††∗ Corresponding author<EMAIL_ADDRESS>
# Renormalization group study of marginal ferromagnetism

Andrea Cavagna  Istituto Sistemi Complessi, Consiglio Nazionale delle
Ricerche, UOS Sapienza, 00185 Rome, Italy  Dipartimento di Fisica, Università
Sapienza, 00185 Rome, Italy  INFN, Unità di Roma 1, 00185 Rome, Italy  Antonio
Culla∗ Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, UOS
Sapienza, 00185 Rome, Italy  Tomás S. Grigera  Instituto de Física de Líquidos
y Sistemas Biológicos (IFLySiB) — CONICET and Universidad Nacional de La
Plata, Calle 59 n 789, B1900BTE La Plata, Argentina  CCT CONICET La Plata,
Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina
Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de
La Plata, Argentina  Istituto Sistemi Complessi, Consiglio Nazionale delle
Ricerche, UOS Sapienza, 00185 Rome, Italy

###### Abstract

When studying the collective motion of biological groups a useful theoretical
framework is that of ferromagnetic systems, in which the alignment
interactions are a surrogate of the effective imitation among the individuals.
In this context, the experimental discovery of scale-free correlations of
speed fluctuations in starling flocks poses a challenge to the common
statistical physics wisdom, as in the ordered phase of standard ferromagnetic
models with $\mathrm{O}(n)$ symmetry, the modulus of the order parameter has
finite correlation length. To make sense of this anomaly a novel ferromagnetic
theory has been proposed, where the bare confining potential has zero second
derivative (i.e. it is marginal) along the modulus of the order parameter. The
marginal model exhibits a zero-temperature critical point, where the modulus
correlation length diverges, hence allowing to boost both correlation and
collective order by simply reducing the temperature. Here, we derive an
effective field theory describing the marginal model close to the $T=0$
critical point and calculate the renormalization group equations at one loop
within a momentum shell approach. We discover a non-trivial scenario, as the
cubic and quartic vertices do not vanish in the infrared limit, while the
coupling constants effectively regulating the exponents $\nu$ and $\eta$ have
upper critical dimension $d_{c}=2$, so that in three dimensions the critical
exponents acquire their free values, $\nu=1/2$ and $\eta=0$. This theoretical
scenario is verified by a Monte Carlo study of the modulus susceptibility in
three dimensions, where the standard finite-size scaling relations have to be
adapted to the case of $d>d_{c}$. The numerical data fully confirm our
theoretical results.

##  I Introduction

Ferromagnetic models have been the staple of the statistical physicists’ way
to study collective motion in biological systems, and more generally in active
matter. The seminal Vicsek model of flocking Vicsek  _et al._ (1995) is
essentially a ferromagnetic $\mathrm{O}(n)$ model on the move, where each
particle aligns its orientation to the local neighbours, but instead of being
anchored on a lattice, it actively moves following its own direction. The
corresponding continuous theory formulated by Toner and Tu Toner and Tu (1995,
1998); Tu  _et al._ (1998), is essentially Navier-Stokes hydrodynamics meeting
the Landau-Ginzburg theory of critical phenomena. Beyond these key cases,
models and theories where local effective alignment plus active motion are the
key ingredients, have been used across many alleys of active matter Marchetti
_et al._ (2013). Of course, in most active systems off-equilibrium effects
play a fundamental role in giving a phenomenology different from the standard
framework of statistical physics; among the many examples, a very vivid one is
the emergence of long-range order in the low temperature phase of the Vicsek
model even in two dimensions, due to the off-equilibrium coupling between
polarization and density, which propagates order more effectively than in the
equilibrium case, hence bypassing the Mermin-Wagner impossibility to have
ferromagnetic order in $d=2$ Toner and Tu (1995).

In some other cases, though, the deviations of active systems from standard
ferromagnetic phenomenology seem not principally due to off-equilibrium
effects. In the case of biological systems this is hardly a surprise, given
that being out of equilibrium is but one of the many new hurdles that biology
puts in front of us when modelling living systems. The case of bird flocks is
interesting, from this point of view. Experiments have shown that connected
correlations are scale-free in starling flocks in the wild Cavagna  _et al._
(2010). Flocks are highly ordered systems, hence in the ferromagnetic context
it is reasonable to model them as (active) $\mathrm{O}(n)$ systems in their
low temperature phase (which is essentially what Toner-Tu theory does), where
the Goldstone theorem Goldstone (1961) grants massless transverse modes,
giving scale-free correlations of the orientations fluctuations. The problem,
however, is that starling flocks display long range correlation also of the
speed fluctuations, namely of the modulus of the order parameter. This is an
anomaly in standard equilibrium systems: while the longitudinal fluctuations
(i.e. the fluctuations that, in a Cartesian orthogonal decomposition, are
parallel to the total magnetization), which are massive at the bare level,
become in fact massless after renormalization due to the coupling with the
transverse modes Patashinskii and Pokrovskii (1973); Brézin  _et al._ (1973);
Brézin and Wallace (1973), the modulus is always a massive mode in the ordered
phase, and it therefore has finite correlation length. Moreover, the off-
equilibrium nature of flocks does not seem to play a crucial role in
connection to this anomaly, as both off-equilibrium simulations of self-
propelled particles ruled by standard $\mathrm{O}(n)$ ferromagnetism Hemelrijk
and Hildenbrandt (2015), and the relative theoretical approaches Kyriakopoulos
_et al._ (2016), find that the speed is not a scale-free variable in the
active case. This is probably not surprising, as experiments show that
starling flocks are quasi-equilibrium systems, since —due to the strong
ordering— the reshuffling time of the interaction network is significantly
larger than the local relaxation time of the velocity Mora  _et al._ (2016).
This does not exclude that off-equilibrium effects may emerge when studying
these systems on very long time scales, but this would not explain the scale-
free behaviour of speed fluctuations. Summing up, speed scale-free
correlations are an anomaly that statistical physics should explain with some
new ingredients unrelated to off-equilibrium effects.

The first attempt to explain scale-free speed correlations was done in Bialek
_et al._ (2014), where a maximum entropy model derived directly from the
experimental correlation data in flocks found that a standard $\mathrm{O}(n)$
ferromagnetic potential confining the modulus of the velocity can give scale-
free speed correlations provided that the amplitude $g$ of the potential is
small enough: within a spin-wave expansion (which holds quite well in the
ordered phase of flocks), the modulus correlation length scales as $g^{-1/2}$,
and because flocks are large but finite systems of linear size $L$, if $g\ll
L^{-2}$, one finds scale-free speed correlations over all observable scales
Bialek  _et al._ (2014). The idea of this approach is to reduce the amplitude
$g$ of the whole bare potential, hence reducing its curvature in the modulus
direction, so to boost the correlation length beyond the system’s size $L$;
but because flocks are finite, this does not require $g$ to be strictly zero,
hence a speed-confining potential bounding the theory is always present in the
effective Hamiltonian. This promising theoretical model, however, did not
stand in front of new generation of experimental data, which showed that a
comparison between theory and data crashes at low values of the flocks’ size
$L$ Cavagna  _et al._ (2022): in small groups, the low value of the potential
amplitude $g$ blows the group speed to values that far exceed the natural
reference speed, and —most importantly— disagree with experimental
observations. Essentially, what happens is that by lowering the amplitude $g$
of the whole confining potential, we are not only decreasing the speed mass
(hence increasing its correlation length), but we are at the same time
depressing the bounding capacity of the potential, hence allowing the entropy
to blow the collective speed to unrealistic values, which are indeed
completely absent in the experimental data.

A different approach - still based on ferromagnetism - was proposed in Cavagna
_et al._ (2019), and successfully tested against numerical simulations and -
most importantly - experimental data in Cavagna  _et al._ (2022). The idea of
the new theory is to have zero curvature of the bare potential from the
outset, without the need to decrease the overall amplitude of the bounding
potential. This can be done by switching from the classic $\mathrm{O}(n)$ bare
potential, $V=g\left(1-\bm{\sigma}\cdot\bm{\sigma}\right)^{2}$, which bounds
around one the modulus of the fluctuating variable $\bm{\sigma}$ and which
needs a small $g$ to decrease the second derivative along the modulus, to the
equally simple form $V=\lambda\left(1-\bm{\sigma}\cdot\bm{\sigma}\right)^{4}$,
which has zero second derivative of the modulus irrespective of the value of
the amplitude $\lambda$; because of this always-vanishing curvature, this was
called marginal potential Cavagna  _et al._ (2019). The fact that the bare
mass of the modulus is zero, suggests that the modulus correlations are scale-
free (even in the bulk) exactly at $T=0$, where entropic effects are not
present; on the other hand, upon raising the temperature, fluctuations create
a non-zero curvature (that is a mass, in field-theoretical language), which
decreases the modulus correlation length. A mean-field analysis showed that
this is indeed the case Cavagna  _et al._ (2019): the marginal model has a
finite-temperature phenomenology completely analogous to its $\mathrm{O}(n)$
cousin, with a standard ordering transition at a finite $T_{c}$, but it also
has a new zero-temperature critical point where the modulus correlation length
diverges as $\xi\sim T^{-1/2}$. Hence, in the marginal model, in order to
obtain scale-free correlations in systems of finite size $L$, one simply has
to push the system deeply in the ordered phase and satisfy $T\ll L^{-2}$,
while the fact that the amplitude $\lambda$ is no longer connected to the
modulus correlation means that it can remain finite, hence allowing the
bounding potential to tame the collective speed of the group. Results of self-
propelled particle simulations ruled by the marginal confining potential are
completely compatible with both the theoretical expectations and the
experimental data Cavagna  _et al._ (2022), hence the marginal theory of speed
control is at the moment a reasonable hypothesis to explain scale-free speed
correlations in flocks.

The analytic study of the marginal theory has been limited up to now to the
equilibrium mean-field approximation Cavagna  _et al._ (2019). Hence, to do
theoretical progress one should first go beyond mean-field, performing a
finite-dimensional study still at equilibrium, and finally extend the analysis
beyond the equilibrium case, eventually including self-propulsion terms in the
equations of motion. Here, we deal with the first part of this program, by
writing an effective field theory for the marginal model valid in the deeply
ordered phase where flocks live, namely in the vicinity of the zero
temperature critical point, and by calculating the critical exponents using
the Renormalization Group (RG) in momentum shell Wilson and Fisher (1972);
Wilson and Kogut (1974) at one loop. Apart from the solid methodological
motivation that it is better to first have a complete theoretical grasp of the
equilibrium case before moving to off-equilibrium, the equilibrium theory has
some interest per se. As we have already said, starling flocks are close to
equilibrium, hence the equilibrium theory has great interest, if nothing else
as a reference theory around which developing a future framework for small
deviations from equilibrium. Finally, marginal ferromagnetism has an
interesting zero temperature critical point, which is unusual under many
respects even in the context of equilibrium statistical physics. The strange
mix that we will find of free critical exponents and interacting theory, with
relevant non-Gaussian couplings, will confirm a posteriori that the marginal
theory has some intrinsic theoretical interest.

