# Heat Equation from Exact Renormalization Group Equation (ERGE) at Local
Potential Approximation (LPA).

Phumudzo T. Rabambi
Department of Physics
University of Witswatersrand
Wits, 2050, South Africa
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

By simply applying the Local Potential Approximation (LPA) on the Polchinski’s
Exact Renormalization Group (ERG) flow equation for single Bosonic and
spinless Fermionic fields, and initially considering only the coarse-graining
(blocking) aspect of Wilson’s Renormalization Group program. Within the LPA
limit the Polchinski’s ERG flow equation simplifies into a heat differential
equation for both Bosonic and Fermionic fields. Solving the differential
equations leads to logarithmic interactions (logarithmic vertex function) in
both Bosonic and Fermionic fields at their fixed points.

##  1 Introduction.

Within the Framework of Exact Renormalization Group (ERG) methods [1, 2, 3,
4], with a particular focus on Polchinski’s Exact Renormalization Group
Equation (ERGE) [4], which is the Wilson’s Exact Renormalization Group (ERG)
[1] derived in the field theory context. We study a system described by a
general effective action consisting of single Bosonic and spinless Fermionic
field modes including their interactions. Working in Euclidean momentum space
where the theory is regularized by employing the cutoff functions of generic
form $K_{UV}(p,\Lambda)$ in both fields, whereby the cutoff functions suppress
field modes with momentum $p$ that are above the physical UV (ultra-violet)
cutoff scale $\Lambda$. The Polchinski’s ERGE emanate from requiring that as
the scale $\Lambda$ is lowered by integrating (course-graining) high momentum
modes, the effective action (vertex functions) of the theory should be varied
in such a way that the partition function of the theory remains unchanged. The
variation of the effective action due to lowering of the physical cutoff scale
is governed by the Renormalization Group (RG) equation, which is an exact
differential equation subsequently giving rise to ERG methods. ERG methods in
field theory are exact non-perturbative formulations of RG flow physics
encoding it as a differential variation of physical scale and the effective
action. They describe the evolution of the renormalized effective action along
the RG flow trajectory in terms of differential equations. From the theory
setup, these equations arise from the decimation (integration) of fields modes
momenta, followed by lowering of the momentum cutoff and then rescaling the
action parameters while enforcing the physical observables to remain invariant
by requiring the partition function to remain invariant. Usually the RG flow
has two physical cutoff scales, one is a high energy scale denoted as the UV
cutoff scale, and the second one is a low energy scale denoted as an IR
(infrared) cutoff scale, and the RG flow will normally be a flow from the UV
(high energy) limit towards the IR (low energy) limit. A generic RG flow
equation has the form

$\displaystyle\Lambda\frac{\partial}{\partial\Lambda}S_{\Lambda}[\Phi]=\mathcal{F}[S_{\Lambda}],$
(1)

where $S_{\Lambda}$ is the effective action and $\Phi$ represents a generic
set of fields. An important aspect to solve from the RG flow equations are the
fixed points of the theory, which is the scale invariant continuum limit of
the theory, and the equation to solve to obtain these fixed points is,

$\displaystyle\Lambda\frac{\partial}{\partial\Lambda}S_{\Lambda}[\Phi]=0.$ (2)

Generally solving such an equation specially in the ERGE context can be
challenging, since a handful of ERG’s problems end up as integro-differential
equations [1, 2, 3, 4], with Polchinski’s ERG flow equation being in this
category. Therefore finding an exact non-perturbative general solution from
these kind of equations generally it is a difficult tasks, in practice one
ends up with an infinite tower of equations and has to always truncate the
equations by performing certain approximations. Several non-perturbative
approximation solutions and methods are possible, such as derivative
expansion, which involves expanding the equations in powers of momenta and
this has shown to yield promising results for calculating critical exponents
and renormalization group flows equations in $2\leq d\leq 4$ scalar fields
theories [5, 6, 7, 8]. The simplest non-perturbative approximation from such
power momentum expansion is obtained from taking the leading term (zeroth
order term of the expansion), which is the Local Potential Approximation (LPA)
expansion. Here we will make use of such power momentum expansion, restrict
the computations within the LPA limit. As a start we avoid the consideration
of next to leading order terms and other higher order terms in momentum or
derivative expansion since these terms are known to yield physical solutions
(physical observables) which are scheme dependent (cutoff scale dependent) [8,
9, 10], which makes it difficult to compute physical observables such as the
critical exponents ambiguously. Whilst the advantage with the leading order
solutions is that, the LPA physical observables solutions are scheme
independent and can be computed unambiguously.

It is known and alluded [11] that the Polchinski’s ERG flow equation can be
written as a functional heat equation. However, presently it appears there is
lack of visible available work towards unpacking this alluded notion at a
basic level. Clearly the notion that the Polchinski’s ERG flow equation can be
represented as a functional heat equation hasn’t received much attention, and
consequently not much work has been done to unpack and elucidate this notion
with simple theoretical approaches. Therefore this paper aims to address this,
by providing a concrete connection or relation between the Polchinski’s ERG
flow equation and the well known standard heat equation within the LPA limit.
Utilizing this notion, in one of our main results in section 3 at equation
(40), we show that at low energy after relaxing the rescaling steps of
Wilson’s Renormalization Group program and only implement the coarse-graining
step, subsequently the Polchinski’s ERG flow equation within the LPA limit
simplifies into a standard heat equation,

$\displaystyle\frac{\partial}{\partial t}U(z,t)=\frac{\partial^{2}}{\partial
z\partial z}U(z,t),$ (3)

where $U(z,t)=e^{-V(z,t)}$, $t$ is a dimensionless parameter, $z$ represent
Bosonic fields and $V(z,t)$ is the vertex or interaction function. We will
explain more about the results and how they are obtained in later sections of
the paper. It is worth mentioning that the results are interesting due to
their simplistic nature of relating the Polchinski’s ERG flow equation within
the LPA to a heat equation, this can potentially open a new front to generally
study ERG equations through the available technology and language already well
developed and established in studying heat equations. Subsequently the
solution to this standard heat equation at the fixed point, which is related
to the steady state solution in the language of heat equations, we find that
the fixed point solution is governed by logarithmic interactions,

$\displaystyle V_{*}(z,t)=\ln\big{(}z\sqrt{2\pi}\big{)}+\frac{1}{2},$ (4)

where $V_{*}$ is the interaction function at the fixed point obtained at
$t=\frac{z^{2}}{2\lambda}$. The solution is interesting since fixed points
normally signals a Conformal Field Theory (CFT), therefore at this low energy
limit we have established a logarithmic interaction CFT. Contrary to
perturbing the solution in the vicinity of the Gaussian fixed point [12, 13,
14, 5], here we perturb the solution in the vicinity of the logarithmic fixed
point obtained above. For the perturbation recipe we use a small parameter to
linearize the Polchinski’s ERG flow equation in the vicinity of the
logarithmic fixed point, but here we implement both rescaling and coarse-
graining steps on the Polchinski’s ERG flow equation. This leads to an
infinite set tower of Hermite differentials at each expansion order in
parameter $t$, with the Hermite polynomials being solutions to these
differential equations. Treating the Fermionic fields in similar fashion, in
section 4 at equation (77) we obtain the following analogue of a heat equation
in Grassmann variables,

$\displaystyle\frac{\partial}{\partial
t}U(z,\bar{z},t)=\frac{\partial^{2}}{\partial
z\partial\bar{z}}U(z,\bar{z},t),$ (5)

where $U(z,\bar{z},t)=e^{-V(z,\bar{z},t)}$, $z$ and $\bar{z}$ are Grassmann
variables representing Fermionic fields, and $V(z,\bar{z},t)$ is the
interaction function. The fixed point solution obtained at
$t=\frac{z\bar{z}}{\lambda}$ also yields a logarithmic interaction function,

$\displaystyle V_{*}\big{(}z,\bar{z},t\big{)}=$
$\displaystyle\big{(}1+\ln(z\bar{z})\big{)},$ (6)

which also leads to similar conclusions as in the Bosonic fields case
regarding the fixed point being controlled by a logarithmic interaction CFT.
However, the nilpotent property in Grassmann variables truncates the expansion
in parameter $t$ and renders it finite, which leads to a finite set tower of
Grassmann differential equations.

