# Interface roughening in nonequilibrium phase-separated systems

M. Besse  Service de Physique de l’Etat Condensé, CEA, CNRS Université Paris-
Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France  Sorbonne Université, CNRS,
Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France
G. Fausti  Service de Physique de l’Etat Condensé, CEA, CNRS Université Paris-
Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France  Max Planck Institute for
Dynamics and Self-Organization, 37077 Göttingen, Germany  M.E. Cates  DAMTP,
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WA, UK  B. Delamotte  Sorbonne Université, CNRS, Laboratoire de
Physique Théorique de la Matière Condensée, 75005 Paris, France  C. Nardini
Service de Physique de l’Etat Condensé, CEA, CNRS Université Paris-Saclay,
CEA-Saclay, 91191 Gif-sur-Yvette, France  Sorbonne Université, CNRS,
Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France

(August 30, 2024)

###### Abstract

Interfaces of phase-separated systems roughen in time due to capillary waves.
Due to fluxes in the bulk, their dynamics is nonlocal in real space and is not
described by the Edwards-Wilkinson or Kardar-Parisi-Zhang equations, nor their
conserved counterparts. We show that in the absence of detailed balance, the
phase-separated interface is described by a new universality class that we
term $\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ. We compute the
associated critical exponents via one-loop renormalization group, and
corroborate the results by numerical integration of the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation. Deriving the
effective interface dynamics from a minimal field theory of active phase
separation, we finally argue that the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ universality class
generically describes liquid-vapor interfaces in active systems.

###### pacs:

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The roughening of interfaces is among the best-studied problems in statistical
mechanics Krug (1997); Barabási  _et al._ (1995). Early theoretical
investigations Peters  _et al._ (1979); Plischke and Rácz (1984); Jullien and
Botet (1985) were concerned with the Eden model Eden (1958), proposed to
describe the shape of cell colonies, and with the ballistic deposition model
Family and Vicsek (1985). Kardar, Parisi and Zhang (KPZ) Kardar  _et al._
(1986) discovered an important universality class for growing rough
interfaces, by adding the lowest order nonlinearity to the continuum Edwards-
Wilkinson (EW) model in which height fluctuations are driven by nonconserved
noise and relax diffusively Edwards and Wilkinson (1982). Despite its fame,
KPZ does not describe all roughening interfaces: in the presence of
conservation laws, the KPZ nonlinearity can be forbidden and novel
universality classes arise Krug (1997); Sun  _et al._ (1989a); Caballero  _et
al._ (2018).

A notable category of interfaces, that stands out from all these well-known
universality classes, are those arising in phase-separated systems, which
roughen due to thermally activated capillary waves v. Smoluchowski (1908);
Rowlinson and Widom (2013); Evans (1979); Wertheim (1976); Weeks (1977).
There, because of fluxes in the bulk phases, liquid-vapor or liquid-liquid
interfaces have nonlocal dynamics in real space Bray  _et al._ (2001);
Shinozaki (1993). In the absence of fluid motion, small amplitude capillary
waves with wavenumber ${\mathbf{q}}$ relax by diffusion at a rate
$\tau^{-1}=\sigma\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}^{3}$, with
$\sigma$ proportional to the interfacial tension Bray  _et al._ (2001).
Denoting by $\hat{h}(\mathbf{x},t)$ the height of such an interface above a
$d$-dimensional plane, the dynamics is most conveniently written in Fourier
space for $h({\mathbf{q}},t)$

$\displaystyle\partial_{t}h({\mathbf{q}},t)=-\sigma\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}^{3}h({\mathbf{q}},t)+\sqrt{2D\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}}\eta({\mathbf{q}},t),$
(1)

where $\eta$ is a Gaussian white noise such that

$\displaystyle\langle\eta({\mathbf{q}}_{1},t_{2})\eta({\mathbf{q}}_{2},t_{2})\rangle=\delta({\mathbf{q}}_{1}+{\mathbf{q}}_{2})\delta(t_{1}-t_{2})\,.$
(2)

The linear evolution equation (1) can be written in real space as
$\partial_{t}\hat{h}=G\star\nabla^{2}\hat{h}+\eta_{R}$ where $\star$ stands
for the spatial convolution,
$\langle\eta_{R}(\mathbf{x}_{1},t_{1})\eta_{R}(\mathbf{x}_{2},t_{2})\rangle\sim
2DG(\mathbf{x}_{1}-\mathbf{x}_{2})\delta(t_{1}-t_{2})$ and the kernel $G$ is
long-ranged: $G(\mathbf{x})\sim|\mathbf{x}|^{-d-1}$ at large distances.