##  II The Marginal Ferromagnetic Theory

###  II.1 Microscopic model

The microscopic Hamiltonian of the general ferromagnetic class of models we
study is given by,

$H=\frac{J}{2}\sum_{i,j}^{N}n_{ij}\left(\bm{\sigma}_{i}-\bm{\sigma}_{j}\right)^{2}+\sum_{i}^{N}V\left(\bm{\sigma}_{i}\cdot\bm{\sigma}_{i}\right)\
,$ (1)

where the $\bm{\sigma}_{i}$ are (classical) spins with $n$ components, living
in an external space of $d$ dimensions. The first ferromagnetic term
represents mutual imitation, favouring the spins to have similar orientation
and modulus. In the finite-dimensional case, the adjacency matrix is given by
$n_{ij}=1$ if $i$ and $j$ are nearest neighbours, and $n_{ij}=0$ otherwise;
$N$ is the total number of spins in the system. Spins are soft real variables,
i.e. their modulus is not fixed, hence the bare potential $V$ has the role to
bound the modulus of the spins around a reference value, which we will fix to
$1$. This requirement, together with rotational invariance and the need to
have a maximum at $\sigma=0$, fixes the general form of the bare potential,
$V\sim\left(1-\bm{\sigma}\cdot\bm{\sigma}\right)^{p}$. The case of normal
ferromagnets is given by the $p=2$ standard $\mathrm{O}(n)$ potential,
$V=g\left(1-\bm{\sigma}\cdot\bm{\sigma}\right)^{2}$, whose coarse-grained
field theory gives the classic Landau-Ginzburg Hamiltonian Parisi (1988); this
theory has non-zero bare mass of the modulus, proportional to $g$, hence the
correlation function of the modulus (i.e. speed correlations, in the
biological context) are not scale-free in the low temperature phase, unless
$g$ itself becomes small, which has its own shortcomings, as we discussed in
the Introduction and demonstrated in Cavagna  _et al._ (2022).

The marginal model, on the other hand, is given by the $p=4$ case, namely by
the following bare potential Cavagna  _et al._ (2019, 2022),

$V(\bm{\sigma}\cdot\bm{\sigma})=\lambda\left(1-\bm{\sigma}\cdot\bm{\sigma}\right)^{4},$
(2)

where $\lambda$ is an amplitude. The marginal form is the simplest one with a
flat minimum also in the longitudinal direction, i.e. a minimum with zero
curvature. With this potential, the modulus mode becomes massless at zero
temperature, irrespective of the value of $\lambda$ Cavagna  _et al._ (2019,
2022), hence developing scale-free correlations. We want to investigate this
zero-temperature critical point with the renormalization group Wilson and
Kogut (1974).

###  II.2 From the mean-field case to field theory

The first step in our study is to define a field-theory version of the
marginal model, to which we can then apply the momentum-shell RG method. To do
this we will proceed in a phenomenological way, similar to the Landau-Ginzburg
case, namely we will look for the coarse-grained field theory whose Landau
approximation gives the same results as the mean-field approximation of the
microscopic model Parisi (1988). The mean-field theory of the marginal model
was studied in Cavagna  _et al._ (2019): by setting the adjacency matrix to
$n_{ij}=1/N$ for all pairs, one obtains a fully-connected (or infinite
dimensional) model where the saddle point method can be used to calculate in
the limit of $N\rightarrow\infty$ the partition function of the system. If we
define the magnetization as,

$\bm{m}=\frac{1}{N}\sum_{i}\bm{\sigma}_{i}\ ,$ (3)

and its modulus $m=|\bm{m}|$, the probability distribution of $m$ defines the
mean-field Gibbs free-energy $g(m)$,

$P(m)\sim e^{-Ng(m)}\ .$ (4)

Working at $T\ll 1$ and expanding $g(m)$ near $m=1$ (which is the equilibrium
magnetization at $T=0$), we obtain (see Appendix A for details),

$\displaystyle g(m)=$ $\displaystyle\lambda\left(1-m^{2}\right)^{4}+$
$\displaystyle+$ $\displaystyle
T\left[a_{2}\left(1-m^{2}\right)^{2}+a_{3}\left(1-m^{2}\right)^{3}+...\right]+$
$\displaystyle+$ $\displaystyle
T^{2}\left[a_{1}\left(1-m^{2}\right)+a_{4}\left(1-m^{2}\right)^{2}+...\right],$
(5)

where the $a_{n}$ are $T$-independent constants which are functions of the
parameters $J$ and $\lambda$ of the Hamiltonian. For $T=0$ the free energy
reduces to the same functional form as $V$, Eq. 2, and it thus has a minimum
with zero curvature. So the mean-field Gibbs free energy, in the limit of
vanishing temperature, has a flat minimum, implying a divergent susceptibility
for fluctuations of the modulus of the magnetization. On the other hand, when
$T$ grows, entropic fluctuations generate a non-zero second derivative of the
free energy, hence making the susceptibility finite. This trade-off between
bare potential and entropic fluctuations close to $T=0$ is the origin of the
zero-temperature critical point of the marginal model. This mean-field
scenario was confirmed also in the finite-dimensional case by numerical
simulations on a cubic lattice Cavagna  _et al._ (2019).

We can reorder the terms in Eq. 5, collecting powers of $(1-m^{2})$ and
writing the coefficients to the lowest order in $T$,

$g(m)=a_{1}T^{2}\left(1-m^{2}\right)+a_{2}T\left(1-m^{2}\right)^{2}+\\\
+a_{3}T\left(1-m^{2}\right)^{3}+\lambda\left(1-m^{2}\right)^{4}+\ldots$ (6)

To proceed in defining the field theory, we do not need the actual values of
the coefficient $a_{n}$, as the only relevant thing is that they do not depend
on the temperature $T$. We now promote the magnetization modulus to a
fluctuating field, $m\rightarrow\phi(\bm{x})$. Because we are interested in
the system’s properties near the marginal critical point Cavagna  _et al._
(2019) at $T=0$, where the equilibrium magnetization modulus is $1$, it is
convenient to work with the shifted field, $\varphi(\bm{x})=1-\phi(\bm{x})$,
which is small near the zero-temperature critical point. We stress the fact
that, even if the magnetization modulus is not analytic for $m$ close to $0$,
we are far from this regime since in the low temperature phase $m\simeq 1$.
Following this scheme we have that $(1-m^{2})=(1-m)(1+m)\rightarrow 2\varphi$,
where the numerical factor $2$ will be absorbed into the couplings of the
field theory. Additionally, we ignore the angular degrees of freedom, focusing
only on the modulus fluctuations, because the fluctuations of modulus and
phase are known to be very weakly coupled to each other in the broken-symmetry
phase Brézin and Wallace (1973); Patashinskii and Pokrovskii (1979); Ryder
(1996). Finally, following the standard ferromagnetic procedure, we introduce
a square gradient term, which embodies ferromagnetic interaction by depressing
short-wavelength fluctuations of the field. By keeping powers up to
$\varphi^{4}$ (higher order terms are discussed in Appendix C), we finally
obtain the following Landau free-energy,

$\mathcal{F}=\int\mathrm{d}^{d}x\left\\{\left(\nabla\varphi\right)^{2}+aT\varphi^{2}+cT^{2}\varphi+vT\varphi^{3}+u\varphi^{4}\right\\}\
,$ (7)

so that the probability of a field configuration is,
$P[\varphi]=\exp[-\mathcal{F}/T]/\mathcal{Z}$. In conventional field theories
Goldenfeld (1992) we normally would ignore the factor $1/T$ in the exponential
weight, because near the critical point it contributes a harmless finite
constant $1/T_{c}$ that can be safely reabsorbed in the field and in the
couplings. In our case, however, we must be careful, as we are dealing with a
critical point at $T_{c}=0$, hence $T$ is not a harmless constant. The
temperature is the coefficient of the quadratic term, and it therefore plays
the role of the bare mass; however, note that powers of $T$ appear also in the
other coefficients, not just in the quadratic one, so that when approaching
the critical temperature, all these coefficients vanish. For this reason one
cannot reabsorb the temperature in the other couplings. The most convenient
way to deal with this situation is to define a new field,

$\psi(\bm{x})=\varphi(\bm{x})/\sqrt{T}\ .$ (8)

This rescaling leads to a theory with a regular coefficient of the square
gradient term, and results in a field amplitude that does not vanish for
$T\rightarrow 0$ (see App. B). We will also drop the linear term, which does
not change the critical behavior of the theory (this is justified in App.
C.2), and set the constant $a$ to $1$, which amounts to a harmless
redefinition of the temperature and of the other couplings. We thus end up
with a Landau-Guinzburg theory for $\psi(x)$ such that
$P[\psi]=\exp[-\mathcal{H}]/\mathcal{Z}$, with,

$\mathcal{H}=\int\mathrm{d}^{d}x\left\\{\left(\nabla\psi\right)^{2}+T\psi^{2}+vT^{3/2}\psi^{3}+uT\psi^{4}\right\\}\
.$ (9)

The novelty of this field theory is that powers of $T$, which here plays the
role of the mass (i.e. of the control parameter), appear in all the couplings.
This is unusual in standard field theories, where the bare couplings are
independent of the temperature (or mass), and thus remain finite when the bare
mass vanishes.

Dimensional analysis of Eq. 9 shows that the naive scaling dimensions are (in
momentum units),

$\displaystyle[k]$ $\displaystyle=1$ $\displaystyle[\psi_{k}]$
$\displaystyle=-\frac{d}{2}-1$ $\displaystyle[T]$ $\displaystyle=2$
$\displaystyle[v]$ $\displaystyle=-\frac{d}{2}$ $\displaystyle[u]$
$\displaystyle=2-d,$ (10)

where $\psi_{k}$ is the field in momentum space. We immediately see that for
$d>2$ the naive scaling dimensions of both $v$ and $u$ are negative,
suggesting that for $d=3$ the theory is infrared-free. However, computing the
naive dimensions of the full cubic and quartic coefficients, $vT^{3/2}$ and
$uT$, we find

$\displaystyle[vT^{3/2}]$ $\displaystyle=\frac{6-d}{2}$ $\displaystyle[uT]$
$\displaystyle=4-d,$ (11)

so that for $d=3$ their naive scaling dimension is positive, suggesting
therefore that the theory actually conserves its non-Gaussian couplings in the
infrared limit, so that it is not free. This apparently contradictory
situation needs to be settled by going beyond mere dimensional analysis, that
is by calculating the renormalization group flow equations.

