Bond-impurity-induced bound states in disordered spin-$\frac{1}{2}$ ladders

We discuss the effect of weak bond disorder in two-leg spin ladders on the dispersion relation of the elementary triplet excitations with a particular focus on the appearance of bound states in the spin gap. Both the cases of modified exchange couplings on the rungs and the legs of the ladder are analyzed. Based on a projection on the single-triplet subspace, the single-impurity and small cluster problems are treated analytically in the strong-coupling limit. Numerically, we study the problem of a single impurity in a spin ladder by exact diagonalization to obtain the low-lying excitations. At finite concentrations and to leading order in the inter-rung coupling, we compare the spectra obtained from numerical diagonalization of large systems within the single-triplet subspace with the results of diagrammatic techniques, namely, low-concentration and coherent-potential approximations. The contribution of small impurity clusters to the density of states is also discussed.

Figure 1

Excitation energy $E$ of the lowest level for the full spin ladder Hamiltonian (2). (a) One rung impurity $J_R^{\prime }=J_R+\delta J_R$. (b) One leg impurity $J_L^{\prime }=J_L+\delta J_L$. In all cases, the normalization is fixed to $J_R=1$. Symbols are obtained by extrapolation of Lanczos diagonalization on finite systems. Open circles are for the pure system($\delta J_R=0$ and $\delta J_L=0$) and correspond to the spin gap; the solid line is a [7,6] Padé approximant to the 13th order strong-coupling series (Ref. 9) for the spin gap of the pure ladder. Dashed lines display the analytical result for the position of the bound state in the effective Hamiltonian (13), namely, Eq. (16) [panel (a)] and Eq. (17) [panel (b)].

Figure 2

Sketch of the diagrammatic expansion of the self-energy $\Sigma \left(E\right)$ in the low-concentration limit. $G_k^0\left(E\right)=1∕(E-{ϵ}_k)$ is the free one-triplet Green’s function.

Figure 3

Spectral function $A_k\left(E\right)$ in low-concentration approximation (LCA) for (a) perturbed rung couplings with $\delta J_R=-0.1$, $c=0.01$; (b) perturbed leg couplings with $\delta J_L=0.5$, $c=0.01$. The dashed lines mark the positions of the (anti) bound states from Eqs. (16, 17). $J_R=1$ and $J_L=0.1$ in both cases.

Figure 4

Density of states (DOS) at finite concentration of modified couplings $J_R^{\prime }$: numerical data (NAV, solid line) for spin ladders with $N={10}^3$ rungs and $J_R=1,J_L=0.1,\delta J_R=-0.1$ [concentration: panel (a) $c=0.01$; panel (b) $c=0.1$; panel (c) $c=0.3$]. Dashed line in panel (a): low-concentration approximation (LCA); in panels (b) and (c): CPA.

Figure 5

Density of states (DOS) at finite concentration of modified couplings $J_L^{\prime }$: numerical data (NAV, solid line) for spin ladders with $N={10}^3$ rungs and $J_R=1,J_L=0.1,$ and $\delta J_L=0.5$ [concentration: panel (a) $c=0.01$; panel (b) $c=0.1$; panel (c) $c=0.9$]. Dashed line in panel (a): low-concentration approximation (LCA). Inset of panel (a): structure of the DOS in the vicinity of the bound state. Panel (c): one observes resonance modes inside the band for $c=0.9$; the case shown here is equivalent to $J_L=0.6$, $\delta J_L=-0.5<0$, and $c=0.1$.

Figure 6

Comparison of eigenenergies of bound states induced by rung-impurity clusters in a clean system, derived analytically, with numerical diagonalization of $H_{\mathrm{eff}}$ on large systems with a finite concentration of modified rung couplings. The density of states (DOS) in the vicinity of the one-impurity level is shown. The patterns in parenthesis denote different types of impurity clusters: with $1$, we indicate the relative position of the rung impurities in the cluster sequence. The numerical data (NAV, solid line) correspond to $J_R=1,J_L=0.1$, $\delta J_R=-0.1$ and $c=0.1$. The letters a–j relate the peaks in the DOS to certain impurity clusters, which are listed in the legend.

Figure 7

Matching between the peak structure inside the triplet band obtained numerically at finite concentration of rung impurities (NAV, solid line) and analytically for systems with different types of impurity clusters (dashed line), for $J_R=1$, $J_L=0.01$, $\delta J_R=-0.05$, and $c=0.1$.