Phase diagram of the $\text{spin}-\frac{1}{2}$ triangular $J_1-J_2$ Heisenberg model on a three-leg cylinder

We study the phase diagram of the frustrated Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor spin-exchange coupling, on three-leg ladders. Using the density-matrix renormalization-group method, we obtain the complete phase diagram of the model, which includes quasi-long-range ${120}^{\circ }$ and columnar order, and a Majumdar-Ghosh phase with short-ranged correlations. All these phases are nonchiral and planar. We also identify the nature of phase transitions.

Figure 1

Lattice structure and mapping of the three-leg cylinder to the one-dimensional chain employed by the finite DMRG calculations. Spins sit on vertices. The lattice is tripartite and the sublattices are labeled $A, B$, and $C$. ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$ are lattice vectors.

Figure 2

Energy vs variance for a $30\times 3$ cylinder with $\theta ={25}^{\circ }$. The linear extrapolation gives a good approximation for the exact ground-state energy.

Figure 3

The calculated phase diagram of the $J_1-J_2$ THM on a three-leg cylinder. The phase transitions are indicated to a resolution of $0.5^{\circ }$. All transitions are second order except for $\theta =-\pi /2$, which is first order (marked by a thick black line).

Figure 4

Visualizations representative of different phases of the THM on a cylinder. Thicker lines represent stronger bonds with red indicating antiferromagnetic bonds and blue indicating ferromagnetic.

Figure 5

(a) Ground-state energy of the THM on a $30\times 3$ cylinder. The transition at $\theta =-\pi /2$ is sharp and consistent with a first-order phase transition (cf. Sec. 7). The transition near $\theta \approx {160}^{\circ }$ suggests a second-order phase transition (cf. Sec. 7). Brown stripes indicate phase transitions. (b) Comparison of the energies of Lanczos and DMRG for the THM on a $4\times 3$ lattice. Lanczos results are from Jolicoeur et al. [18]. DMRG and Lanczos show excellent agreement, but the boundary conditions clearly have a significant effect on the energy of this small lattice.

Figure 6

Average of squared magnetization per plaquette, Eq. (5), on a $30\times 3$ cylinder. The vertical turquoise lines denote the partially polarized region at the second-order transition into the ferromagnetic state.

Figure 7

Short-ranged spin-spin correlation functions for the ground state on a $30\times 3$ cylinder. $S_n$ represents the spin operator for the site $n$ according to the lattice numbering in Fig. 1. Dashed red lines are the limits of the correlation functions for spin-$\frac{1}{2}$ particles, $-3/4\le O_s(i,j)\le 1/4$. Brown stripes indicate phase transitions.

Figure 8

Spin-spin correlators for the THM on a $30\times 3$ cylinder for trial path ACA. $R(i,i_0)$ is spatial distance between vertices $i$ and $i_0$ in units of the lattice spacing. Two possible paths are shown in the inset.

Figure 9

iDMRG results for the correlation length. $\eta$ is the correlation length, calculated from the next-leading eigenvalue of the transfer matrix, Eq. (7). Power-law growth $\eta \propto m^{\kappa }$ indicates gapless quantum critical excitations with power-law correlations, while the saturation of the correlation length in the Majumdar-Ghosh region indicates that this phase is short range and gapped.

Figure 10

Vector chirality correlator, $O_c$, Eq. (11), on a $30\times 3$ cylinder. $R_P$ is the distance between the centers of the plaquettes in units of the lattice spacing. The antiferrochiral pattern in the ${120}^{\circ }$ state can be explained by the tripartite symmetry of the lattice. Antiferrochirality is clearly broken in the Majumdar-Ghosh state. Inset: The path for which the vector chirality correlators were calculated. The $P_{+}/P_{-}$ labels indicate the antiferrochiral ordering.

Figure 11

Scalar chirality correlator, $O_t$, Eq. (14), on a $30\times 3$ cylinder. $R_P$ is the distance between the center of plaquettes in units of the lattice spacing. The path chosen to calculate these correlators is the same as the inset of Fig. 10. The rapid reduction of $O_t$ to zero at long range indicates that the phases are nonchiral and planar.

Figure 12

Squared sublattice magnetization of the ${120}^{\circ }$ state on a $30\times 3$ cylinder, $O^{{120}^{\circ }}$. This order parameter correctly identifies the ${120}^{\circ }$ phase, consistent with the spin-spin correlations, Fig. 7.

Figure 13

Difference between energy per bond of partially polarized states and the nonpolarized state on a $30\times 3$ cylinder at $\theta =153{}^{\circ }$, near the critical point ${\theta }_c=152.0{}^{\circ }$. $S$ is the total spin, which is a good quantum number. The curve has its minimum at a nonzero polarization, which indicates that there is a second-order phase transition close to this point.

Figure 14

Magnetization (ground-state spin polarization) around the critical point ${\theta }_c=152.0{}^{\circ }$ on a $30\times 3$ cylinder. The onset of the phase transition is indicated with the brown stripe.

Figure 15

(a) Spin gap extrapolated to the thermodynamic limit. (b) Binder cumulant of the dimer order parameter, Eq. (16), for the columnar to Majumdar-Ghosh phase transition.