##  III Renormalization Group analysis

###  III.1 General RG procedure

We study the zero-temperature critical behaviour of the Hamiltonian Eq. 9
using Wilson’s momentum-shell renormalization group method Wilson (1971). We
present in this section the recursion relations of the RG transformation. The
diagrammatic perturbation theory can be carried out using the tuning parameter
$T$, and the two composite coupling constants,

$\hat{v}=vT^{3/2}\quad,\quad\hat{u}=uT.$ (12)

Formally, then, all diagrams are the same as in the standard Landau-Guinzburg
theory (with a cubic term). However, after having worked out the RG flow
equations for $(T,\hat{v},\hat{u})$, it will be crucial to go back and study
the RG flow of the _original_ parameters $(T,v,u)$ to understand the critical
behavior, which is different from that of standard Landau-Guinzburg in $d=3$.
In fact, neglecting the explicit $T$-dependence of $\hat{v}$ and $\hat{u}$
leads to physical inconsistencies that are already apparent at the level of
the Landau approximation: if one looks for a constant solution,
$\psi(x)=\psi_{0}$ (thus setting to zero the gradient square) and simply
minimizes $\mathcal{H}$ with respect to $\psi_{0}$, one finds that for fixed
$\hat{v}$ and $\hat{u}$ the potential has two minima, one at $\psi_{0}=0$ and
second one at finite value of $\psi_{0}$ with lower energy, giving a first-
order transition phenomenology. Instead, working with $T\to 0$ at fixed $v$
and $u$ keeps the appropriate balance among the coefficients such that the
Landau potential always has just one minimum at $\psi_{0}=0$, which is
consistent with the mean-field scenario.

To do momentum-shell RG one first rewrites the Hamiltonian in momentum space,
introducing an arbitrary ultraviolet cut-off $\Lambda$ (of the order of the
inverse of the nearest-neighbor distance) that makes all perturbative diagrams
well-behaved in the UV limit. The Wilson procedure then consists of two steps
Wilson and Kogut (1974). First one integrates out all the degrees of freedom
in a thin shell $k\in[\Lambda/b,\Lambda]$ with $b>1$ but close to 1, defining,

$\mathcal{H}_{1}[\psi(k\leq\Lambda/b)]\equiv-\log\int\\!\\!D[\psi(\Lambda/b<k\leq\Lambda)]e^{-H[\psi]}.$
(13)

This step is non-trivial because the non-Gaussian terms couple the on-shell
(UV) and off-shell (IR) modes, and must be carried out perturbatively. Once
$\mathcal{H}_{1}$ is found, in the second step the momentum is rescaled $k\to
k/b$ so that the original cut-off is recovered, and the coarse-grained
Hamiltonian is re-written so that it has the same form as the original one,
but with new, renormalized field and coupling constants. As a result of the
two steps we obtain the novel Landau-Ginzburg Hamiltonian,

$\mathcal{H}_{b}\left[\psi_{b}(\bm{k})\right]=\frac{1}{2}\int_{0}^{\Lambda}\\!\\!\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\,\left[\left(k^{2}+T_{b}\right)\psi_{b}(\bm{k})\psi_{b}(-\bm{k})\right]+\\\
+\hat{v}_{b}\int_{0}^{\Lambda}\\!\\!\frac{\mathrm{d}^{d}k_{1}\mathrm{d}^{d}k_{2}}{(2\pi)^{2d}}\,\psi_{b}(\bm{k}_{1})\psi_{b}(\bm{k}_{2})\psi_{b}(-\bm{k}_{1}-\bm{k}_{2})+\\\
+\hat{u}_{b}\int_{0}^{\Lambda}\frac{\mathrm{d}^{d}k_{1}\mathrm{d}^{d}k_{2}\mathrm{d}^{d}k_{3}}{(2\pi)^{3d}}\,\psi_{b}(\bm{k}_{1})\psi_{b}(\bm{k}_{2})\psi_{b}(\bm{k}_{3})\times\\\
\times\psi_{b}(-\bm{k}_{1}-\bm{k}_{2}-\bm{k}_{3})$ (14)

which depends on the renormalized couplings $T_{b},\hat{v}_{b},\hat{u}_{b}$,
and the renormalized field $\psi_{b}(k)$. In order to find these renormalized
parameters we need to turn to the diagrammatic expansion at one loop.

###  III.2 Relevant diagrams and RG relations

The theory has two vertices (Fig. 1), a cubic one with coupling
$\hat{v}=vT^{3/2}$ and a quartic one with coupling $\hat{u}=uT$.

Figure 1: The two vertices of the marginal field theory.

Combining these two vertices we can make one-loop diagrams with an arbitrary
number of external legs, but we evaluate the renormalized couplings only up to
the $\psi^{4}$ term (four external legs). Diagrams with more than four
external legs give a correction to higher order terms that we do not include
in Hamiltonian Eq. 9 because they are all RG-irrelevant (see App. C). We have
two diagrams that contribute to the renormalization of temperature $T$ and
field (Fig. 2), two that enter the renormalization of $vT^{3/2}$ (Fig. 3) and
three that contribute to the renormalization of $uT$ (Fig. 4).

Figure 2: Diagrams that contribute to the renormalization of $T$ and field.
Dashed lines represent fields with momentum $k<\Lambda/b$ (off-shell), while
solid lines represent integrated fields with on-shell momentum
$\Lambda/b<k<\Lambda$. Figure 3: Diagrams contributing to the renormalization
of $\hat{v}=vT^{3/2}$. Figure 4: Diagrams contributing to the renormalization
of $\hat{u}=uT$.

Combining the contributions of all diagrams, the renormalized couplings are
found to be (details in App. C)

$\displaystyle T_{b}={}$ $\displaystyle
b^{2}\left[T+12uTA_{1}-18v^{2}T^{3}A_{2}\right]$ (15a)
$\displaystyle\left(vT^{3/2}\right)_{b}={}$ $\displaystyle
b^{3-d/2}\Big{[}vT^{3/2}-36uvT^{5/2}A_{2}+$
$\displaystyle\qquad+36v^{3}T^{9/2}A_{3}\Big{]}$ (15b)
$\displaystyle\left(uT\right)_{b}={}$ $\displaystyle
b^{4-d}\Big{[}uT-36u^{2}T^{2}A_{2}+216uv^{2}T^{4}A_{3}+$
$\displaystyle\qquad-162v^{4}T^{6}A_{4}\Big{]},$ (15c)

where

$A_{q}=\int_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{1}{(p^{2}+T)^{q}}\simeq
K_{d}\frac{\Lambda^{d}}{(\Lambda^{2}+T)^{q}}\log b\ ,$ (16)

$K_{d}$ is the area of the unit sphere in $d$ dimensions divided by
$(2\pi)^{d}$ and the approximation is valid for a thin shell ($b\simeq 1$).
Finally, from the $k$-dependence of the two-legged diagrams (Fig. 2), the
field renormalization is found as,

$\psi_{b}(\bm{k})=b^{d_{\psi}}\psi(\bm{k}/b)\ ,$ (17)

where the scaling dimension of the field is given by the $k^{2}$ contribution
of the diagram in Fig. 2 \- right (for its detailed expression see Appendix
C),

$d_{\psi}=-1-\frac{d}{2}+\frac{B}{2}v^{2}T^{3}\Lambda^{d-6},$ (18)

and $B$ is a dimensionless numerical constant whose value we will not need in
the following.

###  III.3 The beta functions

We now “unpack” Eqs. 15 to obtain the RG equations for the original coupling
constants, $(T,v,u)$. Moreover, instead of keeping the RG equations in their
iterative form, we will switch to the fairly more compact differential form,
introducing the standard $\beta$-functions for each coupling Goldenfeld
(1992). To do this, one defines the infinitesimal parameter $x\ll 1$, such
that $b=1+x$ and $\log b\approx x$; in this way the $\beta$-function (or flow
function) of a generic parameter $\cal P$ is defined as,

$\beta_{\cal{P}}=\partial{\cal P}/\partial(\log b)=\partial{\cal P}/\partial
x\ .$ (19)

After using Eq. 15 to work out the flow of the original couplings, their
$\beta$-functions become,

$\displaystyle\beta_{T}$
$\displaystyle=2T+12uTK_{d}\Lambda^{d-2}-12uT^{2}K_{d}\Lambda^{d-4},$ (20a)
$\displaystyle\beta_{v}$
$\displaystyle=-\frac{d}{2}v-18uvK_{d}\Lambda^{d-2}+18uvTK_{d}\Lambda^{d-4}$
$\displaystyle-36uvTK_{d}\Lambda^{d-4},$ (20b) $\displaystyle\beta_{u}$
$\displaystyle=(2-d)u-12u^{2}K_{d}\Lambda^{d-2}+12u^{2}TK_{d}\Lambda^{d-4}$
$\displaystyle-36u^{2}TK_{d}\Lambda^{d-4},$ (20c)

where we have written only the leading term and the first correction in $T$.

##  IV Fixed point and critical exponents

From the zeros of the $\beta$ functions Eq. 20 we find that the RG flow has
only one physically meaningful (i.e. with $T\geq 0$ and $u\geq 0$) fixed
point, namely,

$T^{*}=0\quad,\quad v^{*}=u^{*}=0\ .$ (21)

The Jacobian matrix at this fixed point is

$\frac{\partial(\beta_{T},\beta_{v},\beta_{u})}{\partial(T,v,u)}=\begin{pmatrix}2&0&0\\\
0&-d/2&0\\\ 0&0&2-d\end{pmatrix}\equiv\begin{pmatrix}y_{T}&0&0\\\ 0&y_{v}&0\\\
0&0&y_{u}\end{pmatrix},$ (22)

from which we see that $T$ is an unstable direction, as expected, given that
$T$ is the tuning parameter, while both $u$ and $v$ are stable in $d=3$. The
critical manifold is the $T=0$ plane, and $T$ is the (relevant) control
variable that takes the system away from the critical point. The critical
point is $T_{c}=0$, independently of the (bare) value of $u$ and $v$, and
independently of the cutoff $\Lambda$. Notice that, consistently with the
physics of the problem, there is no negative shift of the mass, as there is
instead in the standard Landau-Ginzburg theory Goldenfeld (1992): the zero
temperature bare critical point cannot be reduced further by fluctuations
under renormalization.

###  IV.1 Critical exponents

Critical exponents can be found as usual from the eigenvalues of the Jacobian,
once we linearize the RG transformation near the fixed point Cardy (1996). In
particular, to calculate the exponent $\nu$, defining the divergence of the
modulus correlation length,

$\xi\sim T^{-\nu}$ (23)

we use the fact that $\xi_{b}=\xi/b$, which gives, $\partial\xi/\partial
x=-\xi$ (the correlation length has always scaling dimension $-1$), so that
$\nu^{-1}$ is the scaling dimension of the control parameter, namely it is the
coefficient of the linear term $T$ in the $\beta$-function of the temperature,

$\nu^{-1}=\left.\frac{\partial\beta_{T}}{\partial
T}\right|_{u^{*},v^{*}}=2+12u^{*}K_{d}\Lambda^{d-2}=2\ ,$ (24)

where we have used the fixed point value, $u^{*}=0$. We conclude that the
divergence of the modulus correlation length is ruled by the same critical
exponent as the free theory, $\nu=1/2$. It is important to note that this
result is due to the fact that the coefficient of the linear term $T$ in the
$\beta$-function of the control parameter depends on $u$ and not on $\hat{u}$.
This is the reason why the exponent is free, even though the effective
coupling $\hat{u}=Tu$ is not asymptotically zero. Notice that, had we kept
hidden into $\hat{u}$ the dependence on the temperature in the function
$\beta_{T}$, we would have found a fixed point at a negative value of $T$,
which is clearly unphysical.

