††thanks: Y.S. and M.H. contributed equally to this work.††thanks: Y.S. and
M.H. contributed equally to this work.

# Extraordinary-log Universality of Critical Phenomena in Plane Defects

Yanan Sun  Minghui Hu  Institute for Theoretical Physics, Anhui Normal
University, Wuhu, Anhui 241000, China  Jian-Ping Lv<EMAIL_ADDRESS>Institute for Theoretical Physics, Anhui Normal University, Wuhu, Anhui
241000, China  Key Laboratory of Functional Molecular Solids, Ministry of
Education, Wuhu, Anhui 241000, China

(August 28, 2024)

###### Abstract

There is growing evidence that extraordinary-log critical behavior emerges on
the open surfaces of critical systems in a semi-infinite geometry. Here, using
extensive Monte Carlo simulations, we observe extraordinary-log critical
behavior on the plane defects of O(2) critical systems in an infinite
geometry. In this extraordinary-log critical phase, the large-distance two-
point correlation $G$ obeys the logarithmic finite-size scaling $G\sim({\rm
ln}L)^{-\hat{q}}$ with the linear size $L$, having the critical exponent
$\hat{q}=0.29(2)$. Meanwhile, the helicity modulus $\Upsilon$ follows the
scaling form $\Upsilon\sim\alpha({\rm ln}L)/L$ with the universal parameter
$\alpha=0.56(3)$. The values of $\hat{q}$ and $\alpha$ do not fall into any
known universality class of critical phenomena, yet they conform to the
scaling relation of extraordinary-log universality. We also discuss the
extension of current results to a quantum system that is experimentally
accessible. These findings reshape our understanding of extraordinary-log
critical phenomena.

Critical phenomena have been a subject of long-standing interest, and
universality is recognized as a pillar of modern critical theory Stanley
(1999). When critical phenomena occur on lattices of special geometries Binder
and Hohenberg (1974); Ohno and Okabe (1984); Landau  _et al._ (1989); Diehl
(1997); Pleimling (2004); Deng  _et al._ (2005); Deng (2006); Dubail  _et al._
(2009); Hasenbusch (2011); Zhang and Wang (2017); Ding  _et al._ (2018); Weber
_et al._ (2018); Weber and Wessel (2019); Jian  _et al._ (2021); Metlitski
(2022), nontrivial scenarios of universality may emerge and be connected to a
wide range of modern concepts Grover and Vishwanath ; Parker  _et al._ (2018);
Liu  _et al._ (2021); Dantchev and Dietrich (2023); Cardy ; Andrei  _et al._
(2020); Yu  _et al._ (2022); Padayasi  _et al._ (2022)

The O(2) criticality is a textbook illustration of critical phenomena. Two-
dimensional (2D) and three-dimensional (3D) O(2) critical phenomena have been
found in diverse systems Svistunov  _et al._ (2015). In two dimensions, the
Kosterlitz-Thouless transition is driven by the unbinding of vortex-antivortex
pairs, and the quasi-long-range order can emerge in the low-temperature phase
Kosterlitz (2017). In three dimensions, the O(2) criticality serves as a
testbed for various techniques and theories, such as the “$\lambda$
transition” of helium which has been the subject of experiments in Earth orbit
Barmatz  _et al._ (2007).

Recently, a renormalization-group study predicted the extraordinary-log
universality (ELU) for the surface critical behavior (SCB) of the O($n$) model
with $2\leq n<n_{c}$, where $n_{c}$ is not precisely known Metlitski (2022). A
consequence of this prediction is the logarithmic scaling form of the surface
two-point correlation $g$ as a function of the distance $r$ Metlitski (2022),
which can be reexpressed by the finite-size scaling (FSS) of the large-
distance correlation $G=g(r\rightarrow\infty)$ as

$G\sim({\rm ln}L)^{-\hat{q}},$ (1)

where the exponent $\hat{q}$ depends on $n$, and $L$ denotes the linear system
size. It was also proposed that the helicity modulus $\Upsilon$,
characterizing the response against a twist in boundary conditions Fisher  _et
al._ (1973), scales as

$\Upsilon\sim\frac{2\alpha}{L}({\rm ln}L),$ (2)

where $\alpha$ is a universal renormalization-group parameter. Besides, a
scaling relation for $\hat{q}$ and $\alpha$ is given by Metlitski (2022)

$\hat{q}=\frac{n-1}{2\pi\alpha}.$ (3)

For the SCB, the existence of extraordinary-log critical phase and scaling
relation (3) were confirmed at $n=3$ Parisen Toldin (2021) and $n=2$ Hu  _et
al._ (2021). For $n=2$, estimates of $\hat{q}$ and $\alpha$ were also obtained
from distinct contexts of O(2) criticality Parisen Toldin and Metlitski
(2022); Zhang  _et al._ (2022); Zou  _et al._ (2022); Sun and Lv (2022)111In
the reference [Phys. Rev. B 106, 064420 (2022)], the exponent $\hat{q}$ was
also estimated as $0.60(3)$ and $0.59(3)$, while the universal parameter
$\alpha$ was also estimated as $0.24(4)$ and $0.30(3)$. (Table 1). Hence, the
long-standing contradiction about the extraordinary phases of SCB in $XY$ and
Heisenberg models has been reconciled Deng  _et al._ (2005); Metlitski (2022).

Figure 1: A three-dimensional lattice in infinite geometry with a plane defect
(left panel) and the phase diagram for plane defect of Villain model (right
panel). The $W$ and $K$ axes, which represent interactions inside and outside
the plane defect, respectively, span the phase diagram. The phase diagram
includes the quasi-long-range ordered, ordered, and disordered phases. It also
contains, at the bulk critical point $K_{c}$, the plane-defect critical
phenomena in bulk O(2) and extraordinary-log critical universality classes,
for $W=K_{c}$ and $W>K_{c}$, respectively. A critical line of Kosterlitz-
Thouless-like (KT-like) transition is also included.

It is unknown whether the ELU may be achieved in a wider context of critical
systems beyond SCB, despite the recent advances Metlitski (2022); Parisen
Toldin (2021); Hu  _et al._ (2021); Parisen Toldin and Metlitski (2022); Zhang
_et al._ (2022); Zou  _et al._ (2022); Sun and Lv (2022). Given that the SCB
is associated with systems in semi-infinite geometries, this question is
essential. In the context of O(2) critical systems, where plane defects are
present (Fig. 1), we address the problem. We demonstrate the presence of ELU
in cases where plane defects have strong intra-plane coupling. The phase
diagram shown in Fig. 1 illustrates the main findings.

