# Sublattice extraordinary-log phase and new special point of the
antiferromagnetic Potts model

Li-Ru Zhang  School of Microelectronics $\&$ Data Science, Anhui University of
Technology, Maanshan, Anhui 243002, China  Chengxiang Ding<EMAIL_ADDRESS>School of Microelectronics $\&$ Data Science, Anhui University of Technology,
Maanshan, Anhui 243002, China  Wanzhou Zhang  College of Physics and
Optoelectronics, Taiyuan University of Technology, Shanxi 030024, China  Long
Zhang<EMAIL_ADDRESS>Kavli Institute for Theoretical Sciences and CAS
Center for Excellence in Topological Quantum Computation, University of
Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

We study the surface criticality of a three-dimensional classical
antiferromagnetic Potts model, whose bulk critical behaviors belongs to the XY
model because of emergent O(2) symmetry. We find that the surface
antiferromagnetic next-nearest neighboring interactions can drive the
extraordinary-log phase to the ordinary phase, the transition between the two
phases belongs to the universality class of the well-known special transition
of the XY model. Further strengthening the surface next-nearest neighboring
interactions, the extraordinary-log phase reappears, but the main critical
behaviors are dominated on the sublattices of the model; the special point
between the ordinary phase and the sublattice extraordinary-log phase belongs
to a new universality class.

###### pacs:

03.67.Bg, 03.65.Ud, 05.30.Rt

##  I Introduction

For a system at the critical point of a phase transition, the correlation
function and order parameter and some other physical quantities exhibit power-
law scaling behaviors, which are called critical behaviors. It is a hot topic
in statistical physics with a long history of research.

At the critical point, the critical behaviors not only manifest in the bulk
but also on the surface, which is called the surface critical
behaviorsbinder1983 . Depending on the strength of the surface interactions,
the surface critical behaviors can be richer than the bulk critical behaviors.
Generally, there are three types of surface critical phases, dubbed ‘ordinary
phase’, ‘special point’, and ‘extraordinary phase’. Ordinary phase refers to
the case when the surface interactions are not too strong, and the surface
critical behavior are purely induced by the bulk critical state; extraordianry
phase refers to the case when the surface interactions are strong enough, the
surface has become ordered or critical with logarithmic scaling behaviors
(extraordinary-log phase); the special point is the transition point between
the ordinary phase and the extraordinary phase. Typical examples can be found
in the classical O($n$) spin modelsOn ; O4 ; O3sp ; XYlog .

The research of surface critical behaviors also has long history, and the
interest in this field are renewed by the recent works in quantum spin
modelsLong2017 ; Ding2018 . A series of related studies greatly promote the
interest in this fieldWeber2018 ; Weber2019 ; Max2022 ; Zhu2021 ; Weber2021 ;
Jian2021 ; Toldin2021 ; Ding2021 ; Yu2021 ; Zhu2021-2 ; Max2022a ; Ding2022 ;
clock ; Lv2022a ; Lv2022b , which includes the finding of the extraordinry-log
phaseMax2022 , a great breakthrough in the research of critical phenomena.
Such type of critical phase, characterized by the logarithmic decaying of
correlation function, has already been numerically found in both the classical
spin modelsO3sp ; XYlog ; Ding2022  and also the quantum spin modelsLv2022a ;
Lv2022b .

In our recent workDing2022 , we find the extraordinary-log phase in the
classical antiferromagnetic Potts model, which has a bulk critical point
belongs to the XY model because of the emergent O(2) symmeryDing2016 ,
although the spin symmetry of the model is discrete. Tunning the surface
nearest neighboring (NN) interactions lead to a surface phase diagram similar
to that of the XY model. More importantly, by adding ferromagnetic next-
nerearest neighboring(NNN) interactions to the surface, a phase transition
from the extraordinary-log phase to the ordered phase is found, whose critical
behaviors are very different from the traditional ordered-disordered phase
transition. In the current paper, with the purpose of exploring new surface
critical behaviors, we add antiferromagnetic NNN interactions to the surface,
we find that the antiferromagnetic NNN interactions can drive the surface from
the extraordinary-log phase to an ordinary phase, and then drive the system to
a new extraordinary-log phase whose main critical behaviors are dominated on
the sublattices if the strength of the NNN interactions are strong enough. The
universality class of the special point between the ordianry phase and the
sublattice extraordinary-log phase is different from the well-known special
point of the XY model.

