Analytical and numerical studies of disordered spin-1 Heisenberg chains with aperiodic couplings

We investigate the low-temperature properties of the one-dimensional spin-1 Heisenberg model with geometric fluctuations induced by aperiodic but deterministic coupling distributions, involving two parameters. We focus on two aperiodic sequences, the Fibonacci sequence and the 6-3 sequence. Our goal is to understand how these geometric fluctuations modify the physics of the (gapped) Haldane phase, which corresponds to the ground state of the uniform spin-1 chain. We make use of different adaptations of the strong-disorder renormalization-group (SDRG) scheme of Ma, Dasgupta, and Hu, widely employed in the study of random spin chains, supplemented by quantum Monte Carlo and density-matrix renormalization-group numerical calculations, to study the nature of the ground state as the coupling modulation is increased. We find no phase transition for the Fibonacci chain, while we show that the 6-3 chain exhibits a phase transition to a gapless, aperiodicity-dominated phase similar to the one found for the aperiodic spin-1/2 XXZ chain. Contrary to what is verified for random spin-1 chains, we show that different adaptations of the SDRG scheme may lead to different qualitative conclusions about the nature of the ground state in the presence of aperiodic coupling modulations.

Figure 1

Decimation procedure for a pair of $S=\frac{1}{2}$ spins.

Figure 2

Decimation procedure for a pair of $S=1$ spins according to the second approach.

Figure 3

Decimation procedure for a mixed-spin pair according to the second approach.

Figure 4

Decimation procedure for a pair of ferromagnetically coupled spin-$\frac{1}{2}$ objects according to the second approach.

Figure 5

Decimation procedure for a pair of $S=1$ spins according to the third approach.

Figure 6

(a) Coupling distribution of the spin-1 Fibonacci-Heisenberg chain. (b) Effective chain obtained from the first SDRG approach after a single lattice sweep.

Figure 7

Renormalization of the spin-1 Fibonacci-Heisenberg chain according to the second SDRG approach. (a) The original chain. (b) Spin-1 pairs connected by (strong) $J_b$ bonds are replaced by spin-$\frac{1}{2}$ pairs. (c) Spin-$\frac{1}{2}$ pairs are decimated, yielding effective couplings between remaining $S=1$ spins.

Figure 8

Renormalization of the spin-1 Fibonacci-Heisenberg chain according to the third SDRG approach. (a) The original chain. (b) Spin-1 pairs connected by (strong) $J_b$ bonds are replaced by spin-$\frac{1}{2}$ pairs, are further-neighbor couplings are produced. (c) Spin-$\frac{1}{2}$ pairs are decimated, yielding effective couplings between remaining $S=1$ spins.

Figure 9

Magnetic susceptibility as a function of temperature for the spin-1/2 Fibonacci-Heisenberg chain. Solid lines correspond to the SDRG prediction for different coupling ratios $r=J_b/J_a$. QMC results (symbols) were obtained using chains with 90 sites and open boundary conditions. Error bars are smaller than symbol size.

Figure 10

Temperature dependence of the magnetic susceptibility for the spin-1 Fibonacci-Heisenberg chain with coupling ratio $r=10$. Solid (dashed) line corresponds to the SDRG prediction according to the second (third) approach, while symbols correspond to QMC results for different chain sizes $L$. QMC error bars are smaller than symbol size.

Figure 11

Temperature dependence of the magnetic susceptibility for the spin-1 Fibonacci-Heisenberg chain, similar as Fig. 10, but with a coupling ratio $r=20$.

Figure 12

Spin 1 chain modulated by the Fibonacci sequence: gap $\Delta =E_0(S_T=2)-E_0(S_T=0/1)$ between the lowest-lying quintuplet $S_T=2$ state energy $E_0(S_T=2)$ and the ground-state energy $E_0$ (which is either in the $S_T=0$ singlet or $S_T=1$ triplet sector, depending on the chain parity), as a function of coupling ratio $r=J_b/J_a$, for different system sizes $L$.

Figure 13

Spin 1 chain modulated by the Fibonacci sequence: string order correlation function $\langle O^z(x=|i-j\left|\right)\rangle$ as a function of distance $x$ taken starting from the quarter-chain point $i=L/4$ up to the maximum value $x=L/2$, for a $L=376$ chain and different aperiodicity strengths $r=J_b/J_a$.

Figure 14

(a) Coupling distribution of the spin-1 Heisenberg chain according to the 6-3 sequence. (b) Effective chain obtained from the first SDRG approach after a single lattice sweep.

