Depinning in the quenched Kardar-Parisi-Zhang class II: Field theory

The Martin-Siggia-Rose MSR action corresponding to Eq. (1) reads

	$\displaystyle{\cal S}[u,\tilde{u}]$	$\displaystyle=$	$\displaystyle\int_{x,t}\tilde{u}(x,t)\Big{\{}\eta\partial_{t}u(x,t)-c\nabla^{2}u(x,t)$
			$\displaystyle-\lambda\left[\nabla u(x,t)\right]^{2}+m^{2}\big{[}u(x,t){-}w\big{]}\Big{\}}$
			$\displaystyle-\frac{1}{2}\int_{x,t,t^{\prime}}\tilde{u}(x,t)\Delta_{0}\big{(}u(x,t)-u(x,t^{\prime})\big{)}\tilde{u}(x,t^{\prime}).$

The field $\tilde{u}(x,t)$ is an auxiliary field introduced to enforce Eq. (1) and called the response field. The last term is obtained by averaging $\mathrm{e}^{\int_{x,t}\tilde{u}(x,t)F(x,(u(x,t))}$ over $F(x,u)$, using its variance (2). Perturbation theory is constructed by expanding around the free theory obtained by setting $\lambda\to 0$ and $\Delta_{0}(u)\to 0$ in Eq. (II.1). The free response function is the response of the field $u(x+x^{\prime},t+t^{\prime})$ to an additional force acting at $x^{\prime}$, $t^{\prime}$,

	$\displaystyle R(x,t)$	$\displaystyle:=$	$\displaystyle\left<\frac{\delta u(x+x^{\prime},t+t^{\prime})}{\delta f(x^{\prime},t^{\prime})}\right>$		(7)
		$\displaystyle=$	$\displaystyle\left<{\delta u(x+x^{\prime},t+t^{\prime})}{\tilde{u}(x^{\prime},t^{\prime})}\right>.$

In Fourier space it reads

	$\displaystyle R(k,t)$	$\displaystyle=$	$\displaystyle\left<\tilde{u}(k,t)u(-k,t^{\prime})\right>$		(8)
		$\displaystyle=$	$\displaystyle\theta(t^{\prime}>t)\frac{1}{\eta}\mathrm{e}^{-(ck^{2}+m^{2})(t^{\prime}-t)/\eta}$
		$\displaystyle=$	$\displaystyle{\parbox{34.14322pt}{{ \leavevmode\hbox to32.45pt{\vbox to4pt{\pgfpicture\makeatletter\hbox{\hskip 2.0pt\lower-2.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.0pt}{0.0pt}\pgfsys@curveto{2.0pt}{1.10458pt}{1.10458pt}{2.0pt}{0.0pt}{2.0pt}\pgfsys@curveto{-1.10458pt}{2.0pt}{-2.0pt}{1.10458pt}{-2.0pt}{0.0pt}\pgfsys@curveto{-2.0pt}{-1.10458pt}{-1.10458pt}{-2.0pt}{0.0pt}{-2.0pt}\pgfsys@curveto{1.10458pt}{-2.0pt}{2.0pt}{-1.10458pt}{2.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } {{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{30.45276pt}{0.0pt}\pgfsys@curveto{30.45276pt}{1.10458pt}{29.55734pt}{2.0pt}{28.45276pt}{2.0pt}\pgfsys@curveto{27.34818pt}{2.0pt}{26.45276pt}{1.10458pt}{26.45276pt}{0.0pt}\pgfsys@curveto{26.45276pt}{-1.10458pt}{27.34818pt}{-2.0pt}{28.45276pt}{-2.0pt}\pgfsys@curveto{29.55734pt}{-2.0pt}{30.45276pt}{-1.10458pt}{30.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } {{}}{}{{}}{}{{}} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }{{}} {{}} {{{ {\pgfsys@beginscope{} {} {} {} \pgfsys@moveto{1.99997pt}{0.0pt}\pgfsys@lineto{-1.19998pt}{1.59998pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{-1.19998pt}{-1.59998pt}\pgfsys@fill\pgfsys@endscope}} }{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.4943pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}.\qquad$