Table 1: For the surface and plane-defect critical phenomena of O($n$)
critical systems in semi-infinite and infinite geometries, respectively, two
classes of extraordinary-log universality have been identified. The critical
exponent $\hat{q}$ and the universal parameter $\alpha$ quantitatively
characterize the extraordinary-log universality. Surface critical phenomena
---
Bulk universality |  model |  $\hat{q}$ |  $\alpha$ |  year
O(3) |  O(3) $\phi^{4}$ Parisen Toldin (2021) |  2.1(2) |  0.15(2) |  2020
O(3) $\phi^{4}$ Parisen Toldin and Metlitski (2022) |  – |  0.190(4) |  2021
O(2) |  $XY$ Hu  _et al._ (2021) |  0.59(2) |  0.27(2) |  2021
O(2) $\phi^{4}$ Parisen Toldin and Metlitski (2022) |  – |  0.300(5) |  2021
Potts Zhang  _et al._ (2022) |  0.60(2) |  – |  2022
clock Zou  _et al._ (2022) |  0.59(1) |  0.26(2) |  2022
Villain Sun  _et al._ (2022) |  0.58(2) |  0.28(1) |  2022
Plane-defect critical phenomena ( this work)
Bulk universality |  model |  $\hat{q}$ |  $\alpha$ |  year
O(2) |  Villain |  0.29(2) |  0.56(3) |  2023
$XY$

We start with a plane-defect Villain model with the Hamiltonian

$\mathcal{H}=\frac{1}{2}\sum^{\Delta\mathcal{J}=0}_{\bf\langle
rr^{\prime}\rangle}\frac{\mathcal{J}^{2}_{{\bf rr^{\prime}}}}{C_{{\bf
rr^{\prime}}}}$ (4)

on simple-cubic lattices, where $C_{\bf rr^{\prime}}$ is a variable for the
pairs of nearest-neighbor sites ${\bf r}$ and ${\bf r^{\prime}}$.
$\mathcal{J}_{{\bf rr^{\prime}}}\in\\{0,\pm 1,\pm 2,...\\}$ parameterizes the
directed flows along bonds. As in the standard Villain model Cha  _et al._
(1991); Wallin  _et al._ (1994); Alet and Sørensen (2003); Šmakov and Sørensen
(2005); Witczak-Krempa  _et al._ (2014); Chen  _et al._ (2014), the flows are
non-divergent and constitute closed directed loops. Periodic boundary
conditions are imposed in each of the [100] ($x$), [010] ($y$), and [001]
($z$) directions. To involve a plane defect, we specify a plane that is
perpendicular to $z$ direction and contains $L^{2}$ sites. The values of
$C_{\bf rr^{\prime}}$ obey

$C_{\bf rr^{\prime}}=\left\\{\begin{array}[]{cc}W&\,\,{\bf r}\,\,{\rm
and}\,\,{\bf r^{\prime}}\in{\rm plane\,\,defect},\\\ K&\,\,{\bf r}\,\,{\rm
or}\,\,{\bf r^{\prime}}\notin{\rm plane\,\,defect}.\end{array}\right.$ (5)

By formulating a worm Monte Carlo algorithm based on the original algorithm in
Ref. Prokof’ev and Svistunov (2001), we simulate model (4). We design two
update schemes, which are described in Appendix A, based on biased random
walks in an extended state space involving the source and sink of flows. While
an update scheme can run on the entire simple-cubic lattice, the other update
scheme only works in the plane defect. The latter scheme allows the sampling
of some quantities that characterize plane-defect critical phenomena.

Figure 2: The two-point correlation $G$ (a), the scaled susceptibility
$\chi_{s}L^{-2}$ (b) and the scaled helicity modulus $\Upsilon L$ (c) for the
extraordinary-log critical phase of the plane-defect Villain model. In panels
(a) and (b), the horizontal coordinates are written as ${\rm ln}(L/l_{0})$,
where $l_{0}$ are taken from preferred least-squares fits, and the plots are
further made on log-log scales. The slopes $-0.29$ and $0.56$ of dashed lines
stand for $-\hat{q}$ and $\alpha$, respectively.

We fix $K$ at $K_{c}=0.333\,067\,04$, the Villain model’s bulk critical point
that was previously estimated by two of us and our colleagues Xu  _et al._
(2019). With $W=0.5$, $1$, $3$ and $7$, we extensively simulate the strong-$W$
regime of plane defect. The system size $L$ ranges from $4$ to $128$. At
$L=128$, the total number of generated closed loops for each $W$ in the worm-
algorithm simulations is about $2.9\times 10^{10}$. In the plane-defect update
scheme, we extract the two-point correlation $g(\delta x,\delta y)$
[$g(0,0)\equiv 1$] unbiasedly from the distribution of the distances $(\delta
x,\delta y)$ between source and sink. We define the large-distance two-point
correlation $G$ by

$G=[g(0,L/2)+g(L/2,0)]/2.$ (6)

If Eq. (1) holds, a fitting ansatz of $G$ reads

$G=a_{0}[{\rm ln}(L/l_{0})]^{-\hat{q}},$ (7)

with $a_{0}$ a non-universal constant and $l_{0}$ a reference length.
Throughout this paper, we test scaling ansatzes against Monte Carlo data by
least-squares fits. We monitor the evolution of $\chi^{2}$ with the minimum
size $L_{\rm min}$ involved in fitting. In principle, one searches the
smallest $L_{\rm min}$ relating to the $\chi^{2}$ per degree of freedom (DoF)
$\chi^{2}/{\rm DoF}=\mathcal{O}(1)$, which does not decrease drastically upon
further increasing $L_{\rm min}$. Practically one prefers the fits with
$\chi^{2}/{\rm DoF}\approx 1$. We should not trust a single fit—conclusions
will be made by comparing all preferred fits. For each $W$, we find that the
estimate of $\hat{q}$ extrapolates to $\hat{q}\approx 0.29$. More precisely,
for $W=0.5$, $1$, $3$ and $7$, we obtain $\hat{q}=0.308(2)$, $0.301(2)$,
$0.289(5)$ and $0.28(1)$ as well as $\chi^{2}/{\rm DoF}\approx 2.8$, $0.1$,
$2.8$ and $1.5$ respectively, with $L_{\rm min}=16$. Based on these
observations, by fixing $\hat{q}=0.29$, we obtain $l_{0}=3.15(2)$, $0.684(2)$,
$0.0153(2)$ and $0.0000100(3)$ as well as $\chi^{2}/{\rm DoF}\approx 2.4$,
$1.4$, $0.8$ and $0.7$, with $L_{\rm min}=48$, $32$, $64$ and $48$, for
$W=0.5$, $1$, $3$ and $7$, respectively. Details of fits are given in Appendix
B. Notice that the estimates of $\hat{q}$ from various $W$ ($W>K_{c}$) are
consistent, indicating the universality of the logarithmic FSS form (7).

In the plane-defect update scheme, the susceptibility $\chi_{s}$ of plane
defect is sampled via $\chi_{s}=\langle n_{s}\rangle$, where $n_{s}$ is the
number of the movements of source and sink between consecutive hits to the
state space of model (4). Borrowing the insights into FSS from the ELU of SCB,
we have the scaling formula

$\chi_{s}=a_{1}L^{2}[{\rm ln}(L/l_{0})]^{-\hat{q}},$ (8)

with $a_{1}$ a non-universal constant. For $W=0.5$, the fit with $L_{\rm
min}=16$ yields $\hat{q}=0.322(1)$ having a huge $\chi^{2}/{\rm DoF}$
($\chi^{2}/{\rm DoF}\approx 7.0$), which reduces to $\chi^{2}/{\rm DoF}\approx
2.7$ with $L_{\rm min}=32$ and $\hat{q}=0.309(3)$. For $W=1$, $3$ and $7$, we
obtain $\hat{q}=0.310(1)$, $0.295(3)$ and $0.29(1)$ as well as $\chi^{2}/{\rm
DoF}\approx 1.7$, $3.2$ and $3.0$ respectively, with $L_{\rm min}=16$.
Further, when $\hat{q}=0.29$ is fixed, we obtain $l_{0}=4.08(3)$, $0.866(3)$,
$0.01909(9)$ and $0.0000119(3)$ as well as $\chi^{2}/{\rm DoF}\approx 1.1$,
$0.9$, $1.4$ and $0.3$, with $L_{\rm min}=64$, $48$, $32$ and $48$, for
$W=0.5$, $1$, $3$ and $7$, respectively. Similar to that found for $G$, as $W$
increases from $0.5$ to $7.0$, $l_{0}$ decreases by several orders of
magnitude. Meanwhile, the estimates of $\hat{q}$ from $\chi_{s}$ are close to
those from $G$. These observations confirm the scaling formula (8), which
corresponds to a logarithmic scaling of two-point correlation.

