# Disorders can induce continuously varying universal scaling in driven
systems

Astik Haldar<EMAIL_ADDRESS><EMAIL_ADDRESS>Theory Division,
Saha Institute of Nuclear Physics, HBNI, Kolkata 700064, India  Abhik Basu
<EMAIL_ADDRESS><EMAIL_ADDRESS>Theory Division, Saha Institute of
Nuclear Physics, HBNI, Kolkata 700064, India

###### Abstract

We elucidate the nature of universal scaling in a class of quenched disordered
driven models. In particular, we explore the intriguing possibility of whether
coupling with quenched disorders can lead to continuously varying universality
classes. We examine this question in the context of the Kardar-Parisi-Zhang
(KPZ) equation, with and without a conservation law, coupled with quenched
disorders having distributions with pertinent structures. We show that when
the disorder is relevant in the renormalisation group sense, the scaling
exponents can depend continuously on a dimensionless parameter that defines
the disorder distribution. This result is generic, and holds for quenched
disorders with or without spatially long ranged correlations, as long as the
disorder remains a “relevant perturbation” on the pure system in the
renormalisation group sense and a dimensionless parameter naturally exists in
its distribution. We speculate on its implications for generic driven systems
with quenched disorders, and compare and contrast with the scaling displayed
in the presence of annealed disorders.

##  I Introduction

Classification of the physics of nonequilibrium systems at long time and
length scales into universality classes remains a theoretically challenging
task. The standard universality classes in critical dynamics are quite robust
to detailed-balance violating perturbations Täuber  _et al._ (2002). In
contrast, genuine nonequilibrium dynamic phenomena, having non-Gibbsian steady
states, are found to be rather sensitive to all kinds of perturbations.
Notable examples include driven diffusive systems Schmittmann and Zia (1972).
For instance, for the Kardar-Parisi-Zhang (KPZ) equation, anisotropic
perturbations are found to be relevant in all dimensions $d\geq 2$ Täuber and
Frey (2002).

Equilibrium systems either in the vicinity of critical points or in broken
symmetry phases for systems with continuous symmetries show universal scaling
that depend only on the spatial dimension $d$ and the symmetry of the order
parameter (e.g., Ising, XY, nematic etc) Fisher (1983), but are independent of
the material parameters that define the model. Prominent exceptions are the
two-dimensional (2D) XY model and its related models, where the long
wavelength universal properties are controlled not by a fixed point but a
fixed line, and as a result, the scaling exponents of the relevant correlation
functions exhibit a continuous dependence on the stiffness parameters Chaikin
and Lubensky (2000). Furthermore, equilibrium dynamics close to critical
points also show dynamic universality though the dynamic scaling exponents,
which characterise the time-dependence of the unequal-time correlation
functions, and now may also depend upon the presence or absence of
conservation laws and the non-dissipative terms in the dynamical equations
Hohenberg and Halperin (1977). Genuinely nonequilibrium systems can also show
the surprising features of continuously varying nonequilibrium universal
properties, which are paramatrised by dimensionless parameters that naturally
appear in the formulations of the dynamical models. Well-known examples
include stochastically forced fully developed three-dimensional
magnetohydrodynamic turbulence Basu (2004) and stochastically coupled Burgers
equations Basu and Frey (2004, 2009). In all these examples, the relevant
dimensionless parameters appear in the definition of the noise variances. More
recently, it has been suggested that a nearly phase-ordered collection of
diffusively mobile, active XY spins on a substrate can be stable, and the
relevant scaling exponents that describe the phase and density correlation
functions can vary continuously with certain anharmonic coupling constants
that appear in the hydrodynamic equations of motion Haldar  _et al._ (2021a,
b).

Disorder is known to affect the large-scale, long time universal properties of
condensed matter systems. Depending upon the time-scales, disorders are
classified into two classes - quenched and annealed. Quenched disorders are
frozen in time and do not thermalise even in equilibrium systems. In contrast,
annealed disorders time-evolve and thermalise in the long time limit in
equilibrium systems. Effects of quenched disorder on the universal properties
of equilibrium systems are well-documented. For example, quenched disorders
that locally affect the critical temperature lead to new universality classes
different from the corresponding pure model; see Refs. Krey (1977); Weinrib
and Halperin (1983); Mukherjee and Basu (2020). Unlike quenched disorders,
annealed disordered equilibrium systems may be viewed as pure systems
supplemented by additional thermal degrees of freedom. This may result into
different static and dynamic universality, depending upon the systems under
consideration Chaikin and Lubensky (2000); Banerjee and Basu (2015); Banerjee
_et al._ (2015).

Effects of quenched disorders on nonequilibrium systems can be strikingly more
complex. For instance, the random Gaussian-distributed quenched columnar
disordered Kardar-Parisi-Zhang (KPZ) equation, which in one dimension (1D)
reduces to the periodic totally asymmetric simple exclusion process (TASEP)
with short-ranged Gaussian-distributed quenched disordered hopping rates is
affected by the disorder only in certain special limits of the model
parameters (in 1D which is equivalent to the half-filled limit in the TASEP
language), leading to new universal scaling behaviour in all dimensions $d$.
Else, pure KPZ universality holds Haldar and Basu (2020); Haldar (2021). In a
recent study, it has been shown that the conserved KPZ (CKPZ) equation, when
coupled with a particular choice of quenched disorder can modify the scaling
behaviour of the pure CKPZ equation in some cases Mukherjee (2021).
Nonetheless, the question of disorder-induced continuous universality in
driven models remains largely unexplored till the date.

In this article, we explore the possibility of disorder-induced continuous
universality in driven models. In the absence of any general theoretical
framework for nonequilibrium systems, it is useful to study simple models
where such questions can be explored systematically by using analytically
tractable calculations. Insights drawn from such studies should be useful to
enhance general understanding of scaling in nonequilibrium setting. To that
end, in this work, we have studied two models - one with a conservation law,
the conserved KPZ (CKPZ) equation and the other without any conservation, the
KPZ equation, both coupled with “orientational” disorders, which couples with
the local gradient of the height field. By carefully choosing the disorder
distributions, we show that the resulting universality classes, when they
depend upon the quenched disorders, can vary continuously with a dimensionless
parameter that characterises the disorder distributions. In addition, we
briefly study a version of annealed disordered CKPZ equation, and show that
similar continuously varying scaling exponents can be found under certain
circumstances. The remainder of the paper is organised as follows. In Section
II, we review the pure KPZ and CKPZ universality classes. Next in Section III,
we introduce the disordered KPZ and CKPZ equations that we use here. In
Section IV, we elucidate the scaling properties of the quenched disordered KPZ
and CKPZ equations. In Section V, we summarise our results. We have used one-
loop dynamic renormalisation group (RG) calculations for our work. We discuss
some of the technical details in Appendix for interested readers. We briefly
analyse the case of annealed disordered CKPZ equation in Appendix.

##  II Equations for fluctuating surfaces

We consider a fluctuating surface without overhangs that may be moving (i.e.,
growing) on average, or not moving. Microscopically, these growth processes
are described by local dynamics or local update rules, and are generic
examples of nonequilibrium driven systems. An interface in $d+1$-dimensional
hyperspace is characterised by variations of height $h({\bf x},t)$, where
${\bf x}(x_{1},x_{2},....,x_{d})$ is a position vector in the $d$-dimensional
hypersurface. We consider fluctuating surfaces having dynamics unaffected by
their absolute heights with respect to specific base planes. In other words,
the dynamics is invariant under constant shifts of $h$. The height
fluctuations display dynamic scaling Barabási and Stanley (1995), and the
associated time-dependent correlation function in the steady states has a
scaling form

$\displaystyle C(r,t-t^{\prime})\equiv\langle[h({\bf x},t)-h({\bf
x^{\prime}},t^{\prime})]^{2}\rangle=r^{2\chi_{h}}\varphi\left(\frac{r^{z}}{t-t^{\prime}}\right)$
(1)

in the long wavelength limit, where $r\equiv|\bf x-x^{\prime}|$; $\chi_{h}$
and $z$ are roughness exponent and dynamical exponent respectively, which
classify the universality class; $\varphi$ is a dimensionless scaling
function. While the values of the scaling exponents are independent of the
model parameters, they of course vary from one universality class to another.
Two such well-known universality classes are those associated with the KPZ and
CKPZ equations, which we review briefly below.

###  II.1 KPZ universality class

The KPZ equation, originally proposed as a surface growth model Kardar  _et
al._ (1986), is the paradigmatic nonequilibrium model that shows
nonequilibrium phase transitions at $d>2$. It is given by

$\displaystyle\frac{\partial h}{\partial
t}=\nu_{1}\nabla^{2}h-\frac{\lambda_{1}}{2}(\bm{\nabla}h)^{2}+\xi_{h}.$ (2)

Here, $\nu_{1}>0$ is a diffusivity, and $\lambda_{1}$ is a nonlinear
coefficient. Furthermore, $\xi_{h}$ is noise with Gaussian distribution and
zero mean, that is added to describe the inherent stochasticity of the
dynamics. It is a white noise, since $h$ in (2) obeys a non-conserved
dynamics. Stochastic noise $\xi_{h}$ has a variance

$\displaystyle\langle\xi_{h}({\bf x},t)\xi_{h}({\bf
x^{\prime}},t^{\prime})\rangle=2D_{1}\delta^{d}({\bf
x-x^{\prime}})\delta(t-t^{\prime}),$ (3)

where $D_{1}>0$ is the noise strength. In addition to its application as a
surface growth model, (2) serves as the active or nonequilibrium hydrodynamic
model for a smoothly varying phase, when there is no other hydrodynamic
variable present Altman  _et al._ (2015), and an active fluid membrane without
momentum conservation Cagnetta  _et al._ (2020) .

Equation (2) is invariant under the transformation ${\bf x}\rightarrow{\bf
x}-\lambda_{1}{\bf a}t,\,t\rightarrow t,\,h\rightarrow h+{\bf a\cdot
x}+\frac{\lambda_{1}}{2}|{\bf a}|^{2}t$, known as the tilt invariance Barabási
and Stanley (1995). This in turn gives an exact exponent relation
$\chi_{h}+z=2$ Barabási and Stanley (1995); Frey and Täuber (1994). In 1D the
scaling exponents are found exactly as a consequence of the tilt invariance
(that holds at all dimensions) and the Fluctuation-Dissipation-Theorem that
holds only at 1D Barabási and Stanley (1995). This gives dynamic exponent
$z=1/2$ and roughening exponent $\chi_{h}=3/2$ exactly, corresponding to a 1D
rough surface. In 2D, which is the lower critical dimension of this model,
there is only a “rough” phase which is perturbatively inaccessible.
Furthermore, the KPZ equation has smooth phase above 2D for low enough noise,
the scaling property of which is identical to the linear Edward-Wilkinson (EW)
equation Barabási and Stanley (1995) with $z=2,\chi_{h}=\frac{2-d}{2}$. As the
noise strength is increased, the KPZ equation undergoes a phase transition
from a smooth to a perturbatively inaccessible rough phase in dimension $d>2$.

###  II.2 CKPZ universality class

The CKPZ equation, which is essentially the conserved analogue of the KPZ
equation, forms a universality class distinct from the KPZ equation. This of
course is not surprising, since in nonequilibrium systems, the presence or
absence of conservation laws not only affect the dynamic scaling exponents,
they can in principle affect the static exponents (e.g., the roughness
exponent) as well, in contrast to equilibrium systems. In the CKPZ equation,
the height field $h({\bf x,t})$ of an interface of a volume conserving system,
a single-valued function measured with respect to an arbitrary base plane,
follows a generic conservation laws:

$\displaystyle\partial_{t}h=-\bm{\nabla}\cdot{\bf J},$ (4)

where ${\bf J}$, the current, has the following form Sun  _et al._ (1989)

$\displaystyle{\bf
J}=\bm{\nabla}\left[\nu_{2}\nabla^{2}h-\frac{\lambda_{2}}{2}(\bm{\nabla}h)^{2}\right].$
(5)

We note that current $\bf J$ in (5) is constructed in such a way that ${\bf
J}({\bf k=0},t)=0$, where $\bf k$ is a wavevector. With this, the CKPZ
equation takes the form Sun  _et al._ (1989)

$\displaystyle\frac{\partial h}{\partial
t}=-\nabla^{2}\left[\nu_{2}\nabla^{2}h-\frac{\lambda_{2}}{2}(\bm{\nabla}h)^{2}\right]+\eta_{h}.$
(6)

Here, $\eta_{h}$ is a conserved noise that models stochastic nature of
dynamics. It is assumed to be zero-mean and Gaussian distributed with a
variance

$\displaystyle\langle\eta_{h}({\bf x},t)\eta_{h}({\bf
x^{\prime}},t^{\prime})\rangle=2D_{2}(-\nabla^{2})\delta^{d}({\bf
x-x^{\prime}})\delta(t-t^{\prime}).$ (7)

In the linear limit, i.e., with $\lambda_{2}=0$, Eq. (6) reduces to the
Mullins-Herring (MH) equation for linear MBE processes Mullins (1957); Herring
(1950). Equation (6) is not tilt invariant, and hence, it has no exact
exponent relation in contrast to the KPZ equation Janssen (1997). Results from
one-loop RG study shows that $d=2$ is the upper critical dimension, and below
2D, in an $\epsilon=2-d$ expansion one finds the scaling exponents as
$z=4-\frac{\epsilon}{3},\,\chi_{h}=\frac{\epsilon}{3}$. This corresponds to a
rough phase. At 2D the interface is logarithmically rough and above 2D, the
surface is smooth, with long wavelength scaling properties statistically
identical to those obtained from the linear MH equation. Unlike the KPZ
equatiom, the CKPZ equation does not admit a smooth-to-rough transition for
strong coupling 111A generalised CKPZ equation has recently been proposed that
contains an additional nonlinear term which is as relevant (in a RG sense) as
the existing nonlinear term in (6). This admits a roughening transition; see
F. Caballero et al, Strong Coupling in Conserved Surface Roughening: A New
Universality Class?, Phys. Rev. Lett. 121, 020601 (2018). We do not discuss
that here..