The paper is organised as follows. In section 2 we discuss the derivation of
Polchinski’s ERG flow equation for a system containing both the single Bosonic
and spinless Fermionic fields, and only considering the coarse-graining step
of Wislon’s Renormalization Group program. In section 3 we additionally
consider the rescaling step of Wilson’s Renormalization Group program on the
Polschinki’s ERG equation, we apply the rescaling step and take the LPA limit
of the Polchinski’s ERG flow equation, then the differential equation
emanating from Polchinki’s ERG flow equation at the LPA limit is solved,
followed by finding the eigenvalues, eigenfunctions and the critical exponents
for the Bosonic fields sector. Then section 4 deals with solving the
Polchinski’s ERG flow equation within the LPA limit in the Fermionic fields
sector.

##  2 Derivation of Polchinski’s ERGE for Bosonic and Fermionic Fields.

The first part of this section provides a review of some useful materials and
known results regarding the Polchinski’s ERG flow equation. At the end we use
these known results to derive results of our own which we will use later on in
the paper. Our own final derivation results which we are after and are worth
noting are given in equation (20) and equation (26), these results represents
Polchinski’s ERG flow equation between interacting Bosonic and spinless
Fermionic. Readers familiar with the material of this section can just note
the mentioned equations above, before deciding to skip this section and move
on to the next sections.

ERGE equations are derived from two steps which subsequently follow one
another repetitively. The first step is a stepwise partial infinitesimal
integration of momentum modes, followed by the second step which involves
rescaling of variables (operators). In the field theory context, the first
step involving the path integral approach has been well elaborated and derived
in [4], which consequently leads to Polchinski’s ERG flow equation, and the
second step in field theory context is well covered in [14, 15, 8] pertaining
to implementing rescaling on Polchinski’s ERG flow equation. Here we will
derive a similar Polchinki’s ERG flow equation for an interacting system of
spinless Fermionic and Bosonic fields. For simplicity, we begin by presenting
the UV-regularized generating functional of connected Greens function for a
simple self-interacting Bosonic field [16, 17, 18], and as a start the
Fermionic fields will be omitted from the discussion of general features of
the flow equation, and they will be included later. We begin by a bare theory
regularized in the UV by the cutoff scale $\Lambda_{0}$ through the
propagator, which also includes a running infrared cutoff scale $\Lambda$, and
the UV-regularized generating function of connected Greens function will be

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=$ $\displaystyle\int D\phi
e^{-\int_{p}\frac{1}{2}\phi(p)G^{-1}_{B}(\frac{p}{\Lambda})\phi(-p)+\int_{p}J(p)\phi(-p)-S_{int}^{\Lambda_{0}}[\phi]},$
(7) ,

where on the RHS we have $J(p)$ as the external source field coupled to the
Boson field $\phi(p)$, the kinetic part of the theory is represented by the
inverse propagator $G_{B}^{-1}\big{(}\frac{p}{\Lambda}\big{)}$ and
$S_{int}^{\Lambda_{0}}[\phi]$ represent the interactions part of the bare
theory. Generically $G_{B}\big{(}\frac{p}{\Lambda}\big{)}$ is written as
follows

$\displaystyle
G_{B}\bigg{(}\frac{p}{\Lambda}\bigg{)}=\frac{K_{B}\big{(}\frac{p^{2}}{\Lambda^{2}}\big{)}}{p^{2}},$
(8)

where $K_{B}\big{(}\frac{p^{2}}{\Lambda^{2}}\big{)}$ is the UV regulating
cutoff function, which is responsible for suppressing high energy momentum
modes, the function falls rapidly as
$\frac{p^{2}}{\Lambda^{2}}\rightarrow\infty$ and it behaves as follows,

$\displaystyle K_{B}\bigg{(}\frac{p^{2}}{\Lambda^{2}}\bigg{)}\rightarrow
1\qquad\text{for}\qquad p^{2}\ll\Lambda^{2}$ (9) $\displaystyle
K_{B}\bigg{(}\frac{p^{2}}{\Lambda^{2}}\bigg{)}\rightarrow
0\qquad\text{for}\qquad p^{2}\gg\Lambda^{2},$

with $\Lambda=\Lambda_{0}e^{-t}$, $\Lambda_{0}$ being the UV regulator scale
which is held fixed, and parameter $t$ being the label that parametrizes the
RG flow and it is dimensionless and usually is a ratio of $\Lambda$ and
$\Lambda_{0}$. Normally the Exact Renormalization Group flow employs two
momentum scales, the first scale is the UV regulator scale $\Lambda_{0}$,
which remains fixed during the flow, the second scale is the infrared (IR)
flow scale $\Lambda$, which controls the flow down to where high momentum
modes have been integrated out, with all modes with $|p|>\Lambda$ being
already eliminated (integrated). What is important in the flow is the ratio of
both scales $\Lambda_{0}$ and $\Lambda$, and this ratio can be captured by the
dimensional parameter $t=\ln\big{(}\frac{\Lambda_{0}}{\Lambda}\big{)}$. Then
during the renormalization group flow starting from the UV ($t=0$,
$\Lambda=\Lambda_{0}$) flowing down to the IR ($t\rightarrow\infty$,
$\Lambda\rightarrow 0$), $\Lambda$ is reduced from $\Lambda_{0}$ to zero while
$S_{int}[\phi,\Lambda]$ evolves in such a way that the UV regularized
generating function remains invariant, and this is achieved if
$S_{int}[\phi,\Lambda]$ satisfies the renormalization group equation given in
equation (14) below, which is the Polchinski’s ERG flow equation obtained in
[4].

Alternatively in terms of functional derivatives acting on
$S_{int}^{\Lambda_{0}}[\phi]$, the UV-regularized generating function above
can also be written as follows [16, 17, 18],

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=$ $\displaystyle\int D\phi
e^{-\int_{p}\frac{1}{2}\phi(p)G^{-1}_{B}(\frac{p}{\Lambda})\phi(-p)+\int_{p}J(p)\phi(-p)-S_{int}^{\Lambda_{0}}[\phi]}$
(10) $\displaystyle=$ $\displaystyle
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{D_{\Lambda}^{\Lambda_{0}}}e^{-S_{int}^{\Lambda_{0}}[\phi]}\bigg{|}_{\phi=G_{B}J(p)},$

where

$\displaystyle D_{\Lambda}^{\Lambda_{0}}=$
$\displaystyle\frac{1}{2}\int_{p}G_{B}\bigg{(}\frac{p}{\Lambda}\bigg{)}\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}$
(11) $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}G_{B}\bigg{(}\frac{p}{\Lambda}\bigg{)}\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}.$

From the functional integral above, an effective interaction
$S_{int}[\phi,\Lambda]$ can now be defined as,

$\displaystyle
e^{-S_{int}[\phi,\Lambda]}=e^{\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}^{\Lambda_{0}}[\phi]},$
(12)

where $S_{int}[\phi,\Lambda]$ consists of all Feynman diagrams generated by
the interactions present in $S_{int}^{\Lambda_{0}}[\phi]$ which are obtained
in the bare theory of the action. However the internal propagators involve
both the UV cutoff $\Lambda_{0}$ and the infrared cutoff $\Lambda$. Now,
taking a differential with respect to the infrared cutoff scale $\Lambda$ in
equation (12) above as shown below,

$\displaystyle\frac{\partial
S_{int}[\phi,\Lambda]}{\partial\Lambda}e^{-S_{int}[\phi,\Lambda]}=\frac{\partial}{\partial\Lambda}\bigg{(}e^{\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}^{\Lambda_{0}}[\phi]}\bigg{)},$
(13)

the following differential flow equation is obtained below,

$\displaystyle-\frac{S_{int}[\phi,\Lambda]}{\partial\Lambda}=\frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}\frac{\partial
G_{B}\big{(}\frac{p}{\Lambda}\big{)}}{\partial\Lambda}\bigg{(}\frac{\delta^{2}S_{int}[\phi,\Lambda]}{\delta\phi(p)\delta\phi(-p)}-\frac{\delta
S_{int}[\phi,\Lambda]}{\delta\phi(p)}\frac{\delta
S_{int}[\phi,\Lambda]}{\delta\phi(-p)}\bigg{)}$ (14)
$\displaystyle-\Lambda\frac{\partial
S_{int}[\phi,\Lambda]}{\partial\Lambda}=\frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta^{2}S_{int}[\phi,\Lambda]}{\delta\phi(p)\delta\phi(-p)}-\frac{\delta
S_{int}[\phi,\Lambda]}{\delta\phi(p)}\frac{\delta
S_{int}[\phi,\Lambda]}{\delta\phi(-p)}\bigg{)},$

where $G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}=\Lambda\frac{\partial
G_{B}(\frac{p}{\Lambda})}{\partial\Lambda}$, and the equation obtained above
is the Polchinski’s ERG flow equation [4]. Building from these results above,
we will now add spinless Fermionic fields with propagator
$G_{F}\big{(}\frac{p}{\Lambda}\big{)}$ to the functional integral as follows,

