Low-Energy Properties of Aperiodic Quantum Spin Chains

We investigate the low-energy properties of antiferromagnetic quantum $XXZ$ spin chains with couplings following two-letter aperiodic sequences, by an adaptation of the Ma-Dasgupta-Hu renormalization-group method. For a given aperiodic sequence, we argue that, in the easy-plane anisotropy regime, intermediate between the $XX$ and Heisenberg limits, the general scaling form of the thermodynamic properties is essentially given by the exactly known $XX$ behavior, providing a classification of the effects of aperiodicity on $XXZ$ chains. As representative illustrations, we present analytical and numerical results for the low-temperature thermodynamics and the ground-state correlations for couplings following the Fibonacci quasiperiodic structure and a binary Rudin-Shapiro sequence, whose geometrical fluctuations are similar to those induced by randomness.

Figure 1

Left end of the Fibonacci $XXZ$ chain. Dashed (solid) lines represent $J_a$ ($J_b$) bonds. An effective coupling $J_b^{\prime }$ is induced between spins separated by only one singlet pair, while $J_a^{\prime }$ connects spins separated by two singlet pairs. Apart from a few bonds close to the chain ends, the effective couplings also form a Fibonacci sequence.

Figure 2

Behavior of $\chi (T)$ for aperiodic $XX$ chains, obtained from the MDH method (symbols) and from numerical diagonalization on chains with ${10}^4–{10}^5$ sites (curves), for two values of $\rho =J_a/J_b$. (a) Fibonacci couplings. The slope of the curves depends on $\rho$, reflecting the marginal character of the aperiodicity. (b) Rudin-Shapiro couplings. The inset plots the inverse square root of $T\chi (T)$ versus $T$ (in log scale) with $\rho =1/4$, showing that for points corresponding to the smallest gaps in each generation the random-singlet phase result $\chi (T)\sim 1/T{ln}^{2}T$ is reproduced (dotted line).

Figure 3

Left end of the first three generations of the Rudin-Shapiro chains. Thick lines indicate strong bonds. Shaded blocks contribute effective spins when renormalized; white blocks form singlets.

Figure 4

GS correlations for chains with RS couplings, obtained from extrapolation of numerical MDH results for chains with $2^{16}$ to $2^{20}$ sites. (a) $C^{xx}(r)$ (upper solid curve) and $C^{zz}(r)$ (lower solid curve) for the $XX$ chain. The dotted curve is proportional to $r^{-3/2}$. (Curves offset for clarity.) Inset: dominant $C^{xx}$ correlations, fitted by a law of the form $r{C}^{xx}(r)=y_0+y_1\ln r$ (dashed curve). (b) $C(r)$ for the Heisenberg chain (solid curve). The dotted curve is proportional to $1/r$.