Aperiodic quantum XXZ chains: Renormalization-group results

We report a comprehensive investigation of the low-energy properties of antiferromagnetic quantum XXZ spin chains with aperiodic couplings. We use an adaptation of the Ma-Dasgupta-Hu renormalization-group method to obtain analytical and numerical results for the low-temperature thermodynamics and the ground-state correlations of chains with couplings following several two-letter aperiodic sequences, including the quasiperiodic Fibonacci and other precious-mean sequences, as well as sequences inducing strong geometrical fluctuations. 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. We also discuss the nature of the ground-state structures and their comparison with the random-singlet phase characteristic of random-bond chains.

Figure 1

Left end of the Fibonacci XXZ chain with $J_a<J_b$. Dashed (solid) lines represent weak (strong) bonds, while circles indicate the positions of the spins. Apart from a few bonds close to the chain ends, the effective couplings also form a Fibonacci sequence.

Figure 2

Left end of the Fibonacci chains with $J_a>J_b$. Effective spins form in the first lattice sweep, giving rise to a Fibonacci chain with the roles of the weak and strong bonds interchanged, exactly as in the original lattice in Fig. 1.

Figure 3

(Color online) Specific heat of the Fibonacci XXZ chains for $J_a∕J_b=\frac{1}{10}$ and three different values of the uniform anisotropy $\Delta$, as given by the “independent-singlet” approximation.

Figure 4

(Color online) Ground-state correlations as a function of the distance between spins for the Fibonacci XX chain. The curves are obtained from numerical diagonalization of closed chains with 2584 sites. For $J_a∕J_b=\frac{1}{10}$ (lower curves), dominant correlations correspond to distances $r_j=1$, 3, 13, 55 and 233, for which $C^{xx}$ and $C^{zz}$ are nearly equal, as predicted by the MDH method (circles), and decay as $1∕r$ (dotted curve). Larger coupling ratios lead to a slower decay of $C^{xx}$ (and a faster decay of $C^{zz}$), as seen for $J_a∕J_b=\frac{1}{2}$ (upper curve, offset for clarity).

Figure 5

Left end of the silver-mean chain with $J_a<J_b$. Effective couplings $J_b^{\prime }$ correspond to the original $J_a$ bonds, while $J_a^{\prime }$ connects spins separated by one singlet pair. Apart from the leftmost bond, the effective couplings also form a silver-mean sequence.

Figure 6

Left end of the silver-mean chain with $J_a>J_b$. Effective spins form in the first lattice sweep, producing the same structure as the third generation for $J_a<J_b$.

Figure 7

(Color online) Ground-state pair correlations in the silver-mean XX (a) and Heisenberg (b) chains, for $\rho =\frac{1}{4}$ (upper curves) and $\rho =\frac{1}{10}$ (lower curves), obtained from the second-order MDH scheme (chains with 8119 sites). Solid and dashed curves correspond to average and typical correlations, respectively.

Figure 8

Decay exponents of $C^{xx}\left(r_j\right)$ and $C_{\mathrm{typ}}^{xx}\left(r\right)$ for the XX silver-mean chain as a function of the coupling ratio $\rho =J_a∕J_b$, obtained from both the second-order MDH scheme (chains with 8819 sites) and the free-fermion method (chains with 3363 sites). Errors bars are at most the same size as the symbols themselves.

Figure 9

First five generations of the bronze-mean XXZ chain with $J_a<J_b$, each showing the leftmost 24 sites. The numbers indicate the positions of the sites in the original chain. Encircled blocks contribute effective spins when renormalized. The labels on the right denote the effective bonds in each generation. The attractor of the block distribution is a two-cycle, reached at the second generation. All bonds to the right of the horizontal arrow follow the same sequence in the second (third) and fourth (fifth) generations.

Figure 10

Ground-state correlation functions of the XX (a) and Heisenberg (b) bronze-mean chains, obtained from the zeroth-order MDH scheme (chains with 2 010 601 sites).

Figure 11

First three generations of the XXZ chain with couplings following the sequence in Sec. 5D, for $J_a<J_b$, showing an effective-spin hierarchy. Solid lines indicate strong bonds; for clarity, weak bonds are not represented. Shaded blocks contribute effective spins when renormalized, while white blocks form singlets. A third-generation effective spin represents three second-generation and nine first-generation spins.

Figure 12

First two generations of the XXZ chain with couplings following the sequence in Sec. 5E, for $J_a<J_b$. The numbers indicate the position of the spins in the original chain. The first lattice sweep generates a fixed-point bond distribution with four different effective couplings ${\tilde{J}}_a$ through ${\tilde{J}}_d$.

