Correlation amplitude and entanglement entropy in random spin chains

Using strong-disorder renormalization group, numerical exact diagonalization, and quantum Monte Carlo methods, we revisit the random antiferromagnetic $XXZ$ spin-$1∕2$ chain focusing on the long-length and ground-state behavior of the average time-independent spin-spin correlation function $C\left(l\right)=\upsilon l^{-\eta }$. In addition to the well-known universal (disorder-independent) power-law exponent $\eta =2$, we find interesting universal features displayed by the prefactor $\upsilon ={\upsilon }_o∕3$, if $l$ is odd, and $\upsilon ={\upsilon }_e∕3$, otherwise. Although ${\upsilon }_o$ and ${\upsilon }_e$ are nonuniversal (disorder dependent) and distinct in magnitude, the combination ${\upsilon }_o+{\upsilon }_e=-1∕4$ is universal if $C$ is computed along the symmetric (longitudinal) axis. The origin of the nonuniversalities of the prefactors is discussed in the renormalization-group framework where a solvable toy model is considered. Moreover, we relate the average correlation function with the average entanglement entropy, whose amplitude has been recently shown to be universal. The nonuniversalities of the prefactors are shown to contribute only to surface terms of the entropy. Finally, we discuss the experimental relevance of our results by computing the structure factor whose scaling properties, interestingly, depend on the correlation prefactors.

Figure 1

(Color online) Schematic decimation procedure (see text).

Figure 2

(Color online) Relative difference $\delta =C∕C_u-1$ between the mean correlation function $C$ and the universal prediction $C_u$ as a function of the distance $l$ in the SDRG framework, for various choices of initial disorder, and both $XX$ and isotropic Heisenberg (or $XXX$) models. Chains A, D, F, and H have couplings distributed according to $P_0\left(J\right)$ [see Eq. (2.22)] with $(J_{\min },{\vartheta }_0)$ equal to (0.5,1), (0,3), (0,1), and (0,0.3), respectively. The results for chains ${\mathrm{F}}_{XX}$ and ${\mathrm{H}}_{XX}$ (omitted for clarity) are statistically indistinguishable from those for chain ${\mathrm{D}}_{XX}$. Error bars (not shown for clarity) are of the order of the statistical data fluctuations. Lines are guides to the eyes.

Figure 3

(Color online) Dependence of $-l^2{C}^{zz}\left(l\right)$ on the spin separation $l$ in the $XX$ model for three different probability distributions of the couplings: a box distribution $({\vartheta }_0=1)$ with $J_{\min }=0$, a box distribution with $J_{\min }=1∕4$, and a binary distribution with $J_{\min }=1∕10$ [see Eqs. (3.1, 3.2)]. The orange solid line corresponds to the disorder-free prediction (Ref. 4) $-l^2{C}_c^{zz}=1∕{\pi }^2$. For short-length scales, all chains approach the behavior of the uniform system. After a disorder-dependent crossover length, the data approach the universal prediction of Eq. (2.21), which is indicated by the black dashed line. Statistical fluctuations increase with $l$, and so results for $l⩾300$, as well as error bars, are omitted for clarity.

Figure 4

(Color online) Dependence of $l^2\mid C^{xx}\left(l\right)\mid$ on the spin separation $l$ for the $XX$ model with the same coupling distributions as in Fig. 3. The short-distance behavior, as in the uniform system, corresponds to a power law with exponent ${\eta }_c=1∕2$. The random-singlet nature of the ground state governs the long-distance behavior, with $\eta =2$, but nonuniversal prefactors. The orange solid line corresponds to the clean-system transverse correlation function (Ref. 5) ${(-1)}^{l}0.14709∕\sqrt{l}$. The black dashed line is the prediction for the disordered system given in Eq. (2.21) (see text).

Figure 5

(Color online) The summed transverse correlation function $C_{\text{sum}}^{xx}\left(l\right)=C^{xx}\left(l\right)+C^{xx}(l+1)$ as a function of the distance $l$ (for odd $l$) for $XX$ chains and the same coupling distributions as in the previous figure. Although for sufficiently strong disorder (circles and triangles) the curves approach a power law, corresponding to the random-singlet exponent $\eta =2$ and to a prefactor $({\upsilon }_o^x+{\upsilon }_e^x)∕3\simeq -1∕12$, the less disordered system (squares) shows a different prefactor. Again, results for $l>400$ and error bars are omitted for clarity.

Figure 6

(Color online) Correlation function ${(-1)}^l{C}^{zz}\left(l\right)$ in the ground state of isotropic random AF Heisenberg spin-$1∕2$ chains [Eq. (3.4)] of length $L_0=200$ with disorder strength $W=0.75$. The quantum Monte Carlo results were obtained at $T∕\overline{J}=1.5\times {10}^{-5}$ and averaged over $N_{\text{samples}}=500$ realizations. When the distance $l$ between spins is even (circles), the asymptotic regime is described by $C^{zz}\left(l\right)\simeq 0.9∕l^2$ (black, solid line), whereas for odd $l$ (squares), the best fit gives $C^{zz}\left(l\right)\simeq -0.98∕l^2$ (red, dashed line).

Figure 7

(Color online) Summed correlation function $C_{\text{sum}}^{zz}\left(l\right)=\mid C^{zz}\left(l\right)+C^{zz}(l+1)\mid$ in the ground state of isotropic random AF Heisenberg spin-$1∕2$ chains [Eq. (3.4)] of length $L_0=200$ with disorder strength $W=0.75$ (red, squares) and $L_0=100$ with $W=1$ (orange, diamonds). The quantum Monte Carlo results were obtained at $T∕\overline{J}=1.5\times {10}^{-5}$ for $W=0.75$ and $T∕\overline{J}=4\times {10}^{-6}$ for $W=1$ and averaged over $N_{\text{samples}}=500$ realizations for each disorder strength. The dashed line is the $1∕\left(12l^2\right)$ prediction.