The phase diagram of the model is shown in Fig. 1 and the paper is organized
as follows: In Sec. II, we introduce the model and the method; in Sec. III, we
present the numerical results, which include the critical properties of the
two special points, the ordinary phase, and the new extraordinary-log phase.
We conclude our paper in Sec. IV.

Figure 1: Surface phase diagram of the antiferromagnetic Potts model (1)with
$T=T_{c}^{\rm bulk}=1.22603$ and $J_{\rm s}=5$.

##  II Model and Method

The Hamiltonian for the three-state Potts model with surface NNN
antiferromagnetic interactions on a simple cubic lattice is defined as

$\displaystyle\mathcal{H}=J\sum\limits_{\langle
i,j\rangle}\delta_{\sigma_{i},\sigma_{j}}+J_{s}\sum\limits_{{\langle
i,j\rangle}^{s}}\delta_{\sigma_{i},\sigma_{j}}+J_{s}^{\prime}\sum\limits_{{\langle\langle
i,j\rangle\rangle}^{\prime}}\delta_{\sigma_{i},\sigma_{j}},$ (1)

where $\langle i,j\rangle$, ${\langle i,j\rangle}^{s}$ and ${\langle\langle
i,j\rangle\rangle}^{\prime}$ denote the bulk NN, the surface NN, and the
surface NNN sites, respectively. All the interactions are antiferromagnetic,
and the spin $\sigma_{i}$ can be mapped to unit vectors on the plane:

$\displaystyle\vec{\sigma}_{i}=(\cos\theta_{i},\sin\theta_{i}),$ (2)

with $\theta_{i}=2\pi\sigma_{i}/3$ and $\sigma_{i}=1,2,3$. The bulk phase
transition of the model belongs to the XY universality because of emergent
O(2) symmetryDing2016 , and the critical point has been refined by extensive
Monte Carlo simulations in Ref. Ding2022, , which is $T_{c}=1.22603$. When
there is no NNN interaction and the strength of the NN interactions
$J_{s}>2.04119$, the surface is in the extraordinary-log phaseDing2022 .

For simulations of the model, we adopt a combination of the local update
(Metropolis algorithm) and the geometric clustering algorithmgeo , making us
able to study the system with size up to $L=128$. The surface variables we
sampled include the surface squared (staggered) magnetization $m_{\rm s1}^{2}$
and the surface magnetic susceptibility $\chi_{\rm s1}$,which are defined as,

$\displaystyle m_{\rm s1}^{2}=\langle\mathcal{M}_{\rm s1}^{2}\rangle,$ (3)
$\displaystyle\chi_{\rm s1}=L^{2}(\langle\mathcal{M}_{\rm
s1}^{2}\rangle-\langle|\mathcal{M}_{\rm s1}|\rangle^{2}),$ (4)

where $\mathcal{M}_{\rm s1}$ is defined as,

$\displaystyle\mathcal{M}_{\rm s1}$ $\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\sum\limits_{\vec{R}}(-1)^{x+y+z}\vec{\sigma}_{\vec{R}}.$
(5)

Here $\vec{R}=(x,y,z)$ is the coordination, for the surface, $z$ should be 1
or $L$, and $L^{2}$ is the number of sites of the surface. We also sample the
sublattice squared (staggered) magnetization $m^{2}_{\rm s1A}$, which are
defined similar to (3), but the sites are restricted to be in the sublattice A
of the surface. (The square surface is bipartite, which can be divided into
two equivalent sublattices)

We also sample the surface specific heat $C_{\rm v1}$,

$\displaystyle C_{\rm v1}$ $\displaystyle=$ $\displaystyle
L^{2}(\langle\mathcal{E}_{\rm 1}^{2}\rangle-\langle\mathcal{E}_{\rm
1}\rangle^{2})/T^{2},$ (6)

where $T$ is the temperature, $\mathcal{E}_{\rm 1}$ is the microscopic energy
density of the surface.