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=\int D\psi D\bar{\psi}D\phi
e^{-\frac{1}{2}\int_{p}\phi(p)G_{B}^{-1}(\frac{p}{\Lambda})\phi(-p)-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}J(p)\phi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)-S_{int}^{\Lambda_{0}}[\phi,\bar{\psi},\psi]}$
(15)

where $S_{int}^{\Lambda_{0}}[\phi,\bar{\psi},\psi]$ is the bare interaction
part of the theory for the Bosonic and Fermionic fields, for simplicity we
denote the bare action as
$S_{int}^{\Lambda_{0}}[\phi,\bar{\psi},\psi]=S_{int}[\phi,\bar{\psi},\psi]$.
Applying equation (10) to the Bosonic fields on the functional integral above
we obtain,

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=\int D\psi D\bar{\psi}D\phi
e^{-\frac{1}{2}\int_{p}\phi(p)G_{B}^{-1}(\frac{p}{\Lambda})\phi(-p)-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}J(p)\phi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)-S_{int}[\phi,\bar{\psi},\psi]}$
(16) $\displaystyle=$ $\displaystyle\int D\psi
D\bar{\psi}e^{-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)}$
$\displaystyle\times
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}e^{-S_{int}[\phi,\bar{\psi},\psi]}}\bigg{|}_{\phi=G_{B}(\frac{p}{\Lambda})J(p)}.$

Apply the same equation (10) to the remaining Fermionic fields in the
functional integral we obtain

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=\int D\psi D\bar{\psi}D\phi
e^{-\frac{1}{2}\int_{p}\phi(p)G_{B}^{-1}(\frac{p}{\Lambda})\phi(-p)-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}J(p)\phi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)-S_{int}[\phi,\bar{\psi},\psi]}$
(17) $\displaystyle=$ $\displaystyle
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}\bigg{[}e^{-\int_{p}\bar{\eta}(p)G_{F}\eta(-p)}e^{-\int_{p}G_{F}\frac{\delta}{\delta\bar{\psi}(p)}\frac{\delta}{\delta\psi(-p)}}e^{-\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}[\phi,\bar{\psi},\psi]}\bigg{]}\bigg{|}_{\phi=G_{B}J,\bar{\psi}=G_{F}\eta,\psi=G_{F}\bar{\eta}}.$

Now we take the effective interaction action
$S_{int}[\phi,\bar{\psi},\psi,\Lambda]$ from the functional integral above as,

$\displaystyle
e^{-S_{int}[\phi,\bar{\psi},\psi,\Lambda]}=e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{-\int_{p}\bar{\eta}(p)G_{F}(\frac{p}{\Lambda})\eta(-p)}e^{-\int_{p}G_{F}(\frac{p}{\Lambda})\frac{\delta}{\delta\bar{\psi}(p)}\frac{\delta}{\delta\psi(-p)}}e^{-\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}[\phi,\bar{\psi},\psi]},$
(18)

and the flow equation for the theory is obtained by taking the derivative with
respect the scale $\Lambda$,

$\displaystyle\frac{\partial}{\partial\Lambda}e^{-S_{int}[\phi,\bar{\psi},\psi,\Lambda]}=\frac{\partial}{\partial\Lambda}\big{(}e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{-\int_{p}\bar{\eta}(p)G_{F}(\frac{p}{\Lambda})\eta(-p)}e^{-\int_{p}G_{F}(\frac{p}{\Lambda})\frac{\delta}{\delta\bar{\psi}(p)}\frac{\delta}{\delta\psi(-p)}}e^{-\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}[\phi,\bar{\psi},\psi]}\big{)},$
(19)

and in the end the results simplifies to the following equation,

$\displaystyle-\Lambda\frac{\partial S_{int}}{\partial\Lambda}=$
$\displaystyle\int_{p}\Lambda\frac{\partial
G_{F}(\frac{p}{\Lambda})}{\partial\Lambda}\bigg{(}\frac{\delta
S_{int}}{\delta\bar{\psi}(p)}\frac{\delta
S_{int}}{\delta\psi(-p)}-\frac{\delta^{2}S_{int}}{\delta\bar{\psi}(p)\delta\psi(-p)}\bigg{)}+\frac{1}{2}\int_{p}\Lambda\frac{\partial
G_{B}(\frac{p}{\Lambda})}{\partial\Lambda}\bigg{(}\frac{\delta
S_{int}}{\delta\phi(p)}\frac{\delta
S_{int}}{\delta\phi(-p)}-\frac{\delta^{2}S_{int}}{\delta\phi(p)\delta\phi(-p)}\bigg{)}$
(20) $\displaystyle=$
$\displaystyle\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S_{int}}{\delta\bar{\psi}(p)}\frac{\delta
S_{int}}{\delta\psi(-p)}-\frac{\delta^{2}S_{int}}{\delta\bar{\psi}(p)\delta\psi(-p)}\bigg{)}+\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S_{int}}{\delta\phi(p)}\frac{\delta
S_{int}}{\delta\phi(-p)}-\frac{\delta^{2}S_{int}}{\delta\phi(p)\delta\phi(-p)}\bigg{)},$

which is the Polchinski ERG flow equation for an interaction theory between
the Bosonic and spinless Fermionic fields. The effective interaction $S_{int}$
sometimes referred to as the interaction function or vertex function will be
further defined as follows,

$\displaystyle S_{int}[\phi,\bar{\psi},\psi,\Lambda]=$
$\displaystyle\sum_{m,n}^{\infty}\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\int\prod_{j=1}^{n}\frac{d^{d}p_{j}d^{d}q_{j}}{(2\pi)^{2d}}\phi(k_{1})\cdots\phi(k_{m})\big{(}\bar{\psi}(p_{1})\psi(q_{1})\big{)}\cdots\big{(}\bar{\psi}(p_{n})\psi(q_{n})\big{)}$
(21)
$\displaystyle\times\delta^{d}(\sum_{i}^{m}k_{i}+\sum_{i}^{n}p_{i}+\sum_{i}^{n}q_{i})\mathcal{L}_{m,n}(k_{1},\cdots,k_{m};p_{1},\cdots,p_{n};q_{1},\cdots,q_{n},\Lambda).$

The first step of Wilson’s Renormalization Group program is now complete. The
first step already performed above, involves integrating out the high momentum
modes of the fields, which is known as coarse-graining or blocking, this
generates an equivalent effective action with reduced physical cutoff scale.
Now, this has to be followed by the second step which is rescaling of the
theory’s action parameters (variables), by rescaling the momenta back to the
original physical cutoff scale including appropriate rescaling of fields in
order for the resulting equations to be cutoff scale independent, which also
ensures the equations (theory) to have fixed points solutions relating to
scale-invariant, massless renormalized theory. In the first step, the fields
$\phi(p)$ are integrated over the range $e^{-t}<|p|\leq 1$ such that the
partition function is left invariant, and the second step involves rescaling,
a change by a factor of $e^{-t}$ in all linear dimensions to restore the
original scale $\Lambda$ of the system, i.e $p\rightarrow p^{\prime}=e^{t}p$,
where parameter $t$ is an infinitesimal parameter.