Graphically this is represented by an arrow from $\tilde{u}$ to $u$. The (microscopic) disorder is represented by two dots connected by a dashed line, whereas the KPZ vertex is a dot with two incoming lines with bars for the derivatives, and one outgoing one. Examples for diagrams correcting the disorder are given on Fig. 3. For an introduction into functional perturbation theory we refer to section 3 of Wiese2021 . Note that the disorder is corrected by the KPZ force, and what we loosely call the renormalized disorder is more precisely the renormalized force correlator, which contains contributions from the KPZ term (see Section II.3 for a detailed discussion). Non-trivial correlations necessitate at least one “disorder” vertex $\Delta_{0}(u)$. As an example, the leading order to the equal-time 2-point function is

	$\displaystyle\left<u(k,0)u(-k,0)\right>$	$\displaystyle=$			(9)
		$\displaystyle=$	$\displaystyle\left[\int_{t}R(k,t)\right]^{2}\Delta_{0}(0)$
		$\displaystyle=$	$\displaystyle\frac{\Delta_{0}(0)}{(ck^{2}+m^{2})^{2}}.$

The arrows represent the response function $R$, the dotted line the effective force correlator $\Delta_{0}(u)$. We assume an upper critical dimension of $d_{\rm c}=4$ as in qEW. Simulations show that in $d=3$ the interface is still rough MukerjeeBonachelaMunozWiese2022 , so the upper critical dimension is above $3$. Noting that physical realizations can only be constructed in integer dimensions, the remaining open question is whether the 4-dimensional system is at its upper critical dimension, or potentially above, see section III.5.4. Note that an interface in anharmonic depinning is less rough than in qEW; this excludes an upper critical dimension larger than four.

Scaling arguments were given in the companion paper MukerjeeBonachelaMunozWiese2022 . We recall the main results here. The static 2-point function is defined as

$$\frac{1}{2}\overline{\langle[u(x)-u(y)]^{2}}\simeq\left\{\begin{array}[]{c}A|x-y|^{2\zeta},~{}|x-y|\ll\xi_{m},\\ Bm^{-2\zeta_{m}},~{}~{}~{}~{}|x-y|\gg\xi_{m}.\end{array}\right.$$

(10)

The average is taken over different disorder configurations (there are no thermal fluctuaions). $\zeta$ is the standard roughness exponent. In contrast to qEW, there is a new exponent $\zeta_{m}>\zeta$. The reason is that the elasticity $c$ renormalizes and thus its anomalous dimension gives rise to another exponent. The quantity $\xi_{m}$ in Eq. (10) is the correlation length created by the confining potential. Every length parallel to the interface scales as $x$ or $\xi_{m}$, whereas in the perpendicular direction it scales as $u\sim x^{\zeta}\sim\xi_{m}^{\zeta}$. To estimate $\xi_{m}$, we take $x=\xi_{m}$ in Eq. (10), obtaining $\xi_{m}^{2\zeta}\sim m^{-2\zeta_{m}}$. As a consequence

$$\xi_{m}\sim m^{-\frac{\zeta_{m}}{\zeta}}.$$

(11)

Note that $\xi_{m}\not\sim\frac{1}{m}$ as for qEW. Fig. 2 shows a scaling collapse of the 2-point function with these scalings.

Define $\psi_{\lambda}$, $\psi_{c}$ and $\psi_{\eta}$ to be the anomalous dimensions of $\lambda$, $c$ and $\eta$ in units of $m^{-1}$,

	$\displaystyle\psi_{\rm c}$	$\displaystyle:=$	$\displaystyle-m\partial_{m}\ln(c),$		(12)
	$\displaystyle\psi_{\lambda}$	$\displaystyle:=$	$\displaystyle-m\partial_{m}\ln(\lambda),$		(13)
	$\displaystyle\psi_{\eta}$	$\displaystyle:=$	$\displaystyle-m\partial_{m}\ln(\eta).$		(14)

In order to relate them to the standard scaling exponents $\zeta$, $\zeta_{m}$ and $z$, we first need to define $z$. It is given by the temporal spread of the perturbation in the surface in the 2-point function as

$$\frac{1}{2}\overline{[u(x,t)-u(x,t^{\prime})]^{2}}\sim|t-t^{\prime}|^{2\zeta/z}.$$

(15)