The surface correlation function is defined as

$\displaystyle
C_{\parallel}(r)=\langle\vec{\sigma}_{i}\cdot\vec{\sigma}_{i+r}\rangle,$ (7)

where the site $i$ and $i+r$ are restricted to be on the surface. The surface
correlation length $\xi_{\rm 1}$ and “structure factor” $F_{1}$ are defined as

$\displaystyle\xi_{\rm 1}=\frac{(m^{2}_{\rm s1}/F_{\rm
1}-1)^{1/2}}{2\sqrt{\sum\limits_{i=1}^{d}\sin^{2}(\frac{k_{i}}{2})}}\;,$ (8)
$\displaystyle F_{\rm
1}=\frac{1}{L^{4}}\Big{\langle}\big{|}\sum\limits_{\vec{R}}(-1)^{x+y+z}e^{i\vec{k}\cdot\vec{R}}\vec{\sigma}_{\vec{R}}\big{|}^{2}\Big{\rangle}$
(9)

where $\vec{k}$ is the “smallest wavevector” along the $x$ direction–i.e.,
$\vec{k}\equiv(2\pi/L,0)$. In the disordered phase, the correlation length
$\xi_{1}$ is finite and the correlation ratio $\xi_{1}/L$ decreases to zero,
while in the ordered phase or extraordinary-log phase $\xi_{1}/L$ diverges
rapidly due to the rapid disappearance of the “structure factor” $F_{1}$. The
correlation ratio $\xi_{1}/L$ in the critical phase has a finite nonzero value
in the thermodynamic limit. Therefore, the correlation ratio $\xi_{1}/L$ is a
good tool for locating the critical point of the phase transition.

##  III Results

###  III.1 Two special points

By the Monte Carlo method we simulate the antiferromagnetic Potts model (1)
with open boundary along the $z$-direction and periodic boundaries in the
$x$\- and $y$\- directions. At the bulk critical point $T=T_{c}^{\rm
bulk}=1.22603$Ding2022 , setting the system surface initially to the
extraordinary-log phase ($J_{s}=5$), varying the surface NNN interactions
$J^{\prime}_{s}$, we can find two special points. As shown in Fig. 2(a), the
surface magnetic susceptibility $\chi_{s1}$ of the system shows two peaks at
$J^{\prime(1)}_{\rm sc}\approx 0.85$ and $J^{\prime(2)}_{\rm sc}\approx 3.3$,
where the former peak diverges faster. The behaviors of the surface squared
magnetization $m_{\rm s1}^{2}$ and the surface structure factor $F_{1}$, as
shown in Figs. 2(b) and (c), also demonstrate the two transitions. We can see
that the $m_{\rm s1}^{2}$ tends to zero in the region between the two critical
points even for small system sizes, the system is in a disordered phase, which
is different from the well known “ordinary phase”; we will discuss the
properties of such phase specifically in the next subsection. The specific
heat $C_{\rm v1}$, as shown in Fig. 2(d), demonstrates a rapidly diverging
peak at the second phase transition point, however, it does not show any
obvious peak for the first transition.

Figure 2: Surface critical behavior of the antiferromagnetic Potts model (1)
at $T=T_{c}^{\rm bulk}=1.22603$, $J_{s}=5$: (a) surface susceptibility
$\chi_{\rm s1}$, (b) surface squared magnetization $m^{2}_{\rm s1}$, (c)
surface structure factor $F_{1}$, and (d) surface specific heat $C_{\rm v1}$.

In order to quantitatively determine the two critical points, we investigate
the surface correlation ratio $\xi_{1}/L$, as shown in Fig. 3, which obviously
shows the two critical points. In the vicinity of the critical point,
$\xi_{1}/L$ satisfies the finite-size scaling (FSS) formula:

$\displaystyle\xi_{1}/L=a_{0}+\sum\limits_{k=1}^{k_{\rm
max}}a_{k}(J^{\prime}_{\rm s}-J^{\prime}_{\rm sc})^{k}L^{ky_{s}}+bL^{y_{1}},$
(10)

where $y_{s}>0$ is the critical exponent, $J^{\prime}_{\rm sc}$ is the
critical point , $y_{1}<0$ is the correction-to-scaling exponent, $a_{0}$,
$a_{k}$, and $b$ are unknown parameters.