The second exponent we are interested in is the anomalous dimension of the
space correlation function, $\eta$, defined by its scaling form near the
critical point Cardy (1996),

$C(k)=k^{-2+\eta}f(k\xi)\ .$ (25)

From the renormalization of the field thorough the RG transformation we can
write a self-consistency equation for the correlation function,

$\langle\psi(k)\psi(k^{\prime})\rangle=\\\
b^{-2d_{\psi}^{*}}\langle\psi_{b}(bk)\psi_{b}(bk^{\prime})\rangle$ (26)

and by using the standard relation,
$(2\pi)^{d}\delta(k+k^{\prime})C(k)=\langle\psi(k)\psi(k^{\prime})\rangle$, we
obtain,

$C(k)=b^{-2d_{\psi}^{*}-d}C(bk),$ (27)

from which we can read the anomalous dimension,

$\eta=2+d+2d^{*}_{\psi}\ ,$ (28)

where $d_{\psi}^{*}$ is the dimension $d_{\psi}$ evaluated at the fixed point.
From Eq. 18 we find then,

$\eta=2+d-2-d+\left.Bv^{2}T^{3}\Lambda^{-3}/2\right|_{T^{*},v^{*},u^{*}}=0.$
(29)

We conclude that both critical exponents take their free-theory values,

$\nu=1/2\quad,\quad\eta=0\ .$ (30)

It might seem surprising to obtain these values in $d=3$, where it is known
that the cubic and quartic Landau-Ginzburg terms are relevant in the RG sense.
However, our result is a consequence of the peculiar way in which the
quadratic, cubic and quartic coefficients are tied together in this theory. If
one goes back to look for fixed points in the composite couplings, Eqs. 15,
one does find a Wilson-Fisher-like fixed point, but it is nonphysical for this
case because - as we have already noted - it would require $T^{*}<0$. One can
verify that, for any starting point $(T,\hat{v},\hat{u})$ with positive
couplings and near $T=0$, the flow always stays in the region with $T>0$,
which is evident considering the flow in $(T,u,v)$ space, where $T=0$ is the
critical manifold.

We should remark that, unlike the usual $\lambda\phi^{4}$ theory, here the
critical exponent $\eta$ does pick up corrections at one loop, coming from the
diagram built by combining two $\psi^{3}$ vertices (which has two external
legs and a non-zero external momentum on internal lines, see appendix figure
60). However, this correction vanishes due to the Gaussian nature of the fixed
point that rules the critical exponents in this case. For this reason, higher
order corrections to the anomalous dimension $\eta$ will also vanish.

###  IV.2 Critical region

The critical point of this theory is rather pathological, since at $T=0$ all
but the gradient terms vanish. Hence, we wish to understand whether there is
some finite neighbourhood of the critical point where the free critical
exponents calculated above can actually be observed. In other words, we must
estimate the size of the critical region, i.e. the region outside which one
expects noticeable departures from the power laws with the fixed-point values
of the exponents. To do this we need to go beyond the linear approximation of
the flow near the fixed point. Hence, we go back to the $\beta$ functions Eq.
20 and rewrite them keeping terms up to $O(T)$,

$\displaystyle\frac{dT}{dx}$
$\displaystyle=\beta_{T}=2T\left(1+6u\Lambda^{d-2}\right),$
$\displaystyle\frac{dv}{dx}$
$\displaystyle=\beta_{v}=-\frac{d}{2}v-18uv\Lambda^{d-2},$ (31)
$\displaystyle\frac{du}{dx}$
$\displaystyle=\beta_{u}=(2-d)u-12u^{2}\Lambda^{d-2},$

where we have set $K_{d}=1$ to simplify the notation. These equations can be
solved exactly. In $d=3$ we obtain,

$\displaystyle T(x)$ $\displaystyle=T_{0}(12\Lambda u_{0}+1)e^{2x}-12\Lambda
T_{0}u_{0}e^{x}$ $\displaystyle v(x)$
$\displaystyle=\frac{v_{0}e^{-\frac{3}{2}x}}{\left(12\Lambda u_{0}+1-12\Lambda
u_{0}e^{-x}\right)^{3/2}}$ (32) $\displaystyle u(x)$
$\displaystyle=\frac{u_{0}e^{-x}}{12\Lambda u_{0}+1-12\Lambda u_{0}e^{-x}},$

where $T_{0}$, $v_{0}$ and $u_{0}$ are the physical (i.e. bare) values of the
theory’s parameters, that is the starting points, at $x=0$, of the RG
transformation.

The critical power-law behaviour ruled by the RG fixed point can actually be
observed only if the flow carries the irrelevant (stable) variables close
enough to their fixed point while still remaining in the region of $T$ where
the linear approximation is valid; therefore, to estimate bounds for the
critical region we follow the flow using Eq. 32 and check whether or not at
the end of the flow the linear approximation is still valid. We start the flow
at $v_{0}\sim O(1)$ and $u_{0}\sim O(1)$, thus selecting a particular theory,
and at some $T_{0}$ such that the physical correlation length is much larger
than the lattice spacing, $\xi_{0}\gg 1/\Lambda$. The flow cannot be continued
beyond the point where the correlation length approaches the lattice spacing,
so we require $\xi(x_{stop})\simeq 1/\Lambda$. If we are in the critical
region, then $\xi_{0}\sim T_{0}^{-1/2}$ so the stop condition implies,

$\displaystyle T_{0}\sim\Lambda^{2}e^{-2x_{stop}},\qquad\text{or}\qquad
e^{x_{stop}}\sim\Lambda T_{0}^{-1/2}\ .$ (33)

We now require that at $T(x_{stop}),u(x_{stop}),v(x_{stop})$ the linear
approximation remains valid, which we can check by evaluating the $\beta$
functions Eq. 31 and comparing them with the linear approximation. From Eq. 31
we see that this needs $u(x_{stop})\ll 1$, which inserting the value of
$x_{stop}$ in Eq. 32 gives the condition,

$T_{0}\ll u_{0}^{-2}.$ (34)

For the validity of the result $\eta=0$ we need that $d_{\psi}$ from Eq. 18 at
$x_{stop}$ does not differ from $d_{\psi}^{*}$. This requires
$v^{2}(x_{stop})T^{3}(x_{stop})\ll 1$, that is,

$T_{0}\ll v_{0}^{-4/3}.$ (35)

Conditions Eq. 34 and Eq. 35 tell us that, for any reasonable value of
physical couplings $v_{0}$, $u_{0}$, we can choose a small enough —but finite—
physical temperature $T_{0}$, below which the theory will be in the critical
regime with free exponents. Considering that any reasonable values of the bare
physical parameters will always be of order one, conditions Eq. 34 and Eq. 35
tell us that the theory will have a rather comfortable critical region above
$T_{c}=0$. These calculations can be generalized for any $d>2$, hence we
conclude that the marginal theory is infrared-free Ryder (1996) with an upper
critical dimension $d_{c}=2$.

If we want to check if the conditions Eq. 34 and Eq. 35 are reasonable for
actual finite-size implementations of the marginal model and compare the
results with experiments we can see the work Cavagna  _et al._ (2022). With
just a single set of parameters with a low enough temperature it is possible
to reproduce scale-free correlations for all the experimental systems,
obtaining also a magnetization (that in Cavagna  _et al._ (2022) is called
polarization) which is compatible with the experimental ones Cavagna  _et al._
(2022). The actual critical exponents may be influenced by non-equilibrium
dynamical effects Cavagna  _et al._ (2013) but the scale-free phenomenology is
the same for data, Self Propelled Particle (SPP) simulations Cavagna  _et al._
(2022) and the equilibrium model here presented.

##  V Finite size scaling and numerical validation

Figure 5: Marginal model modulus susceptibility. The modulus susceptibility of
the marginal model, computed from Monte Carlo via Eq. 42, is shown for various
sizes. Left panel: susceptibility $\chi$ vs. temperature. Right panel:
rescaled susceptibility vs. the scaling variable $L^{d}T$ (Eq. 41). Error bars
are smaller than the symbol size. The collapse is very good, confirming that
the $T=0$ critical point is infrared-free and that the marginal field theory,
Eq. 9 correctly describes the modulus mode of the microscopic model Eq. 1.

In order to check the validity of the theoretical calculations we resort to
simulations and finite-size scaling to investigate the marginal critical point
at $T=0$. We first recall the basic results of finite-size scaling theory
above the upper critical dimension, since this case is different from the more
usual situation where finite-size scaling is applied, i.e. below the critical
dimension.

For conventional ferromagnetic/paramagnetic critical points in three
dimensions, the finite-size scaling for the susceptibility has the general
form Cardy (2012),

$\chi=L^{\gamma/\nu}f((T-T_{c})L^{1/\nu}),$ (36)

where $f(x)$ is a scaling function and $T_{c}$ is the critical temperature.
$\gamma$ and $\nu$ are the usual critical exponents Patashinskii and
Pokrovskii (1979). However, since our theory is infrared-free for $d=3$,
hyperscaling does not hold Brézin (1982) and Eq. 36 is not valid. To find the
correct scaling we start, following Brézin and Zinn-Justin (1985), from the
Landau-Ginzburg Hamiltonian Eq. 9 in its Landau approximation for a finite
system,

$\mathcal{H}=L^{d}\left\\{T\psi_{0}^{2}+vT^{3/2}\psi_{0}^{3}+uT\psi_{0}^{4}\right\\}$
(37)

where $\psi_{0}$ is a space-homogeneous field which represents the zero mode
of the theory. This amounts to neglecting diagrams with loops, which can be
shown not to contribute to the scaling Brézin and Zinn-Justin (1985). At 0
loops the susceptibility is given by,

$\chi\sim\frac{\int\mathrm{D}\psi_{0}\
\psi_{0}^{2}e^{-\mathcal{H}}}{\int\mathrm{D}\psi_{0}\ e^{-\mathcal{H}}}\ .$
(38)

Since we want to evaluate the integrals above via a saddle point it is
convenient to change variable $\psi_{0}\rightarrow\psi_{0}/(L^{d/2}T^{1/2})$
and write the action Eq. 37 as

$\mathcal{H}=\psi_{0}^{2}+\frac{v}{L^{d/2}}\psi_{0}^{3}+\frac{u}{L^{d}T}\psi_{0}^{4}.$
(39)

Then the susceptibility can be written as

$\chi=L^{d}f\left(\frac{v}{L^{d/2}},\frac{u}{L^{d}T}\right).$ (40)

For fixed $v$ and $u$, and for $L$ large enough such that we can ignore the
dependence of the function $f$ on its first argument we obtain,

$\chi=L^{d}f\left(L^{d}T\right).$ (41)

We therefore conclude that the marginal theory has an anomalous finite-size
scaling behaviour due to the fact that its critical point is on the basin of
attraction of an infrared-free fixed point. In general, infrared-free theories
(e.g. $\lambda\phi^{4}$ for $d>4$, which is studied for example in Wittmann
and Young (2014)) have an anomalous scaling that is usually
$\chi=L^{d/2}f((T-T_{c})L^{d/2})$ Brézin and Zinn-Justin (1985). For the
marginal model, however, the peculiar dependence on $T$ of the couplings leads
to a different scaling form, Eq. 41. One can include in this discussion higher
order terms of the marginal field theory Hamiltonian, but it can be easily
verified that their contribution is subleading with respect to $1/(L^{d}T)$.