With these definitions at the fixed point we can derive

	$\displaystyle\frac{\zeta_{m}}{\zeta}$	$\displaystyle=1+\frac{\psi_{\rm c}}{2},$		(16)
	$\displaystyle\zeta_{m}$	$\displaystyle=\psi_{\rm c}-\psi_{\lambda},$		(17)
	$\displaystyle z$	$\displaystyle=\frac{\zeta}{\zeta_{m}}(2+\psi_{\eta}).$		(18)

The first relation is obtained from $x^{-1}\sim q\sim m/\sqrt{c}\sim m^{1+\psi_{\rm c}/2}$, implying $x^{2\zeta}\sim m^{-(2+\psi_{\rm c})\zeta}\equiv m^{-2\zeta_{m}}$. The second follows from $\lambda u\sim c$. The last one is obtained from $\eta/t\sim m^{2}$, implying $t\sim m^{-2-\psi_{\eta}}\sim x^{(2+\psi_{\eta})\zeta/\zeta_{m}}$.

In Eq. (3) we had defined the renormalized (effective) force correlator as

$$\Delta(w-w^{\prime}):=m^{4}L^{d}\overline{(u_{w}-w)(u_{w^{\prime}}-w^{\prime})}^{c}.$$

(19)

The definition of $u_{w}$ is given in Eq. (4). This is the same definition as the one used for qEW LeDoussalWiese2006a ; MiddletonLeDoussalWiese2006 ; RossoLeDoussalWiese2006a ; BonachelaAlavaMunoz2008 . Integrating the equation of motion (1) over space for a configuration $u_{w}(x):=u(x,t)$ at rest yields

$$m^{2}(w-u_{w})+\frac{1}{L^{d}}\int_{x}\underbrace{\lambda\left[\nabla u_{w}(x)\right]^{2}+F\big{(}x,u_{w}(x)\big{)}}_{\mbox{total force}}=0.$$

(20)

Thus the correlator in Eq. (19) measures fluctuations of the total force. Only for qEW ($\lambda=0$) this equals the force exerted by the disorder. To be specific, let us define

	$\displaystyle F_{w}$	$\displaystyle:=$	$\displaystyle\frac{1}{L^{d}}\int_{x}F\big{(}x,u_{w}(x)\big{)},$		(21)
	$\displaystyle\Lambda_{w}$	$\displaystyle:=$	$\displaystyle\frac{1}{L^{d}}\int_{x}\lambda[\nabla u_{w}(x)]^{2}.$		(22)

A configuration at rest then has

$$m^{2}(w-u_{w})+F_{w}+\Lambda_{w}=0.$$

(23)

Our goal is to compare observables with objects in the field theory. What is calculated there is the effective action, or more precisely its 2-time contribution. (In the statics this would be the 2-replica term.) It is the sum of all connected 2-time diagrams, i.e. with two external $\tilde{u}$ fields. To 1-loop order, these are shown in Fig. 3. The 2-point function $\overline{uu}^{c}$ is obtained to all orders by contracting the 2-time contribution to the effective action with two response functions. While in real space this is a convolution, in momentum and frequency space this is simply a multiplication with the response function $R(k,\omega)$. According to Eq. (19) it is to be evaluated at momentum $k=0$ and frequency $\omega=0$. Recall that the response function $R(x,t)$ is the response of the observable $u(x,t)$ to a small uniform kick in force $f$ at $(x,t)=(0,0)$. Since the center of mass follows the center of the driving parabola $w$,

$$f_{\rm c}=\overline{w-u_{w}}=\mbox{const}\quad\Rightarrow\quad\partial_{w}\overline{u_{w}}=1.$$

(24)

Thus a uniform kick $f=m^{2}\delta w$ leads to a response for the center of mass according to $u_{w}\rightarrow u_{w}+\delta w=u_{w}+\frac{f}{m^{2}}$. As a result, the integrated response function is given by

$$\int_{t}R(k=0,t)\equiv\frac{1}{L^{d}}\int_{x}\int_{t}R(x,t)=\frac{1}{m^{2}}.$$

(25)

This is equivalent to $R(k=0,\omega=0)=m^{-2}$.