Figure 13

(Color online) Log-log plots of the specific heat $c\left(T\right)$ and susceptibility $\chi \left(T\right)$ as functions of temperature for XXZ chains with couplings following the marginal tripling sequence, for $J_a∕J_b=\frac{1}{10}$ and three different values of the uniform anisotropy $\Delta$.

Figure 14

First two generations of the binary Rudin-Shapiro XXZ chain, for $J_a<J_b$. The first lattice sweep generates eight different effective couplings, ${\tilde{J}}_a$ through ${\tilde{J}}_h$, labeled in the figure by the letters $a\text{–}h$. Further renormalization does not change the bond distribution. Starting from the second generation, only blocks coupled by ${\tilde{J}}_b$ (thick lines) and ${\tilde{J}}_c$ (thin lines) bonds are renormalized. (For clarity, weaker bonds are not drawn in the picture.)

Figure 15

Pattern of three-spin blocks and isolated spins leading to the effective-spin hierarchy in Rudin-Shapiro XXZ chains. Thick lines indicate strong bonds. Shaded blocks contribute effective spins when renormalized; white blocks form singlets.

Figure 16

Pattern of five-spin and three-spin blocks leading to the effective-spin hierarchy in Rudin-Shapiro XXZ chains. Thick lines indicate strong bonds. Shaded blocks contribute effective spins when renormalized; white blocks form singlets.

Figure 17

Ground-state correlations for chains with Rudin-Shapiro couplings, obtained from extrapolation of numerical MDH results for chains with $2^{16}$ to $2^{20}$ sites. (a) $C^{xx}\left(r\right)$ (upper solid curve) and $C^{zz}\left(r\right)$ (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, corresponding to distances $r_j=\left(2\right)4^{j-1}$, fitted by a law of the form $r{C}^{xx}\left(r\right)=y_0+y_1\ln r$ (dashed curve). (b) $C\left(r\right)$ for the Heisenberg chain (solid curve). The dotted curve is proportional to $1∕r$.

Figure 18

First two generations of the 6-3 sequence, discussed in Sec. 6B, for $J_a<J_b$. In the second generation, dashed lines indicate effective ${\tilde{J}}_a$ bonds, while thick and thin solid lines denote effective ${\tilde{J}}_b$ and ${\tilde{J}}_c$ bonds. In subsequent generations, the ${\tilde{J}}_c{\tilde{J}}_a$ pairs change to ${\tilde{J}}_a{\tilde{J}}_c$, and the first ${\tilde{J}}_a$ becomes ${\tilde{J}}_c$, but the bond distribution is otherwise unchanged.

Figure 19

Thermal dependence of the specific heat (a) and susceptibility (b) of the XX chain with couplings following the 6-3 sequence of Sec. 6B, obtained for $J_a∕J_b=\frac{1}{4}$ from both numerical diagonalization of chains with 46 875 sites (circles) and the MDH scheme (solid curves). The inset in (b) presents a log-linear plot of $T$ versus ${\left(\chi T\right)}^{-\omega }$, with $\omega =\ln 2∕\ln 5$, showing that for $T\simeq {\Lambda }_j$, corresponding to the specific-heat maxima, the susceptibility satisfies the scaling form $\chi \sim T^{-1}{\mid \ln T\mid }^{-1∕\omega }$ (dashed line); at intermediate temperatures, a Curie-like behavior is observed.

Figure 20

(Color online) Ground-state correlations of the XX chain with couplings following the 6-3 sequence of Sec. 6B, for two different values of the coupling ratio $\rho =J_a∕J_b$, as obtained by numerical diagonalization of chains with 1874 sites. Peaks in the curves correspond to the characteristic distances $r_j=9$, 45, and 225.

Figure 21

Leftmost 24 sites in the first six generations of XXZ chains with couplings following the fivefold-symmetry sequence, discussed in Sec. 6C, for $J_a<J_b$. The attractor of the bond distribution is a two-cycle, reached after three lattice sweeps. Second-generation bonds are denoted by $J_{\alpha }$ through $J_{\delta }$; third- and fifth-generation couplings are labeled $a$ through $f$, while $A$ through $G$ label fourth- and sixth-generation bonds. Couplings $J_G$ only occur much farther along the chains. Starting from the fourth generation, lines indicate blocks to be renormalized.

Figure 22

Renormalization step involving the decimation of a two-spin block.

Figure 23

Renormalization step involving a three-spin block.