##  III Disordered KPZ and CKPZ equations

We now minimally couple the KPZ Eq. (2) and CKPZ Eq. (6) equations with
“orientational” quenched disorder to address the question on universality we
raised above. We call it an orientational disorder, since it, a random
quenched disordered vector field ${\bf V}({\bf x})$, whose statistics is given
below, couples with the fluctuation in the local orientation of the height
field given by ${\bm{\nabla}}h$. Further, $V_{i}$ can in general have both
irrotational and solenoidal parts. Quenched vector $V_{i}$ is assumed to be
zero-mean, Gaussian-distributed with a variance

$\displaystyle\langle V_{i}({\bf x})V_{j}({\bf
x^{\prime}})\rangle=\left[2D_{T}P_{ij}+2D_{L}Q_{ij}\right]|{\bf
x-x^{\prime}}|^{-\alpha}.$ (8)

Here $P_{ij}$ and $Q_{ij}$ are the transverse (solenoidal) and longitudinal
(irrotational) projection operators respectively. In the Fourier space, these
are given by

$\displaystyle P_{ij}({\bf
k})=\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}},\,Q_{ij}({\bf
k})=\frac{k_{i}k_{j}}{k^{2}}.$ (9)

Thus, $P_{ij}({\bf k})$ and $Q_{ij}({\bf k})$, respectively, project any
vector they operate on to directions normal and parallel to $\bf k$. Evidently
at 1D, $P_{ij}\equiv 0$ identically. Noise strengths $D_{T}$ and $D_{L}$ are
positive definite. The exponent $\alpha$ paramatrises the measure of how long
range or how spatially correlated the quenched disorder is; we take
$0\leq\alpha<d$ (in this range the short distance cutoff implied in the
Fourier transforms can be smoothly taken to zero). In the Fourier space the
disorder correlation (8) takes the form:

$\displaystyle\langle V_{i}({\bf k},\omega)V_{j}({\bf
k^{\prime}},\omega^{\prime})\rangle=\left[2D_{T}P_{ij}+2D_{L}Q_{ij}\right]k^{-\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\delta^{d}({\bf
k+k^{\prime}})\delta(\omega^{\prime})\delta(\omega),$ (10)

where $\mu\equiv d-\alpha>0$ for spatially long-ranged correlated disorders.
In the more familiar short-ranged disorder case

$\langle V_{i}({\bf x})V_{j}({\bf
x^{\prime}})\rangle=\left[2D_{T}P_{ij}+2D_{L}Q_{ij}\right]\delta^{d}(\bf{x-x^{\prime}})$
(11)

in real space. In the Fourier space, this gives (10) with $\mu=0$. Clearly,
when $D_{L}=D_{T}$, the rhs of (11) is proportional to $\delta_{ij}$ Mukherjee
(2021). Further, in the extreme limits when $D_{T}=0$, the quenched vector
field ${\bf V}$ is irrotational, whereas for $D_{L}=0$, it is solenoidal.

In the next Sections, we study the effects of the coupling of $V_{i}$ with the
KPZ and CKPZ equations, and explore their universal scaling properties, in
particular, the dependence on the dimensionless ratio $\gamma\equiv
D_{T}/D_{L}$.

###  III.1 KPZ equation with quenched disorder

Given the status of the KPZ equation as a paradigmatic nonequilibrium model
that shows phase transitions, it is of great theoretical interest to study
whether or not quenched disorder can modify the universality classes of the
pure system. The quenched disordered version of the KPZ equation - the KPZ
equation with minimally coupled orientational quenched disorder that we use is

$\displaystyle\frac{\partial h}{\partial
t}=\nu_{1}\nabla^{2}h-\frac{\lambda_{1}}{2}(\bm{\nabla}h)^{2}+\kappa_{1}(\bm{V}\cdot\bm{\nabla}h)+\xi_{h}.$
(12)

The disorder-dependent nonlinear term with coefficient $\kappa_{1}$ is the
leading order nonlinear term that respects the invariance under a constant
shift of $h$ (see Ref. Kakin  _et al._ (2021) for a similar coupling). It
breaks the tilt invariance of the pure KPZ equation. As a result, Eq. (12)
lacks any exact exponent identity, in direct contrast with the pure KPZ
equation. Further, the sign of $\kappa_{1}$ is arbitrary. Equation (12)
reduces to the pure KPZ equation (2) for $\kappa_{1}=0$. The additive annealed
noise $\xi_{h}$ is assumed to be zero-mean, Gaussian-distributed with a
variance given by (3). Just as the pure KPZ equation (2) is the nonequilibrium
hydrodynamic model of a phase in the absence of any other hydrodynamic
variables, (12) is the nonequilibrium hydrodynamic model of a smoothly varying
phase field in the presence of orientational quenched disorder in the absence
of any other hydrodynamic variables, or of an active fluid membrane with
quenched disorder with broken tilt invariance and without momentum
conservation. Thus this study should provide insight to the role of quenched
disorders in these systems.

###  III.2 CKPZ equation with quenched disorder

To study the CKPZ equation minimally coupled with orientational quenched
disorder, we use the disordered version of the CKPZ equation proposed and
studied in Ref. Mukherjee (2021). The form of the current $\bf J$
corresponding to this disordered CKPZ equation reads

$\displaystyle{\bf
J}=\bm{\nabla}\left[\nu_{2}\nabla^{2}h-\frac{\lambda_{2}}{2}(\bm{\nabla}h)^{2}-\kappa_{2}(\bm{V}\cdot\bm{\nabla}h)\right].$
(13)

The coefficient $\kappa_{2}$ is the coupling constant of the disorder-
dependent leading order nonlinear term that respects the invariance under a
constant shift of $h$; $\kappa_{2}$ can take any sign. The choice of disorder
coupling that current ${\bf J}({\bf k}\rightarrow 0,t)\rightarrow 0$ at the
thermodynamic limit. With $\kappa_{2}=0$, the model reduces to the pure CKPZ
equation (6). With (13), the disordered CKPZ equation reads

$\displaystyle\partial_{t}h=\nabla^{2}\left[-\nu_{2}\nabla^{2}h+\frac{\lambda_{2}}{2}(\bm{\nabla}h)^{2}+\kappa_{2}(\bm{V}\cdot\bm{\nabla}h)\right]+\eta_{h}.$
(14)

The noise $\eta_{h}$ satisfy the same as Eq. (7).

In the next section we analyse the scaling properties of these disordered
models.

##  IV Universal scaling in the disordered models

The universality classes of the pure KPZ and CKPZ equations are well-
established. We now set out to find whether the quenched disorder is a
relevant perturbation on these universality classes, and if so, what the new
universality classes are. The nonlinear terms present in (12) and (14)
preclude any exact analysis of the problem, and necessitate use of
perturbative approaches. The naïve perturbation theory produces diverging
corrections to the model parameters in the long wavelength limit. We use here
the dynamic RG framework to systematically handle these diverging corrections
in the long wavelength limit. We outline this method below. It is convenient
to express the stochastically driven equations (12) and (14) as path integrals
over configurations of $h({\bf r},t)$ and its dynamic conjugate field
$\hat{h}({\bf r},t)$ Bausch  _et al._ (1976); Täuber (2014), subject to the
distribution of the quenched disorders as specified above. The momentum shell
RG procedure consists of integrating over the short wavelength Fourier modes
of $h({\bf r},t)$, $\hat{h}({\bf r},t)$ and $V_{i}({\bf r})$, followed by
rescaling of lengths and times Täuber (2014). In particular, we follow the
usual convention of initially restricting the wavevectors to be within a
bounded spherical Brillouin zone: $|{\bf k}|<\Lambda$. However, the precise
value of the upper cutoff $\Lambda$ has no effect on our final results. The
fields $h({\bf r},t)$, $\hat{h}({\bf r},t)$ and $V_{i}({\bf r})$ are separated
into the high and low wave vector parts $h({\bf r},t)=h^{<}({\bf
r},t)+h^{>}({\bf r},t)$, $\hat{h}({\bf r},t)=\hat{h}^{<}({\bf
r},t)+\hat{h}^{>}({\bf r},t)$ and $V_{i}({\bf r})=V_{i}^{<}({\bf
r})+V_{i}^{>}({\bf r})$, where $h^{>}({\bf r},t)$, $\hat{h}^{>}({\bf r},t)$
and $V_{i}({\bf r})^{>}$ have support in the large wave vector (short
wavelength) range $\Lambda e^{-l}<|{\bf k}|<\Lambda$, while $h^{<}({\bf
r},t)$, $\hat{h}^{<}({\bf r},t)$ and $V_{i}^{<}({\bf r})$ have support in the
small wave vector (long wavelength) range $|{\bf k}|<e^{-l}\Lambda$; $b\equiv
e^{l}>1$. We then integrate out $h^{>}({\bf r},t)$, $\hat{h}^{>}({\bf r},t)$
and $V_{i}^{>}({\bf r})$ perturbatively in the anhamornic couplings, which can
only be done perturbatively; as usual, this resulting perturbation theory of
$h^{<}({\bf r},t)$, $\hat{h}^{<}({\bf r},t)$ and $V_{i}^{<}({\bf r})$ can be
represented by Feynman graphs, with the order of perturbation theory reflected
by the number of loops in the graphs we consider; see, e.g., Refs. Haldar and
Basu (2020); Haldar (2021). This procedure allows us to calculate the RG flow
equations which give the stable fixed points of the disordered KPZ and CKPZ
equations, which in turn give the associated scaling exponents. In the next
Sections, we analyse the RG flow equations and elucidate the scaling exponents
for the quenched disordered KPZ and CKPZ equations separately. Interested
readers will find the relevant one-loop Feynman diagrams and other details of
the intermediate steps in Appendix C.

###  IV.1 Quenched disordered KPZ equation

The dynamic exponent and roughness exponent in the linear limit of (12) are
$z=2$ and $\chi_{h}=\frac{2-d}{2}$, respectively, which are unsurprisingly
identical to their values in the linear limit of the pure KPZ equation; see
also Appendix B.1. We now study the effects of nonlinear terms on long time
and long wavelength scaling behavior of the linear theory by using one loop
perturbative RG methods, whose basic steps are outlined above.

We define dimensionless effective coupling constants
$g_{1}=\frac{\lambda_{1}^{2}D_{1}}{\nu^{3}}\tilde{K}_{d}$,
$g_{2}=\frac{\kappa_{1}^{2}D_{T}}{\nu_{1}^{2}}\tilde{K}_{d}$ and
$g_{3}=\frac{\kappa_{1}^{2}D_{L}}{\nu_{1}^{2}}\tilde{K}_{d}$, where
$\tilde{K}_{d}=\frac{\int d\Omega_{d}}{(2\pi)^{d}}$ contains the angular
contribution coming from the $d$-dimensional volume integral. There are no
fluctuation corrections to $D_{L}$ and $D_{T}$. Under rescaling of space and
time, we rescale the fields in such a way that $D_{L}$ and $D_{T}$ do not
scale. The ratio $\gamma=\frac{D_{T}}{D_{L}}\equiv\frac{g_{2}}{g_{3}}$ is
marginal in this theory, as we shall see below. Therefore, we do not need to
separately study the RG flows of $g_{2}$ and $g_{3}$; solving anyone of them
suffices, as the other can be eliminated in terms of $\gamma$. For convenience
we choose to work with $g_{1}$ and $g_{3}$. Under the rescaling of space, time
and the fields as given in Appendix C.2, effective coupling $g_{1}$ scales as
$\exp[(2-d)\ell]$, whereas both $g_{2}$ and $g_{3}$ scale as
$\exp[(2+\mu-d)\ell]$ under these rescalings. Both the pure ($g_{1}$) and
disorder ($g_{2}$ or $g_{3}$) nonlinearities naïvely scale the same way for
short-ranged disorder ($\mu=0$), making them compete with each other. Whether
the resulting nonequilibrium steady state is controlled by $g_{1}$, or $g_{2}$
or $g_{3}$, or all of them, can only be ascertained by a RG treatment. In
contrast, for spatially long-ranged disorders $\mu>0$, near a RG fixed point
controlled by $g_{2}$ or $g_{3}$, whereas $g_{1}$ is irrelevant in the RG
sense. These can already be seen from the one-loop fluctuation corrections of
the model parameters that with short-range quench disorder the corrections
originating from the pure nonlinear term $\lambda_{1}$ have the same naïve
infra-red divergence as those originating from the disorder nonlinear term
$\kappa_{1}$. Thus, both types of the fluctuation-corrections must be retained
for a RG treatment. In contrast, for long-ranged disorder, the former class is
necessarily less infra-red divergent than the latter class (see Appendix C
equations C.1.1). Therefore, in this case only the latter class of
fluctuation-corrections is to be retained, and the former class discarded
being less divergent, in the spirit of the RG procedure.