We finally need to remember the field-theoretic definition of the effective action $\Gamma$: It is obtained from the corresponding expectation values by amputation of the response function, which is equivalent to dividing by the response function (in Fourier representation). Due to Eq. (25) this is nothing but multiplication with $m^{2}$, once for each of the two external fields $u$. This gives the factor of $m^{4}$ in Eq. (19), and Eq. (19) is nothing but the 2-time contribution to the effective action $\Gamma$, equivalent to the renormalized force correlator $\Delta(w)$. It is the $(k=0,\omega=0)$ mode of the full effective force correlator in the field theory for depinning.

Having established that Eq. (19) is the proper definition of the renormalized $\Delta(w)$, it is still instructive to study the correlations of all three forces appearing in Eq. (20). To this aim, let us define in addition to Eq. (19)

	$\displaystyle\Delta_{FF}(w-w^{\prime})$	$\displaystyle:=$	$\displaystyle L^{d}\overline{F_{w}F_{w^{\prime}}}^{\rm c},$		(26)
	$\displaystyle\Delta_{F\Lambda}(w-w^{\prime})$	$\displaystyle:=$	$\displaystyle L^{d}\overline{F_{w}\Lambda_{w^{\prime}}}^{\rm c},$		(27)
	$\displaystyle\Delta_{\Lambda\Lambda}(w-w^{\prime})$	$\displaystyle:=$	$\displaystyle L^{d}\overline{\Lambda_{w}\Lambda_{w^{\prime}}}^{\rm c}.$		(28)

A measurement of these quantities is shown below in Fig. 10.

Let us finally give the scaling dimensions,

$$\Delta(0)\sim m^{4}\xi_{m}^{d}\left[u_{w}-w\right]^{2}\sim m^{4-d\frac{\zeta_{m}}{\zeta}-2\zeta_{m}}.$$

(29)

The scaling of the argument of $\Delta(w)$ is given by

$$w\simeq u\sim m^{-\zeta_{m}}.$$

(30)

These scalings are reflected in the FRG flow equations derived below in Eq. (55).

For TL92 in $d=1$, the scaling of a blocked interface at depinning is given by directed percolation TangLeschhorn1992 ; AmaralBarabasiBuldyrevHarringtonHavlinSadr-LahijanyStanley1995 ; BarabasiGrinsteinMunoz1996 ; Hinrichsen2000 ; AraujoGrassbergerKahngSchrenkZiff2014 ; Dhar2017 . In table 1 we summarize the exponents obtained this way, which guide us in the construction and tests of the FRG. Details are given in MukerjeeBonachelaMunozWiese2022 .

In dimensions $d\geq 2$ directed percolation paths are 1-dimensional, whereas the interface is $d$-dimensional. As a result, the mapping to DP no longer exists, and one has to introduce directed surfaces BarabasiGrinsteinMunoz1996 . The exponents we find in $d=2$ and $d=3$ are summarized on table 3 (page 3).

To guide our field-theoretical work, we first checked in dimension $d=1$ that the scaling exponents given in table 1 account for the measured values of $\psi_{\rm c}$ and $\psi_{\lambda}$ given in Eqs. (12)-(13). To this aim, a novel algorithm was designed MukerjeeBonachelaMunozWiese2022 to measure $\psi_{\rm c}$ and $\psi_{\lambda}$ by imposing a spatial modulation in the background-field configuration $w$. The simulations were performed for three different models, all in the qKPZ universality class: the cellular automaton TL92 TangLeschhorn1992 , anharmonic depinning RossoKrauth2001b ; MukerjeeBonachelaMunozWiese2022 , and a direct simulation of Eq. (1) MukerjeeBonachelaMunozWiese2022 . The best results were achieved for anharmonic depinning, thanks to an efficient algorithm for its evolution RossoKrauth2001b .

With the novel algorithm designed in MukerjeeBonachelaMunozWiese2022 , we measured the effective couplings $\lambda$ and $c$, as a function of $m$. In Fig. 4 (left) we show their flow as a function of $m$. To be specific, what we measure (left), and what is predicted from DP via table 1 (right) is

	$\displaystyle\psi_{\rm c}^{d=1}$	$\displaystyle=$	$\displaystyle 1.31(4),\qquad\psi_{\rm c}^{\rm DP}=1.30752(2),$		(31)
	$\displaystyle\psi_{\lambda}^{d=1}$	$\displaystyle=$	$\displaystyle 0.28(3),\qquad\psi_{\lambda}^{\rm DP}=0.26133(2).$		(32)