The interface in passive phase-separated fluids must be described, at least in
the stationary state, by an equation that respects detailed balance. In this
case the interface is a subset of degrees of freedom within a thermally
equilibrated state of the full, phase-separated, system and hence is itself in
equilibrium. The fluctuation-dissipation theorem then means that any
nonlinearity that enters Eq. (1) is of the form
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}\,\delta\mathscr{F}_{I}/\delta
h_{{\mathbf{q}}}$ for some free energy functional $\mathscr{F}_{I}[h]$. A
simple dimensional analysis argument then shows that there exists no
nonlinearity correcting Eq. (1) which is relevant in the renormalization group
(RG) sense, for interfaces of dimension $d=1,2$. Hence the stationary dynamics
of diffusive (i.e., without momentum conservation) interfaces between phase-
separated passive liquids is described by mean-field critical exponents.
Dimensional analysis Täuber (2014) of Eq. (1) then gives $z=3$ and
$\chi=(z-d-1)/2$, in terms of which spatial and temporal correlations scale as
$\langle\hat{h}(\mathbf{x},t)\hat{h}(\mathbf{x}^{\prime},t)\rangle\sim|\mathbf{x}-\mathbf{x}^{\prime}|^{2\chi}$
and
$\langle\hat{h}(\mathbf{x},t)\hat{h}(\mathbf{x},t^{\prime})\rangle\sim|t-t^{\prime}|^{2\chi/z}$.

We shall show in the following that this conclusion changes in phase-separated
systems that lack detailed balance. Important examples are found in active
matter, where elemental units such as self-propelled colloids, bacteria or
cells, extract non-thermal energy from the environment and dissipate it to
self-propel Marchetti  _et al._ (2013). Active liquid-vapor systems are known
to display a phenomenology impossible in equilibrium Gompper  _et al._ (2020);
for example, phase separation arises even in the absence of any attraction
among particles Tailleur and Cates (2008); Fily and Marchetti (2012);
Stenhammar  _et al._ (2014); Cates and Tailleur (2015). Moreover, steady-state
currents can be present Tjhung  _et al._ (2018), and phase-separated states
can be sustained even when the interfacial tension is negative Tjhung  _et
al._ (2018); Fausti  _et al._ (2021); Bialké  _et al._ (2015); Caballero and
Marchetti (2022). In the simplest cases their large-scale interfacial
properties are nonetheless similar to those in passive fluids Cates and
Tailleur (2015); Wittkowski  _et al._ (2014); Nardini  _et al._ (2017);
Partridge and Lee (2019); Solon  _et al._ (2018); Gnan and Maggi (2022). In
this situation, it was recently assessed on the basis of numerical simulations
of particle models Lee (2017); Sussman  _et al._ (2018), and field-theoretical
analysis Fausti  _et al._ (2021), that capillary wave theory holds. Indeed,
some of us have recently shown that small amplitude, long wavelength capillary
waves obey Eq. (1), although activity changes the capillary interfacial
tension (and can make it negative in some parameter regimes, not considered
here) Fausti  _et al._ (2021).

In this Letter we show that when the underlying dynamics breaks detailed
balance, as happens generically in active systems, a new universality class
describes the roughening of the liquid-vapor interface. This is because a new
nonlinearity, which is RG-relevant for interfacial dimension $d\leq 2$, can
appear in Eq. (1). We study the ensuing equation, that we term
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ, by one-loop RG
analysis perturbatively for small $\varepsilon=2-d$. We compute the critical
exponents and confirm our conclusions by numerical simulations of the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation in $d=1$. We
further argue that the $\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ
universality class includes the liquid-vapor interface of active systems
undergoing bulk phase separation. We do so by considering Active Model B+
(AMB+) Tjhung  _et al._ (2018), a minimal continuum description of active
systems undergoing phase separation, and showing how
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ emerges as the
associated equation for the liquid-vapor interface.