Figure 3: Surface correlation ratio $\xi_{1}/L$ of the antiferromagnetic Potts
model (1) with $T=T_{c}^{\rm bulk}=1.22603$ and $J_{\rm s}=5$, the dashed line
positions in (a) are $J^{\prime(1)}_{\rm sc}=0.8478$ and $J^{\prime(2)}_{\rm
sc}=3.3326$; (b) and (c) are the local enlargements of the vicinity of the two
critical points.

For the first transition, the data fitting of $\xi_{1}/L$ according to (10)
with $y_{1}=-1$ gives $J^{\prime(1)}_{\rm sc}=0.8478(9)$ and $y_{\rm
s}=0.59(4)$. We can see that the critical exponent $y_{s}$ coincides with the
result of the well-known special transition of the XY modelOn ; in order to
confirm such transition is in the same universality of the XY model, we study
the scaling behaviors of the squared magnetization $m_{\rm s1}^{2}$ and the
correlation function $C_{\parallel}(L/2)$, at the critical point, they satisfy
the following FSS formulas

$\displaystyle m^{2}_{\rm s1}L^{2}=c+L^{2y_{\rm h1}-2}(a+bL^{y_{1}})$ (11)
$\displaystyle C_{\parallel}(L/2)=L^{-1-\eta_{\parallel}}(a+bL^{y_{1}})$ (12)

where $y_{\rm h1}$ and $\eta_{\parallel}$ are the critical exponents. The data
fitting, with $y_{1}=-1$, gives $y_{\rm h1}=1.698(4)$ and
$\eta_{\parallel}=-0.399(5)$. We can see that the values of $y_{\rm h1}$ and
$\eta_{\parallel}$ satisfy the scaling formulasc1 ; sc2

$\displaystyle\eta_{\parallel}=d-2y_{h1},$ (13)

where $d=3$ is the space dimension of the system. This means that $m_{s1}^{2}$
and $C_{\parallel}$ should decay with the same power as the increasing of the
system size, which is demonstrated in Fig. 4(a). We can also see that the
values of $y_{\rm h1}$ and $\eta_{\parallel}$ coincide with those of the well-
known special point of the XY modelOn , i.e., they are in the same
universality class.

We also investigate the scaling behaviors of the sublattice squared
magentizaton $m^{2}_{\rm s1A}$, which is the same as $m^{2}_{\rm s1}$,
specifically, $m^{2}_{\rm s1A}$ satisfies

$\displaystyle m^{2}_{\rm s1A}L^{2}=c+L^{2y_{\rm h1A}-2}(a+bL^{y_{1}}),$ (14)

where we have written the exponent as $y_{\rm h1A}$. The data of $m^{s}_{\rm
s1A}$ is also shown in Fig. 4(a). We can see that $m^{2}_{s1A}$ shows a
perfect power-lay scaling, however, this is misleading, such power-law scaling
comes from the nonsingular part, which is the first term of the RHS of Eq.
(14). Because in the current case $y_{h1A}<1$, $m_{\rm s1A}^{2}$ is dominated
by the term $c/L^{2}$. This is also the reason why we do not write the scaling
form as $m^{2}_{\rm s1A}=c/L^{2}+L^{2y_{\rm h1A}-4}(a+bL^{y_{1}})$. The log-
log plot of $m^{s}_{\rm s1A}L^{2}-c$ versus $L$ is also included in Fig. 4(a),
and the data fitting according to Eq. (14) gives $y_{\rm h1A}=0.26(3)$.