This confirms our scaling analysis and allows us to measure as shown on Fig. 4 the dimensionless amplitude

$$\mathcal{A}:=\frac{\Delta(0)}{|\Delta^{\prime}(0^{+})|}\frac{\lambda}{c}.$$

(33)

The ideas behind this definition is that the KPZ term has one field more than the elastic term. Thus the ratio $\lambda/c$ has the inverse dimension of a field, which is compensated by the first ratio. That ${\cal A}$ converges to the same value for two different models gives strong evidence that qKPZ is the effective theory, and that a fixed point of the renormalization-group flow is reached. In $d=1$, this ratio reads

$${\cal A}^{d=1}=1.10(2).$$

(34)

The last points to verify is that we can measure the effective-force correlator $\Delta(w)$, that different models in the qKPZ class have the same $\Delta(w)$, and that this function is close to, but distinct from the one for qEW. This is shown in Fig. 12.

Let us remind how anharmonic elastic terms generate a KPZ term at depinning LeDoussalWiese2002 : To this purpose consider a standard elastic energy, supplemented by an additional anharmonic (quartic) term (setting $c=1$ for simplicity),

$${\cal H}_{\mathrm{el}}[u]=\int_{x}\frac{1}{2}\left[\nabla u(x)\right]^{2}+\frac{c_{4}}{4}\left[\left(\nabla u(x)\right)^{2}\right]^{2}.$$

(35)

The corresponding terms in the equation of motion read

	$\displaystyle\partial_{t}u(x,t)$	$\displaystyle=$	$\displaystyle\nabla^{2}u(x,t)+c_{4}\nabla\left\{\nabla u(x,t)\left[\nabla u(x,t)\right]^{2}\right\}$		(36)
			$\displaystyle+...$

Since the r.h.s. of Eq. (36) is a total derivative, it is surprising that a KPZ-term can be generated in the limit of a vanishing driving velocity. This puzzle was solved in Ref. LeDoussalWiese2002 , where the KPZ term arises by contracting the non-linearity with one bare disorder (we drop the index on $\Delta_{0}$ from now on for simplicity of notation),

	$\displaystyle\!\!\!\delta\lambda$	$\displaystyle=$			(37)
		$\displaystyle=$	$\displaystyle-\frac{c_{4}}{p^{2}}\int_{t>0}\int_{{t^{\prime}>0}}\int_{k}\mathrm{e}^{-(t+t^{\prime})(k^{2}+m^{2})}\left[k^{2}p^{2}+2(kp)^{2}\right]$
			$\displaystyle\qquad\qquad\qquad\quad\times\Delta^{\prime}\big{(}u({x,t+t^{\prime}})-u({x,0})\big{)}.\qquad$

As $u(x,t+t^{\prime})-u(x,0)\geq 0$, the leading term in Eq. (37) can be written as

$\displaystyle\delta\lambda=-\frac{c_{4}}{p^{2}}\int\limits_{t>0}\int\limits_{t^{\prime}>0}\int\limits_{k}\mathrm{e}^{-(t+t^{\prime})(k^{2}+m^{2})}[k^{2}p^{2}{+}2(kp)^{2}]\Delta^{\prime}(0^{+}).$

Integrating over $t,t^{\prime}$ and using the radial symmetry in $k$ yields

$$\delta\lambda=-c_{4}\left(1+\frac{2}{d}\right)\int_{k}\frac{\Delta^{\prime}(0^{+})k^{2}}{(k^{2}+m^{2})^{2}}.$$

(39)

This shows that in the FRG a KPZ term is generated from the non-linearity. As $-\Delta^{\prime}(0^{+})>0$, its amplitude is positive. The integral (39) has a strong UV divergence, thus the generation of this term happens at small scales, similar to the generation of the critical force, see appendix A.3.

Here we summarize the 1-loop contributions to $c$, $\lambda$, $\eta$ and $\Delta$. This is almost the same calculation as in Ref. LeDoussalWiese2002 , with a little twist: Since we work in a massive scheme, many of the cancelations in LeDoussalWiese2002 no longer exist. We remind that this change in scheme was forced upon us by our decision to measure the effective parameters of the theory, necessitating to drive with a confining potential. We believe that this is also much closer to real experiments. It is a scheme widely used for perturbative RG for the Ising model in $d=3$, pioneered by G. Parisi and used up to 7 loop-order by B. Nickel and collaborators ParisiBook ; Parisi1980 ; BakerNickelGreenMeiron1976 ; NickelMeironBaker1977 ; BakerNickelMeiron1978 . As discussed above, we think of this fixed-dimension renormalization scheme as an expansion around the $d=0$ qEW fixed point. The diagrams from the perturbation in $\lambda$ are given in Figs. 5-7.