From the FSS analyses of $G$ and $\chi_{s}$, we estimate the value of
$\hat{q}$ as $\hat{q}=0.29(2)$. We plot $G$ and $\chi_{s}L^{-2}$ versus ${\rm
ln}(L/l_{0})$ in Figs. 2(a) and (b) respectively, where $l_{0}$ takes above-
mentioned values from preferred fits. The mutually consistent scaling forms
and critical exponents are hence illustrated.

We analyze the helicity modulus $\Upsilon$, which is defined through the
fluctuations of winding number $\mathcal{W}_{x}$ of directed flows along a
periodic direction (say $x$ direction),

$\Upsilon=\langle\mathcal{W}_{x}^{2}\rangle/L.$ (9)

As indicated in Ref. Sun  _et al._ (2022), such an estimator of $\Upsilon$ is
effective in the analyses of ELU. For an extraordinary-log critical phase,
$\Upsilon$ is expected to scale as $\Upsilon L\propto{\rm ln}L$. This behavior
is roughly illustrated by the Monte Carlo data of $\Upsilon$ in Fig. 2(c).
Further, we perform least-squares fits to

$\leavevmode\nobreak\ \Upsilon L=\alpha({\rm ln}L)+b+cL^{-\omega},$ (10)

where $\alpha$ is a universal parameter, whereas $b$ and $c$ are non-
universal. Unlike the scaling form $\Upsilon L\sim 2\alpha({\rm ln}L)$ [Eq.
(2)], which applies to the SCB that involves two open surfaces, the prefactor
$2$ is now removed due to the uniqueness of plane defect. $\omega$ denotes the
exponent of leading finite-size corrections. Accordingly, we estimate $\alpha$
by the least-squares fits to Eq. (10). We adopt $\omega=0.789$, considering
bulk irrelevant fields Guida and Zinn-Justin (1998); Hasenbusch (2019), and
compare the fits with those using $\omega=1$. We find that the inclusion of
correction term stabilizes the fitting. For $W=0.5$, $1$, $3$ and $7$, we
obtain $\alpha=0.555(3)$, $0.562(4)$, $0.580(6)$ and $0.57(1)$ as well as
$\chi^{2}/{\rm DoF}\approx 0.2$, $0.7$, $2.0$ and $2.2$ respectively, with
$L_{\rm min}=16$. Comparing the fits, we estimate $\alpha=0.56(3)$.

There is a lack of a known critical universality class that the values of
$\hat{q}$ and $\alpha$ fall into. The scaling relation between $\hat{q}$ and
$\alpha$ is then investigated. We test the scaling relation (3) with $n=2$,
which was previously verified merely for SCB (Table 1). Figure 3 displays the
above-obtained fitting results of $\hat{q}$ and $\alpha$ versus $\chi^{2}/{\rm
DoF}$. Next, from each of the estimates of $\hat{q}$ and $\alpha$, via the
equation $\alpha\hat{q}=1/(2\pi)$, we obtain an estimate of $\alpha$ and
$\hat{q}$, respectively. The estimates converted through the equation are also
included in Fig. 3. We note that the results of $\hat{q}$ and $\alpha$ from
direct fits and conversions are consistent. For $\chi^{2}/{\rm DoF}\approx 1$,
such consistency is apparent. Hence, we obtain strong evidence of the scaling
relation $\alpha\hat{q}=1/(2\pi)$ for an ELU of plane-defect critical
phenomena.

Figure 3: Test of the scaling relation $\alpha\hat{q}=1/(2\pi)$ for the
extraordinary-log critical phase of the plane-defect Villain model. Circles
denote the results from least-squares fits to FSS formulae, whereas squares
denote the results converted from fitting results via
$\alpha\hat{q}=1/(2\pi)$. The estimated critical exponent $\hat{q}=0.29(2)$
and universal parameter $\alpha=0.56(3)$ are denoted by the shadows centered
at the red and black lines, respectively.

We have demonstrated that ELU exists in a strong-$W$ regime with $K=K_{c}$.
One also expects the occurrence of a transition at $W=K_{c}$ and $K=K_{c}$ in
the 3D O(2) universality class.

We proceed to investigate critical phenomena on the plane defect for
$K<K_{c}$. Figure 4(a) shows $\Upsilon L$ versus $W$ at $K=0.2$. For
$W\gtrapprox 0.7$, $\Upsilon L$ tends to be independent of $L$ in the
$L\rightarrow\infty$ limit and extrapolates to a $W$-dependent non-trivial
value. These observations indicate a critical phase for the plane defect. As
shown in Fig. 4(b), at $W\approx 0.73$, $G$ scales as $G\propto L^{-\eta}$
with $\eta=1/4$, which is reminiscent of the anomalous dimension of the
Kosterlitz-Thouless transition. It is natural to ask whether there is a
critical phase that features quasi-long-range order with $0<\eta<1/4$. Figure
4(c) demonstrates, with $W=1$, that $\Upsilon L$ is invariant for $K<K_{c}$.
Meanwhile, the exponent $\eta$ is nearly invariant ($\eta\approx 0.16$) in the
parameter regime [Fig. 4(d)]. Our results suggest that, as $K\rightarrow 0$,
the plane-defect criticality reduces drastically to an essential 2D behavior.
The phase diagram, shown in Fig. 1, is then established. It calls for a
sophisticated renormalization-group study to reveal the distribution and
properties of fixed points.

Figure 4: Critical phenomena of the plane-defect Villain model with $K<K_{c}$.
The scaled helicity modulus $\Upsilon L$ (a) and the scaled two-point
correlation $GL^{1/4}$ (b) versus $W$ for $K=0.2$. $\Upsilon L$ (c) and
$GL^{0.16}$ (d) versus $K$ for $W=1.0$. Dashed lines represent phase
transition points.

The relevance of O(2) criticality to various systems of interest determines
its significance Svistunov  _et al._ (2015). One may ask whether the current
findings hold true for a different O(2) critical system. Here, we study a
plane-defect critical $XY$ model and investigate the extraordinary-log
critical phase. The model Hamiltonian reads

$\mathcal{H}=-{\sum_{\langle{\bf r}{\bf r^{\prime}}\rangle}}C_{{\bf r}{\bf
r^{\prime}}}\vec{S}_{\bf r}\cdot\vec{S}_{\bf r^{\prime}}\;,$ (11)

where $\vec{S}_{\bf r}=(S_{\bf{r}}^{a},S_{\bf{r}}^{b})$ represents 2D vectors
of unit length. As defined by Eq. (5), $C_{{\bf r}{\bf r^{\prime}}}$ denotes
the coupling strength for sites ${\bf r}$ and ${\bf r^{\prime}}$. We set
$K=K_{c}$, where $K_{c}=1/2.2018441$ is the bulk critical point obtained in
Ref. Xu  _et al._ (2019). We simulate the model with $W=1$, $3$ and $7$ up to
the linear size $L=128$, using a cluster Monte Carlo algorithm Wolff (1989).
We sample the helicity modulus $\Upsilon=\frac{1}{L^{3}}(\langle
E\rangle-\langle T^{2}\rangle)$ with $E=\sum_{\bf{r}}C_{{\bf
r}({\bf{r}+\bf{e}_{x}})}\vec{S}_{\bf{r}}\cdot\vec{S}_{\bf{r}+\bf{e}_{x}}$ and
$T=\sum_{\bf{r}}C_{{\bf
r}({\bf{r}+\bf{e}_{x}})}(S_{\bf{r}}^{a}S_{\bf{r}+\bf{e}_{x}}^{b}-S_{\bf{r}}^{b}S_{\bf{r}+\bf{e}_{x}}^{a})$,
as well as the two-point correlation
$G=\langle\vec{S}_{\bf{r}}\cdot\vec{S}_{\bf{r^{\prime}}}\rangle$ with
$\bf{r}^{\prime}-\bf{r}$ $=(L/2,0)$. For $G$ and $\Upsilon$, we perform FSS
analyses according to Eqs. (7) and (10), respectively. Figure 5 illustrates
the estimates of $\hat{q}$ and $\alpha$, which are compatible with those of
the plane-defect Villain model. A specially designed $\chi^{2}$ test for
scaling relation of $\hat{q}$ and $\alpha$ as well as the details of fits are
presented in Appendix C.