Figure 4: (a)) log-log plot of $C_{\parallel}(L/2)$, $m^{2}_{\rm s1}$,
$m^{2}_{\rm s1A}$, and $m^{2}_{\rm s1A}L^{2}-c$ versus $L$ at the first
transition point $J^{\prime(1)}_{\rm sc}=0.8478$; (b)log-log plot of
$C_{\parallel}(L/2)$, $m^{2}_{\rm s1}$ and $m^{2}_{\rm s1A}$ at the second
transition point $J^{\prime(2)}_{\rm sc}=3.3326$.

For the second special point, the data fitting gives $J^{\prime(2)}_{\rm
sc}=3.3326(8)$ and $y_{\rm s}=1.44(4)$. We can see that the critical exponent
$y_{\rm s}$ is obviously different from the first one; in order to determine
the universality class of this transition, we investigate the scaling
behaviors of the squared magnetization $m_{\rm s1}^{2}$ and the correlation
function $C_{\parallel}(L/2)$; at the critical point, they satisfy the FSS
formulas (11) and (12). By the data fitting, we get $y_{\rm h1}=1.56(2)$ and
$\eta_{\parallel}=-0.59(3)$; these are obviously different from the results of
the first transition. Furthermore, we can see that they do not satisfy the
scaling law (13), i.e., $m_{\rm s1}^{2}$ and $C_{\parallel}(L/2)$ decay with
different powers, which are demonstrated in Fig. 4(b). In order to understand
such discrepancy and the nature of this phase transition, we investigate the
sublattice squared (staggered) magnetization $m^{2}_{\rm s1A}$, which is also
shown in Fig. 4(b). The data fitting are performed in a similar way according
to Eq. (14), which gives $y_{\rm h1A}=1.80(2)$. We can see that
$\eta_{\parallel}$ and $y_{\rm h1A}$ satisfy the scaling law (13), i.e.,
$C_{\parallel}$ and $m_{\rm s1A}^{2}$ decay with the same power, as shown in
Fig. 4(b). This result reveals the sublattice nature of such transition. In
summary, the second transition is characterized by the following critical
exponents

$\displaystyle y_{\rm s}$ $\displaystyle=$ $\displaystyle 1.44(4),$ (15)
$\displaystyle y_{\rm h1}$ $\displaystyle=$ $\displaystyle 1.56(2),$ (16)
$\displaystyle y_{\rm h1A}$ $\displaystyle=$ $\displaystyle 1.80(2),$ (17)
$\displaystyle\eta_{\parallel}$ $\displaystyle=$ $\displaystyle-0.59(3).$ (18)

###  III.2 The ordinary phase

For the intermediate region
$J^{\prime(1)}_{sc}<J^{\prime}_{s}<J^{\prime(2)}_{sc}$, the surface is in an
ordinary phase. As shown in Fig. 5(a), the surface squared magnetization
$m^{2}_{s1}$ shows a very good power-law scaling, which is the effect of the
nonsingular term $c/L^{2}$ because $y_{\rm h1}<1$ here. This situation is very
similar to the case of the $m^{2}_{\rm s1A}$ at the first special point . In
Fig. 5(a), we also show the log-log plot of $|m^{2}_{\rm s1}L^{2}-c|$ versus
$L$, which also indicates a good power-law scaling; the data fitting according
to Eq. (11) gives $y_{h1}=0.77(2)$, which is consistent with the result of the
ordinary phase of the XY model.

For the sublattice squared magnetization $m^{2}_{\rm s1A}$, the nonsigular
part is also dominated by the $L^{-2}$ scaling, as shown in Fig. 5(a),
however, we can not perform an effective fitting of $m^{2}_{\rm s1A}L^{2}$
according to the scaling formula (14), because $m^{2}_{\rm s1A}L^{2}$
converges to the constant $c$ too fast, as shown in Fig. 5(b); it can be
compared to the data of $m^{2}_{\rm s1}L^{2}$, which shows a very good
asymptotic behavior, also shown in Fig. 5(b).