To assess which nonlinearities might modify Eq. (1), let us first recall the
symmetries that have to be respected. First, we should impose invariance under
rotations and translations, as well as under a shift in the origin of the
reference frame, which translates to $h({\mathbf{q}},t)\to
h({\mathbf{q}},t)+(2\pi)^{d}\delta({\mathbf{q}})C$ in Fourier space, for any
$C\in\mathbb{R}$. Second, the total amounts of liquid and vapor do not change
during roughening, implying that the ‘total height’
$\int_{\mathbf{x}}\leavevmode\nobreak\ \hat{h}$ is constant. We further assume
that chiral symmetry (invariance under $\mathbf{x}\to-\mathbf{x}$) is not
broken. Under these symmetries, the KPZ nonlinearity is forbidden, as it does
not conserve the total height. Moreover, the nonlinearities captured in
existing models of conserved surface growth, such as cKPZ Sun  _et al._
(1989b); Krug (1997) and cKPZ+ Caballero  _et al._ (2018) are found to be RG-
irrelevant by dimensional analysis.

Inspecting Eq. (1) it appears evident that
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$ plays the role of a
mobility and it is thus natural to consider nonlinearities in the form

$\displaystyle\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}\int_{\sum_{i=1}^{n}{\mathbf{q}}_{i}={\mathbf{q}}}g({\mathbf{q}}|{\mathbf{q}}_{1},...,{\mathbf{q}}_{n})h({\mathbf{q}}_{1})...h({\mathbf{q}}_{n}),$
(3)

where the integral is over ${\mathbf{q}}_{1},...,{\mathbf{q}}_{n}$ with the
constraint $\sum_{i=1}^{n}{\mathbf{q}}_{i}={\mathbf{q}}$ (nonlinearities
depending on the frequencies $\omega,\omega_{1},...,\omega_{n}$ could be
considered as well without changing any conclusion). Note that the prefactor
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$ ensures that the total
height of the interface is conserved. Assuming that $g$ is analytic in all its
arguments, and imposing the symmetries mentioned above, the most relevant
nonlinearity that can modify Eq. (1) is
$g({\mathbf{q}}|{\mathbf{q}}_{1},{\mathbf{q}}_{2})=\lambda_{1}i{\mathbf{q}}_{1}\cdot
i{\mathbf{q}}_{2}$. This term is relevant for $d<d_{c}=2$. We are thus led to
investigate the following equation, that we term
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ:

$\displaystyle\partial_{t}h=-\sigma\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}^{3}h+\frac{\lambda_{1}}{2}\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}\mathcal{F}[|\nabla\hat{h}|^{2}]+\sqrt{2D\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}}\eta,$
(4)

where $\mathcal{F}[\cdot]$ stands for the Fourier transform.

We next study the $\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ
equation by RG analysis to one loop, perturbatively in $\varepsilon=2-d>0$.
Within the Martin-Siggia-Rose formalism Täuber (2014), the action $S$
associated with Eq. (4) reads

$\displaystyle
S=\int_{Q}\tilde{h}(-Q)\left(G_{0}^{-1}(Q)h(Q)+A(Q)-D\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}\tilde{h}(Q)\right)$
(5)

where $\tilde{h}$ is a response field, $Q=(\omega,{\mathbf{q}})$,
$G_{0}(Q)^{-1}=-i\omega+\sigma\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}^{3}$
and
$A(Q)=(\lambda_{1}/2)\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}\int_{Q_{1}+Q_{2}=Q}{\mathbf{q}}_{1}\cdot{\mathbf{q}}_{2}h(Q_{1})h(Q_{2})$.

We show in Fig. 1 the one-particle-irreducible diagrams to one loop associated
with the action in Eq. (5) and compute them in sup . The diagram in Fig. 1(a)
gives a nonvanishing contribution that renormalizes $\sigma$, while that of
Fig. 1(b) only gives irrelevant contributions to the renormalization of the
noise. Diagrams in Fig. 1(c) and 1(d) exactly cancel, as suggested by
generalizing the argument of Janssen (1997). (We have confirmed this by
explicitly computation.) We furthermore show in sup  that the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation is stable
under one-loop perturbative RG flow: any nonlinearity there generated is in
the form of Eq. (3), with $g$ analytic in its arguments, and no linear term
more relevant than
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}^{3}h({\mathbf{q}})$ is
generated.

Figure 1: One-loop diagrams for the renormalization of the propagator (a),
noise (b) and nonlinearity (c,d).