Figure 8

(Color online) The magnitude of the transverse $C^{xx}$ and longitudinal $C^{zz}$ correlation functions between ${\mathbf{S}}_0$ and ${\mathbf{S}}_j$ as a function of $j$ (for $j\ne 0$). Main panels (a), (b), and (c) correspond to different samples drawn from different disorder distributions [see Eq. (3.1)] whose parameters $(J_{\min },{\vartheta }_0)$ are (0.25,1), (0,1), and (0,0.7), respectively. Insets (i) and (ii) show $C^{xx}$ and $C^{zz}$ between spins ${\mathbf{S}}_i$ and ${\mathbf{S}}_j$ as a function of $j$ for different values of $i$ between $-3$ and 3 and for values of $j$ around $j^{*}$. In the SDRG picture for these particular realizations, ${\mathbf{S}}_0$ and ${\mathbf{S}}_{j^{*}}$ should couple in a singlet state where $j^{*}$ equals 97, 439, and 297 in panels (a), (b), and (c), respectively. The lines in the insets are guides to the eyes.

Figure 9

(Color online) The various types of singlets. Black circles denote isolated original spins. Blue squares denote original spins belonging to effective three-spin clusters.

Figure 10

(Color online) Ground state of the infinite-disordered AF spin-$1∕2$ chain. The entanglement entropy between the subsystem inside the box of length $l$ and the rest of the chain is equal to the number of singlets shared by them. In this case, $S\left(l\right)=5s_0$, with $s_0$ being the entanglement entropy of a singlet pair when one of the spins is traced out.

Figure 11

(Color online) Schematic entropy counting. (a) When the singlet length (in this case $l_s=5$) is greater than the subsystem length $(l_A=3)$, there are $l_A$ different subsystems whose right boundaries are crossed by the singlet. (b) Otherwise, when the subsystem length $l_B$ is greater than $l_s$ (in this case, $l_B=4$ and $l_s=3$), there are $l_s$ different subsystems whose left boundaries are crossed by the singlet.

Figure 12

(Color online) Entanglement entropy of random $XX$ chains with $L_0=1000$ sites, and disorder of the form of Eq. (3.5) with, from top to bottom, $W=0$ (no disorder, open circles), $W=0.25$ (cyan, circles), $W=0.5$ (orange, triangles), $W=0.75$ (red, squares), and $W=1$ (green, diamonds). These are exact diagonalization results averaged over ${10}^3⩽N_{\text{samples}}⩽{10}^4$ realizations. The black dashed lines are fits of the form $S\left(l\right)=(1∕3)\ln 2\ln l+b$ with $b=1.048,0.925,0.876,0.849$ for $W=0.25,0.5,0.75,1$. In the inset, the constant $b$ is also plotted versus ${\delta }^{-1}$, where ${\delta }^2=\overline{{\left(\ln J\right)}^2}-{\left(\overline{\ln J}\right)}^2$ and naively fitted to $b\left(\delta \right)=0.812+0.03456∕\delta$.

Figure 13

(Color online) The longitudinal structure factor in the $XX$ model for various disordered chains of lengths up to $L_0=4000$. As can be seen, they are practically indistinguishable. Inset (i) highlights ${\mathcal{S}}^z$ for small $q$ in which the curves become somewhat distinguishable near $q=0.06\pi$. Moreover, they slightly deviate from the clean-system prediction ${\mathcal{S}}_c^z=q$ (dotted line) tending to a universal form ${\mathcal{S}}^z=\kappa \mid q\mid$ (dashed line), with $\kappa ={\pi }^2∕12$ (see text). Inset (ii) shows ${\mathcal{S}}^z$ near $q=\pi ∕2$, where disorder is irrelevant. See Eqs. (3.1, 3.2) for the definition of the disorder parameters (here, ${\Omega }_0=1$ and ${\vartheta }_0=0$).

Figure 14

(Color online) The transverse structure factor in the $XX$ model for various disordered chains of lengths up to $L_0=4000$. Inset (i) highlights ${\mathcal{S}}^z$ for small $q$, where ${\mathcal{S}}^x\left(0\right)\approx 0.194$, 0.159, 0.128 for the box distributions with $J_{\min }=1∕4$, $J_{\min }=0$, and the binary distribution, respectively. In all cases, ${\mathcal{S}}^x\left(q\right)-{\mathcal{S}}^x\left(0\right)\sim \mid q\mid$. Inset (ii) shows the behavior of ${\mathcal{S}}^x$ near $q=\pi$. It follows the characteristic divergence of the clean-system prediction (dotted line) until a crossover vector $q_c$ above which it saturates to a constant (see text).

Figure 15

(Color online) The structure factor as a function of $q$ for different disorder parameters [see Eq. (3.5)] in the isotropic Heisenberg model for chains of lengths up to $L_0=200$. Inset (i) shows ${\mathcal{S}}^{\alpha }$ for $q⪡1$ and compares it with the (field-theoretical) clean-system prediction ${\mathcal{S}}_c^{\alpha }\to K\mid q\mid$, with $K=1∕2$ being the Luttinger liquid parameter, and the disordered system prediction ${\mathcal{S}}^{\alpha }\to \kappa \mid q\mid$, with universal $\kappa ={\pi }^2∕12$. Inset (ii) highlights ${\mathcal{S}}^{\alpha }$ near the AF peak $q\approx \pi$. The dashed line is the clean-system prediction, namely, ${\mathcal{S}}_c^{\alpha }\to -\lambda \ln (1-q∕\pi )$, with $\lambda =-\pi ∕2$ (see text).