Note that at 1D the strength $D_{T}=0$, it implies $g_{2}=0$ and $\gamma=0$.
The relevant one loop Feynman diagrams are shown in Appendix C.1. The
differential RG recursion relations for the model parameters are:

$\displaystyle\frac{dD_{1}}{d\ell}=D_{1}\left[z-d-2\chi_{h}+\frac{g_{1}}{4}+2g_{3}\right],$
(15a)
$\displaystyle\frac{d\nu_{1}}{d\ell}=\nu_{1}\left[z-2+\frac{2-d}{4d}g_{1}+2g_{3}\left(\gamma\frac{d-1}{d}+\frac{\mu-d}{2d}\right)\right],$
(15b)
$\displaystyle\frac{d\lambda_{1}}{d\ell}=\lambda_{1}\left[z+\chi_{h}-2+2g_{3}\left(\gamma\frac{d-1}{d}-\frac{1}{d}\right)\right],$
(15c)
$\displaystyle\frac{d\kappa_{1}}{d\ell}=\kappa_{1}\left[z-1+\frac{\mu-d}{2}-\frac{2}{d}g_{3}\right],$
(15d)

along with $dD_{L}/dl=0=dD_{T}/dl$. The flow equations (15) can be used to
find the RG flow of the dimensionless couplings $g_{1}$ and $g_{3}$. We get

$\displaystyle\frac{dg_{1}}{d\ell}=g_{1}\left[2-d+\frac{g_{1}}{4}\left(4-\frac{6}{d}\right)-g_{3}\left\\{2\gamma\left(1-\frac{1}{d}\right)+\left(\frac{3\mu+4}{d}-5\right)\right\\}\right],$
(16a)
$\displaystyle\frac{dg_{3}}{d\ell}=g_{3}\left[2+\mu-d-\frac{g_{1}}{2}\left(\frac{2}{d}-1\right)-g_{3}\left\\{4\gamma\left(1-\frac{1}{d}\right)+2\left(\frac{\mu+2}{d}-1\right)\right\\}\right].$
(16b)

Below we discuss the scaling behavior for short range and long range
correlated disorders separately using the flow equation (16a) and (16b).

####  IV.1.1 Short range disorder

For the short range disorder case, we set $\mu=0$ as discussed earlier. The RG
flow equations of $g_{1}$ and $g_{3}$ are

$\displaystyle\frac{dg_{1}}{dl}=$ $\displaystyle
g_{1}\bigg{[}2-d+\frac{g_{1}}{4}\left(4-\frac{6}{d}\right)$ $\displaystyle-
g_{3}\left\\{2\gamma\left(1-\frac{1}{d}\right)-\left(5-\frac{4}{d}\right)\right\\}\bigg{]},$
(17a) $\displaystyle\frac{dg_{3}}{dl}=$ $\displaystyle
g_{3}[2-d+\frac{g_{1}}{2}\left(1-\frac{2}{d}\right)$ $\displaystyle-
g_{3}\left\\{4\gamma\left(1-\frac{1}{d}\right)+2\left(\frac{2}{d}-1\right)\right\\}].$
(17b)

Flow Eqs. (17a) and (17b) show that both $g_{1}$ and $g_{3}$ have the same
critical dimension $2$. The flow equations in (17) at 2D have the form

$\displaystyle\frac{dg_{1}}{dl}=g_{1}\left[\frac{g_{1}}{4}-g_{3}(\gamma-3)\right];\,\frac{dg_{3}}{dl}=-2\gamma
g_{3}^{2}.$ (18)

In equations (18) (0,0) is the only fixed point. Further, $g_{3}$ flows to
zero in the long RG time limit. In fact, the fixed point (0,0) is stable along
the $g_{3}$-direction, whereas unstable along the $g_{1}$-direction. We
further find that for any “initial conditions”
$g_{1}(\ell=0)>0,\,g_{3}(\ell=0)>0$, the flow ultimately runs away along the
$g_{1}$-axis to infinity suggesting the existence of a perturbatively
inaccessible phase. Since $g_{3}$ flows to zero, we are tempted to speculate
that this inaccessible phase is statistically identical to perturbatively
inaccessible rough phase of the 2D KPZ equation.

The Gaussian fixed point (0,0) is unstable for dimension at or below $d=2$ and
it is the only FP at 2D. For any dimension higher than $d=2$, the Gaussian
fixed point is stable and the nonlinear terms are irrelevant near it. As a
result, the scaling properties are described by linear theory. For $d>2$,
there is at least one strong coupling fixed point corresponding to a rough
phase, which is perturbatively inaccessible. While this is qualitatively
similar to the pure KPZ equation, due to the nonperturbative nature of the
rough phase, we cannot tell whether this rough phase is statistically same as
the rough phase of $d>2$ pure KPZ equation implying irrelevance of the
quenched disorder, or different, in which case, quenched disorder is relevant.
In the latter case, we speculate the yet unknown scaling properties are to be
parametrised by the parameter $\gamma$. Numerical solutions of (12), or
simulations of suitably constructed discrete models equivalent to (12) should
be able to shed further light on this issue. Unsurprisingly, we further
conclude that $d=2$ is the lower critical dimension of (12), same as for the
pure KPZ model.

We now briefly consider the scaling properties at $d=1$ by performing the RG
at $d=1$, instead of expanding about the critical dimension Mukherjee (2021);
Barabási and Stanley (1995). Remembering $\gamma=0$, the flow equations of
$g_{1}$ and $g_{3}$ are

$\displaystyle\frac{dg_{1}}{dl}=$ $\displaystyle
g_{1}\left[1-\frac{g_{1}}{2}+g_{3}\right],$ (19a)
$\displaystyle\frac{dg_{3}}{dl}=$ $\displaystyle
g_{3}\left[1-\frac{g_{1}}{2}-2g_{3}\right].$ (19b)

The Fixed points are found by setting $\frac{dg_{1}}{dl}=0=\frac{dg_{3}}{dl}$
and corresponding stability are given below.

  1. 1.

$(0,0)$: globally unstable.

  2. 2.

$(0,\frac{1}{2})$: stable in the $g_{3}$-direction and unstable in the
$g_{1}$-direction.

  3. 3.

$(2,0)$: stable in the $g_{1}$-direction, and marginally stable in the
$g_{3}$-direction. At this stable fixed point, from Eq. (15a) and (15b) we get
the scaling exponents $z=3/2$ and $\chi_{h}=1/2$, which characterises the KPZ
universality class behavior.

The RG flow diagram is pictorially shown in Fig. 1.

We thus conclude that the short range orientational quenched disorder is
irrelevant in (12), ultimately giving the well-known 1D KPZ universality
class.

Figure 1: Schematic flow diagram in the $g_{1}-g_{3}$ plane for the disordered
KPZ equation at 1D, showing the irrelevance of the short ranged quenched
disorder (see text). Here, $(2,0)$ is the stable fixed point.

####  IV.1.2 Long range correlated disorder

In this Section, we study the effects of spatially correlated or long range
disorder (i.e., $0<\alpha<d$, i.e., $\mu>0$) on the universal scaling
properties of the KPZ equation. The flow equations (16a) and (16b) show the
critical dimension of $g_{1}=2$ but for $g_{2}$ is $2+\mu$. Thus, coupling
$g_{1}$ has a critical dimension lower than $g_{3}$. Therefore, $g_{1}$ is
subleading to $g_{3}$, and the leading scaling behavior in the asymptotic long
wavelength limit should be controlled by $g_{3}$ (if the disorder is relevant
in the RG sense). Since $g_{1}$ is subleading, the renormalised disordered
equation motion is invariant under inversion of $h$, a symmetry absent
microscopically and also in the rough phase of the pure KPZ equation. Thus,
this symmetry under inversion of $h$ appears as an emergent symmetry of the
model in the asymptotic long wavelength limit. The flow Eq. (16b) for $g_{3}$
is

$\displaystyle\frac{dg_{3}}{dl}=g_{3}\left[2+\mu-d-
g_{3}\left\\{4\gamma(1-\frac{1}{d})+2\left(\frac{\mu+2}{d}-1\right)\right\\}\right].$
(20)

In order to calculate the scaling exponents at dimensions $d<2+\mu$, we use
${\cal O}(\epsilon_{1})$ expansion, where $d=2+\mu-\epsilon_{1}$,
$\epsilon_{1}>0$. To the lowest order in ${\cal O}(\epsilon_{1})$, the flow
equation (20) for $g_{3}$ takes the form:

$\displaystyle\frac{dg_{3}}{d\ell}=g_{3}\left[\epsilon_{1}-\frac{1+\mu}{2+\mu}4\gamma
g_{3}\right].$ (21)

Equation (21) reveals that $g_{3}^{*}=0$ is an unstable fixed point, but

$g_{3}^{*}=\frac{\epsilon_{1}}{4\gamma}\frac{2+\mu}{1+\mu}$ (22)

for a non-zero $\gamma$ is a stable fixed point. We evaluate the exponents at
this stable fixed point. We find the dynamic exponent

$z=2-\frac{\epsilon_{1}}{2\gamma}\frac{\gamma(1+\mu)-1}{(1+\mu)},$ (23)

which can be more or less than $2$, its value in the linear theory, implying
that nonlinear disorder effects can induce slower or faster than ordinary
diffusive relaxation of fluctuations, with $\gamma(1+\mu)=1$ is the boundary
between the two kinds of behaviour. We further get the roughness exponent

$\chi_{h}=\frac{\epsilon_{1}}{4}[1+\frac{3+\mu}{\gamma(1+\mu)}]-\frac{\mu}{2}.$
(24)

Thus, both $z$ and $\chi_{h}$ explicitly vary with $\gamma$. In the limit of
$\gamma\rightarrow\infty$, i.e., for $D_{T}\gg D_{L}$, $z=2-\epsilon_{1}/2$
and $\chi_{h}=\epsilon_{1}/4-\mu/2$. We focus on the $d=2$ case. From the
definition of $\epsilon_{1}$, $d=2$ implies $\epsilon_{1}=\mu$. From (57), we
then obtain

$\chi_{h}=\frac{\mu}{4}\bigg{[}-1+\frac{3+\mu}{\gamma(1+\mu)}\bigg{]},$ (25)

which can be positive or negative, depending upon $\gamma$. In fact, for a
large $\gamma$, $\chi_{h}<0$. By setting $\chi_{h}=0$, we get $\gamma_{c}$,
the critical value of $\gamma$:

$\gamma_{c}=(3+\mu)/(1+\mu).$ (26)

Now, $\chi_{h}<0$, which for a fluctuating surface means a “smooth surface”,
means $\langle h^{2}({\bf x},t)\rangle$ is independent of the system size $L$
for large $L$. This, when $h$ is interpreted as a phase, implies long range
orientational order in 2D, result not possible in equilibrium at 2D due to the
well-known Mermin-Wagner theorem (MWT) Mermin and Wagner (1966). Similarly,
for an active membrane, this result means the membrane should be statistically
flat, an impossibility in equilibrium again due to MWT. Since in 2D,
$\epsilon_{1}=\mu$ and $\mu=d-\alpha$, the maximum value of $\epsilon_{1}$ can
be 2 ($=\mu$ at 2D), by using this value we find
$\chi_{h}=\frac{1}{2}[-1+\frac{5}{3\gamma}]$, which can be made negative for
$\gamma>5/3$.

For $\epsilon_{1}<0$, i.e., $d>2+\mu$, $g_{3}=0$ is the only fixed point which
is stable. We therefore conclude that $d=2+\mu$ is the upper critical
dimension of this model.