Adopting the approach used in [14, 15] to apply rescaling on Polchinski’s ERG
flow equation shown earlier in equation (20), results in Polchinski’s original
equation having some additional terms coming from applying rescaling on the
original equation parameters, below we demonstrate how the additional terms
emanate. First, for simplicity we drop the Fermionic fields from equation
(21), then the action $S_{int}$ becomes,

$\displaystyle
S[\phi,\Lambda]=\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}p_{i}}{(2\pi)^{d}}\mathcal{L}_{m,0}(p_{1},\cdots,p_{m},\Lambda)\phi(p_{1})\cdots\phi(p_{m})\delta^{d}(p_{1}+\cdots+p_{m}).$
(22)

Under rescaling the momentum and Bosonic fields will change by a factor
$s=e^{-t}$ in all linear dimensions, and they respectively transform as,

$\displaystyle p=(1+t)p$ (23)
$\displaystyle\phi(s^{-1}p^{\prime})=(1+t)^{d-d_{\phi}}\phi(p),$

and applying this form of rescaling on Polchinski’s ERG flow equation (20) for
a case of single self-interacting Bosonic fields leads to the full rescaled
Polchinski’s ERG equation,

$\displaystyle\big{(}\Lambda\partial_{\Lambda}+d_{\phi}\bigtriangleup_{\phi}+\bigtriangleup_{\partial_{\phi}}-d\big{)}S[\phi,\Lambda]=\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,\Lambda]}{\delta\phi(p)}\frac{\delta
S[\phi,\Lambda]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,\Lambda]}{\delta\phi(p)\delta\phi(-p)}\bigg{)}$
(24)
$\displaystyle\big{(}\partial_{t}+d_{\phi}\bigtriangleup_{\phi}+\bigtriangleup_{\partial_{\phi}}-d\big{)}S[\phi,t]=\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,t]}{\delta\phi(p)}\frac{\delta
S[\phi,t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,t]}{\delta\phi(p)\delta\phi(-p)}\bigg{)},$

where $\partial_{t}=\Lambda\frac{\partial}{\partial\Lambda}$ is the time
derivative,
$\Delta_{\partial_{\phi}}=d+\int_{p}\phi_{p}p\cdot\partial_{p}\frac{\delta}{\delta\phi_{p}}$,
$\Delta_{\phi}=\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}$,
$d_{\phi}=\frac{1}{2}(d-2+\eta_{\phi})$ and $\eta_{\phi}$ is the anomalous
dimension of the scalar field $\phi(p)$. The details regarding how the results
in equation (24) were obtained are provided in the Appendix. Treating the
Fermionic fields in similar fashion, in the case of self-interacting spinless
Fermionic fields, now $S_{int}$ will be $S[\psi,\bar{\psi},\Lambda]$, and the
final derivation results for the Polchinski’s ERG flow equation is,

$\displaystyle\big{(}\partial_{t}-d+d_{\psi\bar{\psi}}\Delta_{\psi\bar{\psi}}+\Delta_{\partial_{\psi\bar{\psi}}}\big{)}S[\psi,\bar{\psi},t]=\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\psi(p)}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\bar{\psi}(-p)}-\frac{\delta^{2}S[\psi,\bar{\psi},t]}{\delta\psi(p)\delta\bar{\psi}(-p)}\bigg{)}.$
(25)

where
$\Delta_{\psi\bar{\psi}}=\int_{p}\psi(p)\frac{\delta}{\delta\psi(-p)}+\int_{p}\bar{\psi}(p)\frac{\delta}{\delta\bar{\psi}(-p)}$,
$\Delta_{\partial_{\psi\bar{\psi}}}=d+\int\frac{d^{d}p}{(2\pi)^{d}}\psi(p)p\cdot\frac{\partial}{\partial
p}\frac{\delta}{\delta\psi(-p)}+\int\frac{d^{d}p}{(2\pi)^{d}}\bar{\psi}(p)p^{\cdot}\frac{\partial}{\partial
p}\frac{\delta}{\delta\bar{\psi}(-p)}$,
$d_{\psi\bar{\psi}}=\frac{1}{2}(d-1+\eta_{\psi})$ and $\eta_{\psi}$ is the
anomalous dimensions of the Fermionic fields. Now, the general rescaled
Polchinski’s ERG flow equation for the interactions between the single Bosonic
and spinless Fermionic fields together will be,

$\displaystyle\big{(}\partial_{t}+d_{\psi\bar{\psi}}\Delta_{\psi\bar{\psi}}+\Delta_{\partial_{\psi\bar{\psi}}}+$
$\displaystyle
d_{\phi}\bigtriangleup_{\phi}+\bigtriangleup_{\partial_{\phi}}-d\big{)}S[\phi,\psi,\bar{\psi},t]=\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\psi(p)}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\bar{\psi}(-p)}-\frac{\delta^{2}S[\phi,\psi,\bar{\psi},t]}{\delta\psi(p)\delta\bar{\psi}(-p)}\bigg{)}$
(26)
$\displaystyle+\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\phi(p)}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,\psi,\bar{\psi},t]}{\delta\phi(p)\delta\phi(-p)}\bigg{)}.$

This is the final derivation we were after in this section. We have derived
the Polchinski’s ERG flow equation for the single Bosonic and spinless
Fermionic fields. We chose spinless Fermionic fields since their computations
are much easier to deal with here, and the results can be generalized to
Fermionic fields with spins.

##  3 Bosonic Field Local Potential Approximation.

The Polchinski’s ERG flow equation derived in the previous section in equation
(26) is an integro-differential equation, in general this type of differential
equation is challenging to solve, especially at non-perturbative level. In
Renormalization Group one can handle such type of differential equations by
making use of perturbative expansion either in powers of small coupling
parameter epsilon ($\epsilon$) expansion or $\frac{1}{N}$ large $N$ parameter
expansion. At non-perturbative level and in the absence of expansion
parameters, one way to tackle this differential equation is to do some
approximation or truncation by reducing the number of degrees of freedom.
Performing derivative or momentum expansion of the effective action is one
possible approximation that can be undertaken, therefore we will take the
effective action $S_{int}$ from equation (21) with only Bosonic fields,

$\displaystyle
S[\phi,\Lambda]=\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\mathcal{L}_{m,0}(k_{1},\cdots,k_{m},\Lambda)\phi(k_{1})\cdots\phi(k_{m})(2\pi)^{d}\delta^{d}(k_{1}+\cdots+k_{m}),$
(27)

and expand the vertex or coupling function $\mathcal{L}_{m,0}$ in terms of
$k^{2}$ momentum, which is a derivative expansion in position space,

$\displaystyle\mathcal{L}_{m,0}(k_{1},\cdots,k_{m},\Lambda)=\mathcal{L}_{m,0}(0,\cdots,0,\Lambda)+\frac{1}{m(m-1)}(k_{1}^{2}+\cdots+k_{m}^{2})\mathcal{L}_{m-2,0}(0,\cdots,0,\Lambda)+\mathcal{O}(k^{4})+\cdots.$
(28)

It then follows that the effective action $S[\phi,\Lambda]$ will be an
expansion in $k^{2}$ momentum,

$\displaystyle S[\phi,\Lambda]=$
$\displaystyle\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\bigg{(}\mathcal{L}_{m,0}+\frac{k_{1}^{2}+\cdots+k_{m}^{2}}{m(m-1)}\mathcal{L}_{m-2,0}+\mathcal{O}(k^{4})+\cdots\bigg{)}\phi(k_{1})\cdots\phi(k_{m})(2\pi)^{d}\delta^{d}(k_{1}+\cdots+k_{m})$
(29) $\displaystyle=$
$\displaystyle\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\bigg{(}V(\phi,\Lambda)+W(\phi,\Lambda)+\mathcal{O}(k^{4})\bigg{)}$

where
$V(\phi,\Lambda)=\mathcal{L}_{m,0}(0,\cdots,0,\Lambda)\phi(k_{1})\cdots\phi(k_{m})(2\pi)^{d}\delta^{d}(k_{1}+\cdots+k_{m})$
is a local term which contains no derivatives, this term can be viewed as a
local potential term, while the $W(\phi,\Lambda)$ term and other higher order
expansion terms are non-local, these terms have derivatives acting on them in
position space. To project out only the local potential term from the action
$S[\phi,\Lambda]$ we can make use of the Hasenfratz and Hazenfratz projection
operator $P(z)$ [5]. When this projection operator is applied on the action
$S[\phi,\Lambda]$ it only selects the local potential term as follows,

$\displaystyle P(z)S[\phi,\Lambda]=$ $\displaystyle
e^{z\frac{\partial}{\partial\phi(0)}}S[\phi,\Lambda]\bigg{|}_{\phi=0}$ (30)
$\displaystyle=$
$\displaystyle\sum_{m}\mathcal{L}_{m,0}(0,0,\cdots,0,\Lambda)z^{m}\delta^{d}(0)$
$\displaystyle=$ $\displaystyle V(z,\Lambda)\delta^{d}(0),$

where $V(z,\Lambda)$ is a local potential function. Then the action of the
operator $P(z)$ on the term $\frac{\delta S[\phi,\Lambda]}{\delta\phi}$ yields
the results,

$\displaystyle\int_{p}P(z)\frac{\delta
S[\phi,\Lambda]}{\delta\phi(p)}\bigg{|}_{\phi=0}=$
$\displaystyle\int_{p}e^{z\frac{\partial}{\partial\phi(0)}}\frac{S[\phi,\Lambda]}{\delta\phi(p)}\bigg{|}_{\phi=0}$
(31) $\displaystyle=$ $\displaystyle V^{\prime}(z,\Lambda)\delta^{d}(p),$

where $V^{\prime}(z,\Lambda)$ is the derivative of the local potential
function $V(z,\Lambda)$ with respect to $z$. Therefore, when the operator
$P(z)$ acts on the entire rescaled Polchinski’s ERG flow equation as follows,