Figure 15

Self-similar coupling distribution obtained from the first SDRG approach for subsequent lattice sweeps. Singlets are formed between spins connected by the strong effective bonds $J_1^{\left(i\right)}$.

Figure 16

First step of renormalization of the spin-1 Heisenberg chain with couplings following the 6-3 sequence according to the second SDRG approach. (a) The original chain. (b) Spin-1 pairs connected by (strong) $J_b$ bonds are replaced by spin-$\frac{1}{2}$ pairs. (c) Spin-$\frac{1}{2}$ pairs are decimated, yielding effective couplings between remaining $S=1$ spins.

Figure 17

Self-similar coupling distribution obtained from the second SDRG approach for subsequent RG steps. (a) Effective chain consisting only of $S=1$ spins. (b) Strongly-connected spin-1 pairs form spin-$\frac{1}{2}$ pairs. (c) Spin-$\frac{1}{2}$ pairs are decimated, giving rise to a new effective chain, again consisting only of spin-1 pairs, with an invariant coupling distribution.

Figure 18

First step of the renormalization of the spin-1 Heisenberg chain with couplings following the 6-3 sequence, according to the third SDRG approach. (a) The original chain. (b) Spin-1 pairs connected by (strong) $J_b$ bonds are replaced by spin-$\frac{1}{2}$ pairs, are further-neighbor couplings are produced. (c) Spin-$\frac{1}{2}$ pairs are decimated, yielding effective couplings between remaining $S=1$ spins.

Figure 19

Self-similar coupling distribution obtained from the third SDRG approach for subsequent RG steps. (a) Effective chain consisting only of $S=1$ spins. (b) Strongly connected spin-1 pairs form spin-$\frac{1}{2}$ pairs, and further-neighbor bonds are formed. (c) Spin-$\frac{1}{2}$ pairs are decimated, giving rise to a new effective chain, again consisting only of spin-1 pairs, with an invariant coupling distribution.

Figure 20

Temperature-dependence of the magnetic susceptibility for the spin-$\frac{1}{2}$ Heisenberg chain with aperiodic couplings following the 6-3 sequence. Solid lines are the SDRG predictions for various coupling ratios $r=J_b/J_a$, while symbols indicate the corresponding QMC results obtained for $L=75$ sites.

Figure 21

Temperature dependence of the magnetic susceptibility for the spin-1 Heisenberg chain with aperiodic couplings following the 6-3 sequence with coupling ratio $r=5$. Solid (dashed) line corresponds to the SDRG prediction according to the second (third) approach, while symbols correspond to QMC results for different chain sizes $L$. QMC error bars are smaller than symbol size.

Figure 22

Temperature dependence of the magnetic susceptibility for the spin-1 Heisenberg chain with aperiodic couplings following the 6-3 sequence, similar as Fig. 21, but with a coupling ratio $r=10$.

Figure 23

Spin-1 chain modulated by the $6\text{−}3$ sequence: gap $\Delta =E_0(S_T=2)-E_0(S_T=01)$ between the lowest-lying quintuplet $S_T=2$ state energy $E_0(S_T=2)$ and the ground-state energy $E_0$ (which is either in the $S_T=0$ singlet or $S_T=1$ triplet sector, depending on the chain parity), as a function of coupling ratio $r=J_b/J_a$, for different system sizes $L$.

Figure 24

Spin-1 chain modulated by the $6\text{−}3$ sequence: string order correlation function $\langle O^z(x=|i-j\left|\right)\rangle$ as a function of distance $x$ taken starting from the quarter-chain point $i=L/4$ up to the maximum value $x=L/2$, for a $L=376$ chain (top) and a $L=105$ chain (bottom) and different aperiodicity strengths $r=J_b/J_a$.

Figure 25

SDRG approach as applied to the spin-1 Fibonacci-Heisenberg chain with $L=14$ sites. After sweeping over the effective chain in (b), all spins are eliminated.

Figure 26

SDRG approach as applied to the spin-1 Fibonacci-Heisenberg chain with $L=22$ sites. Sweeping over the effective chain in (b) removes all effective $J_b$ bonds, leaving a single effective $J_a$ bond, as shown in (c).

Figure 27

SDRG approach as applied to the spin-1 Fibonacci-Heisenberg chain with $L=35$ sites. After sweeping over the effective chain in (b), the central spin remains.