# Scaling theory of continuous symmetry breaking under advection

Harukuni Ikeda<EMAIL_ADDRESS>Department of Physics,
Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan

###### Abstract

In this work, we discuss how the linear and non-linear advection terms modify
the scaling behavior of the continuous symmetry breaking and stabilize the
long-range order, even in $d=2$ far from equilibrium, by means of simple
scaling arguments. For an example of the liner advection, we consider the
$O(n)$ model in the steady shear. Our scaling analysis reveals that the model
can undergo the continuous symmetry breaking even in $d=2$ and, moreover,
predicts the upper critical dimension $d_{\rm up}=2$. These results are fully
consistent with a recent numerical simulation of the $O(2)$ model, where the
mean-field critical exponents are observed even in $d=2$. For an example of
the non-linear advection, we consider the Toner-Tu hydrodynamic theory, which
was introduced to explain polar-ordered flocks, such as the Vicsek model. Our
simple scaling argument reproduces the previous results by the dynamical
renormalization theory. Furthermore, we discuss the effects of the additional
non-linear terms discovered by the recent re-analysis of the hydrodynamic
equation. Our scaling argument predicts that the additional non-linear terms
modify the scaling exponents and, in particular, recover the isotropic scaling
reported in a previous numerical simulation of the Vicsek model. We discuss
that the critical exponents predicted by the naive scaling theory become exact
in $d=2$ by using a symmetry consideration and similar argument proposed by
Toner and Tu.

††preprint: AIP/123-QED

## Introduction.—

One of the most famous no-go theorems in equilibrium statistical mechanics is
the Mermin-Wagner theorem Mermin and Wagner (1966), which prohibits continuous
symmetry breaking in $d=2$ dimensions. However, the Mermin-Wagner theorem does
not hold far from equilibrium, and indeed, there are several out-of-
equilibrium systems showing continuous symmetry breaking even in $d=2$ Vicsek
_et al._ (1995); Bassler and Rácz (1995); Reichl  _et al._ (2010); Loos  _et
al._ (2023); Dadhichi  _et al._ (2020); Galliano  _et al._ (2023); Ikeda
(2023a); Ikeda and Kuroda (2023); Ikeda (2023b). A famous prototypical
numerical model that breaks the Mermin-Wagner theorem is the so-called Vicsek
model Vicsek  _et al._ (1995). The model consists of $XY$ spins flaying with a
constant speed along their magnetic direction, which mimics the flocking
behavior among living things such as birds and bacteria Vicsek  _et al._
(1995); Nishiguchi  _et al._ (2017). Interestingly, the numerical simulation
of the model in $d=2$ showed that the model undergoes continuous symmetry
breaking from the disordered phase, where the mean velocity of the spins
vanishes $\left\langle\bm{v}\right\rangle=0$, to the ordered phase, where
$\left\langle\bm{v}\right\rangle\neq 0$ Vicsek  _et al._ (1995).

In 1995, Toner and Tu proposed a hydrodynamic theory to explain the long-range
order of the Vicsek model in $d=2$ Toner and Tu (1995). The Toner-Tu theory in
1995 (TT95) successfully explained the existence of the order phase of the
Vicsek model in $d=2$. However, subsequent re-analysis of the hydrodynamic
theory by J. Toner reported several additional terms that were not considered
in the original theory TT95 Toner and Tu (1995). These terms hinder the exact
calculations of the critical exponents for compressible fluid Toner (2012).
Recently, an extensive numerical simulation of the Vicsek model has been
performed. The numerical results of the critical exponents are indeed
inconsistent with those of TT95. In particular, the numerical results suggest
almost isotropic scaling behavior Mahault  _et al._ (2019), while TT95
predicts anisotropic scaling Toner and Tu (1995). One of the purposes of this
work is to reconcile this discrepancy by taking into account the effects of
the additional non-linear terms.

In the hydrodynamic theory TT95, the key ingredient to stabilize the long-
range order in $d=2$ is the non-linear advection $(\bm{v}\cdot\nabla)\bm{v}$,
which leads to the super-diffusion and facilitates the relaxation of the
Goldstone mode Toner and Tu (1995); Minami and Nakano (2022). Interestingly, a
recent numerical simulation of the $O(n)$ model with steady shear revealed
that the linear advection $(\bm{v}\cdot\nabla)\bm{\phi}$, where $\bm{\phi}$
denotes the order parameter, can also stabilize the long-range order and
allows the continuous symmetry breaking in $d=2$ Nakano  _et al._ (2021). The
numerical result also reported that the critical exponents of the model agree
with the mean-field theory (linear analysis) even in $d=2$. Our other goal is
to explain this surprising result.