$\displaystyle
P(z)\bigg{(}\partial_{t}+d_{\phi}\triangle_{\phi}+\triangle_{\partial}-d\bigg{)}S[\phi,t]=P(z)\bigg{(}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,t]}{\delta\bar{\phi}(p)}\frac{\delta
S[\phi,t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,t]}{\delta\bar{\phi}(p)\delta\phi(-p)}\bigg{)}\bigg{)},$
(32)

it then leads to the following results,

$\displaystyle\bigg{(}\partial_{t}+d_{\phi}\triangle_{\phi}+\triangle_{\partial}-d\bigg{)}V(z,t)=G_{B}^{\prime}(0)V^{\prime
2}(z,t)-V^{\prime\prime}(z,t)\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)},$
(33)

where now $d_{\phi}=\frac{d-2}{2}$ and the anomalous dimension $\eta_{\phi}=0$
in the LPA limit since there is no momentum dependence in the zeroth order of
the derivative expansion. Further simplification leads to the following
differential equation,

$\displaystyle\partial_{t}V(z,t)=$
$\displaystyle\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}V^{\prime\prime}(z,t)-G_{B}^{\prime}(0)V^{\prime
2}(z,t)-d_{\phi}zV^{\prime}(z,t)+dV(z,t),$ (34)

and writing $V(z,t)$ and $z$ as follows,

$\displaystyle V(z,t)\rightarrow$
$\displaystyle\frac{V(z,t)\int_{p}G_{B}^{\prime}(\frac{p}{\Lambda})}{G_{B}^{\prime}(0)},$
(36) $\displaystyle z\rightarrow$ $\displaystyle
z\sqrt{\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}},$

then the differential equation above becomes,

$\displaystyle\partial_{t}V(z,t)=V^{\prime\prime}(z,t)-V^{\prime
2}(z,t)-d_{\phi}zV^{\prime}(z,t)+dV(z,t).$ (37)

To find the fixed points of the equation above where $V(z,t)=V_{*}(z,t)$ we
have to calculate $\partial_{t}V(z,t)=0$. We begin with a simple approach,
first we opt to remove the rescaling terms from the Polchinski’s ERG flow
equation. We will drop the terms $-d_{\phi}zV^{\prime}(z,t)+dV(z,t)$ from
equation (37), these terms emanate from applying rescaling on the Polchinski’s
ERG flow equation. Dropping of rescaled terms results in the following
Polchinski’s ERG flow equation,

$\displaystyle\partial_{t}V(z,t)=V^{\prime\prime}(z,t)-V^{\prime 2}(z,t)$ (38)
$\displaystyle\noindent\partial_{t}e^{-V(z,t)}=\frac{\partial^{2}}{\partial
z\partial z}e^{-V(z,t)}.$ (39)

Let $U(z,t)=e^{-V(z,t)}$, then $V(z,t)=-\ln(U(z,t))$, and the differential
equation above will now be written as follows

$\displaystyle\partial_{t}U(z,t)=\frac{\partial^{2}}{\partial z\partial
z}U(z,t),$ (40)

which resembles a homogeneous differential equation for the heat equation, and
the solution is of the form,

$\displaystyle U(z,t)=\frac{1}{\sqrt{4\pi\lambda t}}e^{-\frac{z^{2}}{4\lambda
t}},$ (41) $\displaystyle V(z,t)=-\ln(U(z,t)).$

The fixed points of this equation are obtained from computing
$\partial_{t}U_{*}(z,t)=0$ or $\partial_{t}V_{*}(z,t)=0$, which gives the
following solution,

$\displaystyle\partial_{t}V_{*}(z,t)=0$ (42) $\displaystyle\Rightarrow
t=\frac{z^{2}}{2\lambda}\qquad\qquad\qquad\qquad$ $\displaystyle\Rightarrow
U_{*}(z,t)=\frac{1}{z\sqrt{2\pi}}e^{-\frac{1}{2}}$ $\displaystyle\Rightarrow
V_{*}(z,t)=\ln\big{(}z\sqrt{2\pi}\big{)}+\frac{1}{2}.$

The solution indicates that at the fixed point we have a logarithmic potential
or interaction (vertex) function $V_{*}(z,t)=\ln(z\sqrt{2\pi})+\frac{1}{2}$.
Which could give an indication that the fixed point is governed by the
logarithmic CFT (Conformal Field Theory), and when $t\rightarrow 0$ we see
that $U(z,t)\rightarrow 0$ and $V(z,t)\rightarrow\infty$. Now, using the full
rescaled Polchinski’s ERG flow equation in equation (37),

$\displaystyle\partial_{t}V(z,t)=V^{\prime\prime}(z,t)-V^{\prime
2}(z,t)-d_{\phi}zV^{\prime}(z,t)+dV(z,t),$ (43)

and perturbing around the fixed point solution to find non trivial fixed
points solution at order $\mathcal{O}(g_{b})$ where the perturbed solution
$V_{p}(z,t)$ is given as,

$\displaystyle V_{p}(z,t)=$ $\displaystyle V(z,t)+g_{b}\delta V(z,t)$ (44)
$\displaystyle=$ $\displaystyle V(z,t)+g_{b}\sum_{n}f_{n}(z)e^{\lambda_{n}t}$

where $V(z,t)$ is the homogeneous solution obtained in equation (41), and
$g_{b}\ll 1$ is a small interaction constant or term. Plugging $V_{p}(z,t)$
into the full rescaled Polchinski’s differential equation obtained within the
LPA limit in equation (37),

$\displaystyle\partial_{t}V_{p}(z,t)=V^{\prime\prime}_{p}(z,t)-V^{\prime
2}_{p}(z,t)-d_{\phi}zV^{\prime}_{p}(z,t)+dV_{p}(z,t),$ (45)

at order $\mathcal{O}(g_{b})$ the differential equation below is obtained,

$\displaystyle e^{\lambda t}\frac{\partial^{2}f(z)}{\partial z\partial
z}=e^{\lambda t}\bigg{(}(\lambda-d)f(z)+d_{\phi}z\frac{\partial f(z)}{\partial
z}+2\frac{\partial V(z,t)}{\partial z}\frac{\partial f(z)}{\partial
z}\bigg{)},$ (46)

expanding the last term on the RHS as shown below,

$\displaystyle e^{\lambda t}\frac{\partial V(z,t)}{\partial z}\frac{\partial
f(z)}{\partial z}=\frac{z}{2}\bigg{(}\frac{1}{\lambda t}+1+\frac{\lambda
t}{2!}+\cdots\bigg{)}\frac{\partial f(z)}{\partial z},$ (47)

and then collecting the zeroth order ($\mathcal{O}(t^{0})$) terms in expansion
parameter $t$ from the differential equation (46) we get the following
differential equation below,

$\displaystyle\frac{\partial^{2}f(z)}{\partial z\partial z}=$
$\displaystyle(\lambda-d)f(z)+d_{\phi}z\frac{\partial f(z)}{\partial
z}+2\frac{\lambda}{\lambda}\frac{z}{2}\frac{\partial f(z)}{\partial z}$ (48)
$\displaystyle=$
$\displaystyle(\lambda-d)f(z)+\big{(}d_{\phi}+1\big{)}z\frac{\partial
f(z)}{\partial z}$ $\displaystyle=$
$\displaystyle(\lambda-d)f(z)+c_{\phi\lambda}z\frac{\partial f(z)}{\partial
z},$

where $c_{\phi\lambda}=d_{\phi}+1$. Writing the function $f(z)$ as
$f(z)\rightarrow\frac{2}{c_{\phi\lambda}}f(\sqrt{\frac{c_{\phi\lambda}}{2}}z)$
[12, 13, 14, 5], then the differential equation above simplifies to,

$\displaystyle\frac{\partial^{2}f(\tilde{z})}{\partial\tilde{z}\partial\tilde{z}}=2\frac{\lambda-d}{c_{\phi\lambda}}f(\tilde{z})+2\tilde{z}\frac{\partial
f(\tilde{z})}{\partial\tilde{z}},$ (49)

where $\tilde{z}=\sqrt{\frac{c_{\phi\lambda}}{2}}z$. This is a Hermite
differential equation, which is analogous to the Hermite differential equation
also obtained in [12, 13, 5, 14], and the Hermite polynomial which is the
solution to this differential equation is given as follows,