As to the scaling behavior of the surface correlation function
$C_{\parallel}(L/2)$, we find that in the short-range region, it decays very
fast, with a power of about $L^{-5}$; however, in the long-range region, it
decays slower with a power of about $L^{2y_{h1}-4}=L^{-2.46}$. This is also
illustrated in Fig. 5(a). It should be noted that the dashed line in Fig. 5(a)
is not obtained by fitting the long-range part data of $C_{\parallel}(L/2)$
but only a self-consistent check with $y_{h1}=0.77$, which is got from the
fitting of $m^{2}_{s1}$. The data in the long-range region is too small which
makes it too difficult to give accurate data in Monte Carlo simulations.

Figure 5: (a) Log-log plot of the correlation function $C_{\parallel}(L/2)$,
the squared magnetization $m^{2}_{\rm s1}$, the sublattice squared
magnetization $m^{2}_{\rm s1A}$, and $|m^{2}_{\rm s1A}L^{2}-c|$ for the
disordered phase, with $J^{\prime}_{\rm s}=2.5$ and $c=3.68$; (b) the data of
$m^{2}_{\rm s1}L^{2}$ and $m^{2}_{\rm s1A}L^{2}$.

###  III.3 Sublattice extraordinary-log phase

Figure 6: Log-log plot of $C_{\parallel}(L/2)$, $m^{2}_{\rm s1}$, and
$m^{2}_{\rm s1A}$ versus $\ln(L/L_{0})$, with $J^{\prime}_{\rm s}=3.8$ and
$L_{0}=0.519$, 0.724, 0.295 for the three variables repectively. The slopes of
the lines of $C_{\parallel}(L/2)$ and $m^{2}_{\rm s1A}$ are $q$=0.59, and the
slope of $m^{2}_{\rm s1}$ is $q_{1}=1.9$.

We also try to investigate the properties of the phase with $J^{\prime}_{\rm
s}>J^{\prime(2)}_{\rm sc}$, in this region, the efficiency of the algorithm
decreases a lot, which prevents us to get good data for large systems.
However, according to the already known results in our previous workDing2022
that strong enough surface NN interactions will lead to an extraordinary-log
phase, we infer that the phase should be an extraordinary-log one,
furthermore, because the NNN interactions are already very strong, we infer
that the extraordinary-log scaling may be dominated on the sublattices. To
confirm such inference, we investigate the the scaling behaviors of the
surface correlation function $C_{\parallel}(L/2)$ and the sublattice squared
magnetization $m_{\rm s1A}^{2}$, we find that they satisfy the logarithmic
decaying formulas

$\displaystyle C_{\parallel}(L/2)$ $\displaystyle=$ $\displaystyle
a\cdot[\ln(L/L_{0})]^{-q},$ (19) $\displaystyle m^{2}_{\rm s1A}$
$\displaystyle=$ $\displaystyle a\cdot[\ln(L/L_{0})]^{-q}.$ (20)

The value of the deacying exponent $q$ is the same for $C_{\parallel}(L/2)$
and $m^{2}_{\rm s1A}$, which is $q=0.59(3)$, it is consistent with that of the
XY model. An illustrative plot of $C_{\parallel}(L/2)$ and $m^{2}_{\rm s1A}$
is shown in Fig. 6.

We also investigate the squared magnetization $m_{\rm s1}^{2}$, we find that
it also satisfy a logarithmical scaling formula

$\displaystyle m^{2}_{\rm s1}=a\cdot[\ln(L/L_{0})]^{-q_{1}},$ (21)

however, the decaying exponent is found to be $q_{1}=1.9(2)$, which is much
different from $q$; this result is also plotted in Fig. 6.

In order to further understand the properties of such sublattice
extraordinary-log phase, we compute the surface structure factor

$\displaystyle
F(\vec{k}))=\frac{1}{L^{4}}\Big{\langle}\big{|}\sum\limits_{\vec{R}}e^{i\vec{k}\cdot\vec{R}}\vec{\sigma}_{\vec{R}}\big{|}^{2}\Big{\rangle}$
(22)

in the full momentum space ($k_{x}$, $k_{y}$). The result is shown in Fig. 7,
in which we can see that the points ($0,\pm\pi$) and ($\pm\pi,0$) are light,
which means the system has a strip order (in finite system), this is in
correspondence to the sublattice squared magnetization $m^{2}_{\rm s1A}$. The
points ($\pm\pi,\pm\pi$) are also light but weaker than the points
($0,\pm\pi$) and ($\pm\pi,0$), this is in correspondence to the squared
magnetization $m^{2}_{\rm s1}$.