At $d=1$, in which case $D_{T}=0$ necessarily, giving $\gamma=0$. Then (21)
reads

$\displaystyle\frac{dg_{3}}{d\ell}=g_{3}\epsilon_{1}.$ (27)

For 1D, we set $\epsilon_{1}=1+\mu$. The only fixed point here is
$g_{3}^{*}=0$, which is unstable. The lack of a stable fixed point does not
allow us to extract the scaling properties. It is likely to be an artifact of
the one loop study. Notice that at any $d$-dimension for $\gamma\rightarrow
0$, both  $z$ and $\chi_{h}$ diverge in (23) and (24) respectively, which is
unphysical. To investigate the $\gamma=0$ case further, we use (20) and
perform a fixed dimension RG, which is similar in spirit with the RG for the
1D KPZ equation Barabási and Stanley (1995); see also Refs. Mukherjee (2021);
Škultéty and Honkonen (2021). In that scheme Eq. (20) gives at $d$-dimension
the stable fixed point

$\displaystyle
g_{3}^{*}=\frac{d}{2}\cdot\frac{2+\mu-d}{2\gamma(d-1)+2+\mu-d}.$ (28)

At this fixed point, the scaling exponents can be calculated from Eqs. (15a)
and (15b). We obtain

$\displaystyle
z=2-\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{2[2\gamma(d-1)+2+\mu-d]},$
$\displaystyle\chi_{h}=\frac{2-d}{2}-\frac{(2+\mu-d)[2\gamma(d-1)+\mu+d]}{4[2\gamma(d-1)+2+\mu-d]}.$
(29)

Thus for $\gamma=0$,

$\displaystyle z=2-\frac{\mu-d}{2},$ (30)
$\displaystyle\chi_{h}=\frac{2-d}{2}-\frac{\mu+d}{4},$ (31)

which are perfectly well-behaved. In particular, at 1D, we find
$g_{3}^{*}=1/2,z=2-\frac{\mu-1}{2}$ and $\chi_{h}=1-\frac{\mu+1}{4}$. Further,
using (29) we find in 2D $\chi_{h}=-\frac{\mu(2\gamma+2+\mu)}{4(2\gamma+\mu)}$
which gives a smooth surface whatever value of $\mu$ and $\gamma$. Generally
then, regardless of the scheme we used, we uncover explicit
$\gamma$-dependence of $z$ and $\chi_{h}$. This clearly shows parametrisation
of the universality classes by $\gamma$.

In general, an alert reader will easily notice that the roughness exponent in
Eq. (24) from $\epsilon_{1}$ expansion RG, or in Eq. (29) from fixed dimension
RG method reveals that by tuning $\mu$, it can be made positive or negative
for $d<2+\mu$. Thus, by changing $\mu$, the surface can be made rough
($\chi_{h}>0$) or smooth ($\chi_{h}<0$). The transition line in the $\mu-d$
plane can be found by setting $\chi_{h}=0$. By using (24), we find the
equation for the line

$\displaystyle d=2+\mu-\frac{2\mu\gamma(1+\mu)}{\gamma(1+\mu)+3+\mu},$ (32)

in the $\epsilon_{1}$-expansion calculation, and by using (29), the line is
given by

$\displaystyle
2-d=\frac{(2+\mu-d)[2\gamma(d-1)+\mu+d]}{2[2\gamma(d-1)+2+\mu-d]},$ (33)

in the fixed dimension RG calculation. On the line $\chi_{h}=0$, the surface
is logarithmically rough. The dynamic exponent $z$ on that line unsurprisingly
depends on the precise equation of the line $\chi_{h}=0$. We show for two
differenty method: $z=2+\frac{\mu[1-\gamma(1+\mu)]}{3+\mu+\gamma(1+\mu)}$ by
using eq. (32) in $\epsilon_{1}$ expansion case and
$z=d+\frac{d(2+\mu-d)}{2\gamma(d-1)+2+\mu-d}$ by using (33) in fixed dimension
RG analysis.

We plot the values of the scaling exponents as obtained from fixed dimension
RG and $\epsilon_{1}$ expansion as functions of $\gamma$ in Fig. 2 below.

Figure 2: (color online) A plot of the scaling exponents of the quenched
disordered KPZ equation, comparing between the results found from one-loop
$\epsilon_{1}$ expansion and fixed dimension RG, showing the
$\gamma$-dependence of the scaling exponents. Here, suffix $\epsilon_{1}$
indicates scaling exponents obtained from an $\epsilon_{2}$-expansion, whereas
suffix $FD$ indicates fixed dimension RG results. These are plotted for $d=2$
and $\mu=1$.

It is theoretically interesting to calculate the scaling exponents at the
critical dimension $d=2+\mu$. The flow equation of $g_{3}$ is at $d=2+\mu$
reads [see Eq. (21)]

$\displaystyle\frac{dg_{3}}{d\ell}=-\frac{1+\mu}{2+\mu}4\gamma g_{3}^{2}.$
(34)

Equation (34) shows that $g_{3}(\ell)$ is marginally irrelevant; $g_{3}(\ell)$
flows towards $g_{3}^{*}=0$ with increasing of $\ell$. Coupling $g_{3}(\ell)$
takes the form $g_{3}(\ell)=\frac{g_{3}(0)}{1+g_{3}(0)\ell
4\gamma(\frac{1+\mu}{2+\mu})}$, and for large renormalisation group time i.e.,
$\ell\rightarrow\infty$,
$g_{3}(\ell)\simeq\frac{2+\mu}{4\gamma(1+\mu)}\frac{1}{\ell}$. This shows that
time-scale $t$ no longer shows simple scale with length-scale $r$, giving
breakdown of conventional dynamic scaling

$t\sim r^{2}(\log(r/a_{0}))^{-\frac{\gamma(1+\mu)-1}{2\gamma(2+\mu)}}.$ (35)

Thus, the extent of breakdown of dynamic scaling depends on $\gamma$, and can
be faster or slower than ordinary diffusion. We now calculate the variance of
$h$ at the critical dimension $d=2+\mu$. We get

$\displaystyle\langle h^{2}({\bf
x},t)\rangle\simeq\int_{1/L}^{1/a_{0}}{d^{2+\mu}{\bf
q}}~{}q^{-2}[\log(1/q)]^{-\frac{\gamma(1+\mu)-(3+\mu)}{2\gamma(2+\mu)}}.$ (36)

Here, $L$ and $a_{0}$ are linear system size and small-scale cutoff,
respectively. Clearly, $\langle h^{2}({\bf x},t)\rangle$ is bounded: it does
not diverge for large $L$. For $d>2+\mu$, i.e., above the upper critical
dimension, since $g_{3}=0$ is the stable FP, the asymptotic long wavelength
limit scaling is identical to the linear theory.

We thus find that with short range disorder although the disorder coupling
constant $g_{3}$ is naïvely as relevant as the coupling $g_{1}$ of the pure
KPZ equation, in a one-loop theory $g_{3}$ is irrelevant in the RG sense at
all dimensions. For long range correlated disorder $g_{3}$ is relevant while
$g_{1}$ becomes irrelevant, resulting into scaling exponents depending
explicitly on $\gamma$. Lastly, for $\mu<0$ quenched disorder is irrelevant in
all dimensions. Hence, the model belongs to the standard KPZ universality
class. In fact, this model with $\mu<0$ is statistically identical to $\mu=0$
in the long wavelength limit.

We present the phase diagram of the disordered KPZ equation in the $\mu-d$
plane showing the possible phases in Fig. 3

Figure 3: (color online) Schematic phase diagram of the quenched disordered
KPZ equation in $\mu-d$ plane. The broken blue line corresponds to
$\chi_{h}=0$, dividing the disorder-induced rough and smooth phases, and
depends on $\gamma$; here, this line is schematically drawn for $\gamma=1$).

###  IV.2 Quenched disordered CKPZ equation

We now investigate the universal scaling properties of the quenched disordered
CKPZ equation. As in our study of the disordered KPZ equation above, we
consider both short-ranged and long-ranged disorders separately. In the linear
limit of (14), we find dynamic exponent $z=4$ and roughness exponent
$\chi_{h}=\frac{2-d}{2}$, which are unsurprisingly identical to the scaling of
the linearlised pure CKPZ equation; see Appendix B.2 for detailed
calculations. The nonlinear terms can affect the linear theory theory scaling,
if they are relevant in a RG sense. For instance, the pure CKPZ equation has
an upper critical dimension two, meaning for dimension $d<2$, the nonlinear
effects modify the linear theory scaling in the asymptotic long wavelength
limit. In the disordered case, as for the quenched disordered KPZ equation,
the pertinent questions are, whether disorder is relevant in the RG sense, and
if so, what are scaling properties of the resulting universality class. We
systematically address these issues by using one-loop perturbative RG, whose
details are available in the Appendix C.

Similar to our studies above on the quenched disordered KPZ equation, we
define two dimensionless coupling constants
$\varTheta_{1}=\frac{\lambda_{2}^{2}D_{2}}{\nu_{2}^{3}}\tilde{K}_{d}$,
$\varTheta_{2}=\frac{\kappa_{2}^{2}D_{T}}{\nu_{2}^{2}}\tilde{K}_{d}$ and
$\varTheta_{3}=\frac{\kappa_{2}^{2}D_{L}}{\nu_{2}^{2}}\tilde{K}_{d}$. Further,
we set $D_{L}$ and $D_{T}$ do not scale, accordingly we define a ratio
$\gamma=\frac{D_{T}}{D_{L}}\equiv\frac{\varTheta_{2}}{\varTheta_{3}}$. Note
that at 1D the strength $D_{T}=0$, which implies $\gamma=0$, as before.
Furthermore, naïve rescaling of space, time and the fields, as shown in
Appendix C.2 show that the under these rescalings, effective couplings
$\varTheta_{1}$ scales as $\exp[(2-d)\ell]$, whereas both $\varTheta_{2}$ and
$\varTheta_{3}$ scale as $\exp[(2+\mu-d)\ell]$. As in the RG treatment of the
disordered KPZ equation, in the RG analysis for the CKPZ equation, both the
pure ($\varTheta_{1}$) and disorder ($\varTheta_{2}$ or $\varTheta_{3}$)
nonlinearities naïvely scale the same way for short-ranged disorder, making
them compete with each other and control the nonequilibrium steady states. In
contrast, for spatially long-ranged disorders $\mu>0$, $\varTheta_{2}$ or
$\varTheta_{3}$ control the behavior of stedy states, $\varTheta_{1}$ is
irrelevant in the RG sense . The one-loop fluctuation-corrections generating
from pure $\lambda_{2}$ and disorder $\kappa_{2}$ nonlinearities show these
two types of infrared divergences (see Appendix C equations 74). With short-
range disorder, both types have the same divergences, but for long range
disorder the leading divergences originate from the disorder vertices.
Needless to say, as for the disordered KPZ equation studied above, in all
these cases the parameter $\gamma$ is marginal, and has no fluctuation-
corrections. This means there is no need to study the flows of $\varTheta_{2}$
and $\varTheta_{3}$ separately; it is enough to study the flow of one of them
parametrised by $\gamma$. In what follows below, we choose to work with
$\varTheta_{1}$ and $\varTheta_{3}$.

The RG recursion relation of the model parameters for general $\mu\geq 0$ are
(see Appendix C for more details):

$\displaystyle\frac{dD_{2}}{d\ell}=D_{2}\left[z-d-2-2\chi_{h}\right],$ (37a)
$\displaystyle\frac{d\nu_{2}}{d\ell}=\nu_{2}\left[z-4+\frac{4-d}{4d}\varTheta_{1}+2\varTheta_{3}\left(\gamma\frac{d-1}{d}+\frac{\mu-d}{2d}\right)\right],$
(37b)
$\displaystyle\frac{d\lambda_{2}}{d\ell}=\lambda_{2}\left[z+\chi_{h}-4+2\varTheta_{3}\left(\gamma\frac{d-1}{d}-\frac{1}{d}\right)\right],$
(37c)
$\displaystyle\frac{d\kappa_{2}}{d\ell}=\kappa_{2}\left[z-3+\frac{\mu-d}{2}-\frac{2}{d}\varTheta_{3}\right].$
(37d)

The flow equations of the dimensionless coupling constants $\varTheta_{1}$ and
$\varTheta_{3}$ are then calculated by using (37). These are given by

$\displaystyle\frac{d\varTheta_{1}}{d\ell}=\varTheta_{1}\left[2-d-\frac{4-d}{4d}3\varTheta_{1}-\varTheta_{3}\left(2\gamma\frac{d-1}{d}+\frac{4+3\mu-3d}{d}\right)\right],$
(38a)
$\displaystyle\frac{d\varTheta_{3}}{d\ell}=\varTheta_{3}\left[2+\mu-d-\frac{4-d}{2d}\varTheta_{1}-\varTheta_{3}\left\\{\frac{d-1}{d}4\gamma+2\left(\frac{2+\mu-d}{d}\right)\right\\}\right].$
(38b)

We separately discuss the short and long range disorder cases below.