$\displaystyle
f(z)=f_{k}^{(0)}(\tilde{z})=\sum_{r=0}^{\frac{k}{2}}\frac{(-1)^{r}k!}{r!(k-2r)!}(2\tilde{z})^{k-2r},$
(50)

with the Hermite polynomial eigenvalues $\lambda=\lambda_{k}^{(0)}$ given as,

$\displaystyle 2\frac{d-\lambda_{k}^{(0)}}{c_{\phi\lambda}}=2k-1,\quad
k=0,1,2,3,\cdots$ (51)
$\displaystyle\Rightarrow\lambda_{k}^{(0)}=-\frac{(2k-1)d}{4}+d,$

where the superscript $(0)$ indicates that these are zeroth order expansion in
parameter $t$ eigenvalues and eigenfunctions. The Hermite polynomials are the
eigenfunctions or eigenoperators of the theory at this limit. Expanding
equation (46) again to first order in parameter $t$, the Hermite differential
equation reads as follows,

$\displaystyle\frac{\partial^{2}f(z)}{\partial z\partial z}=$
$\displaystyle(\lambda-d)f(z)+d_{\phi}z\frac{\partial f(z)}{\partial
z}+2\frac{\lambda}{2\lambda}\frac{z}{2}\frac{\partial f(z)}{\partial z}$ (52)
$\displaystyle=$ $\displaystyle
2\frac{\lambda-d}{c_{\phi\lambda}}f(z)+2z\frac{\partial f(z)}{\partial z},$

where this time
$f(z)\rightarrow\frac{2}{c_{\phi\lambda}}f\bigg{(}\sqrt{\frac{c_{\phi\lambda}}{2}}z\bigg{)}$
and here $c_{\phi\lambda}=d_{\phi}+\frac{1}{2}$. The corresponding Hermite
polynomial at this first order in expansion parameter $t$ will be the same as
the previous one given in equation (50),

$\displaystyle
f(z)=f_{k}^{(1)}(z)=\sum_{r=0}^{\frac{k}{2}}\frac{(-1)^{r}k!}{r!(k-2r)!}(2z)^{k-2r},$
(53)

with the eigenvalues $\lambda=\lambda_{k}^{(1)}$ for the Hermite polynomial at
this order given as,

$\displaystyle\lambda_{k}^{(1)}=-\frac{(2k-1)(d-1)}{4}+d,\quad
k=0,1,2,3,\cdots.$ (54)

Then at the $n^{th}$ order in expansion parameter $t$ we get the following
Hermite differential,

$\displaystyle\frac{\partial^{2}f(z)}{\partial z\partial z}=$ $\displaystyle
2\frac{\lambda-d}{c_{\phi\lambda}}f(z)+2z\frac{\partial f(z)}{\partial z},$
(55)

where now $c_{\phi\lambda}=d_{\phi}+\frac{1}{(n+1)}$, and the Hermite
polynomial is the same as before, with the eigenvalues
$\lambda=\lambda_{k}^{(n)}$ at $n^{th}$ order given as,

$\displaystyle\lambda_{k}^{(n)}=-\frac{(2k-1)(d-2+\frac{2}{n+1})}{4}+d.$ (56)

Then at $n\rightarrow\infty$,

$\displaystyle\lambda_{k}^{(\infty)}=-\frac{(2k-1)(d-2)}{4}+d.$ (57)

Take note that at order $\frac{1}{t}$ expansion we have set the function
$f^{(-1)}(z)=constant$, since when expanding (46) we find that,

$\displaystyle 0=\frac{z}{\lambda t}\frac{\partial f^{(-1)}(z)}{\partial z}$
(58) $\displaystyle\Rightarrow f^{(-1)}(z)=constant.$

Putting together all the expanded terms above originating from equation (46)
we arrive at the solution,

$\displaystyle V_{p}(z,t)=$ $\displaystyle V(z,t)+g_{b}e^{\lambda t}f(z)$ (59)
$\displaystyle=$ $\displaystyle V(z,t)+g_{b}\bigg{(}\frac{1}{\lambda
t}f^{(-1)}(z)+\lambda_{k}^{(0)}f_{k}^{(0)}(z)+t\lambda_{k}^{(1)}f_{k}^{(1)}(z)+\frac{t^{2}}{2!}\lambda_{k}^{(2)}f_{k}^{(2)}(z)+\cdots+\frac{t^{n}}{n!}\lambda_{k}^{(n)}f_{k}^{(n)}(z)\bigg{)},$
$\displaystyle=$ $\displaystyle V(z,t)+\frac{g_{b}}{\lambda
t}f^{(-1)}_{k}(z)+\sum_{n=0}\frac{\lambda_{k}^{(n)}t^{n}}{n!}f_{k}^{(n)}(z).$

Then at the logarithmic fixed point at $t=\frac{z^{2}}{2\lambda}$, the
solution above becomes,

$\displaystyle V_{p}(z,t)=$
$\displaystyle\ln\big{(}z\sqrt{\pi}\big{)}+\frac{1}{2}+g_{b}\bigg{(}\frac{2}{z^{2}}f^{(-1)}(z)+\lambda_{k}^{(0)}f_{k}^{(0)}(z)+\frac{z^{2}}{2\lambda}\lambda^{(1)}_{k}f_{k}^{(1)}(z)+\cdots+\frac{z^{2n}}{(2\lambda)^{n}n!}\lambda_{k}^{(n)}f_{k}^{(n)}(z)\bigg{)},$
(60) $\displaystyle=$
$\displaystyle\ln\big{(}z\sqrt{\pi}\big{)}+\frac{1}{2}+2\frac{g_{b}}{z^{2}}f^{(-1)}(z)+\sum_{n=0}\frac{\lambda_{k}^{(n)}}{n!}\frac{z^{2n}}{(2\lambda)^{n}}f_{k}^{(n)}(z).$

The results obtained here show that perturbing the interaction/vertex function
in the vicinity of a logarithmic fixed point where we first linearize around
the homogeneous solution as shown in equation (44), plugging this interaction
function into the full rescaled Polchinski’s ERG flow equation as shown in
equation (45), the solution leads to an infinite set tower of Hermite
differential equation in expansion parameter $t$. The solution to these
Hermite differential equations is a Hermite polynomial, but the Hermite
eigenvalues for each Hermite differential equation are different at each order
in expansion parameter $t$ as demonstrated in equations (51), (54) and (56).

Now that the fixed points for the Bosonic system have been determined, the
natural step that follows should be the computation of the systems critical
exponents. Here we will calculate the critical exponents $\nu$ and $w$, where
$\nu$ is the correlation length exponent and is obtained from the highest
positive eigenvalue ($\nu=\frac{1}{\lambda}$), while $w$ is the first
correction-to-scaling and is obtained from the less negative eigenvalue. In
instances where there are an infinity many negative eigenvalues, then $w_{i}$
will be $i^{th}$ correction-to-scaling whereby $w_{1}$ will denote the first
correction-to-scaling, $w_{2}$ denoting the second correction-to-scaling etc.
Here we are also going to calculate the critical points at each order in
expansion parameter $t$. Starting with the zero order expansion in parameter
$t$, at $d=4$, the eigenvalues for the eigenfunctions will be,

$\displaystyle\lambda_{k}^{(0)}=$ $\displaystyle-\frac{(2k-1)d}{4}+4$ (61)
$\displaystyle=$ $\displaystyle-(2k-1)+4,\qquad k=0,1,2,\cdots,$

and the non-negative eigenvalues are

$\displaystyle\lambda_{0}^{(0)}=5,\quad\lambda_{1}^{(0)}=3,\quad\lambda_{2}^{(0)}=1.$
(62)

The corresponding eigenfunctions for these non-negative eigenvalues
respectively will be,

$\displaystyle f_{0}^{(0)}=$ $\displaystyle 1,$ (63) $\displaystyle
f_{1}^{(0)}=$ $\displaystyle 2z,$ $\displaystyle f_{2}^{(0)}=$ $\displaystyle
4z^{2}-2.$

Now, the first two negative eigenvalues are $\lambda_{3}=-1$, $\lambda_{4}=-3$
and the corresponding eigenfunctions respectively are,

$\displaystyle f_{3}^{(0)}=$ $\displaystyle 8z^{3}-12z,$ (64) $\displaystyle
f_{4}^{(0)}=$ $\displaystyle 16z^{4}-48z^{2}+12.$