We obtain the same diagrams as in LeDoussalWiese2002 but with coefficients $a_{i}$ that differ from LeDoussalWiese2002 away from the upper critical dimension. The explicit calculations are given in appendix A. Terms with numerical coefficients only (no $a_{i}$) are those appearing already in qEW.

	$\displaystyle\frac{\delta\eta}{\eta}$	$\displaystyle=$	$\displaystyle-\left[a_{0}\hat{\lambda}\Delta^{\prime}\left(0^{+}\right)+\Delta^{\prime\prime}\left(0^{+}\right)\right]I_{1},$		(40)
	$\displaystyle\frac{\delta c}{c}$	$\displaystyle=$	$\displaystyle-\left[a_{1}\hat{\lambda}\Delta^{\prime}\left(0^{+}\right)+a_{2}\hat{\lambda}^{2}\Delta(0)\right]I_{1},$		(41)
	$\displaystyle\frac{\delta\lambda}{\lambda}$	$\displaystyle=$	$\displaystyle-\left[a_{3}\hat{\lambda}\Delta^{\prime}\left(0^{+}\right)+a_{4}\hat{\lambda}^{2}\Delta(0)\right]I_{1},$		(42)
	$\displaystyle\delta\Delta(u)$	$\displaystyle=$	$\displaystyle\left\{a_{5}\hat{\lambda}^{2}\Delta(u)^{2}-\partial_{u}^{2}\frac{1}{2}\left[\Delta(u)-\Delta(0)\right]^{2}\right\}I_{1},\qquad$		(43)

	$\displaystyle\!\!\!\hat{\lambda}$	$\displaystyle:=$	$\displaystyle\frac{\lambda}{c},$		(44)
	$\displaystyle\!\!\!I_{1}$	$\displaystyle=$	$\displaystyle\int_{k}\frac{1}{(ck^{2}+m^{2})^{2}},$		(45)
	$\displaystyle\!\!\!a_{0}$	$\displaystyle=$	$\displaystyle\frac{d}{4},\qquad a_{1}=1,\qquad a_{2}=\frac{d-1}{3},$		(46)
	$\displaystyle\!\!\!a_{3}$	$\displaystyle=$	$\displaystyle 1,\qquad a_{4}=\frac{d+2}{6},\qquad a_{5}=\frac{d(d+2)}{12}.\qquad$		(47)

The coefficients $a_{i}$ in the limit of $d\to 4$ used by LeDoussalWiese2002 are obtained by setting $d\to 4$, resulting into $a_{i}=1$ for all $i$, except $a_{5}=2$. While this is the standard procedure followed in a dimensional expansion, it misses that in dimension $d=0$ the KPZ term does not exist, thus cannot correct the remaining terms: viscosity $\eta$, and effective force correlator $\Delta(u)$. The factors of $d$ in coefficients $a_{0}$ and $a_{5}$ reflect this physical necessity. No such constraint exists for $c$ and $\lambda$: since they are absent from the equation of motion (1) in $d=0$, their coefficients can well be modified.

As $\lambda$ and $c$ appear in the combination of $\hat{\lambda}=\lambda/c$, the important question is whether this ratio is corrected. This is indeed the case as

$$\frac{\delta\hat{\lambda}}{\hat{\lambda}}=(a_{2}-a_{4})\hat{\lambda}^{2}\Delta(0)I_{1}=-\frac{4-d}{6}\hat{\lambda}^{2}\Delta(0)I_{1}.$$

(48)

Note that this term is negative, and have a power in $\hat{\lambda}$ superior to one. It will therefore stop the RG flow for $\hat{\lambda}$ at large $\hat{\lambda}$, allowing us to close our system of equations!