Figure 5: The two-point correlation $G$ (a) and the scaled helicity modulus
$\Upsilon L$ (b) for the extraordinary-log critical phase of the plane-defect
$XY$ model. In panel (a), the horizontal coordinate is written as ${\rm
ln}(L/l_{0})$, where $l_{0}=1.240$ ($W=1$), $0.0219$ ($W=3$) and $0.0000146$
($W=7$) are taken from preferred least-squares fits, and the plot is further
made on a log-log scale. The slopes $-0.29$ and $0.56$ of dashed lines stand
for $-\hat{q}$ and $\alpha$, respectively.

In the context of O(2) critical systems, we find that the ELU exists in plane
defects of infinite-geometry lattices. The conclusion is based on extensive
Monte Carlo simulations of two models in a broad strong-coupling regime of
plane defects, where the extraordinary-log critical phase features logarithmic
scaling forms of two-point correlation and helicity modulus. The critical
exponent $\hat{q}=0.29(2)$ and the universal parameter $\alpha=0.56(3)$
quantitatively characterize the scaling forms. Both $\hat{q}$ and $\alpha$ are
far from those of SCB, indicating the existence of a new critical universality
class. Surprisingly, the scaling relation (3) is likely to be valid. As a
result, the current findings strongly suggest that ELU may exist in a wider
range of many-body systems than previously thought. In comparison to SCB, the
$\hat{q}$ value of plane-defect criticality is relatively small, leaving more
room for the discovery of ELU in O($n$) models with $n>2$.

Plane defects in O(2) critical bulks could serve as a platform for exotic
critical phenomena, inspiring numerous follow-up studies. Here, we limit
ourselves to quantum ELU. Theoretically, a model of quantum ELU is precious.
On the experimental side, a realization of ELU is awaited. A close classical-
quantum mapping exists between the plane-defect Villain model and the quantum
rotor and Bose-Hubbard models. On this basis, a possible pathway to realizing
quantum ELU is the 2D quantum Hamiltonian $\hat{\mathcal{H}}=-\sum_{\bf\langle
rr^{\prime}\rangle}t_{\bf rr^{\prime}}({\hat{\Phi}}_{\bf
r}^{\dagger}{\hat{\Phi}}_{\bf r^{\prime}}+{\hat{\Phi}}_{\bf
r}{\hat{\Phi}}_{\bf r^{\prime}}^{\dagger})+\frac{U}{2}\sum_{\bf
r}{\hat{n}}_{\bf r}^{2}$, where ${\hat{\Phi}}_{\bf r}^{\dagger}$,
${\hat{\Phi}}_{\bf r}$ and ${\hat{n}}_{\bf r}$ are respectively the bosonic
creation, annihilation and particle number operators at site ${\bf r}$, $U$
represents onsite repulsion, and $t_{\bf rr^{\prime}}$ denotes the bond-
dependent hopping strength realizing an edge defect. Recent advances in
quantum simulations with ultra-cold bosons in optical lattices Greiner  _et
al._ (2002); Baier  _et al._ (2016); Yang  _et al._ (2020) enable the
experimental realization of quantum ELU.

While we were finalizing this paper, an independent work appeared on the ArXiv
Krishnan and Metlitski . Based on a renormalization-group theory, this work
predicts the existence of ELU for a plane defect in the O($n$) model.

###### Acknowledgements.

This work has been supported by the National Natural Science Foundation of
China (Grant No. 12275002).

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##  Appendix A Details of worm Monte Carlo algorithm

The plane-defect Villain model is simulated by an unbiased worm algorithm in
an extended state space of directed flows. The worm algorithm depends on a
biased random walk of source-sink points [usually referred to as Ira ($I$) and
Masha ($M$)], which obeys the detailed balance. A core part of the simulations
relies on iteratively carrying out two update schemes, Algorithm 1 and
Algorithm 2, which run in the whole simple-cubic lattice and the plane defect,
respectively.

Once the source-sink points collide ($I=M$), a closed loop of directed flows
is superposed on the directed-flow configuration. After producing a permanent
quantity of closed loop(s), the update process shifts between Algorithm 1 and
Algorithm 2. Even if Algorithm 1 itself guarantees ergodicity, we include
Algorithm 2 for the efficient simulations in plane defect as well as the
sampling of some quantities that characterize plane-defect critical phenomena.

Algorithm 1 Update in simple-cubic lattice

1\. If $I=M$, randomly pick up a lattice site $I^{\prime}$ in the simple-cubic
lattice, with the probability $1/L^{3}$ for each site. Let $I=M=I^{\prime}$,
${\rm sign}(I)=1$, ${\rm sign}(M)=-1$.

2\. Interchange $I\leftrightarrow M$ and ${\rm sign}(I)\leftrightarrow{\rm
sign}(M)$ with the probability $1/2$.

3\. Randomly pick up a neighbor $I_{n}$ of $I$, with the probability $1/6$ for
each of the neighbors.

4\. Propose to simultaneously move $I\rightarrow I_{n}$ and update the flow
$\mathcal{J}_{II_{n}}$ to $\mathcal{J}^{\prime}_{II_{n}}$:
$\mathcal{J}^{\prime}_{II_{n}}=\mathcal{J}_{II_{n}}+{\rm d_{f}}(I\rightarrow
I_{n}){\rm sign}(I),$ where ${\rm d_{f}}(I\rightarrow I_{n})=\pm 1$ is a fixed
parameter, quantifying the direction of flow along bond-$II_{n}$.

5\. Accept the proposed change with the probability
$p=\left\\{\begin{array}[]{cc}\min[1,e^{\frac{-(\mathcal{J}^{\prime
2}_{II_{n}}-\mathcal{J}^{2}_{II_{n}})}{2W}}]&\,\,I\,\,{\rm
and}\,I_{n}\,\in{\rm plane\,\,defect},\\\
\min[1,e^{\frac{-(\mathcal{J}^{\prime
2}_{II_{n}}-\mathcal{J}^{2}_{II_{n}})}{2K}}]&\,\,I\,\,{\rm
or}\,I_{n}\,\notin{\rm plane\,\,defect}.\end{array}\right.$

Algorithm 2 Update in plane defect

1\. If $I=M$, randomly pick up a lattice site $I^{\prime}$ in the plane
defect, with the probability $1/L^{2}$ for each site. Let $I=M=I^{\prime}$,
${\rm sign}(I)=1$, ${\rm sign}(M)=-1$.

2\. The same as Algorithm 1.

3\. Randomly pick up a neighbor $I_{n}$ of $I$ in the plane defect, with the
probability $1/4$ for each of the neighbors.

4\. The same as Algorithm 1.

