Dimerization and spin decoupling in a two-leg Heisenberg ladder with frustrated trimer rungs

We study the antiferromagnetic spin-half Heisenberg ladder in the presence of an additional frustrating rung spin that is motivated and relevant also for the description of real two-dimensional materials such as the two-dimensional trimer magnet ${\mathrm{Ba}}_4{\mathrm{Ir}}_3{\mathrm{O}}_{10}$. We study the zero-temperature phase diagram, where we combine numerical and analytical methods into an overall consistent description. All numerical simulations are also accompanied by studies of the dynamical spin structure factor obtained via the density matrix renormalization group. Overall, we find in the regime of strong rung coupling a gapped dimerized phase related to competing symmetry sectors in Hilbert space that ultimately results in frustration-driven spin-Peierls transition. In the weak rung-coupling regime, the system is uniform yet shows a gapped spinon continuum together with a sharp coherent low-energy branch that renders the system critical overall. In either case, the additional rung spin quickly gets sidelined and nearly decouples once its bare coupling to the ladder drops somewhat below the direct Heisenberg coupling of the legs.

Figure 1

The model system. (a) Two-leg Heisenberg ladder with trimer rungs and couplings $J\equiv (J_1,J_2,J_3)$. A rung consists of three spin $S=1/2$ sites, where $m=1,2$ are the sites on the legs of the ladder. The presence of the additional center site on a rung (site $m=3$) coupled symmetrically to the leg sites via a finite coupling strength $J_3$ induces frustration. (b) The same model may be seen to represent a hexagonal brick lattice with three-site rungs, using $L_y=2$ with the periodic boundary condition in the vertical direction. The lattice spacing of the three-site unit cell (yellow shaded area) is taken as $a=1$ (horizontally, in either case) or $b$ [vertically, panel (b) only].

Figure 2

Dimerization in the ladder model of Hamiltonian (1) in the strong rung-coupling limit, having $J=(1,J_2,4)$ vs $J_2$ based on DMRG ground-state simulations for arbitrary but fixed $J_2$ for $L=64$ (light thick lines) and $L=128$ (color matched thin dark lines). (a) NN interaction energies along the chain in the system center, showing even (odd) bonds around the system center separately as individual curves in solid (dashed-dotted), respectively. The data includes NN interaction in terms of the original spins with respect to to sites $m=1,2,3$, but also of ${\mathcal{S}}_i\equiv {\mathbf{S}}_i^{\mathrm{tot}}$, as indicated with the legend. (b) Same as (a), also sharing the same legend, but plotting the difference between the even and odd bonds along the chain. The inset shows the average over even and odd bonds, denoted by $\langle \langle \cdots \rangle \rangle$, vs $J_2$ for the $\langle {\mathcal{S}}_i·{\mathcal{S}}_{i+1}\rangle$ data in (a). Here again the color-matched black line refers to $L=128$, whereas the lighter gray line (mostly underneath the black line) refers to $L=64$. The horizontal guide at the bottom of the inset indicates the analytically known expectation value for a plain spin-half Heisenberg chain, $\frac{1}{4}-ln\left(2\right)$ [cf. Eq. (4)]. (c) NN interaction energies within a rung in the system center. Same analysis as in (a), but here the data from even/odd rungs lies indistinguishably on top of each other. (d) Targeting lowest-energy states in global SU(2) spin sectors as indicated in the legend.

Figure 3

Snapshots of NN bond strength $C_{m{m}^{\prime }}^{(i,i^{\prime })}\equiv \langle {\mathbf{S}}_{im}·{\mathbf{S}}_{i^{\prime },m^{\prime }}\rangle$ in the ladder model of Hamiltonian (1) for $J=(1,J_2,4)$ (same as in Fig. 2) with $J_2$ as specified with the panel. The NN bond strength is drawn to scale proportional to the bond width (see the value for the bond at the upper right of each panel for reference). The NN interactions between center spins are shown as semitransparent (light colors) to indicate that no interactions are present in the Hamiltonian for these bonds. All bonds are of the same color, and hence of the same negative sign, thus being antiferromagnetically correlated. The data is for an $L=128$ ladder, showing left boundary, center, and right boundary, with the intermediate ranges cropped as indicated with each horizontal axis.

Figure 4

Lowest-energy eigenstates of model (1) in the strong rung-coupling regime except that here, in addition, also a NN Heisenberg coupling $J_1^{\prime }\in [0,1]$ (horizontal axis) between the center spins was turned on. Therefore, $J_1^{\prime }=0$ well corresponds to the projected two-leg ladder in Eq. (3b). $J_2$ was tuned with $J_1^{\prime }$ as indicated in the panel, such that $J_1^{\prime }=1$ corresponds to the uniform three-leg model with the projected low-energy orbital Hamiltonian as in Eq. (6). While the full, i.e., nonprojected rung state space was present in the simulations, the total weight of the $S=3/2$ multiplet was $\lesssim 0.01$, throughout. Light colors are for $L=64$, whereas darker colors are for $L=128$, similar to Fig. 2. States are color-coded according to their global SU(2) spin sectors as indicated in the legend.

