Adaptive density matrix renormalization group study of the disordered antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain

Using an adaptive strategy which enables the study of quenched disordered system via the density-matrix renormalization-group method, we compute the various ground-state spin-spin correlation measures of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chain with random coupling constants, namely, the mean values of the bulk and of the end-to-end correlations, the typical value of the bulk correlations, and the distribution of the bulk correlations. Our results are in agreement with the predictions of the strong-disorder renormalization-group method. We do not find any hint of logarithmic corrections either in the bulk average correlations, which were recently reported by Shu et al. [Phys. Rev. B 94, 174442 (2016)], or in the end-to-end average correlations. We report the existence of a logarithmic correction on the end-to-end correlations of the clean chain. Finally, we have determined that the distribution of the bulk correlations, when properly rescaled by an associated Lyapunov exponent, is a narrow and universal (disorder-independent) probability function.

Figure 1

The spin-spin correlation function (7) (black squares) as a function of the spin separation $r$ for a chain of $L=500$ sites with open boundary conditions. The blue dashed line in (a) is the simple power-law decay $\sim r^{-1}$. In (b) we plot $C\left(r\right)r$ to emphasize the logarithmic correction. The solid red line is the best fit to the analytical expectation (8) in both panels.

Figure 2

(a) The end-to-end correlation $C_{1,L}=\langle {\mathbf{S}}_1·{\mathbf{S}}_L\rangle$ for system sizes ranging from $L=40$ to $L=500$. (b) Same data as in (a) with $C_{1,L}$ multiplied by $L^2$ in order to highlight the logarithmic prefactor [see Eq. (11)].

Figure 3

The ground-state spin-spin correlation (7) for a single disorder realization $\left\{J_i\right\}$ drawn from (6) for various disorder strengths $D$ (increased in steps by the aDMRG method) and system size $L=10$. The thick solid curve, open symbols, and solid symbols are, respectively, the values obtained using the exact diagonalization, the standard DMRG, and aDMRG methods.

Figure 4

(a) and (b) The entanglement entropy ${\mathcal{S}}_l$ as a function of the system size $l$ [see Eq. (5)] for a single disorder realization of a chain 50 sites long with disorder parameter strength $D=3.0625$. The subsystem $A$ consists of all spins from sites 1 to $l$. In (a), ${\mathcal{S}}_l$ is shown for different numbers of DMRG sweeps ${\delta }_{\text{sweeps}}$ used in the adaptive strategy while the disorder increment $\delta D$ is kept fix. In (b), ${\mathcal{S}}_l$ is shown for different values of $\delta D$ and fixed ${\delta }_{\text{sweep}}$. (c) The total number of states $N$ needed to ensure that the DMRG truncation error is not larger than $T_{\text{err}}={10}^{-10}$ as the disorder strength is increased.

Figure 5

The mean value of the correlation function (7) of the AFM Heisenberg chain for $L=100$ as a function of the spin-spin separation $r$ for various disorder strengths $D$. (a) shows the bare value of $|\overline{C}|$ as a function of the spin-spin separation $r$. In (b), we plot $|\overline{C}|/{\gamma }_D$ as a function of ${\gamma }_{D}r$ in order to obtain a data collapse [see arguments around Eq. (12)]. The black solid line is proportional to the clean correlation (8), and the red dashed line is proportional to $r^{-2}$. The data in (b) are replotted in (c) with the vertical axis multiplied by ${\left({\gamma }_{D}r\right)}^2$. The solid line is a fit to Eq. (2).

Figure 6

(a) Replot of the data in Fig. 5 excluding the data points which do not follow the data collapse. (b) The exponent $\eta$ and the difference $c_{\text{o}}-c_{\text{e}}$ obtained from the best fit of Eq. (2) to the data in (a) (see text).

Figure 7

(a) The average end-to-end correlation $\overline{C_{1,L}}=-\overline{\langle {\mathbf{S}}_1·{\mathbf{S}}_L\rangle }$ of the AFM Heisenberg chain as a function of the system size and for various values of the disorder parameter $D$. The straight lines are power-law fits to $a{L}^{-{\eta }_s}$ from which we obtain ${\eta }_s=1.02\left(7\right)$ for all values of $D$ except for $D=1/16$, in which ${\eta }_s=1.1$. In the clean case, however, the solid line is the fit to Eq. (11) already reported in Fig. 2. (b) A replot of data in (a) with $\overline{C_{1,L}}$ multiplied by $L^{{\eta }_s\left(D\right)}$, with ${\eta }_s\left(D\right)=1+{\delta }_{D,0}$. The solid line is the same in Fig. 2. For an easier comparison, in (b) we have divided $C_{1,L}$ by a factor of 3 in the clean case.

Figure 8

(a) The typical correlation function (13) of the AFM Heisenberg chain for $L=100$ and different values of the disorder strength $D$. The magenta lines are the best fits to Eq. (3), from which the data collapse in (b) is produced. (c) The values of the fitting parameters $A$ and $c_D$ as a function of the disorder parameter $D$.

Figure 9

Normalized histogram of the correlation function $C=|\langle {\mathbf{S}}_{L/4}·{\mathbf{S}}_{3L/4}\rangle |$. In (a), it is rescaled by the parameters $A$ and $c_D$ [see Eq. (3)] given in Fig. 8. (b) shows the same data rescaled by its average and standard deviation. The solid black line is a fit to Eq. (14) (see text). The histogram was built using $2\times {10}^4$ distinct disorder configurations of coupling constants $\left\{J_i\right\}$. The blue dashed and black solid lines are, respectively, the associated distributions for the XX chain [$\mathrm{\Delta }=0$ in (2)] for the longitudinal $C^z$ and transverse $C^x$ spin-spin correlations.