Having obtained the correct scaling form for the marginal model (Eq. 41), we
can test it numerically. We performed Monte Carlo (MC) simulations Barkema and
Newman (2001) on a three-dimensional cubic lattice with periodic boundary
conditions, using the microscopic Hamiltonian Eq. 1, together with the classic
Botzmann weight Barkema and Newman (2001). We used lattices with side $L$
ranging from $10$ to $60$ and temperatures $T$ from $10^{-3}$ to $10^{-8}$,
while the parameters of Eq. 1 and Eq. 2 were fixed to $\lambda=J=1$. We
performed standard Metropolis MC with a temperature-dependent Cartesian
displacement for the spins (since their length is not fixed) such that the
acceptance probability of each move is around $50\%$. We discard the first
$2\times 10^{5}$ MC steps of every simulations, checking every time that we
are well above the equilibration time for that specific simulation. The
modulus susceptibility is computed via the fluctuation-dissipation relation
Amit and Martin-Mayor (1978),

$\chi=\frac{1}{TN}\sum_{i,j}\left(\left<|\bm{\sigma_{i}}||\bm{\sigma_{j}}|\right>-\left<|\bm{\sigma_{i}}|\right>\left<|\bm{\sigma_{j}}|\right>\right),$
(42)

averaging over the MC trajectory. The soundness of the numerical estimates is
checked by using the error analysis presented in Amit and Martin-Mayor (1978),
which makes use of time blocking of data to figure out the adequate simulation
length to prevent error underestimation. We make a small remark for clarity’s
sake: one might be confused by the fact that in the above equation we have
included a prefactor $1/T$, while we omitted it in the computation of the
anomalous finite-size scaling (Eq. 38 and following). This prefactor is
harmless in the usual case, but here, since the critical point is $T=0$, it is
crucial to get it right. However, if we look at the definition of the fields
we find that there is no inconsistency, since the field of Eq. 38 was already
rescaled by the square root of $T$ (see passage from Eq. 7 to Eq. 1). Hence,
if we compute the susceptibility from the field $\psi$ we do not have to
include the prefactor $1/T$ while it must be included when computing it from
the original spins $\bm{\sigma}$.

We show in Fig. 5 the susceptibility for the various system sizes. Using the
scaling variables (right panel), the collapse is quite satisfactory. This
result not only strongly supports the theoretical RG calculations, but also
confirms that indeed the Landau-Ginzburg Hamiltonian Eq. 9 is the correct
effective field theory to describe the modulus mode of the microscopic theory
Eq. 1, validating the approximations we made to obtain the field theory.

##  VI Conclusions

The marginal theory has been introduced as a novel form of speed control in
highly polarised animal groups, where scale-free correlations of both
orientation and speed clash with the standard $\mathrm{O}(n)$ ferromagnetic
scenario in the ordered phase, according to which the correlation length of
the modulus of the order parameter is finite in the whole symmetry-broken
phase. Marginal speed control solves this problem and it reproduces all the
experimental phenomenology Cavagna  _et al._ (2022) by using a bare potential
which has zero second derivative with respect to the modulus of the order
parameter, thus giving a zero-temperature fixed point. The relative
equilibrium field theory has both cubic and quartic vertices, so that a one-
loop RG analysis of the critical exponents is non-trivial; moreover, the
peculiar nature of the $T=0$ critical point demands that the explicit role of
the temperature be treated with care. In the end, the RG flow shows that the
critical exponents regulating correlation length and correlation function have
the free values $\nu=1/2$ and $\eta=0$. This is supported by the anomalous
finite-size scaling of the susceptibility found in Monte Carlo simulations,
which confirm that the marginal theory is free for $d=3$.

Assuming that our theoretical results also hold in the off-equilibrium case
(which is not certain, despite the weak off-equilibrium effects in starling
flocks), one interesting question is whether or not one may observe the free
critical exponents in real instances of bird flocks. As a matter of fact, this
may be quite tricky, at least with the current type of available data.
Previous investigations Cavagna  _et al._ (2013) have shown that the ever-
changing dynamical inflow of information at the boundary of the flocks may
change significantly the bulk decay form of the correlation function, in such
a way to screen completely the underlying critical exponents $\nu$ and $\eta$.
Hence the power law decay of the correlation function (which is linked with
$\eta$ Amit and Martin-Mayor (1978)) computed in the previous studies of
scale-free correlations in starling flocks Cavagna  _et al._ (2010) is not
reproducible with the model we present in this work, which does not take into
account dynamical out-of-equilibrium effects on the boundary of the system
Cavagna  _et al._ (2013). Moreover, it is not possible to measure
independently $\nu$ or $\gamma$ directly from the data Cavagna  _et al._
(2022); cavagna2018physics  since it is not clear how to change the
temperature (or an equivalent control parameter) of a single flock. Hence, to
test the critical exponents of the marginal model in the wild, one would need
a different kind of data, possibly obtained in less perturbed environments
than the currently available ones.

From a field-theoretical point of view, it would be interesting to investigate
further the co-existence of the zero-temperature critical point, $T=0$, which
makes the modulus fluctuations scale-free, and the standard finite critical
point, $T=T_{c}$, where all modes are scale-free. In the symmetry broken
phase, the standard transverse correlation length is infinite, due to the
Goldstone mode; however, there is a finite length scale in this phase, which
regulates the scaling relations below $T_{c}$, namely the Josephson
correlation length, $\xi_{J}$, which diverges at $T_{c}$, but decreases when
lowering the temperature below $T_{c}$ Josephson (1966). At the same time, the
modulus correlation length, $\xi$, increases in the marginal model when going
deeper in the ordered phase. The interplay of these two length scales, which
have opposite behaviour in $T$, and their impact on the scaling properties of
the theory, remains unclear to us and it is possibly worth of further
investigation.

## Acknowledgements

This work was supported by ERC Advanced Grant RG.BIO (contract n.785932) to
ACa. We thank Irene Giardina for discussions. We also thank Víctor Martín
Mayor for illuminating advice about finite-size scaling, and in particular for
pointing out references Brézin (1982) and Brézin and Zinn-Justin (1985).

##  Appendix A Mean Field approximation

###  A.1 Starting point and general idea

We generalize the theoretical analysis of Cavagna  _et al._ (2019) for a model
with $n$-component spins $\bm{\sigma}_{i}$. We want to obtain a closed
approximation for the Gibbs free energy Cavagna  _et al._ (2019),

$g(m)=-Jm^{2}-\bm{m}\cdot\bm{x}_{0}(\bm{m})-\frac{1}{\beta}\ln\int\mathrm{d}\bm{\sigma}\
e^{-\beta\left[S(\sigma)+\bm{x}_{0}(m)\cdot\bm{\sigma}\right]}$ (43)

where $S(\sigma)=J\sigma^{2}+V(\sigma)$ and $\bm{x}_{0}(\bm{m})$ is an
auxiliary variable, defined by the saddle point equation for
$N\rightarrow\infty$ Cavagna  _et al._ (2019), which reads,

$\bm{m}=\frac{\int\mathrm{d}\bm{\sigma}\
\bm{\sigma}e^{-\beta\left[S(\sigma)+\bm{x}_{0}\cdot\bm{\sigma}\right]}}{\int\mathrm{d}\bm{\sigma}\
e^{-\beta\left[S(\sigma)+\bm{x}_{0}\cdot\bm{\sigma}\right]}}.$ (44)

This equation and the integral in Eq. 43 can be solved numerically for any
value of $\beta$ Cavagna  _et al._ (2019), but here we are interested in the
asymptotic form of the free energy for large $\beta$ (or $T\rightarrow 0$),
hence we perform the integrals in Eq. 43 and Eq. 44, using once again the
saddle point method, this time for $\beta\rightarrow\infty$. Since we want all
the corrections up to $O(T^{2})$ we have to expand the exponential in each
integral up to that order. The saddle point equation for the integrals in
$\sigma$ introduces a new player, the saddle point value
$\bm{\sigma}_{0}(\bm{m})$,

$\nabla S(\bm{\sigma})\big{|}_{\bm{\sigma}_{0}(\bm{m})}+\bm{x}_{0}(\bm{m})=0$
(45)

Now we have to solve Eq. 44, which will give us an expression for
$\bm{\sigma}_{0}(\bm{m})$, then use Eq. 45 to find an expression for
$\bm{x}_{0}(\bm{m})$, and eventually plug everything into Eq. 43, in order to
express the explicit dependence of the Gibbs free-energy on the magnetization
(Eq. 5). If we look at Eq. 44 and Eq. 45, we can see that $\bm{m}$,
$\bm{\sigma}_{0}$ and $\bm{x}_{0}$ are parallel (or anti-parallel). Hence it
is convenient to write them as $\bm{m}=m\bm{\hat{w}}$,
$\bm{\sigma}_{0}=\sigma_{0}\bm{\hat{w}}$ and $\bm{x}_{0}=x_{0}\bm{\hat{w}}$,
where $|\bm{\hat{w}}|=1$. This simplifies our saddle-point calculations,
transforming many gradients and Hessian matrices into simple derivatives.