Figure 5

Dynamical structure factor for the two-leg ladder (3) in the strong rung-coupling regime as in Fig. 2, having $J=[1,J_2,4]$ for various $J_2$, is indicated with the left panels top to bottom. The panel labels (a)–(f) each refer to a row that shares the same $J_2$. The left panels show $S_1(k,\omega )$, whereas the right panels show $S_3(k,\omega )$ with the initial spin operator acting on site $m^{\prime }=1$ or 3, respectively (see the text). The corresponding total integrated spectral density is also specified ($I_1$ and $I_3$), as well as the resulting total $I_{\mathrm{tot}}=2I_1+I_3$. The color bar at the top holds for all panels. The spectral data is smoothened with $\delta \omega =0.05$ to remove speckles from linear prediction, except for $J_2=4.3$, which only uses half that value and which also shows a guide for the approximate spin gap at $\omega =0.15$ (dotted line). Panel (a) shows a guide at $\omega \simeq 2.9$ that approximates the upper bound of the dominant spinon band. The weak superimposed wrinkly features as in (f) are attributed to state space truncation within the DMRG, and hence a numerical artefact.

Figure 6

The dimensionless function $B\left(\tilde{h}\right)$ from Eq. (19b). Dashed gray lines indicate asymptotic behavior. The inset shows a weak nonanalyticity at $\tilde{h}=1$ resulting in a vertical slope in main panel.

Figure 7

Static spin expectation values $C_{m{m}^{\prime }}\equiv \langle {\mathbf{S}}_{im}·{\mathbf{S}}_{i{m}^{\prime }}\rangle$ within the same rung $i=0$ in the system center of $L=64$ ladders with open BC for (a) $(J_2=J_3)\le J_1=1$ and (b) vs $J_3$ for fixed smaller $J_2\ll 1$ as indicated in the legend. The color coding in the legend in (a) holds for both panels.

Figure 8

Dynamical structure factor obtained via DMRG for the two-leg ladder (1) in the intermediate to weak rung-coupling regime for $L=64$ rungs [except for row (f), which has $L=128$]. The coupling strength is specified with the panels (panel label and $J$ hold per row). Rows (a)–(c) have decreasing isotropic rung coupling $J_2=J_3$. Rows (c)–(e) have $J_2<J_3\ll 1$ with $\alpha =0, \alpha <0$, and $\alpha >0$, respectively. The remainder of the rows have $\alpha >0$ with increasing rung coupling. Exactly the same analysis as in Fig. 5 otherwise. In the present case, however, $I_{\mathrm{tot}}>1$ indicates that there is also a significant admixture of the $S=3/2$ rung multiplet.

Figure 9

Schematic phase diagram suggested by the SMF analysis vs $J_1/(J_2+J_3)$ and effective orbital field $\tilde{h}\sim \alpha \sim J_2-J_3$. The leg spins are ferromagnetically ($\langle {\mathcal{T}}_i^z\rangle >0$) and antiferromagnetically ($\langle {\mathcal{T}}_i^z\rangle <0$) aligned in the two phases $\alpha \ll -1$ and $\alpha \gg +1$, respectively. They are separated by a phase with spontaneously broken ${\mathbb{Z}}_2$ symmetry where $\langle {\mathcal{T}}^x\rangle \ne 0$.

Figure 10

Local DMRG expectation values as in Eqs. (C1) with a focus on the presence of a nematic phase for the full Hamiltonian (1) for $L=64$ and $J=[1,J_2,4]$, as also analyzed in Fig. 2, having $S^2\equiv S_1+S_2+S_3$. Light colors use the full local spin operators ${\mathbf{S}}_m$, whereas strong colors use the projected spin operators ${\mathbf{S}}_m$, as reflected in Eqs. (C1). The ground state is the same in either case, hence it also includes the local $S=3/2$ multiplet.

Figure 11

Comparison of the orbital expectation value $\langle {\mathcal{T}}_i^z\rangle$ vs $\tilde{h}$ between DMRG and SMF: DMRG data as in Fig. 10, the exact mean-field solution of Eq. (B3a) in the absence of the $\langle {\mathcal{T}}_i^z\rangle$ term (i.e., the Ising model in a transverse field [44]), and the self-consistent mean-field theory for Eq. (B3a).