Phase diagram of the frustrated spin ladder

We revisit the phase diagram of the frustrated spin-1/2 ladder with two competing interchain antiferromagnetic exchanges, rung coupling $J_{\perp }$ and diagonal coupling $J_{\times }$. We suggest, based on the accurate renormalization-group analysis of the low-energy Hamiltonian of the ladder, that marginal interchain current-current interaction plays central role in destabilizing previously predicted intermediate columnar dimer phase in the vicinity of classical degeneracy line $J_{\perp }=2J_{\times }$. Following this insight we then suggest that changing these competing interchain exchanges from the previously considered antiferromagnetic to the ferromagnetic ones eliminates the issue of the marginal interactions altogether and dramatically expands the region of stability of the columnar dimer phase. This analytical prediction is convincingly confirmed by the numerical density-matrix renormalization-group and exact-diagonalization calculations as well as by the perturbative calculation in the strong rung-coupling limit. The phase diagram for ferromagnetic $J_{\perp }$ and $J_{\times }$ is determined.

Figure 1

(Color online) RG flow of couplings $G_{1,2,4,5}$ for the case of antiferromagnetic interchain exchanges $J_{\times }=0.15$ and $J_{\perp }=0.296$. Notations are as follows: $G_1$ (red/dashed), $G_2$ (green/dotted), $-G_4$ (blue/solid), and $G_5$ (magenta/dot-dashed). $\ell$ denotes RG scale.

Figure 2

(Color online) RG flow of couplings $G_{1,2,4,5}$ for the case of ferromagnetic interchain exchanges $J_{\times }=-0.15$ and $J_{\perp }=-0.306$. Notations are as follows: $-G_1$ (red/dashed), $-G_2$ (green/dotted), $-G_4$ (blue/solid), and $G_5$ (magenta/dot-dashed). $\ell$ denotes RG scale.

Figure 3

(Color online) CD operator at the center of the open ladder, $D_{\text{CD}}\left(L/2;L\right)$ as a function of $J_{\perp }$ for $J=1$ and (a) $J_{\times }=-0.2$, (b) $J_{\times }=-0.5$, and (c) $J_{\times }=-1.0$. The symbols represent the data for $L=16$, 24, 32, 48, 64, 96, 128, and 192 from top to bottom. Inset in (c) shows the data for $J_{\times }=-1.0$ and broader regime of $J_{\perp }$, $-8\le J_{\perp }\le 0$.

Figure 4

(Color online) System-size dependence of the CD operator at the center of the open ladder, $D_{\text{CD}}\left(L/2;L\right)$, in a log-log scale for $J=1$ and (a) $J_{\times }=-0.2$, (b) $J_{\times }=-0.5$, and (c) $J_{\times }=-1.0$.

Figure 5

Spin correlations between nearest-neighbor spins (bond energy) along the legs of the ladder, $⟨{\mathbf{S}}_{j,n}\cdot {\mathbf{S}}_{j,n+1}⟩$, for $\left(J,J_{\times },J_{\perp }\right)=\left(1,-1,-3\right)$ and $L=192$. The correlations in the legs $j=1$ and 2 are identical.

Figure 6

(Color online) Slope of the $\text{log}{D}_{\text{CD}}\left(L/2;L\right)-\text{log}L$ plot, ${\Delta }_{\text{CD}}\left(x\right)$, for $J=1$ and (a) $J_{\times }=-0.2$, (b) $J_{\times }=-0.5$, and (c) $J_{\times }=-1.0$.

Figure 7

(Color online) System-size dependence of the SD operator at the center of the open ladder, $D_{\text{SD}}\left(L/2;L\right)$, in a log-log scale for $J=1$ and (a) $J_{\times }=-0.2$, (b) $J_{\times }=-0.5$, and (c) $J_{\times }=-1.0$.

Figure 8

(Color online) $J_{\perp }$ dependence of the $z$ operators for $J=1$ and (a) $J_{\times }=-0.2$, (b) $J_{\times }=-0.5$, (c) $J_{\times }=-0.8$, (d) $J_{\times }=-1.0$. Dark (blue) and light (red) curves represent $z_{\text{rung}}$ and $z_{\text{diag}}$, respectively.

Figure 9

(Color online) Ground-state phase diagram for $J=1$ and the ferromagnetic interchain coupling, $J_{\perp }<0$ and $J_{\times }<0$. Open circles represent the transition points obtained from the DMRG calculation while the other symbols show the CD-Haldane transition points from the analysis of $z$ operators. Dotted lines are guide to eyes. In the schematic pictures for each phase, the bold lines represent singlet pairs and the ellipses stand for the symmetrization of two spins.

Figure 10

(Color online) System-size dependence of the dimer operators at the center of the open ladder in a log-log scale for $J=1$ and $J_{\times }=0.2$; (a) the CD operator $D_{\text{CD}}\left(L/2;L\right)$ and (b) the SD operator $D_{\text{SD}}\left(L/2;L\right)$.

Figure 11

(Color online) Slope of the log-log plots of the dimer operators for $J=1$ and $J_{\times }=0.2$; (a) ${\Delta }_{\text{CD}}\left(x\right)$ and (b) ${\Delta }_{\text{SD}}\left(x\right)$.

Figure 12

(Color online) DMRG data for the dimer operator in the $J_1\text{−}{J}_2$ chain with $J_1=1$; (a) dimer operator $D\left(L/2;L\right)$ at the center of the open chain and (b) slope of the $\text{log}D\left(L/2;L\right)-\text{log}L$ plot, $\Delta \left(x\right)$. The symbols represent the results for $J_2=0.20$, 0.241, 0.30, 0.35, 0.40, and 0.45 from bottom to top.