####  IV.2.1 Short range disorder

When the disorder is short-ranged, we set $\mu=0$. The flow equations of
couplings $\varTheta_{1}$ and $\varTheta_{3}$for the short-ranged disorder
case can be obtained from (37). These are

$\displaystyle\frac{d\varTheta_{1}}{d\ell}=\varTheta_{1}\left[2-d-\frac{4-d}{4d}3\varTheta_{1}-\varTheta_{3}\left(2\gamma\frac{d-1}{d}+\frac{4-3d}{d}\right)\right],$
(39a)
$\displaystyle\frac{d\varTheta_{3}}{d\ell}=\varTheta_{3}\left[2-d-\frac{4-d}{4d}2\varTheta_{1}-2\varTheta_{3}\left\\{2\gamma\frac{d-1}{d}+\frac{2-d}{d}\right\\}\right].$
(39b)

Flow equations (39a) and (39b) show that $\varTheta_{1}$ and $\varTheta_{3}$
both have the same critical dimension $2$, as already argued by using naïve
rescaling of space, time and the fields. Further, the Gaussian fixed point
(0,0) is globally unstable below 2D, whereas this is the only stable fixed
point at or above 2D. That implies that $d=2$ is the upper critical dimension
for this model, same as pure CKPZ; at 2D, couplings $\varTheta_{1}$ and
$\varTheta_{3}$ are marginally irrelevant. Therefore, the linear theory
scaling is expected to be modified by the nonlinear effects at or below 2D,
whereas for $d>2$ the nonlinear couplings are irrelevant and the linear theory
holds in the asymptotic long wavelength limit. To the lowest order in
$\epsilon_{2}\equiv 2-d$, the flow equations (39a) and (39b) reduce to

$\displaystyle\frac{d\varTheta_{1}}{d\ell}=\varTheta_{1}\left[\epsilon_{2}-\frac{3\varTheta_{1}}{4}-\varTheta_{3}\left(\gamma-1\right)\right].$
(40a)
$\displaystyle\frac{d\varTheta_{3}}{d\ell}=\varTheta_{3}\left[\epsilon_{2}-\frac{\varTheta_{1}}{2}-2\gamma\varTheta_{3}\right].$
(40b)

Equations (40a) and (40b) can be used to calculate the fixed points
$(\varTheta_{1}^{*},\varTheta_{2}^{*})$ by setting
$\frac{d\varTheta_{1}}{d\ell}=0=\frac{d\varTheta_{3}}{d\ell}$. We find

  1. 1.

Fixed point 1: (0,0) is globally unstable.

  2. 2.

Fixed point 2: (0,$\epsilon_{2}/2$) unstable in the $\varTheta_{1}$-direction,
but stable in the $\varTheta_{3}$-direction.

  3. 3.

Fixed point 3: ($4\epsilon_{2}/3,0$) unstable in the
$\varTheta_{3}$-direction, but stable in the $\varTheta_{1}$-direction.

  4. 4.

Fixed point 4: The nontrivial fixed point
($\frac{2\epsilon_{2}(1+\gamma)}{2\gamma+1},\frac{\epsilon_{2}}{2(2\gamma+1)}$):
stable along both the $\varTheta_{1}$ and $\varTheta_{2}$ directions, i.e.,
globally stable.

The fixed points and the RG flow lines are shown schematically in Fig. 4.

Figure 4: (color online) Schematic flow diagram at dimension
$d=2-\epsilon_{2},\,\epsilon_{2}>0$ (except 1D) for the short range quenched
disordered CKPZ equation. The small, red circle is the nontrivial fixed point,
which is globally stable.

The globally stable fixed point describes the stable nonequilibrium steady
state of the model. It can be used to calculate the associated scaling
exponents. We find that the dynamic exponent

$z=4-\frac{\epsilon_{2}\gamma}{2\gamma+1},$ (41)

and the roughness exponent

$\chi_{h}=\frac{\epsilon_{2}(1+\gamma)}{2(2\gamma+1)}.$ (42)

From this above expression $\chi_{h}>0$ since $\epsilon_{2}>0$. Thus, it is a
short-range disorder induced rough phase (“SR rough”). We note that as
$\gamma\rightarrow 0$, both $z$ and $\chi_{h}$ approach their linear theory
values, i.e., $z=4,\,\chi=\epsilon_{2}/2$, an unexpected result. In the same
way, we note that at 1D, $\epsilon_{2}=1$ and $\gamma=0$, giving $z=4$,
$\chi_{h}=\frac{1}{2}$, identical to the linear theory results! In the absence
of any general symmetry argument to render the fluctuation corrections to
vanish or turn irrelevant, we believe this is fortuitous, an artifact of the
one-loop perturbation theory. Higher order perturbation theory or numerical
studies should be useful to obtain better quantitative estimates of the
exponents in the limit $\gamma\rightarrow 0$. We re-analyse the 1D case by
using a fixed dimension RG scheme, which is similar in spirit with what one
does for the 1D KPZ equation Barabási and Stanley (1995). To do this, we use
the flow equations (39a) and (39b) directly, and set $d=1$.

The resulting flow equations

$\displaystyle\frac{d\varTheta_{1}}{d\ell}=\left[1-9\varTheta_{1}/4-\varTheta_{3}\right],$
(43)
$\displaystyle\frac{d\varTheta_{3}}{d\ell}=\left[1-3\varTheta_{1}/2-2\varTheta_{3}\right].$
(44)

Solving these two equations, $(1/3,1/4)$ is only stable FP and finds the
exponents $z=4,\chi_{h}=1/2$, again same as the linear theory results,
offering no further insight into the unexpected appearance of the linear
theory results.

The fixed point analysis of the flow equations in 2D, which is the physically
relevant dimension as its pure counterpart is, needs to be done separately, as
2D is the critical dimension. At $d=2$ i.e., $\epsilon_{2}=0$. The resulting
flow equations of $\varTheta_{1}$ and $\varTheta_{3}$ are

$\displaystyle\frac{d\varTheta_{1}}{d\ell}=-\varTheta_{1}\left[\frac{3\varTheta_{1}}{4}+\varTheta_{3}\left(\gamma-1\right)\right].$
(45a)
$\displaystyle\frac{d\varTheta_{3}}{d\ell}=-\varTheta_{3}\left[\frac{\varTheta_{1}}{2}+2\gamma\varTheta_{3}\right].$
(45b)

Thus (0,0) is the only  fixed point. Whether it is a stable or unstable fixed
point can be found only by solving the flow Eqs. (45a) and (45b). In order to
do so, we first define a ratio $\beta=\frac{\varTheta_{3}}{\varTheta_{1}}$.
The flow equation of $\beta$ is

$\displaystyle\frac{d\beta}{d\ell}=\beta\varTheta_{1}\left[\frac{1}{4}-\beta(1+\gamma)\right].$
(46)

Equation (46) shows that

$\beta^{*}=\frac{1}{4(1+\gamma)}=\frac{\varTheta_{3}(\ell)}{\varTheta_{1}(\ell)}$
(47)

gives the fixed point of (46), which in turn gives the separatrix, such that
all initial conditions $\varTheta_{1}(\ell=0),\,\varTheta_{3}(\ell=0)$
maintaining $\varTheta_{3}(\ell=0)/\varTheta_{1}(\ell=0)=\beta^{*}$ will
continue to maintain it under the RG transformations. Furthermore, the
separatrix can be shown to be stable or attractive: We write
$\beta=\beta^{*}+\delta\beta$. To the linear order in $\delta\beta$, we find

$\frac{d\delta\beta}{\delta\ell}=-\beta^{*}\varTheta_{1}(1+\gamma)\delta\beta,$
(48)

showing the attractive nature of the separatrix. Interestingly, as
$\gamma\rightarrow\infty$, $\beta^{*}\rightarrow 0$, meaning that the
separatrix coincides with the $\varTheta_{1}$-axis in the
$\varTheta_{1}-\varTheta_{3}$ plane in that limit. Using the value of
$\beta^{*}$, we can further show that

$\displaystyle\frac{d\varTheta_{1}(\ell)}{d\ell}=-\frac{\varTheta_{1}^{2}(1+2\gamma)}{2(1+\gamma)}<0,$
(49a)
$\displaystyle\frac{d\varTheta_{3}(\ell)}{d\ell}=-\varTheta_{3}^{2}2(1+2\gamma)<0,$
(49b)

giving (0,0) as the globally stable fixed point. This then means
$\varTheta_{1}$ and $\varTheta_{3}$ are marginally irrelevant, with both of
them eventually flowing to zero in the asymptotic long wavelength limit.
Nonetheless, they do so slowly enough to allow for infinite renormalisation of
$\nu_{2}$, leading to logarithmic modulations of the linear theory scaling. We
work this out explicitly below.

Since the separatrix is attractive, we can use
$\varTheta_{3}(l)=\beta^{*}\varTheta_{1}(l)$ in the long wavelength limit in
the flow equation (45a). We can then solve the resulting effective flow
equation for $\varTheta_{1}(l)$ giving

$\varTheta_{1l}=\frac{2(1+\gamma)}{1+2\gamma}\frac{1}{\ell},\;\varTheta_{3l}=\frac{1}{2(1+2\gamma)}\frac{1}{\ell},$
(50)

is the asymptotic limit $\ell\rightarrow\infty$. Since $\ell$ is the logarithm
of a physical length scale, we note that both $\varTheta_{1}(\ell)$ and
$\varTheta_{3}(\ell)$ approach zero logarithmically slowly. Thus, an arbitrary
initial condition $\varTheta_{1}(\ell=0),\,\varTheta_{3}(\ell=0)$ not only
flows to the origin, it also flows towards the attractive separatrix in course
of its flow to the origin under successive RG transformations. The RG flows in
2D are schematically shown in Fig. 5.

Figure 5: Schematic RG flow diagram in 2D ($\epsilon_{2}=0$) for the short-
range quenched disordered CKPZ equation. The origin $(0,0)$ is the only fixed
point, which is stable. The inclined straight line is the separatrix having a
$\gamma$-dependent slope given by Eq. (47).

We can now find the scale-dependent $\nu_{2}(\ell)$ by assuming
$z=4,\,\chi_{h}={0}$. This gives

$\displaystyle\nu_{2}(\ell)\simeq\nu_{2}(0)\ell^{\frac{\gamma}{2\gamma+1}}.$
(51)

However, $D_{2}(\ell)\simeq D_{2}(0)$, since it has no relevant one-loop
fluctuation correction.

These can be used to obtain the variance

$\displaystyle\langle h^{2}({\bf
x},t)\rangle\sim\left[\log(L/a_{0})\right]^{1-\frac{\gamma}{1+2\gamma}},$ (52)

where $L$ and $a_{0}$ are linear system size and small-scale cutoff,
respectively. The logarithmic dependence of $\langle h^{2}({\bf x},t)\rangle$
on the system size $L$ in (52) implies that at $d=2$, the disordered CKPZ
surface is logarithmically rough (“log rough”). We further see that in the
renormalised theory, time-scale $t$ and length-scale $r$ are no longer simply
related by a dynamic scaling exponent. We in fact find

$t\sim r^{4}[\log(r/a_{0})]^{-\frac{\gamma}{1+2\gamma}}$ (53)

for large $r$, signifying breakdown of conventional dynamic scaling due to
logarithmic modulations. The dynamics is logarithmically slower than that in
the linear theory by an extent that depends upon $\gamma$. In the limit
$\gamma=0$ ($D_{T}=0$), $t\sim r^{4}$, same as in the linear theory, which is
likely an artifact of our low order perturbation theory; as $\gamma$ rises,
the dynamics gets slower initially, ultimately saturating at $t\sim
r^{4}\frac{1}{\sqrt{\log(r/a_{0})}}$ as $\gamma\rightarrow\infty$, i.e.,
$D_{T}\gg D_{L}$. In the same limit by using (52), we obtain $\langle
h^{2}({\bf x},t)\rangle\sim\sqrt{\log(L/a_{0})}$.

####  IV.2.2 Long-range correlated disorder

We now study the scaling properties in the presence of long range quenched
disorder, i.e., $\alpha<d$, or $\mu>0$. The flow equations (38a) and (38b)
show that the critical dimensions of $\varTheta_{1}=2$ but for $\varTheta_{3}$
is $2+\mu>0$. Here $\varTheta_{1}$ has critical dimension lower than
$\varTheta_{3}$. Therefore, near a fixed point controlled by $\varTheta_{3}$,
$\varTheta_{1}$ is subleading to $\varTheta_{3}$, and the leading scaling
behavior is described by $\varTheta_{3}$. This is similar to our analysis for
the long range quenched disordered KPZ equation above. Again, as in the
disordered KPZ equation, the disordered CKPZ equation with long range disorder
has invariance under inversion of $h$ as an emergent symmetry in the long
wavelength limit. Hence, Eq. (38b) reduces to

$\displaystyle\frac{d\varTheta_{3}}{dl}=\varTheta_{3}\left[2+\mu-d-\varTheta_{3}\left\\{4\gamma(1-\frac{1}{d})+2\left(\frac{\mu+2}{d}-1\right)\right\\}\right].$
(54)

Notice that Eq. (54) is identical to the flow equation (20) in Sec. IV.1.2,
and hence, the ensuing analysis runs exactly parallel to the one below Eq.
(20) in Sec. IV.1.2. To proceed further, we set $d=2+\mu-\epsilon_{2}$ where
$\epsilon_{2}>0$ and extract the fixed point and the associated scaling
exponents to ${\cal O}(\epsilon_{2})$. The flow equation of $\varTheta_{3}$
reads

$\displaystyle\frac{d\varTheta_{3}}{d\ell}=\varTheta_{3}\left[\epsilon_{2}-\frac{1+\mu}{2+\mu}4\gamma\varTheta_{3}\right].$
(55)

Unsurprisingly, Eq. (55) is same as Eq. (21) in Sec. IV.1.2. Equation (55)
gives $\varTheta_{3}=0$ is an unstable fixed point, but
$\varTheta_{3}=\varTheta_{3}^{*}\equiv\frac{\epsilon_{2}}{4\gamma}\frac{2+\mu}{1+\mu}$
is a stable fixed point for any $\gamma>0$. As before, the scaling exponents
are calculated at the stable fixed point $\varTheta_{3}^{*}$. We find the
dynamic exponent