We thereby obtain in sup  the following RG flow for the reduced coupling
constant $\mathdutchcal{g}=D\lambda_{1}^{2}/\sigma^{3}$

$\displaystyle\Lambda\frac{\mathrm{d}\mathdutchcal{g}}{\mathrm{d}\Lambda}=\varepsilon\mathdutchcal{g}-\frac{3}{8}K_{2}\mathdutchcal{g}^{2},\qquad\varepsilon=2-d.$
(6)

where $\Lambda$ is the momentum scale, $K_{d}=S_{d}/(2\pi)^{d}$ and
$S_{d}=2\pi^{d/2}/\Gamma(d/2)$. As expected from dimensional analysis, the
Gaussian fixed point ($\mathdutchcal{g}^{\star}=0$) is the only one of the RG
flow for $d\geq 2$: here the mean-field exponents exactly describe the
interface properties at large scales. For $d<2$, a new attractive fixed point
emerges at $\mathdutchcal{g}^{\star}=8\varepsilon/(3K_{2})$. At this fixed
point, the critical exponents are

$\displaystyle
z=3-\frac{\varepsilon}{3},\qquad\qquad\chi=\frac{\varepsilon}{3}\,.$ (7)

These exponents describe the new
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ universality class to
one loop, that is, to first order in $\varepsilon$.

To test these predictions, we numerically integrate the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ in $d=1$ using a
pseudo-spectral code with $2/3$ dealiasing procedure. The use of a
pseudospectral code is particularly convenient for the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation because it
has computational complexity $\mathcal{O}(L\log L)$, where $L$ is the system
size, while, given its nonlocality in real space, a finite difference code
would have complexity $\mathcal{O}(L^{2})$. The results presented were
obtained with spatial discretization $\Delta x=1$, time discretization $\Delta
t=10^{-2}$. We quantified the interfacial width via

$\displaystyle
W^{2}(t,L)=\frac{1}{L}\int_{\mathbf{x}}\hat{h}^{2}(\mathbf{x},t),$ (8)

starting from a flat interface and averaging it over noise realizations. Our
RG analysis predicts that, while roughening, $W^{2}\sim t^{2\chi/z}$ and, for
a system of finite size $L$, $W^{2}$ eventually saturates in time to a value
$W_{\infty}^{2}(L)\equiv W(\infty,L)^{2}\sim L^{2\chi}$. Measuring the
interfacial width as a function of time and its saturated value as a function
of $L$ allows us to extract both critical exponents, $z$ and $\chi$.

Figure 2: Interfacial width $W^{2}(t)$ as a function of time for different
system sizes showing a roughening law $W^{2}(t)\sim t^{0.28}$. Parameters used
$\sigma=1.0,D=0.1,\lambda=2.0$, corresponding to the reduced coupling constant
$\mathdutchcal{g}=0.4$. Each curve was obtained by averaging over $500$ noise
realisations. The inset contains the same data but rectified by $t^{0.28}$.

In the case of the linear theory ($\lambda_{1}=0$) we measure, as expected,
$2\chi/z=1/3$ and $2\chi=1$. We then perform simulations with $\lambda_{1}\neq
0$. In Fig. 2 we plot $W^{2}(t,L)$ as a function of time and various system-
sizes both in log-log and in a redressed plot (inset); the latter is a
stringent test of the scaling behavior. Our measurement gives $2\chi/z\simeq
0.28$. We then plot the saturated value of the interfacial width in Fig. 3
both for $\lambda_{1}=0$ and $\lambda_{1}=2$, finding that $2\chi=1$ for
$\lambda_{1}=0$ and $2\chi\simeq 0.78$ for $\lambda_{1}=2$. The values of the
measured critical exponents should be compared with those obtained from the
one-loop RG analysis, Eq. (7), which gives $2\chi/z\simeq 1/4$ and $2\chi=2/3$
in $d=1$. The agreement is very good, given that the RG predictions are
obtained to first order in $\varepsilon$, whereas $\varepsilon=1$ in our
simulations. These numerical results offer strong evidence for the presence of
a perturbatively accessible fixed point for the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation.

Figure 3: $W_{\infty}^{2}(L)\sim L^{2\chi}$ as a function of system size $L$.
Continuous lines: RG predictions either at the Gaussian (blue) or
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ (red) fixed points.
Points: results of numerical integration of the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation with
$\lambda_{1}=0$ (blue) and $\lambda_{1}=2$ (red), corresponding to bare
coupling $\mathdutchcal{g}=0.4$. Other parameters: $\sigma=1.0$ and $D=0.1$.
Error bars are smaller than the symbols’size. Dashed lines are power-law fits.