Figure 7: Surface structure factor $F(\vec{k})$ for the extraordinary phase,
with $J^{\prime}_{s}=3.8$. The system size is $L=16$.

##  IV Conclusion and discussion

In summary, we have tried to add antiferromagnetic NNN interactions to the
surface of the antiferromagnetic Potts model, with the extraordinary-log phase
as the starting point. We find that as the increasing of the NNN interactions,
the system is driven to an ordinary phase in which the correlation function
shares the same exponent of the ordinary phase of the XY model in the long-
range region but decays much faster in the short-range region. Further
strengthening the NNN interactions can drive the surface to a new
extraordinary-log phase whose main properties are dominated on the sublattices
of the surface. The special transition from the disordered phase to the
sublattice extraordinary-log phase belongs to a new universality class that is
different to the well-known special transition of the XY model.

It is well-known that the universality class of a phase transition is related
to the broken symmetry if the phase transiton is between a disordered phase
and a symmtery breaking phase. In the surface criticality, it is shown that
the universality calss of the special transition between a disordered phase
(ordinary phase) and an extraordinary-log phase is also related to the
symmetry of the model, although the extraordinary-log phase does not breaks
the symmetry (without long-range order). For example, the special points of
the classical O(2) and O(3) model belongs to different universality
classesXYlog ; Toldin2021 . In the current paper, it is obvious that the new
special point is also related to the symmetry of the sublattice extraordinary
phase, which not only has an O(2) symmetry of spins but also a $Z_{2}$
symmetry of the lattice.

Surface criticality is a hybrid of the bulk criticality and the surface
physics. In the study of the classical O($n$) model, different surface
critical behaviors are obtained by tuning the surface NN interactions;
especially, the special point is a multicritical point that comes from the
merge of the 3D criticality and the corresponding 2D criticality. For an
antiferromagnetic spin model, NNN interactions may lead to different
universality class in 2D, therefore, adding NNN interactions to the surface
will lead to the interplay between the new 2D criticality and the 3D
criticality, subsequently the surface criticality may be different. This is a
simple way to find new surface criticality; the special point between the
extraordinary-log phase and the ordered phase presented in Ref. Ding2022,  and
the new special point presented in the current paper are two examples of such
purpose. Our results can be helpful in the exploring of new surface critical
behaviors.

## Acknowledgment

C.D. is supported by the National Science Foundation of China under Grants
Number 11975024, the Anhui Provincial Supporting Program for Excellent Young
Talents in Colleges and Universities under Grant Number gxyqZD2019023. L.Z. is
supported by the National Key R&D Program (2018YFA0305800), the National
Natural Science Foundation of China under Grants Numbers 12174387 and
11804337, CAS Strategic Priority Research Program (XDB28000000) and CAS Youth
Innovation Promotion Association. W.Z. was supported by the Hefei National
Research Center for Physical Sciences at the Microscale (KF2021002).