$z=4-\frac{\epsilon_{2}}{2\gamma}\frac{\gamma(1+\mu)-1}{1+\mu}$ (56)

which is less than $4$. Thus, the dynamics is faster than in the linear
theory. Further, the roughness exponent

$\chi_{h}=\frac{\epsilon_{2}}{4}[1+\frac{1}{\gamma(1+\mu)}]-\frac{\mu}{2}.$
(57)

Clearly, both $z$ and $\chi$ depends explicitly on and vary continuously with
$\gamma$. As $\gamma\rightarrow\infty$,
$z=4-\epsilon_{2}/2,\;\chi_{h}=\epsilon_{2}/4-\mu/2$. On the other hand, if
$\gamma\rightarrow 0$, i.e., $D_{T}\ll D_{L}$, both $z$ and $\chi_{h}$
diverge, an unexpected result reminiscent of what we have found in the
analogous analysis for the long range disordered KPZ equation. At 1D,
necessarily $\gamma=0$, the flow of $\varTheta_{3}$ reads same as (27)

$\displaystyle\frac{d\varTheta_{3}}{d\ell}=\varTheta_{3}\epsilon_{2}.$ (58)

Equation (58) surprisingly has no stable FP, so it cannot describe stable
scaling behaviour at 1D, which we believe just as an artifact of our one-loop
calculations. To investigate this further, we perform a fixed dimension RG
akin to the 1D KPZ equation Barabási and Stanley (1995). In this scheme flow
Eq. (54) has a fixed point
$\varTheta_{3}^{*}=\frac{d}{2}\cdot\frac{2+\mu-d}{2\gamma(d-1)+2+\mu-d}$ at a
(fixed) dimension $d$, which is a stable fixed point. We calculate the
exponents at this fixed point are

$\displaystyle
z=4-\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{2[2\gamma(d-1)+2+\mu-d]},$ (59a)
$\displaystyle\chi_{h}=\frac{2-d}{2}-\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{4[2\gamma(d-1)+2+\mu-d]}.$
(59b)

Thus the scaling exponents depend explicitly on $\gamma$. These results reveal
the generally expected existence of a stable scaling regime parametrised by
$\gamma$ different from the linear theory scaling.

We give a plot of the scaling exponents, showing their dependence on the
disorder parameter $\gamma$ in Fig. 6.

For $d=1$ with $\gamma=0$, the value of the fixed point is $\varTheta_{3}=1/2$
and exponents are $z=4-\frac{\mu-1}{2}$ and
$\chi_{h}=\frac{1}{2}-\frac{\mu-1}{4}$ using the values of (59a) and (59b).

Similar to the disordered KPZ equation with long range quenched disorder, both
smooth and rough phases are possible for $d<2+\mu$, as can be seen from Eq.
(57), obtained from an $\epsilon_{1}$ expansion RG, and from Eq. (59b),
obtained by using a fixed dimension RG method. As $\gamma$ is varied, the
system can undergo a transition between a smooth with $\chi_{h}<0$ and a rough
$\chi_{h}>0$ phase. The transition line in the $\mu-d$ plane can be found from
the condition $\chi_{h}=0$, and is parametrised by $\gamma$. The equation of
this line obtained from an $\epsilon_{2}$ expansion RG is given by

$\displaystyle d=2+\mu-\frac{2\mu\gamma(1+\mu)}{\gamma(1+\mu)+1}.$ (60)

The same obtained from a fixed dimension RG scheme is given by

$\displaystyle
2-d=\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{2[2\gamma(d-1)+2+\mu-d]}.$ (61)

As expected and similar to the quenched disordered KPZ equation studied above,
the location of these lines in the $\mu-d$ plane depend explicitly on
$\gamma$. The surface is logarithmically rough on the line $\chi_{h}=0$. The
dynamic exponent $z$ on this line will be
$4-\frac{\mu[\gamma(1+\mu)-1]}{\gamma(1+\mu)+1}$ by using (60) for
$\epsilon_{2}$ expansion case or $2+d$ by using (61) for fixed dimension RG
case.

Figure 6: (color online) The scaling exponents are plotted as functions of
$\gamma$ to compare between the results found from one-loop $\epsilon_{2}$
expansion RG and fixed dimension RG. Here, suffix $\epsilon_{2}$ indicates
scaling exponents obtained from an $\epsilon_{2}$-expansion, whereas suffix
$FD$ indicates fixed dimension RG results. We have set $d=2,\,\mu=1$.

We find that at 2D, $\epsilon_{2}=\mu$ in the $\epsilon_{2}$-expansion method,
and hence from (57) $\chi_{h}=\frac{\mu}{4}[-1+\frac{1}{\gamma(1+\mu)}]$. We
further note that again in the fixed dimension RG method, by using $d=2$,
(59b) gives $\chi_{h}=-\frac{\mu(2\gamma+\mu-2)}{4(2\gamma+\mu)}$. On the
whole, these show that $\chi_{h}$ can be negative depending on values of
$\gamma$ and $\mu$. This means that the surface can be smooth due to presence
of long range disorder instead of logarithmically rough in pure CKPZ case.

Precisely at dimension $d=2+\mu$, the flow equation of $g_{2}$ is

$\displaystyle\frac{d\varTheta_{3}}{d\ell}=-\frac{1+\mu}{2+\mu}4\gamma\varTheta_{3}^{2}.$
(62)

Equation (62), which has the same structure as (34), reveals that
$\varTheta_{3}(\ell)$ is marginally irrelevant and it flows to zero with
increasing of $\ell$; in fact $\varTheta_{3}=0$ is the only fixed point, which
is stable. Furthermore, $\varTheta_{3}(\ell)$ scales as
$\frac{2+\mu}{4\gamma(1+\mu)}\frac{1}{\ell}$ in the limit of large
renormalisation time length ($\ell$), same as the behaviour of $g_{3}(\ell)$
for large $\ell$ in Sec. IV.1.2 above. This behaviour of coupling constant
realises the leading sacling: dynamic exponent $z=4$ and roughness exponent
$\chi_{h}=-\mu/2$. This in turn gives breakdown of conventional dynamic
scaling with

$t\sim r^{4}(\log(r/a_{0}))^{-\frac{\gamma(1+\mu)-1}{2\gamma(2+\mu)}},$ (63)

meaning the dynamics is logarithmically faster than the linear theory.
Furthermore, the extent of “logarithmic speeding up” vis-a-vis the linear
theory depends explicitly on $\gamma$, and also unsurprisingly on $\mu$. The
variance of $h$ at $d=2+\mu$ is

$\displaystyle\langle h^{2}({\bf
x},t)\rangle\simeq\int_{1/L}^{1/a}\frac{d^{2+\mu}{\bf
q}}{q^{2}}[\log(1/q)]^{-\frac{\gamma(1+\mu)-1}{2\gamma(2+\mu)}}.$ (64)

The above expression says that variance does not diverge for large $L$.

So far we have studied $\mu\geq 0$ above. In principle, one may also consider
the case $\mu<0$. This choice makes $\varTheta_{3}$ subdominant to
$\varTheta_{1}$ at all dimensions, making the model identical to the pure CKPZ
equation in the long wavelength limit. In fact, as in the case of quenched
disorder KPZ equation above, the $\mu<0$ case for the CKPZ equation is
statistically identical to the pure ckpz case in the long wavelength limit.
Interestingly, the $\mu=0$ case for the quenched disordered KPZ equation is
statistically identical to the pure KPZ equation in the long wavelength limit;
in contrast, the $\mu=0$ case for the quenched disordered CKPZ equation
belongs to a new universality class different from the pure CKPZ equation.
This amply highlights the role of the conservation law in determining the
universality classes in driven diffusive systems.

We summarise our results on the disordered CKPZ equation in the form of a
phase diagram in Fig. 7 showing all the possible phases in $\mu-d$ plane.

Figure 7: (color online) Schematic phase diagram of the disordered CKPZ
equation in the $\mu-d$ plane. The broken blue line divides disorder induced
rough and smooth phases, and depends on disorder parameter $\gamma$; here this
line is schematically drawn for $\gamma=1$).

We thus obtain the scaling exponents varying continuously with $\gamma$ for
both the short and long range disorder cases.

We summarise our results on the scaling exponents of the quenched disordered
KPZ and CKPZ equations in the following Table 2 and Table 2.

Disordered KPZ equation
---
$\mu=0$: uncorrelated disorder |  $\mu>0$: correlated disorder
|  Dimension |  Exponents |  Possible phases |  Dimension |  Exponents |  Possible phases
---|---|---|---|---|---
$d<2$ |  Pure KPZ result: disorder irrelevant. At $d=1$, $z=3/2$ and $\chi_{h}=1/2$. |  Rough |  $d<2+\mu$ |  $\epsilon_{1}$ expansion: ‘  $\displaystyle z=2-\frac{\epsilon_{1}}{2\gamma}\frac{\gamma(1+\mu)-1}{(1+\mu)},$ $\displaystyle\chi_{h}=\frac{\epsilon_{1}}{4}[1+\frac{3+\mu}{\gamma(1+\mu)}]-\frac{\mu}{2}.$ Fixed dimension RG:  $\displaystyle z=2-\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{2[2\gamma(d-1)+2+\mu-d]},$ $\displaystyle\chi_{h}=\frac{2-d}{2}-\frac{(2+\mu-d)[2\gamma(d-1)+\mu+d]}{4[2\gamma(d-1)+2+\mu-d]}.$ |  Disorder induced rough and smooth possible. At the transition $\chi_{h}=0$ with $z=2+\frac{\mu[1-\gamma(1+\mu)]}{3+\mu+\gamma(1+\mu)}$ and $z=d+\frac{d(2+\mu-d)}{2\gamma(d-1)+2+\mu-d}$ for $\epsilon_{1}$ expansion and fixed dimension RG case respectively.
$d=2=d_{lc}$ |  Pure KPZ result. |  Rough |  $d=2+\mu=d_{uc}$ |  $z=2$, $\chi_{h}=-\mu/2$. |  Smooth
$d>2$ |  EW result: $z=2,\,\chi_{h}=\frac{2-d}{2}$, or the scaling exponents of the pertubatively inaccessible rough. |  Smooth and rough with a roughening transition |  $d>2+\mu$ |  EW result: $z=2,\,\chi_{h}=\frac{2-d}{2}$. |  Smooth

Table 1: Scaling exponents for the disordered KPZ equation as obtained from
both $\epsilon_{1}$-expansion and fixed dimension RG.

Disordered CKPZ equation
---
$\mu=0$: uncorrelated disorder |  $\mu>0$: correlated disorder
|  Dimension |  Exponents |  Possible phases |  Dimension |  Exponents |  Possible phases
---|---|---|---|---|---
$d=1<d_{uc}=2$ |  $\epsilon_{2}$ expansion with $\epsilon_{2}=1$:  $\displaystyle z=4-\frac{\epsilon_{2}\gamma}{2\gamma+1},$ $\displaystyle\chi_{h}=\frac{\epsilon_{2}(1+\gamma)}{2(2\gamma+1)}.$ Fixed dimension RG: $z=4,\,\chi_{h}=1/2$. |  Rough |  $d<2+\mu$ |  $\epsilon_{2}$ expansion: ‘  $\displaystyle z=4-\frac{\epsilon_{2}}{2\gamma}\frac{\gamma(1+\mu)-1}{1+\mu},$ $\displaystyle\chi_{h}=\frac{\epsilon_{2}}{4}[1+\frac{1}{\gamma(1+\mu)}]-\frac{\mu}{2}.$ Fixed dimension RG:  $\displaystyle z=4-\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{2[2\gamma(d-1)+2+\mu-d]},$ $\displaystyle\chi_{h}=\frac{2-d}{2}-\frac{(2+\mu-d)[2\gamma(d-1)+\mu-d]}{4[2\gamma(d-1)+2+\mu-d]}.$ |  Disorder induced rough and smooth possible. At the transition $\chi_{h}=0$ with $z=4-\frac{\mu[\gamma(1+\mu)-1]}{\gamma(1+\mu)+1}$ for $\epsilon_{2}$ expansion case and $z=2+d$ for fixed dimension RG case.
$d=2=d_{uc}$ |  $z=4$ and $\chi_{h}=0$ with logarithmic roughness. |  Logarithmic rough or smooth |  $d=2+\mu=d_{uc}$ |  $z=4$, $\chi_{h}=-\mu/2$ |  Smooth
$d>2$ |  MH result: $z=4,\,\chi_{h}=\frac{2-d}{2}$. |  Smooth |  $d>2+\mu$ |  MH result: $z=4,\,\chi_{h}=\frac{2-d}{2}$. |  Smooth

Table 2: Scaling exponents for the disordered CKPZ equation as obtained from
both $\epsilon_{2}$-expansion and fixed dimension RG.

##  V Summary and outlook

In summary, we have studied the universal scaling properties of the KPZ and
conserved KPZ equations coupled to orientational quenched disorders. We have
chosen a particular form of disorder that couples with the local gradient of
the height field, i.e., the disorder is sensitive to the local height
nonuniformity. It is described by a zero-mean Gaussian distributed quenched
vector field $\bf V$. Vector $\bf V$ can in general have both irrotational and
solenoidal parts. This is reflected in the variance of $\bf V(k)$ having parts
proportional to the transverse $P_{ij}({\bf k})$ and longitudinal projection
operator $Q_{ij}({\bf k})$, with amplitudes $D_{T}$ and $D_{L}$, respectively.
This allows us to define a dimensionless number $\gamma\equiv D_{T}/D_{L}$. We
have studied the scaling properties of the models for both spatially short and
long ranged quenched disorders by using a one-loop perturbative RG scheme.