The first negative eigenvalues closest to zero are used to calculate the
critical exponent $w_{1}$ and $w_{2}$, and we find,

$\displaystyle w_{1}=1,$ (65) $\displaystyle w_{2}=3.$

The largest positive eigenvalue $\lambda_{0}^{(0)}=5$ is used to calculate the
critical exponent $\nu$, and we find,

$\displaystyle\nu=0,2.$ (66)

Moving to the first order expansion in parameter $t$, the eigenvalue formula
for the first order eigenfunction at $d=4$ is,

$\displaystyle\lambda_{k}^{(1)}=$ $\displaystyle-\frac{(2k-1)(d-1)}{4}+d,$
(67) $\displaystyle=$ $\displaystyle-(2k-1)\frac{3}{4}+4.$

The non-negative eigenvalues are, $\lambda_{0}^{(1)}=\frac{19}{4}$,
$\lambda_{1}^{(1)}=\frac{13}{4}$, $\lambda_{2}^{(1)}=\frac{5}{4}$ and their
corresponding eigenfunctions respectively are the same as in (64),

$\displaystyle f_{0}^{(0)}=$ $\displaystyle 1,$ (68) $\displaystyle
f_{1}^{(0)}=$ $\displaystyle 2z,$ $\displaystyle f_{2}^{(0)}=$ $\displaystyle
4z^{2}-2.$

The first two negative eigenvalues are $\lambda_{3}^{(1)}=-\frac{1}{4}$ and
$\lambda_{4}^{(1)}=-\frac{5}{4}$. At this order the critical exponents are as
follows,

$\displaystyle\nu=0.21,$ (69) $\displaystyle w_{1}=0.25,$ $\displaystyle
w_{2}=1.25.$

At the second order in expansion parameter $t$ we find the critical exponents
to be,

$\displaystyle\nu=0.21,\quad w_{1}=0.67,\quad w_{2}=4.67.$ (70)

The process should now be straight forward pertaining to how the critical
exponents can be calculated at each order in expansion parameter $t$,
therefore we will stop the computations at this order. Due to the infinite set
of Hermite differentials equations obtained earlier in (46) where they had the
same Hermite polynomial as a solution but with different eigenvalues at each
order in expansion parameter $t$, this also naturally forces the critical
exponent to have different values at each order in expansion parameter $t$.

##  4 Fermion fields homogeneous and hermite differential equations

For the Fermions we follow similar steps undertaken in the single Bosonic
field computations from the previous section. We start by first writing down
an operator that will extract and return a local potential term, similar to
the one given in Bosonic fields in equation (30). For the Fermions we define
the operator,

$\displaystyle
P(z,\bar{z})X[\psi,\bar{\psi}]=e^{z\frac{\partial}{\partial\psi(0)}+\bar{z}\frac{\partial}{\partial\bar{\psi}(0)}}X[\psi,\bar{\psi}]\bigg{|}_{\psi,\bar{\psi}=0},$
(71)

where $X[\psi,\bar{\psi}]$ is the test functional operator containing
Fermionic fields, and the variables $z$ and $\bar{z}$ are Grassman variables.
Dropping the Bosonic fields from the effective action term $S_{int}$ in
equation (21), $S_{int}$ will now be,

$\displaystyle
S[\psi,\bar{\psi},\Lambda]=\sum_{n=1}\int\prod_{j=1}^{n}\frac{d^{d}p_{j}d^{d}q_{j}}{(2\pi)^{d}}\mathcal{L}_{0,n}(p_{1},\cdots,p_{n};q_{1},\cdots,q_{n},\Lambda)\bar{\psi}(p_{1})\psi(q_{1})\cdots\bar{\psi}(p_{n})\psi(q_{n})\delta^{d}(\sum_{i}^{n}p_{i}+\sum_{i}^{n}q_{i}),$
(72)

and the full rescaled Polschinski’s ERG flow equation for the Fermionic fields
is,

$\displaystyle\bigg{(}\partial_{t}+d_{\psi\bar{\psi}}\triangle_{\psi\bar{\psi}}+\triangle_{\partial\psi\bar{\psi}}-d\bigg{)}S[\psi,\bar{\psi},t]=\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\bar{\psi}(p)}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\psi(-p)}-\frac{\delta^{2}S[\psi,\bar{\psi},t]}{\delta\bar{\psi}(p)\delta\psi(-p)}\bigg{)},$
(73)

where
$\Delta_{\psi\bar{\psi}}=\int_{p}\psi(p)\frac{\delta}{\delta\psi(-p)}+\int_{p}\bar{\psi}(p)\frac{\delta}{\delta\bar{\psi}(-p)}$
and
$\Delta_{\partial_{\psi\bar{\psi}}}=d+\int\frac{d^{d}p}{(2\pi)^{d}}\psi(p)p\cdot\frac{\partial}{\partial
p}\frac{\delta}{\delta\psi(-p)}+\int\frac{d^{d}p}{(2\pi)^{d}}\bar{\psi}(p)p^{\cdot}\frac{\partial}{\partial
p}\frac{\delta}{\delta\bar{\psi}(-p)}$. Acting with the operator
$P(z,\bar{z})$ given in equation (71) on the differential equation above, so
that the operator $P(z,\bar{z})$ extracts and returns only the local potential
terms in $S[\psi,\bar{\psi},t]$ while at the same time the fields $\psi$ and
$\bar{\psi}$ are replaced by the Grassmann variables $z$ and $\bar{z}$
respectively as shown below,

$\displaystyle
P(z,\bar{z})\bigg{(}\partial_{t}+d_{\psi\bar{\psi}}\triangle_{\psi\bar{\psi}}+\triangle_{\partial\psi\bar{\psi}}-d\bigg{)}S[\psi,\bar{\psi},t]=$
$\displaystyle
P(z,\bar{z})\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\bar{\psi}(p)}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\psi(-p)}-\frac{\delta^{2}S[\psi,\bar{\psi},t]}{\delta\bar{\psi}(p)\delta\psi(-p)}\bigg{)}$
(74)
$\displaystyle\partial_{t}V(z,\bar{z},t)+d_{\psi\bar{\psi}}\bigg{(}z\frac{\partial
V(z,\bar{z},t)}{\partial z}+\bar{z}\frac{\partial
V(z,\bar{z},t)}{\partial\bar{z}}\bigg{)}$ $\displaystyle-
dV(z,\bar{z},t)=\bigg{(}G_{F}^{\prime}(0)\frac{\partial
V(z,\bar{z},t)}{\partial z}\frac{\partial
V(z,\bar{z},t)}{\partial\bar{z}}-\frac{\partial^{2}V(z,\bar{z},t)}{\partial
z\partial\bar{z}}\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{)},$

where $V(z,\bar{z},t)=\sum_{n}\mathcal{L}_{0,n}(0,\cdots,0;t)(z\bar{z})^{n}$,
and for generality we are less focused on the fact that the higher degree
terms of $V(z,\bar{z},t)$ will be nilpotent. Making similar transformations as
in the Bosonic fields case where,

$\displaystyle V(z,\bar{z},t)\rightarrow$
$\displaystyle\frac{V(z,\bar{z},t)\int_{p}G_{F}^{\prime}(\frac{p}{\Lambda})}{G_{F}^{\prime}(0)}$
(75) $\displaystyle z\rightarrow$ $\displaystyle
z\sqrt{\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}}$
$\displaystyle\bar{z}\rightarrow$
$\displaystyle\bar{z}\sqrt{\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}},$

and then dropping the terms emanating from rescaling the Polchinski’s ERG flow
equation simplifies to,

$\displaystyle\partial_{t}V(z,\bar{z},t)=\frac{\partial^{2}V(z,\bar{z},t)}{\partial
z\partial\bar{z}}-\frac{\partial V(z,\bar{z},t)}{\partial z}\frac{\partial
V(z,\bar{z},t)}{\partial\bar{z}}.$ (76)

Let $U(z,\bar{z},t)=e^{-V(z,\bar{z},t)}$, then equation (76) becomes,

$\displaystyle\partial_{t}U(z,\bar{z},t)=\frac{\partial^{2}U(z,\bar{z},t)}{\partial
z\partial\bar{z}},$ (77)

which is a Grassmann variable analogue of a homogeneous differential equation
obtained earlier in equation (40), and the solution is of the form,

