Special Transition and Extraordinary Phase on the Surface of a Two-Dimensional Quantum Heisenberg Antiferromagnet

1. We carefully checked the syntax errors and typos.

2. We rewrite the definition of $C_\parallel$ as $C_{\parallel}(r)=\frac{(-1)^r}{L}\sum_{x}\langle\mbf{S}_{(x,1)}\cdot\mbf{S}_{(x+r,1)}\rangle$ in Eq. (2).

3. The following discussion is added to section 4:
In Ref. [17], the extraordinary-log phase was proposed based on the perturbative RG analysis near the normal fixed point at the 1D boundary. Starting from the normal fixed point, where the spins show an infinitesimal long-range order, the spin interactions would be relevant and lead to short-range correlations at a free-standing boundary, but the coupling with the bulk critical modes reverses the RG flow direction and makes the normal fixed point stable. However, the logarithmically slow running towards this fixed point leads to the logarithmic decay of the spin correlation function instead of a long-range order, thus this is dubbed the extraordinary-log universality.
In the whole phase space, the results of our numerical work may be far from the normal surface fixed point, hence different from the extraordinary-log behavior.

4. We add a short paragraph to clarify the question of correction to scaling (two lines above Eq. (9)):
It should be noted that the finite-size scaling corrections arise from two sources, one is the leading correction proportional to $L^{-\omega_1}$, with $\omega_1 = 0.759$ for the current model, which comes from the irrelevant scaling field, another one is the background contribution analytic in $L^{-1}$. In practice, the analytic term $L^{-1}$ cannot be distinguished from $L^{-\omega_1}$ in the fitting procedure due to their close exponents. We also tried the fitting with $\omega_1=1$, the fitting quality is slightly worse and the difference of the results is very small, which falls in the range of the uncertainty of the error bars. Such a strategy has also been applied to all the other data fittings in this and the next subsections, although not explicitly stated.

5. There is a typo of the definition of Qs (Eq. 7), we have fixed it.

6. We revisited the scaling behaviors of $S(\pi)$, $S(\pi+\delta q)$, and $(\xi_s/L)^2$ in the extraordinary phase (ordered), the paragraph after 7 lines of Eq. (27) is revised as:

Furthermore, according to the definition of Eq. (5), in an ordered phase, $S(\pi+\delta q)$ should growth logarithmically (the constant term cancels out after summing, and the integral of the $1/r$ term contribute the logarithmic term), i.e., the data of $S(\pi+\delta q)$ should satisfy the finite-size scaling form

$S(\pi+\delta q)=a+b\log(L);$

combining with scaling of $S(\pi)$ in Eq. (26), we get the scaling formula of the square of the correlation ratio, which is written as

$(\xi_s/L)^2=a+bL/\log (L).$

Figure 10 shows the scaling behaviors of $S(\pi+\delta q)$ and $(\xi_s/L)^2$ in the extraordinary phase, with $J_s=16$, which further demonstrates that there is a long-range AF order.