Our final task is to evidence that the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ nonlinearity is indeed
generically present for the dynamics of interfaces in phase-separated active
systems. To do this, we consider continuum scalar field theories for active
phase separation Wittkowski  _et al._ (2014); Tjhung  _et al._ (2018); Thomsen
_et al._ (2021). These are generalizations of Model B Hohenberg and Halperin
(1977); Chaikin and Lubensky (2000); Bray  _et al._ (2001), the standard
large-scale description of phase separation in diffusive passive fluids
(without momentum conservation). Like Model B they address the dynamics of a
conserved scalar density field $\phi$ in $d+1$ dimensions which switches
steeply from positive to negative values on crossing the interface from liquid
to vapor. Unlike Model B, the active versions take into account that detailed
balance is broken at the microscale. The ensuing minimal theory is called AMB+
Nardini  _et al._ (2017); Tjhung  _et al._ (2018), and includes all terms that
break detailed balance up to order $\mathcal{O}(\nabla^{4},\phi^{2})$ Tjhung
_et al._ (2018):

$\displaystyle\partial_{t}\phi$ $\displaystyle=$
$\displaystyle-\nabla\cdot\left(\mathbf{J}+\sqrt{2DM}\bm{\Lambda}\right),$ (9)
$\displaystyle{\bf J}/M$ $\displaystyle=$
$\displaystyle-\nabla\mu_{\lambda}+\zeta(\nabla^{2}\phi)\nabla\phi,$ (10)
$\displaystyle\mu_{\lambda}[\phi]$ $\displaystyle=$
$\displaystyle\frac{\delta\mathscr{F}}{\delta\phi}+\lambda|\nabla\phi|^{2}.$
(11)

Here $\mathscr{F}=\int
d\mathbf{r}\,\left[f(\phi)+\frac{K(\phi)}{2}|\nabla\phi|^{2}\right]$,
$f(\phi)$ is a double-well local free energy, and $\bm{\Lambda}$ is a vector
of zero-mean, unit-variance, Gaussian white noises. Standard Model B is
recovered at vanishing activity ($\lambda=\zeta=0$), unit mobility $M=1$ and
constant noise level $D$ Hohenberg and Halperin (1977). Here we shall retain
the choice $M=1$, constant $D$ and further assume for simplicity that $K$ is
also constant. Our analysis below assumes that the interfacial tensions
determining the Ostwald process Tjhung  _et al._ (2018) and the relaxation of
capillary waves Fausti  _et al._ (2021), which in active systems can differ,
are both positive so that the system undergoes bulk phase separation rather
than microphase separation.

The linear description of capillary waves, Eq. (1), has been classically
derived for passive fluids starting from Model B Bray  _et al._ (2001), and
more recently for active fluids starting from AMB+ Fausti  _et al._ (2021).
This is done by assuming that $\phi$ evolves quasi-statically with respect to
fluctuations of the interfacial height, equivalent to an ansatz
$\phi(\mathbf{x},y,t)=\varphi(y-\hat{h}(\mathbf{x},t))$, which is exact at
leading order in $h$ and ${\mathbf{q}}$ Fausti  _et al._ (2021). As detailed
in sup , we extend this procedure to obtain the nonlinear terms that correct
Eq. (1):

$\displaystyle\sum_{n=0}^{\infty}\int_{{\mathbf{q}}_{1},\mathbf{x},\mathbf{x}_{1}}(-1)^{n}\frac{(\hat{h}(\mathbf{x})-\hat{h}(\mathbf{x}_{1}))^{n}}{2n!}\mathinner{\\!\left\lvert{\mathbf{q}}_{1}\right\rvert}^{n}\left\\{A_{n}({\mathbf{q}}_{1})\partial_{t}\hat{h}(\mathbf{x})+\zeta
D_{n}({\mathbf{q}}_{1})\mathbf{\nabla}_{\mathbf{x}}^{2}\hat{h}\right\\}e^{-i{\mathbf{q}}\cdot\mathbf{x}_{1}-i{\mathbf{q}}_{1}\cdot\mathbf{x}_{1}+i{\mathbf{q}}_{1}\cdot\mathbf{x}}=-\sigma_{\lambda}{\mathbf{q}}^{2}h_{\mathbf{q}}+\chi_{\mathbf{q}},$
(12)

where the Stratonovich convention is used, expressions for the $h$-independent
factors $A_{n}({\mathbf{q}})$, $D_{n}({\mathbf{q}})$, $\sigma_{\lambda}$ are
given in sup , and $\chi_{\mathbf{q}}$ can be found as a sum of Gaussian
noises sup . To leading order in $h$ and ${\mathbf{q}}$, Eq. (12) reduces to
Eq. (1), where the interfacial tension $\sigma$ is proportional to the
capillary-waves interfacial tension $\sigma_{\rm cw}$ Fausti  _et al._ (2021).