To tackle the above two problems, we perform simple scaling arguments for the
continuous symmetry breaking with advection. Our theory for the linear
advection of the $O(n)$ model predicts that the upper critical dimension below
which the Gaussian fixed point gets unstable is $d_{\rm up}=2$. This explains
the mean-field behavior observed in the numerical simulation in $d=2$ Nakano
_et al._ (2021). For the non-linear advection, our scaling analysis can
reproduce the previous theoretical results reported in TT95 Toner and Tu
(1995). Furthermore, the simplicity of the scaling argument allows us to take
into account the additional non-linear terms, which have not been taken into
account in the original theory TT95 Toner and Tu (1995). The new critical
exponents obtained by this work indeed support the isotropic scaling reported
in the previous numerical simulation of the Vicsek model Mahault  _et al._
(2019). We also argue that the exponents predicted by the native scaling
argument actually become exact in $d=2$.

##  $O(n)$ model in shear.—

We first discuss the effects of the linear advection. We consider the $O(n)$
model in the steady shear Nakano  _et al._ (2021):

$\displaystyle\frac{\partial\phi_{a}}{\partial
t}+\bm{v}\cdot\nabla\phi_{a}=\nabla^{2}\phi_{a}-\frac{\delta
F[\bm{\phi}]}{\delta\phi_{a}}+\xi_{a},$ (1)

where the $n$-component vector $\bm{\phi}=\\{\phi_{1},\cdots,\phi_{n}\\}$
denotes the order parameter,

$\displaystyle F[\bm{\phi}]=\int
d\bm{x}\left[\frac{\varepsilon}{2}\left(\bm{\phi}\cdot\bm{\phi}\right)+\frac{g}{4}\left(\bm{\phi}\cdot\bm{\phi}\right)^{2}\right]$
(2)

denotes the standard $\phi^{4}$ free-energy, and $\xi_{a}$ denotes the thermal
nosie of zero mean and variance:

$\displaystyle\left\langle\xi_{a}(\bm{x},t)\xi_{b}(\bm{x}^{\prime},t^{\prime})\right\rangle=2T\delta_{ab}\delta(\bm{x}-\bm{x}^{\prime})\delta(t-t^{\prime}).$
(3)

The advection term in Eq. (1), $\bm{v}\cdot\nabla\phi_{a}$, is a linear
function of the order parameter. For the velocity field $\bm{v}$, we consider
the simple shear along the $x_{1}$ direction:

$\displaystyle\bm{v}=\dot{\gamma}x_{2}\bm{e}_{1},$ (4)

where $\bm{e}_{1}$ denotes the unit vector along the $x_{1}$ axis. Recently,
an extensive numerical simulation of the model has been performed Nakano  _et
al._ (2021). The numerical result showed that the model undergoes the
continuous symmetry breaking even in $d=2$. Furthermore, the critical
exponents of the transition agree with the mean-field prediction. Here, we
explain this result through a simple scaling argument.

To investigate the large spatio-temporal behavior of the model, we consider
the following scaling transformations Nishimori and Ortiz (2011); Onuki and
Kawasaki (1979) 111One can introduce the proportional coefficients for the
advection and diffusion terms, as in $c_{a}\bm{v}\cdot\nabla\phi_{a}$ and
$c_{d}\nabla^{2}\phi_{a}$, and consider their scaling dimensions. However, in
such a case, the fixed point corresponds to $c_{a}=c_{d}=0$: the model loses
the spatial dependence. We are not interested in such a case. For the same
reason, we also omit the proportional coefficient of $\nabla^{2}\bm{v}$ and
$(\bm{v}\cdot\nabla)\bm{v}$ in the Vicsek model.:

$\displaystyle x_{1}\to b^{\zeta_{1}}x_{1},\ x_{2}\to b^{\zeta_{2}}x_{2},\
x_{i}\to bx_{i}\ i=3,\cdots,d,$ $\displaystyle t\to b^{z}t,\ \phi_{a}\to
b^{\chi}\phi_{a},\ \varepsilon\to b^{y_{\varepsilon}}\varepsilon,\ g\to
b^{y_{g}}g.$ (5)