###  A.2 Computation of necessary terms

We want to expand the Gibbs free-energy Eq. 43 up to $O(T^{2})$ (that is
$1/\beta^{2}$). To accomplish that, we write Eq. 43 expanding the integral in
$\bm{\sigma}$, using the saddle point method, which reads,

$\displaystyle g(m)$
$\displaystyle=-Jm^{2}-x_{0}m-\frac{1}{\beta}\ln\left[\frac{e^{-\beta\left(S_{0}+x_{0}\sigma_{0}\right)}}{\det[S_{\alpha\beta}(\sigma_{0})]}\left(1+\frac{1}{\beta}B_{0}\right)\right]$
$\displaystyle=-Jm^{2}+x_{0}(\sigma_{0}-m)+S_{0}+$ $\displaystyle\ \ \
+\frac{1}{2\beta}\ln\det[S_{\alpha\beta}(\sigma_{0})]-\frac{1}{\beta^{2}}B_{0}$
(46)

where $S_{0}=S(\sigma_{0})$, $S_{\alpha\beta}$ is the Hessian matrix of $S$
and $B_{0}=B(\sigma_{0})$ is the first coefficient of the expansion in
$1/\beta$ of the integral in Eq. 43, which will be computed later on. If we
look at Eq. 44 and Eq. 45, we can write $\sigma_{0}$ and $x_{0}$ as expanding
around $m$,

$\displaystyle\sigma_{0}=m+\frac{1}{\beta}C_{m}+O(1/\beta^{2}),$ (47)
$\displaystyle
x_{0}=-S^{\prime}_{m}-\frac{1}{\beta}S^{\prime\prime}_{m}C_{m}+O(1/\beta^{2}),$
(48)

where $C_{m}=C(m)$ is the first coefficient of the expansion in $1/\beta$,
coming from Eq. 44, that will be computed later;
$S^{\prime}_{m}=S^{\prime}(m)$ and $S^{\prime\prime}_{m}=S^{\prime\prime}(m)$
are respectively the first and second derivative of $S$, from now on this
notation will be used for derivatives. If we plug Eq. 47 and Eq. 48 into Eq.
46 and keep all terms up to order $1/\beta^{2}$, we find,

$g(m)=V(m)+\frac{1}{\beta}\ln\det\left[S_{\alpha\beta}(m)\right]-\frac{1}{\beta^{2}}\Bigg{[}B(m)+\\\
+\frac{1}{2}S^{\prime\prime}(m)C^{2}(m)-\frac{C(m)\left[\det
S_{\alpha\beta}(m)\right]^{\prime}}{2\det\left[S_{\alpha\beta}(m)\right]}\Bigg{]}.$
(49)

Even if we want to compute the free energy up to $O(1/\beta^{2})$ we do not
need to compute the corresponding terms in the expansions Eq. 47 and Eq. 48,
because they cancel out once we substitute them into the free energy. To
compute the term of order $1/\beta$ in Eq. 49 we just need to evaluate the
determinant of the Hessian of the function $S$ at $m$, the Hessian matrix is
diagonal and gives,

$\det
S_{\alpha\beta}=S^{\prime\prime}(m)\left[\frac{S^{\prime}(m)}{m}\right]^{n-1}.$
(50)

If we take the logarithm and expand near $m^{2}\sim 1$ we find,

$\ln\det\left[S_{\alpha\beta}(m)\right]\sim(m^{2}-1)^{2},$ (51)

which is the term of order $T$ in Eq. 5. Going to next order, we can compute
the terms $B(m)$ and $C(m)$ by expanding the integrals of Eq. 43 and Eq. 44
using the saddle point method. After some calculations we find that the
leading order in $(m^{2}-1)$, for the term of order $1/\beta^{2}$ of the Eq.
49 is given by the term $B(m)$, which reads,

$B(m)\sim-\frac{S_{\alpha\beta\mu\nu}(m)}{24}\left<y_{\alpha}y_{\beta}y_{\mu}y_{\nu}\right>\sim-\frac{1}{8}\left[\frac{S^{\prime\prime\prime\prime}}{(S^{\prime\prime})^{2}}+...\right]\\\
\sim\mathrm{const.}+O(m^{2}-1),$ (52)

where $S_{\alpha\beta\mu\nu}$ is the fourth order derivatives tensor of $S$
and the $y_{\alpha}$ are Gaussian distributed variables with,

$\displaystyle\left<y_{\alpha}\right>$ $\displaystyle=0$ (53)
$\displaystyle\left<y_{\alpha}y_{\beta}\right>$
$\displaystyle=\left[\delta_{\alpha\beta}\frac{S^{\prime}(m)}{m}+\frac{m_{\alpha}m_{\beta}}{m^{2}}\left(S^{\prime\prime}(m)-\frac{S^{\prime}(m)}{m}\right)\right]^{-1},$
(54)

therefore we can compute the expected value
$\left<y_{\alpha}y_{\beta}y_{\mu}y_{\nu}\right>$ in Eq. 52 using Wick’s
theorem Zinn-Justin (2002) and the above equation for the covariance. In the
end, we obtain that the first non-vanishing term of order $T^{2}$, apart from
the constant, is of order $(m^{2}-1)$, as we can read in Eq. 5.

##  Appendix B Initial rescaling of fields

We now spell out the initial rescaling of fields that links the free energy
Eq. 7 with the free energy Eq. 9. If we compute with the mean field
approximation the single-particle variance of the spin modulus
$s_{i}=|\bm{\sigma}_{i}|$ we obtain,

$\displaystyle C_{0}=\left<s_{i}^{2}\right>-\left<s_{i}\right>^{2}\sim T,$
(55)

which can also be obtained by computing the connected correlation function in
the Gaussian approximation of Eq. 7,

$\left\langle\varphi(\bm{k})\varphi(\bm{k}^{\prime})\right\rangle^{0}_{c}=\delta(\bm{k}+\bm{k}^{\prime})\frac{T}{k^{2}+aT}\sim\delta(\bm{k}+\bm{k}^{\prime})\frac{C_{0}}{k^{2}+aT}.$
(56)

We see that the $T$ prefactor is problematic, since in the limit of vanishing
temperature the correlation function’s amplitude $C_{0}$ vanishes. We want to
investigate the regime of small $T$ where the modulus correlation length is
large, but we do not want the amplitude of the correlation function itself to
vanish. For this reason it seems natural to define a new field,

$\psi=\frac{\varphi}{\sqrt{T}},$ (57)

such that the correlation function of $\psi$ has a fixed amplitude for every
temperature,

$\displaystyle\left<\psi(\bm{k})\psi(\bm{k}^{\prime})\right>^{0}_{c}=\delta(\bm{k}+\bm{k}^{\prime})\frac{1}{k^{2}+aT}$
(58)

at least in the Gaussian and mean-field approximations. We do not expect great
deviations of $C_{0}$ from the mean field behaviour, given the finding
discussed in the main text that the zero-temperature critical point is ruled
by the Gaussian fixed point.

##  Appendix C Renormalization Group calculations

###  C.1 Diagrams at one-loop

The marginal field-theory Hamiltonian 9 has two non-Gaussian vertices: a cubic
one with coupling $vT^{3/2}$ ($\blacktriangle$) and a quartic one with
coupling $uT$ ($\bullet$) (see Fig. 1). We can combine these two vertices, in
order to form all the possible one-loop diagrams with an arbitrary number of
external legs. Since we evaluate the renormalized couplings only up to the
term $\psi^{4}$, we stop at four external legs. All the diagrams with more
than four external legs give a correction to higher order terms that we do not
include in Eq. 9 because they are RG-irrelevant. The diagrams that give a
contribution to the renormalization of temperature $T$ are,

$\displaystyle\raisebox{-0.5pt}{\includegraphics[width=65.04034pt]{square1.pdf}}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \
=12uT\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{q^{2}+T}$
(59)
$\displaystyle=-18v^{2}T^{3}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)\left[(\bm{k}-\bm{q})^{2}+T\right]}$
(60)

The renormalization of $vT^{3/2}$ comes from

$\displaystyle=-36uvT^{5/2}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)\left[(\bm{k}_{1}-\bm{q})^{2}+T\right]}$
(61)
$\displaystyle=36v^{3}T^{9/2}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)\left[(\bm{k}_{1}+\bm{q})^{2}+T\right]}\times$
$\displaystyle\times\frac{1}{\left[(\bm{k}_{1}+\bm{k}_{2}+\bm{q})^{2}+T\right]}$
(62)

finally the renormalization of $uT$ is due to

$\displaystyle=-36u^{2}T^{2}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)\left[(\bm{k}_{1}+\bm{k}_{2}-\bm{q})^{2}+T\right]}$
(63)
$\displaystyle=216uv^{2}T^{4}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)\left[(\bm{k}_{1}+\bm{k}_{2}-\bm{q})^{2}+T\right]}\times$
$\displaystyle\times\frac{1}{\left[(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3}-\bm{q})^{2}+T\right]}$
(64)
$\displaystyle=-162v^{4}T^{6}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)\left[(\bm{k}_{1}+\bm{q})^{2}+T\right]}\times$
$\displaystyle\times\frac{1}{\left[(\bm{k}_{1}+\bm{k}_{2}+\bm{q})^{2}+T\right]\left[(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3}+\bm{q})^{2}+T\right]}$
(65)

Dashed lines represent fields with momentum $k<\Lambda/b$ (off-shell), while
solid lines represent integrated fields with momentum $\Lambda/b<k<\Lambda$
(on-shell). Since we are interested in the corrections to the couplings of
momentum-independent terms ($\psi^{2}$, $\psi^{3}$ and $\psi^{4}$) we can
compute all these diagrams at zero external momentum and obtain the
corrections of Eqs. 15a, 15b and 15c.

###  C.2 The linear term

We have ignored in the Landau-Ginzburg free energy Eq. 9 a linear term in
$\psi$ that would have read $cT^{3/2}\psi$ (following the mean-field Gibbs
free energy Eq. 6 and using the rescaling $\psi=\varphi/\sqrt{T}$), where $c$
is a constant independent of temperature. We made this choice because the
linear term can be removed with a simple shift of the field by a constant
value. If we include the linear term in the theory, we find that the packed
constant $cT^{3/2}$ is corrected by the diagram,

$\displaystyle\raisebox{-0.5pt}{\includegraphics[width=151.76964pt]{lonely.pdf}}\
\ \ \ \ \ \ \
=3vT^{3/2}\int\limits_{\Lambda/b}^{\Lambda}\frac{\mathrm{d}^{d}q}{(2\pi)^{d}}\frac{1}{\left(q^{2}+T\right)}$
(66)

After the same calculations that we did for the other terms, the $\beta$
function of the parameter $c$ is,

$\beta_{c}=-\frac{c}{2}+3v\Lambda^{d-2}-18uc\Lambda^{d-2},$ (67)

which tells us that at the Gaussian fixed point $v^{*}=u^{*}=c^{*}=0$, the
parameter $c$ is also irrelevant. We also note that the linear term does not
produce any diagram which could contribute to the renormalization of the other
couplings, Eq. 15. Hence the phenomenology that we have described in the main
text does not change, even if we add the linear term. Also in this case the
differential equation $\derivative{c}{x}=\beta_{c}(c,v,u)$ is exactly solvable
and it reads (for $d=3$),

$c(x)=\frac{c_{0}e^{-x/2}+3\Lambda v_{0}e^{-x/2}(1-e^{-x})}{\left[1+12\Lambda
u_{0}(1-e^{-x})\right]^{3/2}},$ (68)

which means that for any starting condition the parameter $c$ flows to $0$.