A final important point to mention is that the confining potential $\sim m^{2}$ is not renormalized. In qEW this is due to the statistical tilt symmetry (STS) Wiese2021 , which can be checked perturbatively: Since the effective force correlator contains $u$ only as a difference $u(x,t)-u(x,t^{\prime})$, no field $u$ without a time derivative can be generated. The same holds true here: since the additional KPZ vertex has additional spatial derivatives, it cannot generate a field $u$ without spatial derivatives. This property is very useful, as we can as in qEW use $m$ as an RG scale, without caveat.

Finally, the critical force is

	$\displaystyle F_{\rm c}$	$\displaystyle=$	$\displaystyle F_{\rm c}^{(1)}+F_{\rm c}^{(2)}$		(49)
		$\displaystyle\simeq$	$\displaystyle\left[\Delta^{\prime}(0^{+})+\frac{d}{2}\hat{\lambda}\Delta(0)\right]\int_{k}\frac{1}{ck^{2}+m^{2}}.$

The first contribution is negative, identical to qEW. The second is positive, and specific to qKPZ. The non-linearity reduces the force needed to depin the interface. This is derived in appendix A.3.

Above we calculated the perturbative corrections. We now derive the corresponding RG relations. Since $m$ is not corrected under renormalization, we use it to parameterize the flow of the remaining quantities. To this aim, first define the dimensionless field as

$$\mathbf{u}:=u\,m^{\zeta_{m}}.$$

(50)

We have $-m\frac{\partial}{\partial m}\Delta(u)=[\delta\Delta]\varepsilon I_{1}$. The integral $I_{1}$ defined in Eq. (45) is evaluated in Eq. (103) of appendix A.1,

$\displaystyle I_{1}$

$\displaystyle:=$

$\displaystyle\int_{k}\frac{1}{(ck^{2}+m^{2})^{2}}=\frac{m^{d-4}}{c^{d/2}}\frac{2\Gamma(1+\frac{\varepsilon}{2})}{\varepsilon(4\pi)^{d/2}}.~{}~{}~{}$

(51)

It scales as

$\displaystyle I_{1}\sim\xi_{m}^{-d}m^{-4},$

(52)

where we remind that

$$\Delta(0)\sim\xi_{m}^{d}m^{4}u^{2}.$$

(53)

The dimensionless renormalized correlator $\tilde{\Delta}(\mathbf{u})$ is then defined in terms of the effective force correlator $\Delta(u)$, such that it absorbs $\varepsilon I_{1}$ as

$$\tilde{\Delta}(\mathbf{u}):=\varepsilon I_{1}m^{2\zeta_{m}}\Delta\left(u=\mathbf{u}m^{-\zeta_{m}}\right).$$

(54)

The explicit $m$-dependent factor in front of $\Delta$ is the scaling dimension given in Eq. (29). This yields the flow equation for the effective dimensionless force correlator,

	$\displaystyle\partial_{\ell}\tilde{\Delta}(u)=$	$\displaystyle\left(4-d\frac{\zeta_{m}}{\zeta}-2\zeta_{m}\right)\tilde{\Delta}(u)+u\zeta_{m}\tilde{\Delta}^{\prime}(u)$		(55)
		$\displaystyle+\frac{d(d+2)}{12}\tilde{\lambda}^{2}\tilde{\Delta}(u)^{2}$
		$\displaystyle-\tilde{\Delta}^{\prime}(u)^{2}-\tilde{\Delta}^{\prime\prime}(u)\big{[}\tilde{\Delta}(u)-\tilde{\Delta}(0)\big{]}.$

Here we defined the dimensionless combination $\tilde{\lambda}$

$$\tilde{\lambda}:=\frac{\lambda}{c}m^{-\zeta_{m}}\equiv\hat{\lambda}m^{-\zeta_{m}}.$$

(56)

Its flow equation is obtained from Eq. (48) as

$$-{m}\partial_{m}{\tilde{\lambda}}=\zeta_{m}\tilde{\lambda}-\frac{4-d}{6}\tilde{\lambda}^{3}\tilde{\Delta}(0).$$

(57)

It has one fixed point $\tilde{\lambda}=0$, and a second non-trivial fixed point at

$$\tilde{\lambda}_{\rm c}=\sqrt{\frac{6\zeta_{m}}{(4-d)\tilde{\Delta}(0)}}.$$

(58)

We can see that in $d=4$ the fixed point disappears as $\tilde{\lambda}$ goes to infinity.