For each of the models, we have calculated the relevant scaling exponents,
i.e., the roughness exponent $\chi_{h}$ and dynamic exponent $z$. For
instance, we find that the disordered KPZ equations with short ranged quenched
disorder belongs to the well-known KPZ equation only, i.e., the disorder is
irrelevant in the RG sense. Unsurprisingly, $d=2$ remains the lower critical
dimension of the model, just as it is for the pure KPZ equation. For long
range disorder, the model no longer belongs to the KPZ universality class; a
new universality class emerges, with the pure KPZ nonlinear term being
irrelevant in the RG sense. Our result on long range orientational order
induced by long range quenched disorder, when $h$ is considered as a phase, is
theoretically intriguing, and should stimulate further work on disordered
active XY models. The asymptotic long wavelength scaling is controlled by the
disorder nonlinearity only, and the associated scaling exponents depend
explicitly on $\gamma$ and $\mu$; the latter parameter characterises the
spatial scaling of the disorder variance. In contrast, for the disordered CKPZ
equation, even with short range disorder we find that a new universality class
emerges at dimension $d\leq 2$; the scaling exponents are calculated by using
a one-loop expansion, which depend explicitly on $\gamma$. Further, $d=2$ is
now the upper critical dimension, as in the pure CKPZ equation. For long range
disorder, the pure CKPZ nonlinearity is irrelevant in the RG sense; the
asymptotic long wavelength properties are controlled by the disorder
nonlinearity only. This is exactly analogous to the universality in the
quenched disordered KPZ with long range disorder. We show that as $\mu>0$ is
varied, both the quenched disordered KPZ and CKPZ equations can have a
transition between a rough phase and a smooth phase. Lastly, the scaling
exponents, which are calculated at the one-loop order, depend explicitly on
$\gamma$ and $\mu$.

Beyond the specific results, we find the surprising generic conclusion that
whenever quenched disorders are relevant (in the RG sense), the parameter
$\gamma$ enters into the expressions of the scaling exponents. Since $\gamma$
can in principle vary continuously (between zero and infinity), so do the
scaling exponents and hence does the universality class itself. We thus find
continuously varying universality, induced by quenched disorder, in the two
models that we considered. Furthermore, the existence of continuously varying
universality appears to be quite robust: it exists with (CKPZ) or without
(KPZ) a conservation law, so long as the quenched disorder itself remains
relevant in the RG sense. We can relate this with the symmetry of the disorder
distribution: $\gamma$ is the parameter that determines the relative strengths
of the transverse and longitudinal parts of the disorder variance. Thus
varying $\gamma$ effectively means variation of the local alignment of $\bf
V({\bf q})$ with respect to $\bf q$. This continuous variation of the
universality class is in fact more robust than our results perhaps imply.
Since the variance of a quenched disordered vector field can generally be a
combination of transverse and longitudinal parts, a dimensionless parameter
like $\gamma$ automatically arises, which remains a free parameter,
continuously varying universal properties are generically expected so long as
the quenched disorder remains relevant in the RG sense. These results may be
tested by designing and simulating appropriate agent-based models coupled to a
vector quenched disorder field with specified distributions. Do these results
apply to annealed disorders as well? As shown in Appendix D, the CKPZ
equations, when driven by vector fields with linear dynamics show this
behaviour. Preliminary results on the KPZ equation driven by an annealed
disorder field also shows similar behaviour (not shown here). This however
comes with a caveat. An annealed field has a dynamics, which is not
necessarily linear in general. We have assumed a linear dynamics for the
annealed field only for simplicity. In general nonlinear terms should be
present, which may or may not be relevant in the RG sense. If it is relevant,
then there is a possibility the dynamics of the annealed disorder field is
renormalized, and that a parameter equivalent to $\gamma$ may acquire specific
RG fixed point value(s), instead of being a free, continuously varying
parameter. Thus it is possible that the resulting universality class of the
height field (or in general a dynamical field driven by the annealed disorder)
will not vary continuously. However, in the event the nonlinear effects in the
dynamics of the annealed field are irrelevant, we expect continuously varying
universality class to follow, as we have already found in the CKPZ equation
with annealed disorder. We therefore conclude that, the existence of
continuously varying universality classes with a vector quench disorder field
is likely to be more generic than its annealed counterparts. Lastly, we
caution the reader that the continuously varying universality class observed
with quenched disorder here does not hold for any kind of quenched disorder,
even if the disorder is relevant in the RG sense. This is because, in order
for this result to hold, there must a free dimensionless parameter like
$\gamma$ in the disorder distribution. Not all quench disorder distributions
may allow this. For instance, there is no scope of this in the recent studies
on quenched columnar disordered KPZ equation Haldar and Basu (2020); Haldar
(2021), since the disorder there is a scalar.

Our results may be verified by direct numerical simulations of the
stochastically driven equations of motion, which may be conveniently done by
using pseudospectral methods Basu and Frey (2009). It would be interesting to
develop equivalent quenched disordered agent-based lattice-gas models, in
terms of a height field or a phase field, which may be studied by using Monte-
Carlo simulations and explore and validate the RG results above. We hope our
studies will provide new impetus in these directions.

Apart from their obvious theoretical interests, our results have experimental
implications as well. For instance, a possible physical origin of the (vector)
quenched disorder is the underlying disordered substrates. The substrates, if
not very carefully prepared, may have different patches, characterised by
different values of the parameter $\gamma$. For sufficiently large patches,
each such patch should be characterised by scaling exponents parametrised by
$\gamma$. If the experimental measurements are done on a region that includes
may such patches having different $\gamma$, the measured correlation functions
are not likely to show any clean scaling, but should rather display a
broadening of the measured values of the scaling exponents, or a kind of
smeared scaling behaviour. We hope our studies here will provide further
impetus to research along these lines in the future.

##  VI Acknowledgement

A.B. thanks the SERB, DST (India) for partial financial support through the
MATRICS scheme [file no.: MTR/2020/000406].

##  Appendix A Generating functional

The generating functional of the disordered KPZ or CKPZ equation has the form
Martin  _et al._ (1973); Zinn-Justin (2002); Bausch  _et al._ (1976);
DeDominicis and Martin (1979)

$\displaystyle\mathbb{Z}=\int_{{\bf
x},t}\mathcal{D}\hat{h}\mathcal{D}h\mathcal{D}Ve^{-\mathnormal{S}[\hat{h},h,V]}.$
(65)

Here $\mathnormal{S}$ is the action functional and it contains two parts: (i)
harmonic ($\mathnormal{S_{H}}$) (ii) anharmonic ($\mathnormal{S}_{A}$). Field
$\hat{h}({\bf x},t)$ is the dynamic conjugate field of $h({\bf x},t)$ Bausch
_et al._ (1976).

We here write the down the explicit forms of the action functionals for the
disordered KPZ and CKPZ equations.

###  A.1 Disordered KPZ equation

The action functional is constructed by using Eq. (12) and averaging over the
Gaussian-distributed noise $\xi_{h}$. We get

$\displaystyle\mathnormal{S^{\text{KPZ}}_{H}}=$ $\displaystyle\int_{{\bf
x},t}\left[-D_{1}\hat{h}\hat{h}+\hat{h}\left(\partial_{t}-\nu_{1}\nabla^{2}\right)h\right]$
$\displaystyle+\int_{\bf
x,x^{\prime}}\frac{V_{i_{x}}V_{j_{x^{\prime}}}}{4[D_{T}P_{ij}+D_{L}Q_{ij}]|x-x^{\prime}|^{-\alpha}},$
(66) $\displaystyle\mathnormal{S^{\text{KPZ}}_{A}}=$ $\displaystyle\int_{{\bf
x},t}-\hat{h}\left[\frac{\lambda_{1}}{2}(\bm{\nabla}h)^{2}+\kappa_{1}({\bf
V}.\bm{\nabla}h)\right],$ (67)

where $S^{\text{KPZ}}_{H}$ and $S^{\text{KPZ}}_{A}$ are, respectively, the
harmonic and anhamornic parts of the action functional.

###  A.2 Disordered CKPZ equation

The action functional is constructed by using (14) and then averaging over the
Gaussian-distributed noise $\eta_{h}$. We get

$\displaystyle\mathnormal{S^{\text{CKPZ}}_{H}}=$ $\displaystyle\int_{{\bf
x},t}\left[-D_{2}(-\nabla^{2})\hat{h}\hat{h}+\hat{h}\left(\partial_{t}+\nu_{2}\nabla^{4}\right)h\right]$
$\displaystyle+\int_{\bf
x,x^{\prime}}\frac{V_{i_{x}}V_{j_{x^{\prime}}}}{4[D_{T}P_{ij}+D_{L}Q_{ij}]|x-x^{\prime}|^{-\alpha}},$
(68) $\displaystyle\mathnormal{S^{\text{CKPZ}}_{A}}=$ $\displaystyle\int_{{\bf
x},t}-\hat{h}\left[\frac{\lambda_{2}}{2}\nabla^{2}(\bm{\nabla}h)^{2}+\kappa_{2}\nabla^{2}({\bf
V}.\bm{\nabla}h)\right],$ (69)

where $S^{\text{CKPZ}}_{H}$ and $S^{\text{CKPZ}}_{H}$ are the harmonic and
anharmonic parts of the action functional.

##  Appendix B Harmonic theory

In this Section, we consider the harmonic parts of the action functionals,
which correspond to the linear parts of the underlying stochastically driven
equations of motion.

###  B.1 Disordered KPZ equation

The propagator and correlator of $h$ in the Fourier space can be obtained from
action (66). These are

$\displaystyle\langle\hat{h}_{-{\bf k},-\omega}h_{{\bf
k},\omega}\rangle=\frac{1}{-i\omega+\nu_{1}k^{2}},$ (70a)
$\displaystyle\langle|h_{{\bf
q},\omega}|^{2}\rangle=\frac{2D_{1}}{\omega^{2}+\nu_{1}^{2}k^{4}}.$ (70b)

We thus get dynamic exponent $z=2$ at all dimensions. The equal-time
correlator in real space

$\displaystyle C(r)\equiv\langle[h({\bf x},t)-h({\bf
x^{\prime}},t)]^{2}\rangle=2\int\frac{d^{d}{\bf
k}}{(2\pi)^{d}}\frac{D_{1}[1-e^{i{\bf k\cdot[x-x^{\prime}]}}]}{\nu_{1}k^{2}},$
(71)

for large $r\equiv|{\bf x-x^{\prime}}|$. Following (1), we conclude that
$\chi_{h}=\frac{2-d}{2}$. Therefore, $\chi_{h}=1/2$ at 1D and $\chi_{h}=0$ at
2D in the Gaussian theory, or in the linearized equation (12).

###  B.2 Disordered CKPZ equation

The bare propagator and correlator of $h$ in Fourier space can be obtained
from the harmonic part of action (68). These are

$\displaystyle\langle\hat{h}_{-{\bf k},-\omega}h_{{\bf
k},\omega}\rangle=\frac{1}{-i\omega+\nu_{2}k^{4}}.$ (72a)
$\displaystyle\langle|h_{{\bf
q},\omega}|^{2}\rangle=\frac{2D_{2}k^{2}}{\omega^{2}+\nu_{2}^{2}k^{8}}.$ (72b)

Here $z=4$, and the equal-time correlator $C(r)$ for this case in real space
can be defined similarly as eq. (71) which gives $\chi_{h}=\frac{2-d}{2}$. It
means $\chi_{h}=1/2$ at 1D and $\chi_{h}=0$ at 2D in the Gaussian theory, or
in the linearised Eq. (14).

##  Appendix C Dynamic renormalisation group (RG) method

We have discussed above about the method of dynamic RG, so, here we present
the intermediate details such as loop diagrams, rescaling, fluctuation
corrections.

###  C.1 One-loop Feynman diagrams

We represent the bare two point functions, the anharmonic terms or the
“vertices” in the action functional and the fluctuation-corrections of the
model parameters by the Feynman diagrams, as shown in Fig. 8. These diagrams
are for both the disordered KPZ and CKPZ equations. However, their expressions
differ depending upon the model in question.

Figure 8: $(i)$ Bare two point functions: a. propagator, b. correlator, c.
disorder correlator. Figures $(ii)-(v)$ give the Feynman diagrams for the one-
loop corrections in the theory; the one-loop diagrams have the same structure
for in both the disordered KPZ and CKPZ equations, though their values differ
(see text) . Figure $(ii)$ give one-loop corrections to the diffusivity
($\nu_{1}$ or $\nu_{2}$), figure $(iii)$ are for the noise strength ($D_{1}$
or $D_{2}$), figure $(iv)$ are for the pure nonlinear terms ($\lambda_{1}$ or
$\lambda_{2}$), and figure $(v)$ correspond to the disordered nonlinear terms
($\kappa_{1}$ or $\kappa_{2}$). The number written before a Feynman diagram is
the symmetry factor of the corresponding diagram.