5\. Accept the proposed change with the probability
$p=\min[1,e^{\frac{-(\mathcal{J}^{\prime
2}_{II_{n}}-\mathcal{J}^{2}_{II_{n}})}{2W}}].$

##  Appendix B Extraordinary-log critical phase of the plane-defect Villain
model

We simulate the extraordinary-log critical phase of plane-defect Villain model
using the worm algorithm, with the lattice sizes $L=4$, $8$, $16$, $32$, $48$,
$64$, $96$ and $128$. The number of generated closed loops ranges from
$9.1\times 10^{8}$ to $7.3\times 10^{9}$ for $L\leq 32$ and ranges from
$1.1\times 10^{10}$ to $2.9\times 10^{10}$ for $48\leq L\leq 128$. The first
one sixth of the generated closed loops are utilized for the thermalization in
each Markov chain.

Here, we analyze the FSS for the extraordinary-log critical phase with
$W=0.5$, $1$, $3$ and $7$ at $K=K_{c}=0.333\,067\,04$. According to a standard
criterion, we prefer the fits with $\chi^{2}/{\rm DoF}\approx 1$ and conclude
by comparing the fits that are stable against gradually increasing $L_{\rm
min}$, which is the size of the minimum system involved. To supplement the
presentation in the main text, we provide more details about the FSS analyses
of $G$, $\chi_{s}$, and $\Upsilon$. The data of these quantities are fitted to
the scaling formulae

$G=a_{0}[{\rm ln}(L/l_{0})]^{-\hat{q}},$ (S1) $\chi_{s}=a_{1}L^{2}[{\rm
ln}(L/l_{0})]^{-\hat{q}},$ (S2)

and

$\Upsilon L=\alpha({\rm ln}L)+b+cL^{-\omega},$ (S3)

respectively, where $\hat{q}$ and $\alpha$ are universal parameters, and
$\omega$ is a correction exponent. $l_{0}$, $a_{0}$, $a_{1}$, $b$ and $c$
represent non-universal constants. The results of least-squares fits are
presented in Tables S1, S2 and S3, for $G$, $\chi_{s}$ and $\Upsilon$,
respectively.

In addition to the quantities discussed in the main text, we take into
consideration the two-point correlation $G^{\prime}$ for the distances
$(\delta_{x},\delta_{y})=(L/4,0)$ and $(\delta_{x},\delta_{y})=(0,L/4)$, which
is defined by

$G^{\prime}=[g(0,L/4)+g(L/4,0)]/2.$ (S4)

For the extraordinary-log critical phase at $W=0.5$, $1$, $3$ and $7$, we
perform FSS analyses according to

$G^{\prime}=a_{0}[{\rm ln}(L/l_{0})]^{-\hat{q}}.$ (S5)

The results of fits are summarized in Table S4. Using the results of $l_{0}$
from preferred fits, we plot $G^{\prime}$ versus ${\rm ln}(L/l_{0})$ in Fig.
S1. The leading FSS behavior of $G^{\prime}$ is similar to that of $G$.

##  Appendix C Extraordinary-log critical phase of the plane-defect $XY$ model

We simulate the extraordinary-log critical phase of plane-defect $XY$ model
using the Wolff cluster algorithm, with the lattice sizes $L=8$, $16$, $32$,
$48$, $64$, $96$ and $128$. The total number of Wolff steps ranges from
$1.6\times 10^{8}$ to $6.4\times 10^{8}$ for $L\leq 32$ and ranges from
$2.0\times 10^{8}$ to $9.6\times 10^{8}$ for $48\leq L\leq 128$. The first one
sixth of the Wolff Monte Carlo steps are utilized for the thermalization in
each Markov chain.

We consider the coupling strengths $W=1$, $3$ and $7$ of plane defect at
$K=K_{c}=1/2.2018441$. The data for $G$ and $\Upsilon$ are fitted according to
Eqs. (S1) and (S3), respectively. Tables S5 and S6 provide a summary of the
outcomes of least-squares fits. Using the results of $\hat{q}$ and $\alpha$
from least-squares fits, a test of the scaling relation
$\alpha\hat{q}=1/(2\pi)$ is given by Fig. S2.

Figure S1: Two-point correlation $G^{\prime}$ versus ${\rm ln}(L/l_{0})$ for
the plane-defect Villain model, where $l_{0}$ are taken from preferred least-
squares fits. The plot is further made on a log-log scale. The slope $-0.29$
of dashed lines stands for $-\hat{q}$. Figure S2: Test of the scaling relation
$\alpha\hat{q}=1/(2\pi)$ for the extraordinary-log critical phase of the
plane-defect $XY$ model. Circles denote the results from least-squares fits to
FSS formulae, whereas squares denote the results converted from fitting
results via $\alpha\hat{q}=1/(2\pi)$. The estimated critical exponent
$\hat{q}=0.29(2)$ and universal parameter $\alpha=0.56(3)$ are denoted by the
shadows centered at the red and black lines, respectively. Table S1: Fits of
the two-point correlation $G$ to $G=a_{0}[{\rm ln}(L/l_{0})]^{-\hat{q}}$ for
the extraordinary-log critical phase of plane-defect Villain model at $W=0.5$,
$1$, $3$ and $7$. We investigate both cases where $\hat{q}$ is free and fixed
at $0.29$.

$W$ | $L_{\rm min}$ | $L_{\rm max}$ |  $\chi^{2}$/DoF | $a_{0}$ | $l_{0}$ | $\hat{q}$
---|---|---|---|---|---|---
0.5 | 4 | 128 | 3119.51/5 | 0.5644(5) | 1.535(4) | 0.3728(5)
8 | 114.42/4 | 0.5116(8) | 2.22(1) | 0.3247(9)
16 | 8.48/3 | 0.494(2) | 2.59(4) | 0.308(2)
32 | 7.97/2 | 0.490(5) | 2.7(1) | 0.304(5)
48 | 4.72/1 | 0.47(1) | 3.3(4) | 0.29(1)
4 | 47488.48/6 | 0.49014(3) | 2.2704(8) | 0.29
8 | 2223.24/5 | 0.47999(5) | 2.815(3) | 0.29
16 | 125.62/4 | 0.47685(8) | 3.016(5) | 0.29
32 | 17.53/3 | 0.4756(1) | 3.10(1) | 0.29
48 | 4.85/2 | 0.4750(2) | 3.15(2) | 0.29
64 | 4.30/1 | 0.4748(3) | 3.16(3) | 0.29
1.0 | 4 | 128 | 345.88/5 | 1.001(1) | 0.418(3) | 0.3241(5)
8 | 34.02/4 | 0.968(2) | 0.500(6) | 0.3105(9)
16 | 0.34/3 | 0.945(4) | 0.57(1) | 0.301(2)
32 | 0.14/2 | 0.94(1) | 0.59(4) | 0.299(5)
48 | 0.0001/1 | 0.93(3) | 0.6(1) | 0.30(1)
4 | 5471.78/6 | 0.92571(5) | 0.6099(4) | 0.29
8 | 649.07/5 | 0.92218(7) | 0.6463(7) | 0.29
16 | 42.88/4 | 0.9198(1) | 0.674(1) | 0.29
32 | 4.16/3 | 0.9189(2) | 0.684(2) | 0.29
48 | 0.23/2 | 0.9186(3) | 0.689(3) | 0.29
64 | 0.04/1 | 0.9184(4) | 0.690(5) | 0.29
3.0 | 4 | 128 | 9.97/5 | 1.518(6) | 0.0138(5) | 0.295(1)
8 | 9.36/4 | 1.51(1) | 0.0143(8) | 0.293(2)
16 | 8.47/3 | 1.49(2) | 0.016(2) | 0.289(5)
32 | 0.09/2 | 1.40(3) | 0.030(7) | 0.268(8)
48 | 0.09/1 | 1.39(7) | 0.03(2) | 0.27(2)
4 | 21.33/6 | 1.4974(1) | 0.01546(4) | 0.29
8 | 11.58/5 | 1.4970(2) | 0.01557(5) | 0.29
16 | 8.49/4 | 1.4967(3) | 0.01567(8) | 0.29
32 | 6.86/3 | 1.4964(3) | 0.0158(1) | 0.29
48 | 1.60/2 | 1.4973(5) | 0.0155(2) | 0.29
64 | 0.81/1 | 1.4978(8) | 0.0153(2) | 0.29
7.0 | 4 | 128 | 4.56/5 | 1.89(3) | 0.000025(5) | 0.273(4)
8 | 4.50/4 | 1.88(5) | 0.00003(1) | 0.272(7)
16 | 4.38/3 | 1.91(8) | 0.00002(1) | 0.28(1)
32 | 1.22/2 | 1.6(1) | 0.0002(2) | 0.24(2)
48 | 0.94/1 | 1.8(3) | 0.0001(2) | 0.26(5)
4 | 18.97/6 | 2.0108(3) | 0.0000109(1) | 0.29
8 | 10.24/5 | 2.0118(5) | 0.0000106(1) | 0.29
16 | 5.71/4 | 2.0125(6) | 0.0000104(2) | 0.29
32 | 5.61/3 | 2.0127(8) | 0.0000103(2) | 0.29
48 | 1.30/2 | 2.014(1) | 0.0000100(3) | 0.29
64 | 0.31/1 | 2.015(2) | 0.0000096(4) | 0.29

