Spin-$\frac{1}{2}$ string correlations and singlet-triplet gaps of frustrated ladders with ferromagnetic legs and alternate ferromagnetic and antiferromagnetic rungs

The frustrated ladder with alternate ferromagnetic exchange $-J_F$ and antiferromagnetic exchange $J_A$ to first neighbors and ferromagnetic exchange $-J_L$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of $2N$ spins-$\frac{1}{2}$ with periodic boundary conditions. The ground state is a singlet $(S=0)$ and the singlet-triplet gap ${\varepsilon }_T$ is finite for the exchanges considered. Spin-$\frac{1}{2}$ string correlation functions $g_1\left(N\right)$ and $g_2\left(N\right)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2\left(\infty \right)>0$ and $g_1\left(\infty \right)=0$. The minimum $N^{*}$ of $g_2\left(N\right)$ is related to the range of ground-state spin correlations. Convergence to $g_2\left(\infty \right)$ is from below, and $g_1\left(N\right)$ decreases exponentially for $N\ge N^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F, J_L$, and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1\text{−}{J}_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-$\frac{1}{2}$ systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekulé VB diagram at $J_L=J_F/2\le J_A$, that is reversed on interchanging $-J_F$ and $J_A$. Inversion symmetry is spontaneously broken in the dimer phase of the $J_1\text{−}{J}_2$ model where the Kekulé diagrams are the doubly degenerate ground states at $J_2/J_1=1/2$.

Figure 1

(a) The F-AF spin-$\frac{1}{2}$ ladder with F exchange $-J_L<0$ between spins $r$ and $r+2$ in either leg, F exchange $-J_F$ in rungs $2r-1, 2r$, and AF exchange $J_A$ in rungs $2r, 2r+1$. (b) Kekulé diagrams $|K1\rangle$ and $|K2\rangle$ with singlet-paired spins $(2r-1,2r)$ and $(2r,2r+1), r=1$ to $N$.

Figure 2

(a) Singlet-triplet gap ${\varepsilon }_T(J_L,J_F)$ vs $J_{-}=J_L-J_F/2$ at constant $J_{+}=J_L+J_F/2$ and system sizes $2N=16$ and 24. The range is $-J_{+}\le J_{-}\le J_{+}$, and ${\varepsilon }_T(0,0)=1$. (b) ${\varepsilon }_T(J_L,J_F)$ at constant $J_{+}=2$ vs $-0.4\le J_{-}\le 0.4$ at $2N=20$, 24, and 32. The crossing points are ${\varepsilon }_T=N({\varepsilon }_F-{\varepsilon }_0)$.

Figure 3

Singlet-triplet gap ${\varepsilon }_T(J_L,J_F)$ vs $J_L$ at constant $J_F$ and system sizes $2N=16$ and 24. The gaps are $<0.005$ at the singlet/F boundary, $2J_L=J_F/(J_F-1)$. The dashed line is $(1-2J_L)\mathrm{\Delta }\left(1\right)/4$ where $\mathrm{\Delta }\left(1\right)=0.4105$ is the Haldane gap [46]; the crosses at $J_L=0$ and $0.25$ are for $J_F=200$ and $2N=64$.

Figure 4

Size dependence of singlet-triplet gaps ${\varepsilon }_T(J_L,J_F)$ at (a) constant $J_L=1.5$, variable $J_F$ and (b) constant $J_F=1.5$, variable $J_L$. The dashed line is the $1/N$ extrapolation from $2N=24$ to $2N=64$ or 96.

Figure 5

Singlet-triplet gaps ${\varepsilon }_T(J_L,J_F)$ vs $1/J_L=-1/J_2$ at constant $J_F=-1,0,0.5$, and 1 at system sizes $2N=16$ in (a) and 24 in (b). Open symbols refer to $J_1\text{−}{J}_2$ models with $J_1=(1-J_F)/2$ and $J_2=-J_F$. The ladder at $J_F=-1$ is a $J_1\text{−}{J}_2$ model. The $1/J_L=0$ gaps are $(1-J_F)/2N$ for both. The $J_F=1$ gap is finite in ladders, zero in $J_1\text{−}{J}_2$ models.

Figure 6

(a) String correlation functions $g_2\left(N\right)$ at system size $2N, J_L=0$, and $J_F=2,5$, and 50; linear extrapolation to string order $g_2\left(\infty \right)$. (b) $g_1\left(N\right)$ for the same $J_L,J_F$ with solid symbols for $N\ge N^{*}$, the minimum of $g_2\left(N\right)$, open symbols for $N<N^{*}$. The lines are Eq. (15) with the indicated $\xi$.

Figure 7

Representative singlet VB diagrams $|q\rangle$ with $N$ lines $(m,n)$ in ladders with $2N$ spins and $C_N$ translational symmetry: $|a2\rangle$ has one line $(2,5)$ of length 3 and $(N-1)$ lines of unit length; $|a1\rangle$ has one line $(1,4)$ of length 3; $|b2\rangle$ has one line $(2,N+1)$ of length $N-1$, the maximum length; $|c\rangle$ has two lines of length $N-1, (2,N+1)$, and $(N+2,1)$.

Figure 8

(a) String correlation functions $g_2\left(N\right)$ at system size $2N$ and $J_F=0,J_L=1,2$, and 3. Linear extrapolation to string order $g_2\left(\infty \right)$. (b) $g_1\left(N\right)$ for the same $J_L,J_F$; solid symbols for $N\ge N^{*}$, open symbols for $N<N^{*}$. The lines are Eq. (15) with the indicated $\xi$.

Figure 9

Size dependence of $g_2\left(N\right)$ at $J_L=0$ and the indicated $J_F$ from $2N=12$ to 96. The string correlation function $\tilde{g}(N/2)$ is for the HAF with $N$ spins-1. The magenta point at $2N=96$ is for $J_F=400$.

Figure 10

Size dependence of the string correlation function $g_{-}\left(N\right)$ of $J_1\text{−}{J}_2$ models with $2N$ spins, $J_1=1$, and $J_2=0,-2$, and $-4$. The $J_2=0$ expression [11] is used at $J_2=-2$ and $-4$.

Figure 11

String correlation function $g_{-}\left(N\right)$ at $J_2=0.2$ in the gapless phase, fitted as in Fig. 10. Solid and dashed lines are $g_{-}\left(N\right)$ and $g_{+}\left(N\right)$ at $J_2=0.35$ and $0.45$ in the gapped dimer phase. The point at $J_2=0.5$ is exact. The lines at $J_2=0.35$ are to guide the eye.