## References

  * (1) K. Binder, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, Vol. 8 (AcademicPress, London, 1983).
  * (2) K. Binder and P. C. Hohenberg, Surface effects on magnetic phase transitions, Phys. Rev. B 9, 2194 (1974).
  * (3) Y. Deng, H.W. J. Blöte, and M. P. Nightingale, Surface and bulk transitions in three-dimensional O($n$) models, Phys. Rev. E 72,016128 (2005).
  * (4) Y. Deng, Bulk and surface phase transitions in the three dimensional O(4) spin model, Phys. Rev. E 73, 056116 (2006).
  * (5) F. P. Toldin, Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class, Phys. Rev. Lett. 126, 135701 (2021).
  * (6) M. Hu, Y. Deng, and J.-P. Lv, Extraordinary-Log Surface Phase Transition in the Three-Dimensional XY Model,Phys. Rev. Lett. 127, 120603 (2021).
  * (7) L. Zhang and F. Wang, Unconventional surface critical behavior induced by a quantum phase transition from the two-dimensional Affleck-Kennedy-Lieb-Tasaki phase to a Néel-ordered phase, Phys. Rev. Lett. 118, 087201 (2017).
  * (8) C. Ding, L. Zhang, and W. Guo, Engineering Surface Critical Behavior of (2 + 1)-Dimensional O(3) Quantum Critical Points, Phys. Rev. Lett. 120, 235701 (2018).
  * (9) L. Weber, F. Parisen Toldin, and S. Wessel, Nonordinary edge criticality of two-dimensional quantum critical magnets, Phys. Rev. B 98, 140403(R) (2018).
  * (10) L. Weber and S. Wessel, Nonordinary criticality at the edges of planar spin-1 Heisenberg antiferromagnets, Phys. Rev. B 100, 054437 (2019).
  * (11) M. A. Metlitski, Boundary criticality of the O(N) model in d=3 critically revisited, SciPost Phys. 12, 131 (2022).
  * (12) W. Zhu, C. Ding, L. Zhang, and W. Guo, Surface critical behavior of coupled Haldane chains, Phys. Rev. B 103, 024412 (2021).
  * (13) L. Weber and S. Wessel, Spin versus bond correlations along dangling edges of quantum critical magnets, Phys. Rev. B 103, L020406 (2021).
  * (14) C.-M. Jian, Y. Xu, X.-C. Wu, and C. Xu, Continuous Néel-VBS quantum phase transition in non-local one-dimensional systems with SO(3) symmetry, SciPost Phys. 10, 033 (2021).
  * (15) F. P. Toldin, M. A. Metlitski, Boundary criticality of the 3d O(N) model: from normal to extraordinary, Phys. Rev. Lett. 128, 215701 (2022).
  * (16) C. Ding, W. Zhu, W.-A. Guo, L. Zhang, Special Transition and Extraordinary Phase on the Surface of a (2+1)-Dimensional Quantum Heisenberg Antiferromagnet, arXiv:2110.04762 (2021).
  * (17) X.-J. Yu, R.-Z. Huang, H.-H. Song, L. Xu, C. Ding, L. Zhang, Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains, Phys. Rev. Lett. 129, 210601 (2022).
  * (18) W. Zhu, C. Ding, L. Zhang, W. Guo, Exotic surface behaviors induced by geometrical settings of two-dimensional dimerized quantum XXZ model, arXiv:2111.12336 (2021).
  * (19) J. Padayasi, A. Krishnan, M. A. Metlitski, I. A. Gruzberg, M. Meineri, The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap, SciPost Phys. 12, 190(2022).
  * (20) L.-R. Zhang, C. Ding, Y. Deng, and L. Zhang, Surface criticality of antiferromagnetic Potts model, Phys. Rev. B 105, 224415 (2022).
  * (21) X. Zou, S. Liu, and W. Guo, Surface critical properties of the three-dimensional clock model, Phys. Rev. B 106, 064420 (2022).
  * (22) Y. Sun, J. Lyu, and J.-P. Lv, Classical-quantum correspondence of special and extraordinary-log criticality: Villain’s bridge, Phys. Rev. B 106, 174516 (2022).
  * (23) Y. Sun and J.-P. Lv, Quantum extraordinary-log universality of boundary critical behavior, Phys. Rev. B 106, 224502 (2022)
  * (24) C. Ding, H. W. J. Böte, Y. Deng, Emergent O(n) symmetry in a series of three-dimensional Potts models, Phys. Rev. B 94, 104402 (2016).
  * (25) J. R. Heringa and H. W. J. Blöte, Geometric cluster Monte Carlo simulation, Phys. Rev. E 57, 4976(1998).
  * (26) M. N. Barber, Scaling Relations for Critical Exponents of Surface Properties of Magnets, Phys. Rev. B 8, 407 (1973).
  * (27) T. C. Lubensky and M.H. Rubin, Critical phenomena in semi-infinite systems. I. $\varepsilon$ expansion for positive extrapolation length, Phys. Rev. B 11, 4533(1975).