Table S2: Fits of the susceptibility $\chi_{s}$ to $\chi_{s}=a_{1}L^{2}[{\rm
ln}(L/l_{0})]^{-\hat{q}}$ for the extraordinary-log critical phase of plane-
defect Villain model at $W=0.5$, $1$, $3$ and $7$.

$W$ | $L_{\rm min}$ | $L_{\rm max}$ |  $\chi^{2}$/DoF | $a_{1}$ | $l_{0}$ | $\hat{q}$
---|---|---|---|---|---|---
0.5 | 4 | 128 | 11202.68/5 | 0.6290(5) | 1.354(3) | 0.4267(4)
8 | 732.74/4 | 0.5423(7) | 2.252(9) | 0.3550(6)
16 | 21.14/3 | 0.507(1) | 2.95(3) | 0.322(1)
32 | 5.31/2 | 0.493(4) | 3.3(1) | 0.309(3)
48 | 3.76/1 | 0.483(8) | 3.7(3) | 0.299(8)
4 | 215291.53/6 | 0.49828(2) | 2.4563(5) | 0.29
8 | 15937.48/5 | 0.48279(4) | 3.348(2) | 0.29
16 | 866.04/4 | 0.47686(6) | 3.780(4) | 0.29
32 | 38.33/3 | 0.4746(1) | 3.972(8) | 0.29
48 | 5.28/2 | 0.4738(2) | 4.04(1) | 0.29
64 | 1.10/1 | 0.4734(3) | 4.08(3) | 0.29
1.0 | 4 | 128 | 2091.94/5 | 1.077(1) | 0.356(2) | 0.3543(5)
8 | 198.77/4 | 1.007(2) | 0.503(4) | 0.3269(7)
16 | 5.21/3 | 0.964(3) | 0.64(1) | 0.310(1)
32 | 3.18/2 | 0.952(9) | 0.69(4) | 0.305(4)
48 | 1.51/1 | 0.93(2) | 0.8(1) | 0.295(8)
4 | 27600.22/6 | 0.92927(4) | 0.7105(4) | 0.29
8 | 3487.84/5 | 0.92361(5) | 0.7773(6) | 0.29
16 | 244.87/4 | 0.91951(9) | 0.832(1) | 0.29
32 | 20.79/3 | 0.9178(1) | 0.857(2) | 0.29
48 | 1.85/2 | 0.9172(2) | 0.866(3) | 0.29
64 | 1.85/1 | 0.9172(3) | 0.866(5) | 0.29
3.0 | 4 | 128 | 146.26/5 | 1.661(6) | 0.0076(2) | 0.323(1)
8 | 25.48/4 | 1.581(9) | 0.0117(5) | 0.308(2)
16 | 9.46/3 | 1.52(2) | 0.016(1) | 0.295(3)
32 | 0.47/2 | 1.44(3) | 0.027(5) | 0.277(6)
48 | 0.41/1 | 1.45(6) | 0.03(1) | 0.28(1)
4 | 1088.78/6 | 1.5013(1) | 0.01733(3) | 0.29
8 | 138.99/5 | 1.4984(1) | 0.01834(5) | 0.29
16 | 12.08/4 | 1.4967(2) | 0.01894(7) | 0.29
32 | 4.14/3 | 1.4963(2) | 0.01909(9) | 0.29
48 | 0.93/2 | 1.4967(3) | 0.0189(1) | 0.29
64 | 0.04/1 | 1.4972(6) | 0.0188(2) | 0.29
7.0 | 4 | 128 | 21.84/5 | 2.19(3) | 0.0000039(8) | 0.312(4)
8 | 9.59/4 | 2.03(5) | 0.000011(3) | 0.293(6)
16 | 8.91/3 | 1.99(7) | 0.000015(8) | 0.29(1)
32 | 1.76/2 | 1.7(1) | 0.0002(2) | 0.24(2)
48 | 0.45/1 | 1.9(3) | 0.00002(6) | 0.28(4)
4 | 53.03/6 | 2.0149(3) | 0.00001168(8) | 0.29
8 | 9.80/5 | 2.0132(4) | 0.0000122(1) | 0.29
16 | 9.04/4 | 2.0130(4) | 0.0000123(1) | 0.29
32 | 8.42/3 | 2.0127(6) | 0.0000124(2) | 0.29
48 | 0.52/2 | 2.0141(8) | 0.0000119(3) | 0.29
64 | 0.21/1 | 2.015(1) | 0.0000118(4) | 0.29

Table S3: Fits of the helicity modulus $\Upsilon$ to $\Upsilon L=\alpha({\rm
ln}L)+b+cL^{-\omega}$ for the extraordinary-log critical phase of plane-defect
Villain model at $W=0.5$, $1$, $3$ and $7$. The symbol ‘-’ means that the
leading correction term is not involved.