For the noise $\xi_{a}$, Eq. (3) implies $\xi_{a}\to
b^{-\frac{z+\zeta_{1}+\zeta_{2}+d-2}{2}}\xi_{a}$ Medina  _et al._ (1989).
Then, Eq. (1) reduces to

$\displaystyle b^{\chi-z}\frac{\partial\phi_{a}}{\partial
t}+b^{\chi+\zeta_{2}-\zeta_{1}}\dot{\gamma}x_{2}\frac{\partial\phi_{a}}{\partial
x_{1}}$ $\displaystyle=b^{\chi-2\zeta_{1}}\frac{\partial^{2}\phi_{a}}{\partial
x_{1}^{2}}+b^{\chi-2\zeta_{2}}\frac{\partial^{2}\phi_{a}}{\partial
x_{2}^{2}}+b^{\chi-2}\sum_{i=3}^{d}\frac{\partial^{2}\phi_{a}}{\partial
x_{i}^{2}}$
$\displaystyle-b^{\chi+y_{\varepsilon}}\varepsilon\phi_{a}-b^{y_{g}+3\chi}g\left|\bm{\phi}\right|^{2}\phi_{a}+b^{-\frac{z+\zeta_{1}+\zeta_{2}+d-2}{2}}\xi_{a}$
(6)

We first discuss the model without shear $\dot{\gamma}=0$. In this case,
assuming that each term in Eq. (6) has the same scaling dimension, we get

$\displaystyle\chi-z=\chi-2\zeta_{1}=\chi-2\zeta_{2}=\chi-2$
$\displaystyle=\chi+y_{\varepsilon}=y_{g}+3\chi=-(z+\zeta_{1}+\zeta_{2}+d-2)/2,$
(7)

leading to

$\displaystyle
z=2,\zeta_{1}=\zeta_{2}=1,\chi=\frac{2-d}{2},y_{\varepsilon}=-2,\ y_{g}=d-4.$
(8)

To see the stability of the ordered phase, one can observe the fluctuation of
the order parameter $\left\langle\delta\phi_{a}^{2}\right\rangle\sim
b^{2\chi}$. For $d\leq 2$, $\chi\geq 0$, and thus, the fluctuation diverges in
the thermodynamic limit $b\to\infty$, which destroys the long-range order
Toner and Tu (1995), as expected from the Mermin-Wagner theorem Mermin and
Wagner (1966). Eq. (5) implies that the coefficient of the non-linear term $g$
scales as $g^{\prime}=b^{-y_{g}}g$ after the scale transformation
$\bm{x}^{\prime}\to b^{-1}\bm{x}$ Nishimori and Ortiz (2011). When $y_{g}>0$,
the non-linear term is irrelevant, and the critical exponents of the Gaussian
fixed point Eq. (8) become exact. The upper critical dimension $d_{\rm up}$
above which the Gaussian fixed point stabilizes is obtained by setting
$y_{g}=0$, leading to $d_{\rm up}=4$ Nishimori and Ortiz (2011). The results
are fully consistent with the scaling analysis of the equilibrium model
Nishimori and Ortiz (2011).

Now, we discuss how the finite shear rate $\dot{\gamma}\neq 0$ changes the
above scaling behavior. If we require all terms in Eq. (6) to have the same
scaling dimension, we obtain seven equations, while there are only six unknown
variables, $\zeta_{1}$, $\zeta_{2}$, $z$, $\chi$, $y_{\varepsilon}$, and
$y_{g}$, which are overdetermined and do not have solutions. So, we here
assume that the advection ${\dot{\gamma}x_{2}\partial_{x_{1}}\phi_{a}\sim
b^{\chi+\zeta_{2}-\zeta_{1}}}$ plays a more dominant role than the diffusion
${\partial_{x_{1}}^{2}\phi_{a}\sim b^{\chi-2\zeta_{1}}}$, which is tantamount
to assuming
${\chi+\zeta_{2}-\zeta_{1}>\chi-2\zeta_{1}\to\zeta_{1}+\zeta_{2}>0}$ and
neglect the term $b^{\chi-2\zeta_{1}}$ in Eq. (6). Requiring that all the
remaining terms in Eq. (6) have the same scaling dimension, we get