Notably, the $\lambda_{1}$-term of Eq. (4) is not present at bare level in Eq.
(12). Furthermore, Eq. (12) contains nonlinearities in the form of Eq. (3)
with $g$ singular. In order to perform the RG analysis of Eq. (12), we
transform it to the Ito convention in sup , following a standard procedure
Gardiner  _et al._ (1985); Lau and Lubensky (2007); Cates  _et al._ (2022). We
then find the canonical dimension of the nonlinear terms and show that the
singular nonlinearities in Eq. (12) are all RG-irrelevant sup . Moreover, by
computing one-loop diagrams we further show in sup  that $\lambda_{1}\neq 0$
is generated under RG from the nonlinearities of Eq. (12). These results
strongly suggest that the liquid-vapor interface of AMB+ belongs to the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ universality class.
However a complete proof would require us to derive the analog of Eq. (12)
allowing for a dependence of $\phi$ on interfacial curvature, and then show
that no relevant singular nonlinearity is generated by the associated RG flow.
This goes beyond the scope of this Letter. Preliminary numerical simulations
of AMB+, which will be presented elsewhere, do however indicate that the
interface roughens accordingly to $W^{2}(t)\sim t^{0.28}$ as expected from the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ universality class.

In conclusion, we have introduced a minimal field theory, termed the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation, to describe
the roughening of interfaces in nonequilibrium phase-separated systems lacking
momentum conservation. The
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ equation differs from
the standard description of roughening interfaces (either EW, KPZ or their
conserved counterparts) because diffusive fluxes in the bulk cause the
interfacial dynamics to be nonlocal in real space. We discovered a nontrivial
fixed point of the RG flow for interfacial dimension $d<2$. This should
control active, phase-separated interfaces in $d+1=2$ bulk dimensions. We
characterized this new universality class, computing its critical exponents by
one-loop RG and numerical simulations. We finally gave evidence that the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ class includes
interfacial roughening in phase-separated active systems, by explicitly
deriving the effective interface equation from a scalar active field theory
for the particle density in $d+1$ bulk dimensions, and showing that the
$\mathinner{\\!\left\lvert{\mathbf{q}}\right\rvert}$KPZ nonlinearity thereby
emerges.

Previous studies of roughening of the liquid-vapor interface in active
particle models concluded that $z\sim 2$ as in the Edwards-Wilkinson
universality class Wysocki  _et al._ (2016); Lee (2017); Patch  _et al._
(2018). We speculate that such disagreement might arise if these particle
models undergo bubbly phase separation Tjhung  _et al._ (2018) instead of bulk
phase separation as addressed here. (Notably, in Patch  _et al._ (2018), vapor
bubbles are indeed visible in the liquid phase). Local events in which a vapor
bubble ‘pops’ through the interface do not conserve the total height
$\int_{\mathbf{x}}\leavevmode\nobreak\ \hat{h}$, and also create overhangs,
evading our description – at least on scales smaller than the largest bubbles
present. More work is needed to clarify this aspect. Beyond active systems,
our results might also describe the roughening of interfaces in other
nonequilibrium phase-separating systems, such as granular materials Oyarte
_et al._ (2013).

###### Acknowledgements.

CN thanks F. Caballero for a discussion in the initial stages of this work. GF
was supported by the CEA NUMERICS program, which has received funding from the
European Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement No 800945. BD acknowledges the support from
the French ANR through the project NeqFluids (grant ANR-18-CE92-0019). Work
funded in part by the European Research Council under the Horizon 2020
Programme, ERC grant agreement number 740269 and by the National Science
Foundation under Grant No. NSF PHY-1748958, NIH Grant No. R25GM067110 and the
Gordon and Betty Moore Foundation Grant No. 2919.02. MEC is funded by the
Royal Society. CN acknowledges the support of an Aide Investissements d’Avenir
du LabEx PALM (ANR-10-LABX-0039-PALM).

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