Asymmetric spin-$\frac{1}{2}$ two-leg ladders: Analytical studies supported by exact diagonalization, DMRG, and Monte Carlo simulations

We consider asymmetric spin-$\frac{1}{2}$ two-leg ladders with nonequal antiferromagnetic (AF) couplings $J_{\parallel }$ and $\kappa J_{\parallel }$ along legs $\left(\kappa \le 1\right)$ and ferromagnetic rung coupling, $J_{\perp }$. This model is characterized by a gap $\Delta$ in the spectrum of spin excitations. We show that in the large $J_{\perp }$ limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at $\kappa =1$, admits a linear scaling for the spin gap $\Delta \sim J_{\perp }$. The second class appears for a strong asymmetry of the coupling along legs, $\kappa J_{\parallel }⪡J_{\perp }⪡J_{\parallel }$ and is characterized by two energy scales: the exponentially small spin gap $\Delta \sim J_{\perp }\text{exp}\left(-J_{\parallel }/J_{\perp }\right)$, and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange $\sim J_{\perp }^2/J_{\parallel }$. We report numerical results obtained by exact diagonalization, density-matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetries. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.

Figure 1

(Color online) (a) Sketch of the spiral staircase Heisenberg ladder. (b) View of the system from the top. For $\theta =0$ the model corresponds to the standard antiferromagnetically coupled spin-$\frac{1}{2}$ Heisenberg ladder with FM rung coupling. The case $\theta =\pi$ corresponds to the 1D $SU\left(2\right)$ symmetric single-pole ladder model.

Figure 2

(Color online) Dynamical spin susceptibility in the weak-coupling limit at $J_{\perp }/J_{\parallel }=0.1$: (a) ladder system $\left(\theta =0\right)$ and (b) single-pole ladder system $\left(\theta =\pi \right)$. In both cases, results are obtained on $2\times 12$ lattices with ED techniques. We choose a broadening $s=0.1J_{\parallel }$. The solid (red) lines represent the bonding spectrum, $S\left(q,\omega \right)$, the dashed (blue) lines corresponds to the antibonding spectrum, $R\left(q,\omega \right)$. The spectral functions in the left panels are normalized to the structure factors $S\left(q,t=0\right),R\left(q,t=0\right)$, respectively; the latter are shown in the right panels.

Figure 3

(Color online) ED results for the width $W$ in the single-pole ladder model $\left(\theta =\pi \right)$. The effective SN interaction yields a bandwidth proportional to $J_{\perp }^2/J_{\parallel }$.

Figure 4

(Color online) Normalized residue as a function of $J_{\perp }/J_{\parallel }$ for the ferromagnetic single-pole ladder. The calculations were carried out with ED on an $2\times 8$ lattice.

Figure 5

(Color online) QMC results of the bonding and antibonding dynamical spin susceptibilities for the ladder $\left(\theta =0\right)$ (two top panels) and single-pole ladder $\left(\theta =\pi \right)$ (two bottom panels) systems at $J_{\perp }/J_{\parallel }=1.0$. $\beta J_{\parallel }=200$ was taken in the simulations.

Figure 6

(Color online) Equal time spin-spin correlation function at $J_{\perp }/J_{\parallel }=0.5$ for the single-pole ladder along both chains. Simulations were done at $\beta J_{\parallel }=5000$ on a $2\times 800$ lattice.

Figure 7

(Color online) Scales in the correlation function [Eq. (26)] on the second leg as a function of $J_{\perp }/J_{\parallel }$. The dotted line is a fit to a linear law and the dashed curve is a fit with $v=0.28J_{\parallel }$. The form of $\Delta$ is discussed at the end of Sec. 4C.