$\displaystyle\chi-z=\chi+\zeta_{2}-\zeta_{1}=\chi-2\zeta_{2}=\chi-2$
$\displaystyle=\chi+y_{\varepsilon}=y_{g}+3\chi=-(z+\zeta_{1}+\zeta_{2}+d-2)/2,$
(9)

leading to

$\displaystyle
z=2,\zeta_{1}=3,\zeta_{2}=1,\chi=-\frac{d}{2},y_{\varepsilon}=-2,\ y_{g}=d-2.$
(10)

The assumption $\zeta_{1}+\zeta_{2}>0$ is satisfied self-consistently. The
anisotropic exponent $\zeta_{1}=3$ is consistent with the linear analysis
Nakano  _et al._ (2021). Since $\chi<0$, the fluctuation of the order
parameter $\left\langle\delta\phi_{a}^{2}\right\rangle\sim b^{2\chi}$ vanishes
in the large scale $b\gg 1$, which allows the long-range order even in $d=2$.
This is in sharp contrast with the equilibrium system without shear
$\dot{\gamma}=0$, where the long-range order can exist only for $d>2$ Mermin
and Wagner (1966). The upper critical dimension is obtained by setting
$y_{g}=0$ Nishimori and Ortiz (2011), leading to $d_{\rm up}=2$, meaning that
Eqs. (10) are exact even in $d=2$. The exponents $\zeta_{1}$, $\zeta_{2}$, and
$y_{\varepsilon}$ are indeed consistent with a recent numerical simulation in
$d=2$ Nakano  _et al._ (2021). However, strictly speaking, the logarithmic
corrections are expected at the upper critical dimension $d=d_{\rm up}=2$
Nishimori and Ortiz (2011). Further numerical studies would be beneficial to
elucidate this point.

## Hydrodynamic theory for the Vicsek model.—

Now we discuss the effects of the non-linear advection of the hydrodynamic
theory of the Vicsek model Vicsek  _et al._ (1995). We first investigate the
minimal hydrodynamic theory for the polar ordered flocks investigated in Ref.
Besse  _et al._ (2022):

$\displaystyle\frac{\partial\bm{v}}{\partial
t}+(\bm{v}\cdot\nabla)\bm{v}=\nabla^{2}\bm{v}-\frac{\delta
F[\bm{v}]}{\delta\bm{v}}+\bm{\xi},$ (11)

where $\bm{v}(\bm{x},t)$ represents the local velocity of the flocks, and the
functional form of the free-energy $F$ is given by Eq. (2). Compared to the
original model proposed by Toner and Tu Toner and Tu (1995), the model Eq.
(11) assumes that the fluid is incompressible and neglects the terms related
to the pressure-gradient forces, such as $\nabla p$ and
$\bm{v}(\bm{v}\cdot\nabla)p$. Some irrelevant terms, such as
$(\bm{v}\cdot\nabla)^{2}\bm{v}$, are also neglected Besse  _et al._ (2022).
Since the velocity field $\bm{v}$ itself plays the role of the order
parameter, Eq. (11) has the non-linear advection term
$(\bm{v}\cdot\nabla)\bm{v}$ 222One can include two more non-linear advection
terms: $(\nabla\cdot\bm{v})\bm{v}$ and $\nabla(\bm{v}\cdot\bm{v})$, which has
the same scaling dimension as $(\bm{v}\cdot\nabla)\bm{v}$ and do not change
the results of the scaling analysis.. In the ordered phase $\varepsilon<0$,
the mean value of the velocity field has a finite value
$\left\langle\bm{v}\right\rangle=v_{0}\bm{e}_{\parallel}$, where
$\bm{e}_{\parallel}$ denotes the unit vector along the mean velocity
$v_{0}=\left|\left\langle\bm{v}\right\rangle\right|$. Now we decompose the
velocity field as ${\bm{v}=\bm{v}_{\parallel}+\bm{v}_{\perp}}$. We consider
the following scaling transformations Toner and Tu (1995):

$\displaystyle x_{\parallel}\to b^{\zeta}x_{\parallel},\ \bm{x}_{\perp}\to
b\bm{x}_{\perp},\ t\to b^{z}t,$ $\displaystyle\bm{v}_{\perp}\to
b^{\chi}\bm{v}_{\perp},\bm{v}_{\parallel}\to
b^{\chi^{\prime}}\bm{v}_{\parallel},\varepsilon\to
b^{y_{\varepsilon}}\varepsilon,\ g\to b^{y_{g}}g.$ (12)