###  C.3 Higher order couplings

To check for the relevance of terms of order higher than $\psi^{4}$ we need to
know the naive scaling dimension of their $T$-independent couplings. To see
that, go back to Eq. 6, which gives the dependence on $T$ of each coupling,
based on the mean-field Gibbs free energy. We find that, before the rescaling
$\psi=\varphi/\sqrt{T}$, the higher order terms can be written as,

$\mathcal{H}_{high}=\frac{1}{T}\int\mathrm{d}^{d}x\
\Bigg{\\{}u_{5}\varphi^{5}+u_{6}\varphi^{6}+\dots+u_{8}\varphi^{8}+\\\
+u_{9}T\varphi^{9}+u_{10}T\varphi^{10}+\dots+u_{n}T\varphi^{n}+\dots\Bigg{\\}},$
(69)

where every $u_{l}$ is a constant independent of $T$. We find this dependence
on $T$ from Eq. 49, where we can see that the lowest order (in $T$) that
generates the terms from $(m-1)^{5}$ up to $(m-1)^{8}$ is the first term (the
bare marginal potential) hence their couplings do not depend on $T$. On the
other hand, the lowest order term that generates powers from $(m-1)^{9}$ and
above is the logarithm of order $T$. Upon rescaling the field we have,

$\mathcal{H}_{high}=\int\mathrm{d}^{d}x\
\Bigg{\\{}u_{5}T^{3/2}\psi^{5}+\dots+u_{8}T^{3}\psi^{8}+\\\
+u_{9}T^{9/2}\psi^{9}+\dots+u_{n}T^{n/2}\psi^{n}+\dots\Bigg{\\}},$ (70)

which can be expressed as,

$u_{n}\psi^{n}\rightarrow\begin{cases}T^{n/2-1}\ \ \ \ \ \ \ \ \ \ \text{for}\
4<n<9\\\ T^{n/2}\ \ \ \ \ \ \ \ \ \ \ \ \ \text{for}\ n\geq 9\end{cases}$ (71)

Using the expressions above we can compute the naive scaling dimensions of the
$u_{n}$ couplings, which are

$\left[u_{n}\right]=\begin{cases}2+d\left(1-\frac{n}{2}\right)\ \ \ \ \ \ \ \
\ \ \text{for}\ 4<n<9\\\ d\left(1-\frac{n}{2}\right)\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \text{for}\ n\geq 9\end{cases}$ (72)

For $d=3$ we see that $[u_{n}]<0$ for all $n>4$, hence the Gaussian fixed
point $v^{*}=u^{*}=u_{n}^{*}=0$, remains stable even after adding higher-order
terms.

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# FluentEditor: text-based speech editing by considering
acoustic and prosody consistency

###### Abstract

Text-based speech editing (TSE) techniques are designed to enable users to
edit the output audio by modifying the input text transcript instead of the
audio itself. Despite much progress in neural network-based TSE techniques,
the current techniques have focused on reducing the difference between the
generated speech segment and the reference target in the editing region,
ignoring its local and global fluency in the context and original utterance.
To maintain the speech fluency, we propose a fluency speech editing model,
termed FluentEditor, by considering fluency-aware training criterion in the
TSE training. Specifically, the acoustic consistency constraint aims to smooth
the transition between the edited region and its neighboring acoustic segments
consistent with the ground truth, while the prosody consistency constraint
seeks to ensure that the prosody attributes within the edited regions remain
consistent with the overall style of the original utterance. The subjective
and objective experimental results on VCTK demonstrate that our FluentEditor
outperforms all advanced baselines in terms of naturalness and fluency. The
audio samples and code are available at
https://github.com/Ai-S2-Lab/FluentEditor.

Index Terms—  Speech Editing, Fluency Modeling, Acoustic Consistency, Prosody
Consistency

##  1 Introduction

Text-based speech editing (TSE) [1] allows for modification of the output
audio by editing the transcript rather than the audio itself. With the rapid
development of the internet, audio-related media sharing has become a
prevalent activity in our daily lives. Note that TSE can bring great
convenience to the audio generation process and be applied to a variety of
areas with personalized voice needs, including video creation for social
media, games, and movie dubbing.

Over the past few years, many attempts adopted text-to-speech (TTS) systems to
build neural network-based TSE models. For example, the CampNet [2] conducts
mask training on a context-aware neural network based on Transformer to
improve the quality of the edited voice. $A^{3}T$ [3] suggests an alignment-
aware acoustic and text pretraining method, which can be directly applied to
speech editing by reconstructing masked acoustic signals through text input
and acoustic text alignment. More recently, the diffusion model has gradually
become the backbone of the NN-based TSE with remarkable results. For example,
EdiTTS [4] takes the diffusion-based TTS model as the backbone and proposes a
score-based TSE methodology for fine-grained pitch and content editing.
FluentSpeech [5] proposes a context-aware diffusion model that iteratively
refines the modified mel-spectrogram with the guidance of context features.

However, during training, the existing approaches just constrain the Euclidean
Distance [6] between the mel-spectrum to be predicted and the ground truth to
ensure the naturalness of TSE. Although they consider the use of contextual
information to mitigate the over-smoothing problem of edited speech, their
objective functions are not designed to ensure fluent output speech [7, 8]. We
consider two challenges to be tackled for effective speech fluency modeling.
1) Acoustic Consistency: the smoothness of the concatenation between the
region to be edited and its neighboring regions should be close to the real
concatenation point [9]. 2) Prosody Consistency: the prosody style of the
synthesized audio in the region to be edited needs to be consistent with the
prosody style of the original utterance [10, 11].

To address the above issues, we propose a novel fluency speech editing scheme,
termed FluentEditor, by introducing the acoustic and prosody consistency
training criterion to achieve natural and fluent speech editing. Specifically,
1) To achieve the acoustic consistency, we design the Acoustic Consistency
Loss $\mathcal{L}_{AC}$ to calculate whether the variance at the boundaries is
close to the variance at the real concatenation points. 2) To achieve the
prosody consistency, we introduce the Prosody Consistency Loss
$\mathcal{L}_{PC}$ to let the high-level prosody features of the synthesized
audio in the region to be edited be close to that of the original utterance.
The high-level prosody features are extracted by the pre-trained GST-based
prosody extractor[11]. The subjective and objective results on the VCTK [12]
dataset show that the acoustic and prosody consistency of the FluentEditor is
significantly better than the advanced TSE baselines, while the proposed
FluentEditor can ensure a high degree of fluency like real speech.

The main contributions of this work can be summarized as follows: 1) We
propose a novel fluency speech editing scheme, termed FluentEditor; 2) We
adopt the diffusion model as the backbone and introduce Acoustic and Prosody
Consistency Losses to conduct the fluency modeling for TSE; 3) The proposed
model outperforms all advanced TSE baselines in terms of naturalness and
fluency.

##  2 FluentEditor: Methodology

We formulate the proposed FluentEditor, a TSE model that ensures speech
fluency by considering acoustic and prosody consistency. We first introduce
the overall workflow, then further elaborate the fluency-aware training
criterion and the run-time inference.

###  2.1 Overall Workflow

As shown in Fig.1, our FluentEditor adopts the mask prediction-based diffusion
network as the backbone, which consists of a text encoder, and a spectrogram
denoiser. The spectrogram denoiser seeks to adopt the Denoising diffusion
probabilistic model (DDPM) to learn a data distribution $p(\cdot)$ by
gradually denoising a normally distributed variable through the reverse
process of a fixed Markov Chain of length $T$.

Assume that the phoneme embedding of the input phoneme sequence is
$X=(X_{1},\ldots,X_{|X|})$ and the acoustic feature sequence for $X$ is
$\hat{Y}=(\hat{Y}_{1},\ldots,\hat{Y}_{|\hat{Y}|})$. The masked acoustic
feature sequence $\hat{Y}_{mask}=Mask(\hat{Y},\lambda)$ is obtained by
replacing the random spans of $\hat{Y}$ with the random vector according to a
$\lambda$ probability. Specifically, the text encoder aims to extract the
high-level linguistic feature $\mathcal{H}_{X}$ for $X$. The spectrogram
denoiser then aggregates the $\mathcal{H}_{X}$ and the condition input $C$ to
guide the reverse process of the diffusion model $\Theta(Y_{t}|t,C)$ ($t\in
T$), where $Y_{t}$ is a noisy version of the clean input $\hat{Y}_{0}$.
Similar to [5], the condition input $C$ consists of the frame-level linguistic
feature $\mathcal{H}^{f}_{X}$, acoustic feature sequence $\hat{Y}$, masked
acoustic feature sequence $\hat{Y}_{mask}$, speaker embedding $e_{spk}$ and
the pitch embedding $e_{pitch}$. In the generator-based diffusion models,
$p_{\theta}(Y_{0}|Y_{t})$ is the implicit distribution imposed by the neural
network $f_{\theta}(Y_{t},t)$ that outputs $Y_{0}$ given $Y_{t}$. And then
$Y_{t-1}$ is sampled using the posterior distribution $q(Y_{t-1}|Y_{t},Y_{0})$
given $Y_{t}$ and the predicted $Y_{0}$.

To model speech fluency, we design acoustic consistency loss
$\mathcal{L}_{AC}$ and prosody consistency loss $\mathcal{L}_{PC}$ on the
basis of the original reconstruction loss, to ensure that the acoustic and
prosody performance of speech generated in the editing area is consistent with
the context and the original utterance. For reconstruction loss, we follow [5]
and employ Mean Absolute Error (MAE) and the Structural Similarity Index
(SSIM) [13] losses to calculate the difference between $Y_{0}$ and the
corresponding ground truth segment $\hat{Y}_{0}$. In the following subsection,
we will introduce $\mathcal{L}_{AC}$ and $\mathcal{L}_{PC}$ in detail.

Fig. 1: The overall workflow of FluentEditor. The total loss function includes
Reconstruction Loss, and Acoustic and Prosody Consistency Losses.

###  2.2 Fluency-Aware Training criterion

####  2.2.1 Acoustic Consistency Loss

The acoustic consistency loss $\mathcal{L}_{AC}$ employs smoothness
constraints at both the left and right boundaries for the predicted acoustic
feature $Y_{0}$. We compare the variance, $\Delta\varrho_{Y_{0}}^{L}$ and
$\Delta\varrho_{Y_{0}}^{R}$, for the left and right boundaries with
$\Delta\hat{\varrho}_{\hat{Y_{0}}}^{L/R}$ of ground truth speech at the
corresponding boundaries to serve as the proxy of the overall smoothness
$\mathcal{L}_{AC}$.

Specifically, $\mathcal{L}_{AC}$ consists of $\mathcal{L}_{AC}^{L}$ and
$\mathcal{L}_{AC}^{R}$, and we use Mean Squared Error (MSE)[14] to measure the
proximity between the target segment and the ground truth:

$\displaystyle\mathcal{L}_{AC}$
$\displaystyle=\mathcal{L}_{AC}^{L}+\mathcal{L}_{AC}^{R}$ (1)
$\displaystyle=\rm{MSE}(\Delta\varrho_{Y_{0}}^{L},\Delta\hat{\varrho}_{\hat{Y_{0}}}^{L})+\rm{MSE}(\Delta\varrho_{Y_{0}}^{R},\Delta\hat{\varrho}_{\hat{Y_{0}}}^{R})$

Note that the Euclidean distance between two adjacent frames is obtained by
the smoothness extractor. Take $\Delta\varrho_{Y_{0}}^{L}$ as an example,

$\vspace{-1mm}\Delta\varrho_{Y_{0}}^{L}=\varrho_{{Y_{0}}^{L}}-\varrho_{Y^{{L}^{pre}}_{0}}\vspace{-2mm}$
(2)

where $Y^{{L}^{pre}}_{0}$ denotes the speech frame preceding the left boundary
of the masked region. In other words, the ending frame of the adjacent non-
masked region is on the left side. To comprehensively capture the statistical
properties of the audio signal, we utilize variance to describe the feature
information of each Mel spectrogram frame, denoted as $\varrho_{{Y_{0}}^{L}}$
and $\varrho_{Y^{{L}^{pre}}_{0}}$.

