Logarithmic delocalization of end spins in the $S=\frac{3}{2}$ antiferromagnetic Heisenberg chain

Using the DMRG method we calculate the surface spin correlation function, $C_L\left(l\right)=⟨S_l^z{S}_{L+1-l}^z⟩$, in the spin $S=\frac{3}{2}$ antiferromagnetic Heisenberg chain. For comparison we also investigate the $S=\frac{1}{2}$ chain with $S=1$ impurity end spins and the $S=1$ chain. In the half-integer spin models the end-to-end correlations are found to decay to zero logarithmically, $C_L\left(1\right)\sim {\left(\log L\right)}^{-2d}$, with $d=0.13\left(2\right)$. We find no surface order, in clear contrast with the behavior of the $S=1$ chain, where exponentially localized end spins induce finite surface correlations. The lack of surface order implies that end spins do not exist in the strict sense. However, the system possesses a logarithmically weakly delocalizing boundary excitation, which, for any chain lengths attainable numerically or even experimentally, creates the illusion of an end spin. This mode is responsible for the first gap, which vanishes asymptotically as ${\Delta }_1\approx \left(\pi v_{S}d\right)∕\left(L\ln L\right)$, where $v_S$ is the sound velocity and $d$ is the logarithmic decay exponent. For the half-integer spin models our results on the surface correlations and on the first gap support universality. Those for the second gap are less conclusive, due to strong higher-order corrections.

Figure 1

Log-log plot of the first gap ${\Delta }_1$ versus $L$. The inset shows the associated effective (local) exponent $\zeta \left(L\right)$ versus $1∕\ln \left(BL\right)$ with $B$ chosen appropriately, see Eq. (16), for each data set.

Figure 2

Length-dependent effective parameters, $B\left(L\right)$ and $d_1\left(L\right)$, determined from two-point fits of the first gap ${\Delta }_1$ using Eq. (12) with $x_1=0$.

Figure 3

Scaled first gap $L{\Delta }_1$ as a function of $1∕\ln \left(BL\right)$, with $B$ chosen individually for each curve according to Eq. (16). The common slope (dotted line) determines the universal value $d_1=0.13\pm 0.02$.

Figure 4

Scaling dimension $x_2$ associated with the scaled second gap ${\Delta }_{2}L$ as a function of $1∕\ln BL$. The value of $B$ is chosen individually for each curve according to Eq. (16). The dotted line indicates the conjectured scaling.

Figure 5

Exponential finite-size scaling of the surface correlation function $C_L\left(l\right)$ of the $S=1$ HAF chain for $l=1,2,3$. Inset shows the behavior of the extrapolated values $C\left(l\right)$ as a function of $l$.

Figure 6

Local fit parameters, $A\left(L\right)$, $\alpha \left(L\right)$, and $C(l,L)$, for Eq. (21) calculated from three-point fits for the ${\mathcal{H}}_{3∕2}$ model.

Figure 7

Local fit parameters, $A\left(L\right)$ and $\mu \left(L\right)$, for Eq. (22) calculated from two-point fits for the ${\mathcal{H}}_{3∕2}$ model.

Figure 8

Local fit parameters, $A\left(L\right)$ and $\mu \left(L\right)$, for Eq. (22) calculated from two-point fits for the ${\mathcal{H}}_{1∕2}^1(\lambda =1)$ model.