Then, Eq. (11) in the space perpendicular to $\bm{e}_{\parallel}$ reduces to

$\displaystyle b^{\chi-z}\frac{\partial\bm{v}_{\perp}}{\partial
t}+b^{2\chi-1}(\bm{v}_{\perp}\cdot\nabla_{\perp})\bm{v}_{\perp}+b^{\chi+\chi^{\prime}-\zeta}(\bm{v}_{\parallel}\cdot\nabla_{\parallel})\bm{v}_{\perp}$
$\displaystyle=b^{\chi-2}\nabla_{\perp}^{2}\bm{v}_{\perp}+b^{\chi-2\zeta}\nabla_{\parallel}^{2}\bm{v}_{\perp}$
$\displaystyle-b^{\chi+y_{\varepsilon}}\varepsilon\bm{v}_{\perp}-b^{y_{g}+\chi+2\chi^{\prime}}g\left|\bm{v}_{\parallel}\right|^{2}\bm{v}_{\perp}$
$\displaystyle-b^{y_{g}+3\chi}g\left|\bm{v}_{\perp}\right|^{2}\bm{v}_{\perp}+b^{-\frac{z+\zeta+d-1}{2}}\bm{\xi}_{\perp}.$
(13)

As before, we assume that the advection
${(\bm{v}_{\perp}\cdot\nabla_{\perp})\bm{v}_{\perp}\sim b^{2\chi-1}}$ plays
the dominant role than the diffusion ${\nabla_{\perp}^{2}\bm{v}_{\perp}\sim
b^{\chi-2}}$, which is tantamount to assuming ${2\chi-1>\chi-2\to\chi>-1}$.
Then, we obtain the following scaling relations:

$\displaystyle\chi-z=\chi+\chi^{\prime}-\zeta=2\chi-1=\chi-2\zeta$
$\displaystyle=y_{\varepsilon}+\chi=y_{g}+3\chi=-(z+\zeta+d-1)/2,$ (14)

leading to

$\displaystyle z=\frac{2(1+d)}{5},\ \zeta=\frac{d+1}{5},\
\chi=\frac{3-2d}{5},$ $\displaystyle\chi^{\prime}=-\frac{d+1}{5},\
y_{\varepsilon}=-\frac{2(1+d)}{5},\ $ (15)

and

$\displaystyle y_{g}=\chi-z-\max[\chi+2\chi^{\prime},3\chi]=\frac{2(d-4)}{5}.$
(16)

The assumption $\chi>-1$ is satisfied self-consistently for $d<4$. The
resultant $z$, $\zeta$, and $\chi$ are consistent with the previous study by
Toner and Tu (TT95) Toner and Tu (1995). The exponents $\chi$ and
$\chi^{\prime}$ both become negative, $\chi<0$ and $\chi^{\prime}<0$, for
$d>3/2$, which allows the long-range order even in $d=2$ Toner and Tu (1995).
Note that they satisfy the scaling relation $\chi^{\prime}=\chi-1+\zeta$,
which is expected from the condition of the incompressible flow
$\nabla\cdot\bm{v}=\nabla_{\parallel}\cdot\bm{v}_{\parallel}+\nabla_{\perp}\cdot\bm{v}_{\perp}=0$
Chen  _et al._ (2016). By setting $y_{g}=0$, we get the upper critical
dimension $d_{\rm up}=4$. The previous renormalization group analysis revealed
that there are no corrections for the exponents $z$, $\zeta$, and $\chi$ for
incompressible fluid Toner and Tu (1995); Toner  _et al._ (2005); Chen  _et
al._ (2018).

Table 1: Critical exponents. Data are taken from Ref Mahault  _et al._ (2019). | $d=2$ |  |  | $d=3$ |  |
---|---|---|---|---|---|---
| TT95 | Vicsek | This work | TT95 | Vicsek | This work
$\chi$ | -1/5 | -0.31(2) | -1/3 | -3/5 | -0.62 | -2/3
$\zeta$ | 3/5 | 0.95(2) | 1 | 4/5 | 1 | 1
$z$ | 6/5 | 1.33(2) | 4/3 | 8/5 | 1.77 | 5/3

Recently, an extensive numerical simulation of the Vicsek model reported that
the critical exponents $\chi$, $\zeta$, and $z$ are inconsistent with TT95,
Eqs. (15), see TABLE 1 Mahault  _et al._ (2019); Nishiguchi (2023). In
particular, the numerical results suggest the almost isotropic scaling
$\zeta\approx 1$ Mahault  _et al._ (2019); Nishiguchi (2023). This discrepancy
would be reconciled by considering the hydrodynamic theory for compressible
fluid Toner (2012). For the compressible fluid, the density fluctuations
$\delta\rho$ are coupled with $\bm{v}$, leading to some additional non-linear
coupling terms between $\bm{v}$ and $\delta\rho$ in Eq. (13) Toner and Tu
(1995); Toner (2012). These non-linear terms were not taken into account in
the previous theory (TT95) Toner and Tu (1995); Toner (2012).

