameter $m^2$ vs $1/L$ on the XC4, XC8, and XC12 cylinders using the fitting formula $m^2(1/L)=m^2(1/L=0)+a/L+b/L^2$. All plots in this figure are for $J_3=0$.

Figure 4

Numerical results for scalar spin chirality. Panel (a) shows correlations between pairs of the smallest triangles ${\Delta }_1$ on the kagome lattice for $J_2=0.2$ and different values of $J_3$ on the YC8-24 cylinder. In (b), the long-distance value of these correlations is extracted and plotted versus $J_3$, which clearly shows the CSL region. Plots (c) and (d) compare the correlations of different types of triangles [defined in the inset of (b)] for $J_2=0.2,J_3=0.24$ on the YC8-24 and YC12-24 cylinders. Panel (e) shows the chiral order parameter $\langle {\chi }_{{\Delta }_i}\rangle$ calculated directly from the complex code for the indicated triangles on the YC8-24 cylinder at $J_2=0.2,J_3=0.3$. The red arrows indicate the order of the three spins in the triple product defining the scalar spin chirality.

Figure 5

Log-linear plot of spin correlations for $J_2=0.2$ on the YC8-24 cylinder. These correlations decay more quickly with $J_3$—a consequence of the transition from the $q=(0,0)$ Néel to the CSL phase. This is seen more clearly from the plot of the long-distance spin correlation $S$ vs $J_3$, shown in the inset.

Figure 6

Numerical studies of the cuboc1 phase. All plots in this figure use $J_2=0.2$. In (a) we show the spin correlations for a central region of the YC12-24 cylinder with $J_3=0.5$, following the same conventions as Fig. 2. The dashed hexagon indicates the 12-site unit cell. Panel (b) shows log-linear plots of the spin correlations for various $J_3$ values on the YC8-24 cylinder (here $J_3=0.24,0.3,0.38,0.4,0.42,0.45,0.5$ for the successive curves with increasing values of the correlations). The inset plots the $J_3$ dependence of the long-distance spin correlation $S$. Plots (c) and (d) compare the spin correlation for $J_3=0.5$ on the YC8-24 and YC12-24 cylinders with different truncation errors $\varepsilon$. The data with plus symbol give the results of an extrapolation to zero truncation error. Panel (e) contrasts the correlations of the chirality on the four different types of triangles (as shown in the inset) well into the cuboc1 phase for $J_3=0.7$ on the XC8-24 cylinder. The correlations on the ${\Delta }_1,{\Delta }_2$ triangles are very small and sometimes change sign. This is probably consistent with zero spontaneous chirality on these triangles in the thermodynamic limit.

Figure 7

Entanglement entropy in the cuboc1 phase. In (a) the entanglement entropy vs the subsystem length $l_x$ for $J_2=0.2,J_3=0.7$ on the XC8-18 cylinder is fitted using the conformal field theory formula $S\left(l_x\right)=(c/6)ln[(L_x/\pi )sin(l_x\pi /L_x)]+g$ with $c=3.0,g=3.3$. In (b), the entanglement entropy is plotted vs $lnx$ $[x=(L_x/\pi )sin(l_x\pi /L_x)]$ for $J_2=0.2,J_3=0.7$ on XC8-16 and XC8-18 cylinders. The dashed line is the fit curve used shown in (a).

Figure 8

Scans of ground-state energy and entanglement entropy of a bipartition into equal halves, as probes of quantum phase transitions. Panels (a) and (b) show the $J_3$ dependence for $J_2=0.2$ on the YC8 cylinder over a region spanning the transition from the CSL to the $q=(0,0)$ Néel phase. No sharp features are observed, suggesting a continuous phase transition. In panels (c) and (d) the same quantities are shown for $J_2=0.2$ on the XC8 cylinder, to probe the transition from the CSL to the cuboc1 phase. In this case an abrupt feature is observed, suggesting a first order transition. In the final two panels, (e) and (f), the system is scanned with $J_2=J_3=J^{\prime }$ on the YC8 cylinder, to study the transition from the time-reversal invariant QSL to the CSL. In this case, the transition again appears continuous.