$W$ | $L_{\min}$ | $L_{\max}$ |  $\chi^{2}$/DoF | $\alpha$ | $b$ | $c$ | $\omega$
---|---|---|---|---|---|---|---
0.5 | 16 | 128 | 805.87/4 | 0.4780(5) | -0.004(2) | - | -
32 | 55.14/3 | 0.503(1) | -0.108(4) | - | -
48 | 5.29/2 | 0.512(2) | -0.148(7) | - | -
64 | 0.16/1 | 0.516(3) | -0.17(1) | - | -
4 | 134.68/5 | 0.5362(8) | -0.298(4) | 1.254(9) | 0.789
8 | 6.96/4 | 0.549(1) | -0.358(6) | 1.47(2) | 0.789
16 | 0.52/3 | 0.555(3) | -0.39(1) | 1.60(6) | 0.789
32 | 0.51/2 | 0.554(7) | -0.38(4) | 1.6(2) | 0.789
48 | 0.35/1 | 0.55(2) | -0.35(9) | 1.3(6) | 0.789
4 | 347.10/5 | 0.5138(7) | -0.177(3) | 1.326(9) | 1
8 | 31.24/4 | 0.530(1) | -0.250(5) | 1.73(2) | 1
16 | 0.69/3 | 0.541(2) | -0.30(1) | 2.12(7) | 1
32 | 0.35/2 | 0.544(6) | -0.32(3) | 2.3(3) | 1
48 | 0.30/1 | 0.54(1) | -0.30(7) | 2.1(9) | 1
1.0 | 16 | 128 | 124.26/4 | 0.5207(8) | 0.633(3) | - | -
32 | 13.83/3 | 0.534(1) | 0.578(6) | - | -
48 | 0.69/2 | 0.540(2) | 0.55(1) | - | -
64 | 0.65/1 | 0.539(3) | 0.55(2) | - | -
4 | 8.56/5 | 0.554(1) | 0.462(5) | 0.74(1) | 0.789
8 | 3.69/4 | 0.558(2) | 0.44(1) | 0.80(3) | 0.789
16 | 2.12/3 | 0.562(4) | 0.42(2) | 0.90(8) | 0.789
32 | 2.01/2 | 0.565(9) | 0.41(5) | 1.0(3) | 0.789
48 | 0.68/1 | 0.54(2) | 0.5(1) | 0.1(8) | 0.789
4 | 31.92/5 | 0.541(1) | 0.532(4) | 0.79(1) | 1
8 | 7.88/4 | 0.548(2) | 0.502(8) | 0.95(4) | 1
16 | 2.31/3 | 0.554(3) | 0.47(2) | 1.2(1) | 1
32 | 1.89/2 | 0.559(7) | 0.45(4) | 1.5(4) | 1
48 | 0.68/1 | 0.54(2) | 0.54(9) | 0.2(1.3) | 1
3.0 | 16 | 128 | 32.79/4 | 0.548(2) | 2.611(7) | - | -
32 | 3.59/3 | 0.557(2) | 2.574(9) | - | -
48 | 3.54/2 | 0.556(4) | 2.58(2) | - | -
64 | 1.01/1 | 0.562(5) | 2.55(2) | - | -
4 | 15.00/5 | 0.574(2) | 2.47(1) | 0.68(2) | 0.789
8 | 6.32/4 | 0.582(4) | 2.43(2) | 0.83(5) | 0.789
16 | 6.10/3 | 0.580(6) | 2.45(3) | 0.8(1) | 0.789
32 | 3.56/2 | 0.56(1) | 2.56(8) | 0.1(5) | 0.789
48 | 2.33/1 | 0.59(4) | 2.4(2) | 1.5(1.4) | 0.789
4 | 26.34/5 | 0.562(2) | 2.539(8) | 0.73(2) | 1
8 | 5.69/4 | 0.573(3) | 2.49(1) | 1.00(7) | 1
16 | 5.67/3 | 0.574(5) | 2.49(3) | 1.0(2) | 1
32 | 3.57/2 | 0.56(1) | 2.57(6) | 0.1(7) | 1
48 | 2.27/1 | 0.59(3) | 2.4(2) | 2.4(2.1) | 1
7.0 | 16 | 128 | 10.00/4 | 0.547(3) | 6.64(1) | - | -
32 | 8.71/3 | 0.551(4) | 6.63(2) | - | -
48 | 0.85/2 | 0.566(7) | 6.56(3) | - | -
64 | 0.56/1 | 0.56(1) | 6.58(5) | - | -
4 | 6.63/5 | 0.574(4) | 6.51(2) | 0.62(5) | 0.789
8 | 6.63/4 | 0.573(7) | 6.51(3) | 0.6(1) | 0.789
16 | 6.52/3 | 0.57(1) | 6.53(6) | 0.5(3) | 0.789
32 | 2.09/2 | 0.63(3) | 6.2(2) | 2.4(9) | 0.789
48 | 0.82/1 | 0.55(7) | 6.6(4) | -0.5(2.7) | 0.789
4 | 7.21/5 | 0.563(4) | 6.56(2) | 0.66(5) | 1
8 | 6.75/4 | 0.566(6) | 6.55(3) | 0.7(1) | 1
16 | 6.72/3 | 0.56(1) | 6.56(5) | 0.7(4) | 1
32 | 2.00/2 | 0.61(2) | 6.3(1) | 3.5(1.3) | 1
48 | 0.81/1 | 0.55(6) | 6.6(3) | -0.8(4.2) | 1

Table S4: Fits of the two-point correlation $G^{\prime}$ to $G^{\prime}=a_{0}[{\rm ln}(L/l_{0})]^{-\hat{q}}$ for the extraordinary-log critical phase of plane-defect Villain model at $W=0.5$, $1$, $3$ and $7$. $W$ | $L_{\rm{min}}$ | $L_{\rm{max}}$ |  $\chi^{2}$/DoF | $a_{0}$ | $l_{0}$ | $\hat{q}$
---|---|---|---|---|---|---
0.5 | 4 | 128 | 26009.53/5 | 0.6413(4) | 1.454(2) | 0.4366(4)
8 | 596.25/4 | 0.5356(5) | 2.638(8) | 0.3471(5)
16 | 12.53/3 | 0.509(1) | 3.23(3) | 0.322(1)
32 | 7.06/2 | 0.502(3) | 3.43(9) | 0.315(3)
48 | 3.72/1 | 0.490(7) | 3.9(3) | 0.304(7)
4 | 359734.02/6 | 0.50273(2) | 2.6074(4) | 0.29
8 | 18915.26/5 | 0.48562(3) | 3.632(2) | 0.29
16 | 1182.19/4 | 0.47942(5) | 4.107(4) | 0.29
32 | 92.33/3 | 0.47727(8) | 4.293(7) | 0.29
48 | 7.96/2 | 0.4762(1) | 4.39(1) | 0.29
64 | 0.40/1 | 0.4757(2) | 4.44(2) | 0.29
1.0 | 4 | 128 | 4332.24/5 | 1.082(1) | 0.415(2) | 0.3568(5)
8 | 57.61/4 | 0.978(2) | 0.684(5) | 0.3153(7)
16 | 2.24/3 | 0.957(3) | 0.77(1) | 0.307(1)
32 | 1.07/2 | 0.949(9) | 0.81(4) | 0.303(4)
48 | 0.34/1 | 0.93(2) | 0.9(1) | 0.296(8)
4 | 36548.61/6 | 0.93147(4) | 0.8081(4) | 0.29
8 | 1746.29/5 | 0.92372(6) | 0.9088(7) | 0.29
16 | 179.34/4 | 0.92094(9) | 0.950(1) | 0.29
32 | 14.67/3 | 0.9193(2) | 0.976(2) | 0.29
48 | 1.00/2 | 0.9188(2) | 0.985(3) | 0.29
64 | 0.77/1 | 0.9187(3) | 0.987(5) | 0.29
3.0 | 4 | 128 | 314.26/5 | 1.611(5) | 0.0121(3) | 0.314(1)
8 | 4.19/4 | 1.490(7) | 0.0234(9) | 0.289(2)
16 | 4.16/3 | 1.49(1) | 0.024(2) | 0.288(3)
32 | 3.23/2 | 1.46(3) | 0.028(5) | 0.282(7)
48 | 2.08/1 | 1.53(8) | 0.019(8) | 0.30(2)
4 | 904.58/6 | 1.49952(9) | 0.02125(3) | 0.29
8 | 4.90/5 | 1.4962(1) | 0.02261(6) | 0.29
16 | 4.47/4 | 1.4963(2) | 0.02257(8) | 0.29
32 | 4.47/3 | 1.4963(3) | 0.0226(1) | 0.29
48 | 2.25/2 | 1.4967(4) | 0.0224(2) | 0.29
64 | 2.10/1 | 1.4969(6) | 0.0223(3) | 0.29
7.0 | 4 | 128 | 29.95/5 | 1.95(2) | 0.000024(4) | 0.281(3)
8 | 12.69/4 | 1.81(4) | 0.00007(2) | 0.261(5)
16 | 5.42/3 | 1.96(8) | 0.00002(1) | 0.28(1)
32 | 1.99/2 | 1.7(1) | 0.0001(1) | 0.25(2)
48 | 0.22/1 | 2.1(5) | 0.00001(2) | 0.30(6)
4 | 35.54/6 | 2.0106(3) | 0.0000155(1) | 0.29
8 | 35.45/5 | 2.0105(4) | 0.0000156(2) | 0.29
16 | 5.87/4 | 2.0121(5) | 0.0000150(2) | 0.29
32 | 5.82/3 | 2.0120(7) | 0.0000150(3) | 0.29
48 | 0.28/2 | 2.014(1) | 0.0000143(4) | 0.29
64 | 0.28/1 | 2.014(1) | 0.0000143(6) | 0.29
Table S5: Fits of the two-point correlation $G$ to $G=a_{0}[{\rm
ln}(L/l_{0})]^{-\hat{q}}$ for the extraordinary-log critical phase of plane-
defect $XY$ model at $W=1$, $3$ and $7$.