For the scaling argument, it is sufficient to estimate the order of magnitude
of the new non-linear terms for $b\gg 1$. Below, we provide a rough
estimation. In principle, one can express $\delta\rho$ as a function of
$\bm{v}$ by solving the equation of continuity
$\partial_{t}\rho=-\nabla\cdot\bm{J}$ Toner and Tu (1995); Toner (2012),
meaning that the additional non-linear terms would also be expressed as
functions of $\bm{v}$. The effects of the density fluctuations appear through
the pressure-gradient forces, such as $\nabla p$ and $\bm{v}(\bm{v}\cdot\nabla
p)$ Toner (2012), implying that the new non-linear terms may also involve at
least one spatial derivative $\nabla_{\parallel}$ or $\nabla_{\perp}$. So, the
least-order non-linear terms would be written as

$\displaystyle(\nabla\cdot\bm{v})\bm{v}_{\parallel{\rm or}\perp},\
(\bm{v}\cdot\nabla)\bm{v}_{\parallel{\rm or}\perp},\ \nabla_{\parallel{\rm
or}\perp}(\bm{v}\cdot\bm{v}).$ (17)

The higher order terms of $\bm{v}_{\parallel{\rm or}\perp}$ and
$\nabla_{\parallel{\rm or}\perp}$ are negligible since these terms have
smaller scaling dimensions. From the scaling point of view, Eqs. (17) yield
contributions proportional to $b^{2\chi-1}$, $b^{\chi+\chi^{\prime}-1}$,
$b^{2\chi^{\prime}-1}$, $b^{2\chi-\zeta}$, $b^{\chi+\chi^{\prime}-\zeta}$, and
$b^{2\chi^{\prime}-\zeta}$. Using Eqs. (15), one can see that most terms are
irrelevant or do not change the scaling. However,

$\displaystyle\nabla_{\parallel}(\bm{v}_{\perp}\cdot\bm{v}_{\perp})\sim
b^{2\chi-\zeta}$ (18)

diverge much faster than the terms in Eq. (13) for $b\gg 1$, which makes the
scaling Eqs. (15) improper. The above discussion showed that the most relevant
contribution is $b^{2\chi-\zeta}$. Note that the argument itself does not
guarantee the existence of such terms. However, fortunately, the non-linear
terms having the same scaling dimension have indeed been found by a more
detailed analysis of the hydrodynamic theory of the compressible fluid by J.
Toner Toner (2012). He reported several non-linear terms coupled to the
density fluctuations $\delta\rho$, such as
$\delta\rho\partial_{\parallel}\bm{v}_{\perp}$ and
$\bm{v}_{\perp}\partial_{\parallel}\delta\rho$, where $\delta\rho$ and
$\bm{v}_{\perp}$ have the same scaling dimension
$\delta\rho\sim\bm{v}_{\perp}\sim b^{\chi}$, implying
$\delta\rho\partial_{\parallel}\bm{v}_{\perp}\sim\bm{v}_{\perp}\partial_{\parallel}\delta\rho\sim
b^{2\chi-\zeta}$ Toner (2012).

Now we discuss the additional terms indeed recover the isotropic scaling
$\zeta=1$. To get the new scaling exponents, we assume that the advection-like
terms along the parallel direction, Eqs. (18), play more dominant roles than
the diffusion along that direction $\nabla_{\parallel}^{2}\bm{v}_{\perp}\sim
b^{\chi-2\zeta}$, which is tantamount to assuming
$2\chi-\zeta>\chi-2\zeta\to\chi+\zeta>0$. Then, we get the new scaling
relations:

