Frustrated quantum magnetism in the Kondo lattice on the zigzag ladder

The interplay between the Kondo effect, indirect magnetic interaction, and geometrical frustration is studied in the Kondo lattice on the one-dimensional zigzag ladder. Using the density-matrix renormalization group, the ground-state and various short- and long-range spin- and density-correlation functions are calculated for the model at half filling as a function of the antiferromagnetic Kondo interaction down to $J=0.3t$, where $t$ is the nearest-neighbor hopping on the zigzag ladder. Geometrical frustration is shown to lead to at least two critical points: Starting from the strong-$J$ limit, where almost local Kondo screening dominates and where the system is a nonmagnetic Kondo insulator, antiferromagnetic correlations between nearest-neighbor and next-nearest-neighbor local spins become stronger and stronger, until at $J_{\mathrm{c}}^{\dim }\approx 0.89t$ frustration is alleviated by a spontaneous breaking of translational symmetry and a corresponding transition to a dimerized state. This is characterized by antiferromagnetic correlations along the legs and by alternating antiferro- and ferromagnetic correlations on the rungs of the ladder. A mechanism of partial Kondo screening that has been suggested for the Kondo lattice on the two-dimensional triangular lattice is not realized in the one-dimensional case. Furthermore, within the symmetry-broken dimerized state, there is a magnetic transition to a ${90}^{\circ }$ quantum spin spiral with quasi-long-range order at $J_{\mathrm{c}}^{\mathrm{mag}}\approx 0.84t$. The quantum-critical point is characterized by a closure of the spin gap (with decreasing $J$) and a divergence of the spin-correlation length and of the spin-structure factor $S\left(q\right)$ at wave vector $q=\pi /2$. This is opposed to the model on the one-dimensional bipartite chain, which is known to have a finite spin gap for all $J>0$ at half filling.

Figure 1

Sketch of the Kondo-lattice model on the one-dimensional zigzag ladder. $-t$ denotes the hopping between neighboring sites along the legs and on the rungs. $J>0$ is the strength of the local antiferromagnetic exchange coupling with the local spin-$1/2$ magnetic moments. Energy units are fixed by choosing $t=1$.

Figure 2

Equivalent sketch of the model, Eq. (1), highlighting the translational symmetries. $a$ is the primitive translation.

Figure 3

Spin-correlation function $\langle {\mathit{S}}_i{\mathit{S}}_j\rangle$ (see color code) for nearest neighbors $i,j$ on the zigzag ladder as obtained for the ground state of the Kondo lattice with $L=52$ sites at half filling and for various coupling strengths $J/t$.

Figure 4

$J$ dependence of the order parameter for dimerization, $O_{\mathrm{D}}$, Eq. (5), in the critical $J$ range for different $L$.

Figure 5

Nearest-neighbor spin correlations as in Fig. 3 but for different $L$ at fixed $J=0.7t$.

Figure 6

Top: Spin correlation function $\langle {\mathit{S}}_i{\mathit{S}}_j\rangle$ for $i=10$ as a function of the distance $\mathrm{\Delta }i\equiv j-i$, obtained for the ground state of the Kondo lattice on the zigzag ladder with $L=60$ sites at half filling and for various coupling strengths $J$ (with $t=1$). Bottom: The same but for the conduction-electron spin correlations $\langle {\mathit{s}}_i{\mathit{s}}_{i+\mathrm{\Delta }i}\rangle$.

Figure 7

$J$ dependence of the spin-structure factor $S\left(q\right)=L^{-1}{\sum }_{i,j}^L{e}^{iq(R_i-R_j)}\langle {\mathit{S}}_i{\mathit{S}}_j\rangle$ (color code on the right) for $L=40$ sites.

Figure 8

Spin-structure factor $S\left(Q\right)$ at $Q=\pi /2$ for different $J$ as functions of $L$ on a logarithmic scale.

Figure 9

Spin-structure factor $S\left(q\right)$ for different $J$ as indicated and for $L=60$ sites.

Figure 10

Right: Spin gap $\mathrm{\Delta }{E}_{\mathrm{S}}\left(L\right)$ as a function of $1/L$ for different $J$ as indicated. Lines represent the results of linear fits $\mathrm{\Delta }{E}_{\mathrm{S}}\left(L\right)-\mathrm{\Delta }{E}_{\mathrm{S}}\left(\infty \right)\propto 1/L$. Left: Values for $\mathrm{\Delta }{E}_{\mathrm{S}}\left(\infty \right)$ obtained by extrapolation to the $1/L\to 0$ limit as a function of $J$. Error bars indicate the uncertainty of the extrapolation. Fitting the $J$ dependence of $\mathrm{\Delta }{E}_{\mathrm{S}}\left(\infty \right)$ (gray line) yields a critical value $J_{\mathrm{c}}^{\mathrm{mag}}=0.84t\pm 0.03t$ (see horizontal red bar).

Figure 11

Correlation length $\xi$ as function of $J$ (for $J>J_{\mathrm{c}}^{\mathrm{mag}}$) at $L=40$. Error bars on $\xi$ result from fitting the DMRG data for the distance dependence of the spin correlation function $\langle {\mathit{S}}_i{\mathit{S}}_j\rangle$. Solid line: data for $\xi$ are fitted by $\xi =\text{const}\times {(J-J_{\mathrm{c}}^{\mathrm{mag}})}^{-\nu }$. Optimal values: $J_{\mathrm{c}}^{\mathrm{mag}}\approx 0.84t, \nu \approx 0.4$, and $\text{const}\approx 1.48$.

Figure 12

$J$ dependence (for $J>J_{\mathrm{c}}^{\mathrm{mag}}$) of the spin-structure factor $S\left(q\right)$ at wave vector $q=\pi /2$ and $L=40$. Error bars reflect the scattering of the data for different bond dimensions. Data are fitted by $S(\pi /2)=\text{const}\times {(J-J_{\mathrm{c}}^{\mathrm{mag}})}^{-\gamma }$. Optimal values: $J_{\mathrm{c}}^{\mathrm{mag}}\approx 0.84t, \gamma \approx 0.15$, and $\text{const}\approx 1.02$.

Figure 13

Finite-size scaling for $S(\pi /2)$ (see text). The data for different $L$ as indicated and for $J=0.85,0.86,...,0.95$ collapse to a single line at $J_{\mathrm{c}}^{\mathrm{mag}}\approx 0.845t, \nu \approx 0.4$, and $\gamma \approx 0.2$.