Figure 8

(Color online) (a) Size and temperature converged values of the spin gap $\Delta$ as a function of the coupling $J_{\perp }/J_{\parallel }$ for various twist angles. The gap is rescaled by $J_{\text{eff}}=\frac{J_{\parallel }}{4}\left[1+{\text{cos}}^2\left(\theta /2\right)\right]$ such that in the large-$J_{\perp }$ limit it converges asymptotically toward the Haldane gap of a AF spin-1 chain ${\Delta }_H/J_{\text{eff}}\simeq 0.41$. (b) Spin gap for the single-pole ladder model, $\theta =\pi$. Lines denote quadratic and exponential fits for the spin gap (see text). (c) Spin gap for SSHL at larger $J_{\perp }$ for different $\theta \text{s}$, see detailed discussion in Sec. 3B.

Figure 9

(Color online) (a) String order parameters ${\mathcal{O}}_s$ and ${\mathcal{O}}_H$ as a function of interleg coupling $J_{\perp }$ for $\theta =\pi$. In the parameter range $J_{\perp }/J_{\parallel }<0.5$ finite-size effects are still present. (b) Finite-size scaling of the order parameters for $\theta =8\pi /9$ $\left(J_{\perp }/J_{\parallel }=0.2\right)$ and $\theta =\pi$ $\left(J_{\perp }/J_{\parallel }=0.3\right)$. The simulations are carried out up to $\beta J_{\parallel }=7000$ and $2\times 800$ spins.

Figure 10

(Color online) DMRG analysis of the single-pole ladder $\left(\theta =\pi \right)$—panel (a) shows the spin gap between the ground states of the $S^z=0$ and $S^z=1$ spin sectors for several system sizes using periodic BCs (double chain, see text). The error bars indicate the convergence with respect to the number of states $D$ kept (with an extrapolation in $1/D\to 0$ with $D\le 5120$). The convergence to the Haldane gap for $J_{\perp }⪢J_{\parallel }$ is reproduced within 1% relative error, within the fitting range at small $J_{\parallel }/J_{\perp }$ indicated by the vertical dotted line. Panel (b) shows spin gap data using open BCs for the ground states of several spin sectors $S^z∊\left\{0,1,2,3,4\right\}$. Keeping up to 2560 states, all energies are converged with negligible uncertainties. The consecutive energy differences (spin gap) ${\Delta }_{S^z,S^z+1}^S$ between the ground states of the spin sectors $S^z$ and $S^z+1$ are plotted vs $J_{\perp }/J_{\parallel }$ for $L=128$ (dashed) and $L=512$ (solid), respectively. The vertical order (top to bottom) of the entries in the legend matches the order of the lines appearing in the panel (top to bottom). The extrapolated spin gap ${\Delta }_0^S$ (gray and black dots for $L=128$ and $L=512$, respectively), as well as ${\Delta }_{\infty }^S$ (crosses), as obtained from a finite-size scaling of ${\Delta }_{1,2}^S$ for systems of length $L∊\left\{128,256,512\right\}$, are shown. Panel (c) shows ${\Delta }_0^S$ from panel (b) for $L∊\left\{128,512\right\}$ and also for $L=256$ together with ${\Delta }_{\infty }^S$ (all for open BC) on a semilog plot again with inverted $x$ axis similar to panel (a). The extrapolated value of the gap for small $J_{\parallel }/J_{\perp }$ for open BC compares well with the exact Haldane gap within 1% relative error already for the smaller system size $L=128$ (fit range up to first vertical dotted guide). Fits to the data for large $J_{\parallel }/J_{\perp }$ are shown in solid and dashed curves. The fit range chosen is again indicated by the two vertical dotted guides, here for $J_{\parallel }/J_{\perp }>1$. Panel (d) summarizes the data obtained with periodic (circles) and open (crosses) boundary conditions DMRG calculations, as compared with QMC data (black circles). The solid line is a fit for large $J_{\parallel }/J_{\perp }$ similar to the one in panel (c). Error bars indicate the respective uncertainty in energy. Inset shows that $\Delta /J_{\perp }$ reaches the maximum at $J_{\perp }\simeq 2J_{\parallel }$.

Figure 11

(Color online) (a) Zigzag path used for the Jordan-Wigner transformation, Eq. (A1) and (b) gauge transformation, leading to Eq. (A3).