$\displaystyle\chi-z=\chi+\chi^{\prime}-\zeta=2\chi-1=2\chi-\zeta$
$\displaystyle=y_{\varepsilon}+\chi=y_{g}+3\chi=-(z+\zeta+d-1)/2,$ (19)

leading to the isotropic scaling exponents

$\displaystyle z=\frac{d+2}{3},\ \zeta=1,\ \chi=\chi^{\prime}=\frac{1-d}{3},$
$\displaystyle y_{\varepsilon}=-\frac{2+d}{3},\ y_{g}=\frac{d-4}{3}.$ (20)

The assumption $\chi+\zeta>0$ is satisfied self-consistently for $d<4$. By
setting $y_{g}=0$, we get the upper critical dimension $d_{\rm up}=4$. For
$d<4$, our theory may not give the correct result near the transition point.
Furthermore, the transition of the Vicsek model is known to be discontinuous
in $d=2$, implying that one can not apply the scaling argument itself Grégoire
and Chaté (2004); Solon  _et al._ (2015). Deep inside the ordered phase, the
fluctuations are predominantly controlled by the Goldstone modes, which do not
change the free-energy $F[\bm{v}]$. Therefore, the non-linear term
proportional to $g$ would be negligible. In TABLE 1, we compare the Eqs. (20)
with the recent numerical results of the Vicsek model in the order phase in
$d=2$ and $3$. The agreement is reasonably good, considering the simplicity of
our theory. Further theoretical Toner and Tu (1995); Toner  _et al._ (2005);
Chen  _et al._ (2016); Sartori and Lee (2019); Tasaki (2020); Di Carlo and
Scandolo (2022); Jentsch and Lee (2023), numerical Grégoire and Chaté (2004);
Mahault  _et al._ (2019), and, hopefully, experimental Nishiguchi  _et al._
(2017); Iwasawa  _et al._ (2021); Nishiguchi (2023) studies would be
beneficial to estimate more accurate values of the critical exponents.

## Exact critical exponents in $d=2$.—

We get almost perfect agreement in $d=2$. This is surprising because (i) the
full hydrodynamic equation involves far more non-linear terms than those
considered in the previous section Toner (2012), and (ii) the naive scaling
argument, in general, does not give the exact critical exponents for non-
linear systems Nishimori and Ortiz (2011). Below, we try to justify this
result by using symmetry consideration Sartori and Lee (2019) and argument in
TT95 Toner and Tu (1995); Toner  _et al._ (2005). First, note that the
fluctuations in the ordered phase are dominated by the Goldstone mode
$\bm{v}_{\perp}$, which does not change the free-energy $F[\bm{v}]$. As
before, we assume that the density fluctuation $\delta\rho$ is expressed as a
function of $\bm{v}\approx\bm{v}_{\perp}$. Then, $\bm{v}_{\perp}$ is only the
relevant quantity. Let us assume that the equation of motion (EOM) of
$\bm{v}_{\perp}$ is written as follows:

$\displaystyle\frac{\partial\bm{v}_{\perp}}{\partial
t}=\bm{f}[\bm{v}_{\perp}]+\bm{\xi}_{\perp},$ (21)

where $\bm{f}$ denotes the restoring force of $\bm{v}_{\perp}$, and
$\bm{\xi}_{\perp}$ denotes the white noise. The restoring force of the
Goldstone mode $\bm{f}$ should vanish in the limit of the small wave number in
the Fourier space, implying that each term in $\bm{f}$ should involve at least
one spatial derivative $\nabla_{\perp{\rm or}\parallel}$. Above the lower
critical dimension, $\bm{v}_{\perp}\sim b^{\chi}$, $\nabla_{\perp}\sim
b^{-1}$, and $\nabla_{\parallel}\sim b^{-\zeta}$ vanish in the large scale
$b\to\infty$. So, $\bm{f}[\bm{v}_{\perp}]$ would be expanded by
$\bm{v}_{\perp}$ and $\nabla_{\perp{\rm or}\parallel}$ as follows Sartori and
Lee (2019):