$W$ | $L_{\rm min}$ | $L_{\rm max}$ |  $\chi^{2}$/DoF | $a_{0}$ | $l_{0}$ | $\hat{q}$
---|---|---|---|---|---|---
1.0 | 8 | 128 | 30.30/4 | 0.7730(7) | 1.059(5) | 0.3022(4)
16 | 10.31/3 | 0.767(1) | 1.10(1) | 0.2990(8)
32 | 0.53/2 | 0.756(4) | 1.20(3) | 0.293(2)
48 | 0.12/1 | 0.76(1) | 1.1(1) | 0.297(7)
8 | 935.43/5 | 0.75322(4) | 1.1966(7) | 0.29
16 | 137.58/4 | 0.75200(6) | 1.222(1) | 0.29
32 | 1.95/3 | 0.75114(9) | 1.240(2) | 0.29
48 | 1.10/2 | 0.7510(1) | 1.242(3) | 0.29
64 | 0.53/1 | 0.7509(3) | 1.247(7) | 0.29
3.0 | 8 | 128 | 6.19/4 | 1.429(4) | 0.0264(6) | 0.2842(9)
16 | 6.19/3 | 1.429(7) | 0.027(1) | 0.284(2)
32 | 1.79/2 | 1.39(2) | 0.033(4) | 0.276(4)
48 | 0.03/1 | 1.33(5) | 0.05(2) | 0.26(1)
8 | 43.42/5 | 1.45397(8) | 0.02298(3) | 0.29
16 | 17.98/4 | 1.4544(1) | 0.02281(5) | 0.29
32 | 12.23/3 | 1.4548(2) | 0.02266(8) | 0.29
48 | 5.41/2 | 1.4553(3) | 0.0224(1) | 0.29
64 | 1.38/1 | 1.4564(6) | 0.0219(3) | 0.29
7.0 | 8 | 128 | 4.47/4 | 1.88(2) | 0.000034(4) | 0.274(2)
16 | 2.97/3 | 1.90(3) | 0.000028(6) | 0.278(4)
32 | 2.23/2 | 1.96(8) | 0.00002(1) | 0.29(1)
48 | 1.89/1 | 2.1(3) | 0.00001(1) | 0.30(4)
8 | 52.38/5 | 1.9919(1) | 0.00001530(6) | 0.29
16 | 12.04/4 | 1.9928(2) | 0.00001495(8) | 0.29
32 | 2.38/3 | 1.9936(3) | 0.0000146(1) | 0.29
48 | 2.07/2 | 1.9938(5) | 0.0000146(2) | 0.29
64 | 1.09/1 | 1.993(1) | 0.0000149(4) | 0.29

Table S6: Fits of the helicity modulus $\Upsilon$ to $\Upsilon L=\alpha({\rm
ln}L)+b+cL^{-\omega}$ for the extraordinary-log critical phase of plane-defect
$XY$ model at $W=1$, $3$ and $7$.

$W$ | $L_{\min}$ | $L_{\max}$ |  $\chi^{2}$/DoF | $\alpha$ | $b$ | $c$ | $\omega$
---|---|---|---|---|---|---|---
1.0 | 8 | 128 | 1202.17/5 | 0.4597(5) | 0.510(1) | - | -
16 | 36.92/4 | 0.498(1) | 0.393(4) | - | -
32 | 4.22/3 | 0.520(4) | 0.31(1) | - | -
48 | 2.55/2 | 0.531(10) | 0.26(4) | - | -
64 | 0.02/1 | 0.56(2) | 0.14(9) | - | -
8 | 1.83/4 | 0.560(3) | 0.11(1) | 1.01(3) | 0.789
16 | 1.81/3 | 0.56(1) | 0.10(5) | 1.0(2) | 0.789
32 | 1.72/2 | 0.57(3) | 0.1(2) | 1.3(8) | 0.789
48 | 0.31/1 | 0.7(1) | -0.6(6) | 5.7(38) | 0.789
8 | 2.69/4 | 0.543(2) | 0.197(9) | 1.14(3) | 1
16 | 1.94/3 | 0.550(9) | 0.16(4) | 1.3(2) | 1
32 | 1.76/2 | 0.56(3) | 0.1(1) | 1.8(12) | 1
48 | 0.28/1 | 0.66(9) | -0.4(5) | 8.7(58) | 1
3.0 | 8 | 128 | 585.80/5 | 0.5086(5) | 2.533(1) | - | -
16 | 18.72/4 | 0.534(1) | 2.454(3) | - | -
32 | 8.07/3 | 0.546(4) | 2.41(1) | - | -
48 | 5.22/2 | 0.562(10) | 2.35(4) | - | -
64 | 5.11/1 | 0.56(2) | 2.37(9) | - | -
8 | 6.19/4 | 0.575(3) | 2.27(1) | 0.66(3) | 0.789
16 | 6.08/3 | 0.57(1) | 2.28(5) | 0.6(2) | 0.789
32 | 5.14/2 | 0.60(3) | 2.1(2) | 1.4(8) | 0.789
48 | 5.13/1 | 0.6(1) | 2.2(6) | 1.1(38) | 0.789
8 | 6.27/4 | 0.564(2) | 2.326(9) | 0.74(3) | 1
16 | 6.25/3 | 0.565(9) | 2.32(4) | 0.8(2) | 1
32 | 5.16/2 | 0.59(3) | 2.2(1) | 1.9(11) | 1
48 | 5.15/1 | 0.58(9) | 2.2(5) | 1.5(57) | 1
7.0 | 8 | 128 | 557.53/5 | 0.5207(5) | 6.530(1) | - | -
16 | 13.86/4 | 0.546(1) | 6.453(3) | - | -
32 | 8.58/3 | 0.554(4) | 6.42(1) | - | -
48 | 1.31/2 | 0.579(10) | 6.32(4) | - | -
64 | 1.28/1 | 0.58(2) | 6.34(9) | - | -
8 | 7.18/4 | 0.585(3) | 6.27(1) | 0.64(3) | 0.789
16 | 6.11/3 | 0.57(1) | 6.32(5) | 0.5(2) | 0.789
32 | 2.15/2 | 0.64(3) | 6.0(2) | 2.0(8) | 0.789
48 | 1.09/1 | 0.5(1) | 6.6(6) | -1.7(37) | 0.789
8 | 6.69/4 | 0.575(2) | 6.328(9) | 0.72(3) | 1
16 | 6.34/3 | 0.570(9) | 6.35(4) | 0.6(2) | 1
32 | 2.07/2 | 0.62(3) | 6.1(1) | 2.9(11) | 1
48 | 1.11/1 | 0.54(9) | 6.5(5) | -2.5(57) | 1