Properties of the random-singlet phase: From the disordered Heisenberg chain to an amorphous valence-bond solid

We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study $S=1/2$ spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.

Figure 1

Qualitative AVBS ground state of an $S=1/2$ spin chain. The open and solid circles represent the two sublattices of the bipartite lattice and the arches indicate singlets (valence bonds). The short valence bonds form ordered domains, between which spinons localize. In the ground state the spinons freeze pairwise into long-bond singlets.

Figure 2

SDRG evolution of the disorder-averaged log-energy scale for the random-$Q$ chain (thicker red curve) and the random-$J$ chain (thinner black curve), both with disorder parameter $d=1$ and chain length $N=8192$. The energy is graphed versus the fraction $n=N_{\mathrm{\Omega }}/N$ of the active (not yet decimated) spins under the action of the RG (thus, $n=2/N$ at the final step). Asymptotically the curves for the both models tend to a power law $-ln\left(\mathrm{\Omega }\right)\sim 1/\sqrt{n}$ in the late stage of the RG, as indicted by the dashed line.

Figure 3

Two different types of loop structures in the transition graph of the overlap between two VB states $\left|v\right.〉$ (upper, black bonds) and $\left|v^{\prime }\right.〉$ (lower, red bonds). The gray and white sites belong to two different sublattices. Type (a) indicates the case where $\left|v\right.〉\ne \left|v^{\prime }\right.〉$ and type (b) is a single-bond structure with $\left|v\right.〉=\left|v^{\prime }\right.〉$. The correlation $C\left(\left|i-j\right|\right)$ for any pair of spins ($i$ and $j$) located in the same loop is a finite constant.

Figure 4

Projection-power convergence of the disorder-averaged long-distance spin correlations in QMC calculations for the (a) the random-$J$ system and (b) the random-$Q$ system, both with $d=1$ and $N=128$. Results are shown for two different trial states; the SDRG state obtained for each individual disorder realization and a translationally invariant amplitude-product state [23, 24] with bond-length ($l$) amplitude $h\left(l\right)=l^{-2}$.

Figure 5

SDRG and QMC results for spin correlations at the largest distance of the random-$J$ model with disorder parameters $d=1,2$. The largest distance is $r_{\text{max}}=N/2$ for the QMC results, and $r_{\text{max}}=N/2-1$ for the SDRG data. The SDRG results have been fitted to the form $C\left(r\right)=\alpha r^{-2}+\beta r^{-4}$, with adjustable constants $\alpha$ and $\beta$. The form $C\left(r\right)\propto r^{-2}{ln}^{1/2}(r/r_0)$ was used in fits to the QMC data.

Figure 6

The correlation functions in Fig. 5 multiplied by $N^2$ shows the presence of a multiplicative logarithmic correction to the $1/N^2$ scaling for the QMC results, which increases with distance. The dashed line and the solid line for $d=1$ are both described by the same correction form given in Eq. (22), but with slightly different fitting parameters; the dashed line for $d=1$ is the best fit for all data points, while the solid line (also shown in Fig. 5) is the best fit to the $N\ge 12$ data. The correction for the SDRG results shows a small enhancement for very small $r$ and fast convergence to a constant, implying a small conventional subleading power-law correction.

Figure 7

Upper panel: QMC results for the spin correlation function at the longest odd distance of the random-$J$ chain with $d=1$ and its two components corresponding to the different types of loop structures in the overlap graph. The dashed line indicates the inverse-square function $1/N^2$. Lower panel: Corrections to the $1/N^2$ scaling made visible by dividing out the leading power law. The logarithmic form in Eq. (22) was used to fit the corrections for the full correlation function and the (a) part. The (b) part of the correction has been fitted to a power-law form $\alpha +\beta N^{-1}$.

Figure 8

(a) Full spin correlations for an $N=128$ random-$J$ chain with $d=1$, incorporating both the mean correlation and the typical correlation in the SDRG calculations. The results are compared with QMC results. The black and red dashed lines show, respectively, the pure $1/N^2$ decay and that form modified by a multiplicative log of the same type used in the fit in Fig. 5. (b) The full SDRG spin correlation at the largest even and odd distances. The dashed line corresponds to the form $1/N^2$.

Figure 9

Four types of loop structures in the transition graph of the overlap $\langle v^{\prime }|v\rangle$ that contribute to dimer correlations. The black and red bonds correspond to $\left|v\right.〉$ and $\left.〈v^{\prime }\right|$, respectively. The gray and white sites belong to two different sublattices. Types (a) and (c) are absent in the SDRG ground state since $\left|v\right.〉=\left|v^{\prime }\right.〉$ and only noncrossing bonds are present, while these cases do contribute in QMC simulations where type (c) with two loops $(i,j)(j+1,i+1)$ can be realized using more than four spins connected with noncrossing bipartite bonds and $\left|v\right.〉\ne \left|v^{\prime }\right.〉$.

Figure 10

SDRG and QMC results for dimer correlations of the random-$J$ model with different disorder parameters $d$. The fitting forms are $D^{*}\left(r\right)=\alpha r^{-4}+\beta r^{-5}$ (SDRG) and $D^{*}\left(r\right)\propto r^{-4}ln(r/r_0)$.

Figure 11

Contribution to the dimer correlation from the loop case labeled (d) in Fig. 9 at the largest distance for the random-$J$ chain with different disorder strength $d$. Each data point was obtained by averaging over more than ${10}^4$ disorder realizations. The solid line indicates a power-law decay as $N^{-6}$, showing consistency with an asymptotic distance dependence $\propto r^{-6}$ for large $d$.

Figure 12

SDRG and QMC results for the spin (a) and dimer (b) correlations in the random-$Q$ model. The functional forms fitted to the data (curves shown) are the same as in the corresponding cases in Fig. 5.

Figure 13

Corrections to the $1/r^2$ scaling of the spin correlations of the random-$Q$ model, made visible by dividing the data in Fig. 12 at $r=N/2$ by the leading power law.

Figure 14

SDRG rules for the $Q_3$ chain. The thick bonds indicate the strongest coupling [a $Q_3$ coupling (top), a $Q_2$ coupling (middle), or a $J$ coupling (bottom)] to be decimated; each pair of spins connected to the thick bonds forms singlet. The decimation procedure leads to an effective coupling between sites $i$ and $j$.

Figure 15

A typical case where unpaired spins occur. The chain is periodic with two ends (in the figure) coupled to each other. The orange (gray) rectangular regions indicate blocks of $Q$ terms that are active (decimated). The black curves are effective $J$ couplings. When the active $Q$ terms are to be decimated, no effective coupling will be generated according to the decimation rule; thus the two spins (indicated by circles) are left unpaired.