$\displaystyle\bm{f}[\bm{v}_{\perp}]$
$\displaystyle=(D_{1}\nabla_{\perp}^{2}+D_{2}\nabla_{\parallel}^{2})\bm{v}_{\perp}+(D_{3}\nabla_{\perp}+D_{4}\nabla_{\parallel})(\nabla_{\perp}\cdot\bm{v}_{\perp})$
$\displaystyle+\lambda_{1}(\bm{v}_{\perp}\cdot\nabla_{\perp})\bm{v}_{\perp}+\lambda_{2}(\nabla_{\perp}\cdot\bm{v}_{\perp})\bm{v}_{\perp}+\lambda_{3}\nabla_{\perp}(\bm{v}_{\perp}\cdot\bm{v}_{\perp})$
$\displaystyle+\nu\nabla_{\parallel}(\bm{v}_{\perp}\cdot\bm{v}_{\perp})+O(\left|\nabla\right|^{4},\left|\bm{v}_{\perp}\right|^{4}),$
(22)

where $D_{i}$, $\lambda_{i}$, and $\nu$ are some constants. The higher order
terms of $\bm{v}_{\perp}$ and $\nabla_{\perp{\rm or}\parallel}$ have smaller
scaling dimensions and would be negligible for $b\gg 1$. In $d=2$,
$\bm{v}_{\perp}$ has just one component, and the equation can be further
simplified as

$\displaystyle\frac{\partial v_{\perp}}{\partial t}$
$\displaystyle=(d_{1}\partial_{\perp}^{2}+d_{2}\partial_{\parallel}^{2}+d_{3}\partial_{\perp}\partial_{\parallel})v_{\perp},$
$\displaystyle+\lambda\partial_{\perp}v_{\perp}^{2}+\nu\partial_{\parallel}v_{\perp}^{2}+\xi_{\perp},$
(23)

where $d_{i}$, $\lambda$, and $\nu$ are some constants. For $\nu=0$, the
critical exponents of Eq. (23) have been calculated exactly Sartori and Lee
(2019), leading to the same results as TT95, Eqs. (15). Below, we derive the
exact critical exponents for $\nu\neq 0$. Since the non-linear terms
proportional to $\lambda$ and $\mu$ are written as the total derivative of
$\parallel$ or $\perp$, the perturbative expansion of these terms can only
yield the terms $\partial_{\parallel{\rm or}\perp}({some\ function})$. This
implies that the perturbative re-normalization calculation for these non-
linear terms does not contribute to the terms that do not involve the spatial
derivatives Toner and Tu (1995); Mahault  _et al._ (2019); Sartori and Lee
(2019). In particular, the scaling behaviors of $\partial_{t}v_{\perp}\sim
b^{\zeta-z}$ and $\xi_{\perp}\sim b^{-(z+\zeta+d-1)/2}$ should remain
unchanged. Assuming $\partial_{t}v_{\perp}\sim\xi_{\perp}$, we get Mahault
_et al._ (2019):

$\displaystyle z=2\chi+\zeta+d-1.$ (24)

A recent numerical simulation of the Vicsek model indeed confirmed the above
scaling relation in $d=2$ Mahault  _et al._ (2019). One can see that Eq. (23)
is invariant under the “Pseudo-Galilean” transformation: $v_{\perp}\to
v_{\perp}+V$, $x_{\perp}\to x_{\perp}+2\lambda Vt$, and $x_{\parallel}\to
x_{\parallel}+2\nu Vt$, which implies the following scaling relations Toner
and Tu (1995); Toner  _et al._ (2005)

$\displaystyle 1=\chi+z,\ \zeta=\chi+z,$ $\displaystyle\to\zeta=1,\ \chi=1-z.$
(25)

Using Eqs. (24) and (25), we reproduce the exponents $z$, $\zeta$, and $\chi$
in Eqs. (20). Since Eq. (23) is the most general form of EOM obtained by the
symmetry consideration, we conclude that $z$, $\zeta$, and $\chi$ calculated
above are the exact critical exponents of the Vicsek model in $d=2$. It is
unclear if a similar argument can be applied to Eq. (21) in $d=3$.

## Summary.—

In summary, we have investigated the continuous symmetry breaking with linear
and non-linear advections by means of simple scaling arguments. Our theory
demonstrated that the linear and non-linear advection terms can generally
reduce the lower-critical dimension and stabilize the long-range order even in
$d=2$, where the Mermin-Wanger theorem prohibits the long-range order in
equilibrium Mermin and Wagner (1966). In particular, our scaling theory, for
the first time, can explain the mean-field behavior of the sheared $O(2)$
model in $d=2$ Nakano  _et al._ (2021) and the isotropic scaling behavior of
the Vicesek model Mahault  _et al._ (2019).

###### Acknowledgements.

We thank H. Nakano, Y. Kuroda, and D. Nishiguchi for valuable discussions and
comments. This work was supported by KAKENHI 23K13031.

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