####  C.1.1 Disordered KPZ equation

The fluctuation-corrections of the model parameters of the disordered KPZ
equation (12) corresponding to the one-loop diagrams in Fig. 8 are given
below.

$\displaystyle{\rm
Fig.~{}}\ref{feynmann}(ii):~{}~{}\nu_{1}^{<}=\nu_{1}\left[1+\frac{2-d}{4d}\frac{\lambda_{1}^{2}D_{1}}{\nu_{1}^{3}}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2}}+\left\\{\frac{d-1}{d}\frac{2\kappa_{1}^{2}D_{T}}{\nu_{1}^{2}}+\frac{\mu-d}{d}\frac{\kappa_{1}^{2}D_{L}}{\nu_{1}^{2}}\right\\}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (73a) $\displaystyle{\rm
Fig.~{}}\ref{feynmann}(iii):~{}~{}D_{1}^{<}=D_{1}\left[1+\frac{\lambda_{1}^{2}D_{1}}{4\nu_{1}^{3}}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2}}+\frac{2\kappa_{1}^{2}D_{L}}{\nu_{1}^{2}}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (73b) $\displaystyle{\rm
Fig.~{}}\ref{feynmann}(iv):~{}~{}\lambda_{1}^{<}=\lambda_{1}\left[1+\left\\{\frac{d-1}{d}\frac{2\kappa_{1}^{2}D_{T}}{\nu_{1}^{2}}-\frac{2}{d}\frac{\kappa_{1}^{2}D_{L}}{\nu_{1}^{2}}\right\\}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (73c) $\displaystyle{\rm
Fig.~{}}\ref{feynmann}(v):~{}~{}\kappa_{1}^{<}=\kappa_{1}\left[1-\frac{2}{d}\frac{\kappa_{1}^{2}D_{L}}{\nu_{1}^{2}}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (73d)

####  C.1.2 Disordered CKPZ equation

The fluctuation-corrections of the parameters in the disordered CKPZ equation
(14) corresponding to the one-loop Feynman diagrams in Fig. 8 are given below.

$\displaystyle{\rm
Fig.~{}}\ref{feynmann}(ii):~{}~{}\nu_{2}^{<}=\nu_{2}\left[1+\frac{4-d}{4d}\frac{\lambda_{2}^{2}D_{2}}{\nu_{2}^{3}}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2}}+\left\\{\frac{d-1}{d}\frac{2\kappa_{2}^{2}D_{T}}{\nu_{2}^{2}}+\frac{\mu-d}{d}\frac{\kappa_{2}^{2}D_{L}}{\nu_{2}^{2}}\right\\}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (74a) $\displaystyle{\rm
Fig.~{}}\ref{feynmann}(iii):~{}~{}D_{2}^{<}=D_{2}+{\rm
subleading~{}corrections}.$ (74b) $\displaystyle{\rm
Fig.~{}}\ref{feynmann}(iv):~{}~{}\lambda_{2}^{<}=\lambda_{2}\left[1+\left\\{\frac{d-1}{d}\frac{2\kappa_{2}^{2}D_{T}}{\nu_{2}^{2}}-\frac{2}{d}\frac{\kappa_{2}^{2}D_{L}}{\nu_{2}^{2}}\right\\}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (74c) $\displaystyle{\rm
Fig.~{}}\ref{feynmann}(v):~{}~{}\kappa_{2}^{<}=\kappa_{2}\left[1-\frac{2}{d}\frac{\kappa_{2}^{2}D_{L}}{\nu_{2}^{2}}\int_{\varLambda/b}^{\varLambda}\frac{d^{d}{\bf
q}}{(2\pi)^{d})}\frac{1}{q^{2+\mu}}\right].$ (74d)

###  C.2 Rescaling

We rescale a Fourier wavevector and frequency as ${\bf k}\rightarrow b{\bf k}$
and $\omega\rightarrow b^{z}\omega$, equivalently in the real space ${\bf
x}\rightarrow{\bf x}/b,\,t\rightarrow t/b^{z}$, where $z$ is the dynamic
exponent. The fields are accordingly rescaled as

$\displaystyle h({\bf k},\omega)=\xi h(b{\bf k},b^{z}\omega);\,\hat{h}({\bf
k},\omega)=\hat{\xi}\hat{h}(b{\bf k},b^{z}\omega).$ (75)

Furthermore, $V_{i}({\bf k},\omega)=b^{z+\frac{d}{2}+\frac{\mu}{2}}V_{i}(b{\bf
k},b^{z}\omega)$, as obtained from Eq. (10).

####  C.2.1 Scaling of the parameters in the disordered KPZ equation

We set $\xi\hat{\xi}=b^{d+2z}$ by demanding the term in action (66) $\int
d^{d}{\bf k}d\omega\hat{h}h\omega$ does not scale under the rescaling of
wavevectors and frequencies. Accordingly the model parameters scale as given
below.

$\displaystyle\nu_{1}^{\prime}=\nu_{1}^{<}b^{z-2},\,D^{\prime}_{1}=D_{1}^{<}b^{z-d-2\chi_{h}},$
$\displaystyle\lambda_{1}^{\prime}=\lambda_{1}^{<}b^{z-2+\chi_{h}},\,\kappa^{\prime}_{1}=\kappa_{1}b^{z-1+\frac{\mu}{2}-\frac{d}{2}},$
$\displaystyle
D_{L}^{\prime}=D_{L}^{<}=D_{L},\,\,D_{T}^{\prime}=D_{T}^{<}=D_{T}.$ (76)

####  C.2.2 Scaling of the parameters in the disordered CKPZ equation

We set $\xi\hat{\xi}=b^{d+2z}$ by demanding the term in the action functional
(68) $\int d^{d}{\bf k}d\omega\hat{h}h\omega$ does not scale under the
rescaling of wavevectors and frequencies. Accordingly the model parameters
scale as given below.

$\displaystyle\nu_{2}^{\prime}=\nu_{2}^{<}b^{z-4},\,D^{\prime}_{2}=D_{2}^{<}b^{z-d-2-2\chi_{h}},$
$\displaystyle\lambda_{2}^{\prime}=\lambda_{2}^{<}b^{z-4+\chi_{h}},\,\kappa_{2}^{\prime}=\kappa_{2}b^{z-3+\frac{\mu}{2}-\frac{d}{2}}$
$\displaystyle
D_{L}^{\prime}=D_{L}^{<}=D_{L},\,\,D_{T}^{\prime}=D_{T}^{<}=D_{T}.$ (77)

##  Appendix D Annealed disorder

We now briefly discuss what happens when the disorder is “annealed” instead of
“quenched”. This means the disorder is no longer frozen in time. At the
simplest level, we model it by a time-dependent fluctuating vector field ${\bf
U}({\bf x},t)$ that has zero mean and is assumed to be Gaussian distributed.
If $\bf U$ is interpreted as a velocity field, then this is reminiscent of the
well-known passive scalar turbulence problem Kraichnan (1994); Adzhemyan  _et
al._ (1998); Banerjee and Basu (2018); Mukherjee and Basu (2021). The Eq. of
motion of $\bf U$ is

$\displaystyle\frac{\partial\bf U}{\partial t}=b\nabla^{2}{\bf U}+{\bf f},$
(78)

where the noise ${\bf f}$ is zero-mean, Gaussian-distributed with a variance

$\displaystyle\langle f_{i}({\bf k},\omega)f_{j}({\bf
k^{\prime}},\omega^{\prime})\rangle$
$\displaystyle=[2a_{1}P_{ij}+2a_{2}Q_{ij}]k^{2-\mu}\delta^{d}({\bf
k-k^{\prime}})\delta(t-t^{\prime}).$ (79)

Here, $\mu=d-\alpha$, as in the main text. Then the velocity autocorrelation
function can be calculated exactly. It in Fourier space is

$\displaystyle\langle U_{i}({\bf k},\omega)U_{j}({\bf
k^{\prime}},\omega^{\prime})\rangle=\frac{2a_{1}P_{ij}+2a_{2}Q_{ij}}{\omega^{2}+b^{2}k^{4}}~{}k^{2-\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\delta^{d}({\bf
k+k^{\prime}})\delta(\omega+\omega^{\prime}),$ (80)

where, $a_{1}/b=2\tilde{D}_{T}$ and $a_{2}/b=2\tilde{D}_{L}$. We are
interested in the large $b$ limit, with $\tilde{D}_{L}$ and $\tilde{D}_{T}$
being finite. In this limit, the annealed disorder autocorrelation in real
space reduces to

$\displaystyle\langle U_{i}({\bf x},t)U_{j}({\bf
x^{\prime}},t^{\prime})\rangle$
$\displaystyle=\left[2\tilde{D}_{T}P_{ij}+2\tilde{D}_{L}Q_{ij}\right]|{\bf
x-x^{\prime}}|^{-\alpha}\delta(t-t^{\prime}).$ (81)

We briefly discuss the scaling properties of the disordered CKPZ equation with
annealed disorder of the type defined above. The one-loop Feynman diagrams are
identical to their quenched disorder counterparts. We focus on the one-loop
diagrams which vanish in the absence of the annealed disorder.

Let us consider the last diagram in Fig. 8(ii) that originates from the
disorder coupling $\kappa_{2}$ and corrects $\nu_{2}$:

$\displaystyle\int_{\bf q}\frac{({\bf k-q})_{i}({\bf
k-q})^{2}q^{2-\mu}(2a_{1}P_{ij}+2a_{2}Q_{ij})_{q}}{bq^{2}[bq^{2}+\nu_{2}({\bf
k-q})^{4}]},$ (82)

where k is an external wavevector and q is an internal wavevector of the one-
loop diagram. Noise strength $D_{2}$ receives no relevant or diverging
corrections.

In Fig. 8(iv) the triangle diagrams that originate from the disorder coupling
$\kappa$ correct $\lambda_{2}$. They are

$\displaystyle{\rm first\;\;triangle\;\;diagram}\sim\int_{\bf
q}q^{6}q^{2-\mu}[2a_{1}P_{im}+2a_{2}Q_{im}]$
$\displaystyle[\frac{1}{bq^{2}[\nu_{2}^{2}q^{8}-b^{2}q^{4}]}-\frac{1}{\nu_{2}q^{4}[b^{2}q^{4}-\nu_{2}q^{8}]}].$
(83) $\displaystyle{\rm second\;\;triangle\;\;diagram}\sim$
$\displaystyle~{}~{}~{}~{}~{}~{}\int_{\bf
q}\frac{q_{m}q_{j}q^{4}q^{2-\mu}[2a_{1}P_{im}+2a_{2}Q_{im}]}{bq^{2}[bq^{2}+\nu_{2}q^{4}]^{2}}.$
(84)

Similarly, the last diagram in Fig. 8(v), coming from the disorder vertex,
corrects $\kappa_{2}$.

$\displaystyle\int_{\bf
q}\frac{q_{m}q_{j}q^{4}q^{2-\mu}[2a_{1}P_{im}+2a_{2}Q_{im}]}{bq^{2}[bq^{2}+\nu_{2}q^{4}]^{2}}.$
(85)

All the expressions of different one-loop diagrams given in
(82),(83),(84),(85) are infrared divergent if $b\ll\nu_{2}q^{2}$ is satisfied.
In fact, these corrections are same as in (74). However, these diagrams give
subleading corrections to the parameters in the opposite limit
$b\gg\nu_{2}q^{2}$.

As the high wavevector modes are being eliminated starting from
$k=\varLambda$, the upper waveveector curoff, there can be two situations:
either $\nu_{2}k^{2}$ dominates over $b$ over a substantial range of (i.e.,
intermediate values of) wavevectors just below $\varLambda$, or $b$ dominates
over $\nu_{2}k^{2}$ in the same wavevector region. Noting that with $\nu_{2}$
gaining positive corrections but $b$ gets none, there can be two distinct
possibilities.

(I) It is possible that as one eliminates the higher wavevector modes, before
$b$ starts dominating, long enough RG time is spent, and the annealed-
disordered CKPZ equation is renormalised. In this case, The corrections of
parameters and corresponding flow equations are same as equations of (37).
Therefore the result is identical to quenched disordered CKPZ equation. In
this case, for even lower wavevector regimes with $b\gg\nu_{2}k^{2}$, there
are no new relevant fluctuation corrections to the model parameters. Hence,
the same universality as the quenched disordered CKPZ equation ensues, with a
continuously varying parameter
$\tilde{\gamma}\equiv\tilde{D}_{T}/\tilde{D}_{L}$ parametrising the
universality class.

(II) In the opposite case, not enough RG time is available to make $\nu_{2}$
substantially renormalise, making $b$ dominate over $\nu_{2}k^{2}$ for low
enough $k$. In this case, further more elimination from the disorder-dependent
diagrams do not produce any relevant corrections to the parameters. However,
the fluctuation corrections from the pure CKPZ nonlinear terms still survive,
ultimately giving the scaling exponents of the pure CKPZ equation. In this
case, naturally, there are no $\tilde{\gamma}$-dependence of the scaling
exponents.

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