Similarly, we compute the smoothness constraint for the right
boundary$\mathcal{L}_{AC}^{R}$, where $Y^{{R}^{nex}}_{0}$ denotes the speech
frame succeeding the right boundary of the masked region, in other words, the
starting frame of the adjacent non-masked region on the right side.

####  2.2.2 Prosody Consistency Loss

The prosody consistency loss $\mathcal{L}_{PC}$ is responsible for capturing
the prosody feature $\mathcal{H}_{Y_{0}}^{P}$ from the predicted region
$Y_{0}$ while also analyzing the overall prosody characteristics
$\hat{\mathcal{H}}_{\hat{Y}}^{P}$ present in the original speech, then employ
the MSE loss to conduct the prosody consistency constraints.

$\vspace{-1mm}\mathcal{L}_{PC}=\text{MSE}(\mathcal{H}_{Y_{0}}^{P},\hat{\mathcal{H}}_{\hat{Y}}^{P})\vspace{-2mm}$
(3)

Note that the prosody features $\mathcal{H}_{Y_{0}}^{P}$ and
$\hat{\mathcal{H}}_{\hat{Y}}^{P}$ are obtained by the pre-trained prosody
extractor. Specifically, the prosody extractor utilizes the reference encoder
[11] of the Global Style Token (GST) [11] model to convert $Y_{0}$ and
$\hat{Y}$ into high-level prosody features with fixed length for easy
comparison.

$\mathcal{H}_{Y_{0}}^{P}=\text{GST}(Y_{0}),\ \ \
\hat{\mathcal{H}}_{\hat{Y}}^{P}=\text{GST}(\hat{Y})\vspace{-1mm}$ (4)

Lastly, following [5], the total loss function is the sum of reconstruction
loss and two new loss functions, $\mathcal{L}_{AC}$ and $\mathcal{L}_{PC}$,
across all non-contiguous masked regions, since the mask region in a sentence
may include multiple non-contiguous segments [5]. In a nutshell,
$\mathcal{L}_{AC}$ and $\mathcal{L}_{PC}$ of the FluentEditor are introduced
to ensure fluent speech with consistent prosody.

###  2.3 Run-time Inference

In run-time, given the original text and its speech, the user can edit the
speech by editing the text. Note that we can manually define modification
operations (i.e., insertion, replacement, and deletion). The corresponding
speech segment of the edited word in the given text is treated as the masked
regions in Fig. 1. Similar to [5], our FluentEditor reads the edited text and
the remaining acoustic feature $\hat{Y}-\hat{Y}_{mask}$ of the original speech
to predict the $Y_{0}$ for the edited word. At last, the $Y_{0}$ and its
context $\hat{Y}-\hat{Y}_{mask}$ are concatenated as the final fluent output
speech.

Table 1: Objective and subjective evaluation results of comparative study. * means the value achieves suboptimal. Method | Objective Evaluation | Subjective Evaluation (FMOS)
---|---|---
|  MCD $(\downarrow)$ |  STOI $(\uparrow)$ |  PESQ $(\uparrow)$ | Insertion | Replacement
Ground Truth |  NA |  NA |  NA |  4.37 $\pm$ 0.05 |  4.42 $\pm$ 0.01
CampNet |  3.85 |  0.53 |  1.38 |  3.89 $\pm$ 0.01 |  3.94 $\pm$ 0.03
$A^{3}T$ |  3.79 |  0.76 |  1.59 |  3.82 $\pm$ 0.03 |  3.83 $\pm$ 0.02
FluentSpeech |  3.50 |  0.79 |  1.93 |  4.02 $\pm$ 0.04 |  4.04 $\pm$ 0.01
FluentEditor (Ours) |  3.47 |  0.81 |  1.85* |  4.25 $\pm$ 0.03 |  4.26 $\pm$ 0.01
Table 2: Objective and subjective results of ablation study. Method |  C-FMOS |  MCD $(\downarrow)$
---|---|---
FluentEditor |  0.00 |  3.47
w/o $\mathcal{L}_{AC}$ |  -0.16 |  3.48
w/o $\mathcal{L}_{PC}$ |  -0.21 |  3.51

##  3 Experiments and Results

###  3.1 Dataset

We validate the FluentEditor on the VCTK [12] dataset, which is an English
speech corpus uttered by 110 English speakers with various accents. Each
recording is sampled at 22050 Hz with 16-bit quantization. The precise forced
alignment is achieved through Montreal Forced Aligner (MFA) [15]. We partition
the dataset into training, validation, and testing sets, randomly with 98%,
1%, and 1%, respectively.

###  3.2 Experimental Setup

The configurations of text encoder and spectrogram denoiser are referred to
[5]. The diffusion steps $T$ of the FluentEditor system is set to 8. Following
GST [11], the prosody extractor comprises a convolutional stack and an RNN.
The dimension of the output prosody feature of the GST-based prosody extractor
is 256. Following [3], we adopt a random selection strategy, with a fixed
masking rate of 80%, for masking specific phoneme spans along with their
corresponding speech frames. The pre-trained HiFiGAN [16] vocoder is used to
synthesize the speech waveform. We set the batch size is 16. The initial
learning rate is set at $2\times 10^{-4}$, and the Adam optimizer [17] is
utilized to optimize the network. The FluentEditor model is trained with 2
million training steps on one A100 GPU.

###  3.3 Evaluation Metric

For subjective evaluation, We conduct a Mean Opinion Score (MOS) [18]
listening evaluation in terms of speech fluency, termed FMOS. Note that FMOS
allows the listener to feel whether the edited segments of the edited speech
are fluent compared to the context. We keep the text content and text
modifications consistent among different models to exclude other interference
factors, only examining speech fluency. Furthermore, Comparative FMOS (C-FMOS)
[18] is also used to conduct the ablation study. For objective evaluation, we
utilize MCD [19], STOI [20], and PESQ [21] to measure the overall quality of
the edited speech.

###  3.4 Comparative Study

We develop four neural TSE systems for a comparative study, that includes: 1)
CampNet [2] propose a context-aware mask prediction network to simulate the
process of text-based speech editing; 2) $\boldsymbol{A^{3}T}$ [3] propose the
alignment-aware acoustic-text pre-training that takes both phonemes and
partially-masked spectrograms as inputs; 3) FluentSpeech [5] takes the
diffusion model as backbone and predict the masked feature with the help of
context speech; and 4) FluentEditor (Ours) designs the acoustic and prosody
consistency losses. We also add the Ground Truth speech for comparison. Note
that two ablation systems, that are “$\boldsymbol{w/o}$
$\boldsymbol{\mathcal{L}_{AC}}$” and “$\boldsymbol{w/o}$
$\boldsymbol{\mathcal{L}_{PC}}$”, are built to validate the two new losses.

###  3.5 Main Results

Objective results: We select 400 test samples from the test set randomly and
report the objective results in the second to fourth columns of Table 1. Note
that we follow [5] and just measure the objective metrics of the masked region
using the reconstructed speech. We observe that our FluentEditor achieves the
best performance in terms of overall speech quality. For example, the MCD and
STOI values of FluentEditor obtain optimal results and PESQ achieves
suboptimal results among all systems. It suggests that the FluentEditor
performs proper acoustic feature prediction for the speech region to be
edited. Note that objective metrics do not fully reflect the human perception
[22], we further conduct subjective listening experiments.

Subjective results: For FMOS evaluation, we selected 50 audio samples from the
test set and invited 20 listeners to evaluate speech fluency. Following [23],
we test the insertion and replacement operations and present the FMOS results
in the last two columns of Table 1. We find that FluentEditor consistently
achieves superior fluency-related perceptual scores. For example, FluentEditor
obtains the top FMOS value of 4.25 for insertion and 4.26 for replacement,
that very close to that of ground truth. This demonstrates the effectiveness
of the fluency-aware training criterion. By considering the acoustic and
prosody consistency constraints, our FluentEditor allows for weakening the
editing traces and improving the prosody performance of the edited speech.

###  3.6 Ablation Study

To further validate the contribution of our $\mathcal{L}_{AC}$ and
$\mathcal{L}_{PC}$ respectively, the subjective and objective ablation
results, of insertion and replacement, are reported in Table 2. We follow the
previous section to prepare the samples and listeners. It’s observed that the
C-FMOS and MCD values of these two ablation systems both drop when we remove
the $\mathcal{L}_{AC}$ and $\mathcal{L}_{PC}$ respectively, indicating that
the acoustic and prosody consistency constraints play a vital role in
enhancing both the naturalness and fluency of the edited speech.

(a) FluentSpeech

(b) FluentEditor

Fig. 2: Visualizations of the generated mel-spectrograms by FluentEditor and
FluentSpeech baseline.

###  3.7 Visualization Analysis

As illustrated in the Fig.2, we visualize the mel-spectrograms produced by
FluentEditor and the FluentSpeech baseline111Due to space limits, we just
report the FluentSpeech baseline. More visualization results and speech
samples are referred to our website:
https://github.com/Ai-S2-Lab/FluentEditor. . The red boxes indicate the random
masked and its reconstructed speech segment of utterance “Scottish Women
appear at Eden Court, Inverness, tonight.”. We can see that FluentEditor can
generate mel-spectrograms with richer frequency details compared with the
baseline, resulting in natural and expressive sounds, which further
demonstrates the effectiveness of acoustic and prosody consistency losses.
Nevertheless, we recommend that the reader listen to our speech samples1 to
visualize the advantages.

##  4 Conclusion

In this paper, we introduce a novel text-based speech editing (TSE) model,
termed FluentEditor, that involves two novel fluency-aware training criterions
to improve the acoustic and prosody consistency of edited speech. The acoustic
consistency loss $\mathcal{L}_{AC}$ to calculate whether the variance at the
boundaries is close to the variance at the real concatenation points, while
the prosody consistency loss $\mathcal{L}_{PC}$ to let the high-level prosody
features of the synthesized audio in the region to be edited be close to that
of the original utterance. The objective and subjective experiments on VCTK
demonstrate that incorporating $\mathcal{L}_{AC}$ and $\mathcal{L}_{PC}$
yields superior results and ensures fluent speech with consistent prosody. In
future work, we will consider the multi-scale consistency and further improve
the FluentEditor architecture.

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