Adaptive density matrix renormalization group for disordered systems

We propose a simple modification of the density matrix renormalization-group (DMRG) method in order to tackle strongly disordered quantum spin chains. Our proposal, akin to the idea of the adaptive time-dependent DMRG, enables us to reach larger system sizes in the strong disorder limit by avoiding most of the metastable configurations, which hinder the performance of the standard DMRG method. We benchmark our adaptive method by revisiting the random antiferromagnetic XXZ spin-1/2 chain for which we compute the random-singlet ground-state average spin-spin correlation functions and von Neumann entanglement entropy. We then apply our method to the bilinear-biquadratic random antiferromagnetic spin-1 chain tuned to the antiferromagnet and gapless highly symmetric SU(3) point. We find the new result that the mean correlation function decays algebraically with the same universal exponent $\varphi =2$ as the spin-1/2 chain. We then perform numerical and analytical strong-disorder renormalization-group calculations, which confirm this finding and generalize it for any highly symmetric $\mathrm{SU}(N$) random-singlet state.

Figure 1

The EE $S$ as a function of the subsystem size $l$ for the random XX chain obtained via exact diagonalization and via the standard and adaptive DMRG methods. We have considered chains of size $L=120$ and different values of disorder strength $D$ (see legends). In (a) the EE is averaged over 2000 disorder realizations. In (b) and (c), the EE is computed for a single disorder realization. The thick brown line in (a) corresponds to the curve $\frac{ln2}{6}lnl+0.62$.

Figure 2

Log-log plot of the arithmetic average correlation function $C_{\text{av}}^z\left(x\right)$ vs. $x$ for (a) the Heisenberg and (b) the XX chains. The system size $L=120$, the disorder strength $D=1$ and we averaged over $1000$ disorder realizations. The DMRG data is obtained using the adaptive strategy. The solid and dashed lines are best fit of the DMRG data.

Figure 3

The arithmetic average correlation functions $C_{\text{av}}^3$ and $C_{\text{av}}^8$ [see Eq. (8)] for the SU(3)-symmetric disordered spin-1 bilinear-biquadratic chain Eq. (7) with $\theta =\frac{\pi }{4}$, system size $L=84$, and disorder strength $D=1$. The continuous blue and dashed green lines are the best fit of $C_{\text{av}}^3$ for $x>10$. They are compatible with the SDRG prediction $C_{\text{av}}\sim x^{-2}$. As in the spin-$1/2$ case, a logarithmic is also compatible with our data. The DMRG data are obtained using the adaptive strategy and averaging over $1000$ disorder realizations.

Figure 4

Sketch of the SU(3) random-singlet state. Sites connected by links are in a singlet state formed by $3,6,9,\cdots$ spins. Notice the links do not overlap. Different colors represent singlets with different number of spins.

Figure 5

Schematic representation of the clustering process of three spins into a singlet state. Colors are for aesthetic purposes only.

Figure 6

Schematic representation of the clustering process of six spins into a singlet state. Colors are for aesthetic purposes only.

Figure 7

The arithmetic mean spin-spin correlation $C_{\text{av}}^{\alpha }$ function as a function of the spin separation $x$ computed for distances $x=1+3n$, with $n\in {\mathbb{N}}_{+}$. For comparison, we also show the singlet spin-spin correlation $C_{\text{av,singlet}}^{\alpha }$ (see text). The system size is $L=3^{12}$ (where periodic boundary conditions were considered), the disorder parameter is $D=1$, and we have averaged over $M=2\times {10}^3$ different disorder realizations.

Figure 8

The probability $Q(t;\rho )$ of finding a spin cluster composed of $t$ original spins at the density scale $\rho$ for various different density values. The data were obtained via the numerical implementation of the SDRG procedure where the parameters used are the same as in Fig. 7. The dashed line is the simplified prediction Eq. (17) with an offset for better comparison. The lines are guides to the eye.