$\displaystyle U(z,\bar{z},t)=ce^{\frac{z\bar{z}}{\lambda t}+\ln(\lambda t)}$
(78) $\displaystyle\Rightarrow$ $\displaystyle
V(z,\bar{z},t)=\frac{z\bar{z}}{\lambda t}+\ln(\lambda t)+\ln(c).$

where $c$ is a constant, for simplicity we set it to a unit, $c=1$. At the
fixed point we have $\partial_{t}U(z,\bar{z},t)=\partial_{t}V(z,\bar{z},t)=0$,
which is calculated as follows,

$\displaystyle\partial_{t}V(z,\bar{z},t)=$
$\displaystyle\frac{\partial^{2}V(z,\bar{z},t)}{\partial
z\partial\bar{z}}-\frac{\partial V(z,\bar{z},t)}{\partial z}\frac{\partial
V(z,\bar{z},t)}{\partial\bar{z}}=0$ (79) $\displaystyle=$
$\displaystyle\frac{1}{\lambda t}-\frac{z\bar{z}}{\lambda^{2}t^{2}}$
$\displaystyle\Rightarrow$ $\displaystyle t=\frac{z\bar{z}}{\lambda}.$

Therefore at the fixed point,

$\displaystyle U\big{(}z,\bar{z},t=\frac{z\bar{z}}{\lambda}\big{)}=$
$\displaystyle e^{1+\ln(z\bar{z})}$ (80) $\displaystyle
V\big{(}z,\bar{z},t=\frac{z\bar{z}}{\lambda}\big{)}=$
$\displaystyle\big{(}1+\ln(z\bar{z})\big{)}.$

Notice that the fixed point interaction (vertex) function $V(z,\bar{z},t)$
obtained above for the Fermionic fields in equation (80) is also a logarithmic
interaction resembling the logarithmic function obtained in equation (42) for
the Bosonic fields. Which also leads to similar conclusions as in the Bosonic
fields case regarding the fixed point being controlled by a logarithmic
interaction CFT. Perturbing around the homogeneous solution as follows,

$\displaystyle V_{p}(z,\bar{z},t)=$ $\displaystyle V(z,\bar{z},t)+g_{f}\delta
V(z,\bar{z},t)$ (81) $\displaystyle=$ $\displaystyle
V(z,\bar{z},t)+g_{f}e^{\lambda t}f(z,\bar{z}),$

where $g_{f}\ll 1$ is the fermion interaction term. At order
$\mathcal{O}(g_{f})$ ignoring $\mathcal{O}(g_{f}^{2})$ terms we get the
following differential equation,

$\displaystyle\frac{\partial^{2}f(z,\bar{z})}{\partial
z\partial\bar{z}}e^{\lambda t}=e^{\lambda
t}\bigg{(}(d-\lambda)f(z,\bar{z})+\frac{\partial f(z,\bar{z})}{\partial
z}\frac{\partial V}{\partial\bar{z}}+\frac{\partial V}{\partial
z}\frac{\partial
f(z,\bar{z})}{\partial\bar{z}}-\frac{1}{2}(d-1)\bigg{(}z\frac{\partial
f(z,\bar{z})}{\partial z}+\bar{z}\frac{\partial
f(z,\bar{z})}{\partial\bar{z}}\bigg{)}\bigg{)},$ (82)

where,

$\displaystyle\frac{\partial
V(z,\bar{z},t)}{\partial\bar{z}}=-\frac{z}{\lambda t},\quad\frac{\partial
V(z,\bar{z},t)}{\partial z}=\frac{\bar{z}}{\lambda t}.$ (83)

A similar differential equation as the one in equation (82) has been obtained
in [19]. Expanding the exponential $e^{\lambda t}$ on both the LHS and RHS,
then collecting the zeroth order terms in parameter $t$ we get the following
differential,

$\displaystyle\frac{\partial^{2}f(z,\bar{z})}{\partial z\partial\bar{z}}=$
$\displaystyle(d-\lambda)f(z,\bar{z})-\frac{\partial f(z,\bar{z})}{\partial
z}z+\bar{z}\frac{\partial
f(z,\bar{z})}{\partial\bar{z}}-\frac{1}{2}(d-1)\bigg{(}z\frac{\partial
f(z,\bar{z})}{\partial z}+\bar{z}\frac{\partial
f(z,\bar{z})}{\partial\bar{z}}\bigg{)}$ (84) $\displaystyle=$
$\displaystyle(d-\lambda)f(z,\bar{z})+\bigg{(}1-\frac{1}{2}(d-1)\bigg{)}z\frac{\partial
f(z,\bar{z})}{\partial
z}+\bigg{(}1-\frac{1}{2}(d-1)\bigg{)}\bar{z}\frac{\partial
f(z,\bar{z})}{\partial\bar{z}}.$

Performing the following transformation,

$\displaystyle f(z,\bar{z})$
$\displaystyle\rightarrow\frac{1}{1-\frac{1}{2}(d-1)}f(z^{\prime},\bar{z}^{\prime})$
(85) $\displaystyle z^{\prime}=z\sqrt{\bigg{(}1-\frac{1}{2}(d-1)\bigg{)}}$
$\displaystyle\bar{z}^{\prime}=\bar{z}\sqrt{\bigg{(}1-\frac{1}{2}(d-1)\bigg{)}},$

then the differential equation above becomes,

$\displaystyle\frac{\partial^{2}f(z^{\prime},\bar{z}^{\prime})}{\partial
z^{\prime}\partial\bar{z}^{\prime}}=\frac{(d-\lambda)}{1-\frac{1}{2}(d-1)}f(z^{\prime},\bar{z}^{\prime})+z^{\prime}\frac{\partial
f(z^{\prime},\bar{z}^{\prime})}{\partial
z^{\prime}}+\bar{z}^{\prime}\frac{\partial
f(z^{\prime},\bar{z}^{\prime})}{\partial\bar{z}^{\prime}}.$ (86)

Since $f(z^{\prime},\bar{z}^{\prime})$ is a Grassmann function, the expansion
of the function will terminate at second order in Grassmann variables
$z^{\prime}$ and $\bar{z}^{\prime}$ because they will be nilpotent. Therefore,
the highest degree polynomial function that can be the solution to the
differential equation above will be,

$\displaystyle
f(z^{\prime},\bar{z}^{\prime})=z^{\prime}\bar{z}^{\prime}+\frac{1}{2}.$ (87)

Plugging this eigenfunction/eigenoperator solution to the differential
equation above we find the eigenvalue $\lambda=\lambda^{(0)}$ at zeroth order
in expansion parameter $t$ to be,

$\displaystyle\lambda^{(0)}=2\bigg{(}1-\frac{1}{2}(d-1)\bigg{)}+d,$ (88)

and therefore at $d=4$, $\lambda^{(0)}=3$. At first order in expansion
parameter $t$ we get the following differential equation from equation (82),

$\displaystyle\frac{\partial^{2}f(z^{\prime},\bar{z}^{\prime})}{\partial
z^{\prime}\partial\bar{z}^{\prime}}=\frac{(d-\lambda)}{\frac{1}{2}-\frac{1}{2}(d-1)}f(z^{\prime},\bar{z}^{\prime})+z^{\prime}\frac{\partial
f(z^{\prime},\bar{z}^{\prime})}{\partial
z^{\prime}}+\bar{z}^{\prime}\frac{\partial
f(z^{\prime},\bar{z}^{\prime})}{\partial\bar{z}^{\prime}},$ (89)

and here we have performed the following transformation,

$\displaystyle f(z,\bar{z})$
$\displaystyle\rightarrow\frac{1}{\frac{1}{2}-\frac{1}{2}(d-1)}f(z^{\prime},\bar{z}^{\prime})$
(90) $\displaystyle
z^{\prime}=z\sqrt{\bigg{(}\frac{1}{2}-\frac{1}{2}(d-1)\bigg{)}}$
$\displaystyle\bar{z}^{\prime}=\bar{z}\sqrt{\bigg{(}\frac{1}{2}-\frac{1}{2}(d-1)\bigg{)}}.$

The same eigenfunction given in equation (87) is the solution to the
differential equation (89), here we find the corresponding eigenvalue
$\lambda=\lambda^{(1)}$ to be,

$\displaystyle\lambda^{(1)}=2\bigg{(}\frac{1}{2}-\frac{1}{2}(d-1)\bigg{)}+d,$
(91)

and at $d=4$, $\lambda^{(1)}=2$. Then at the $n^{th}$ order expansion in
parameter $t$ we get the following differential equation,

$\displaystyle\frac{\partial^{2}f(z^{\prime},\bar{z}^{\prime})}{\partial
z^{\prime}\partial\bar{z}^{\prime}}=\frac{(d-\lambda)}{\frac{1}{n}-\frac{1}{2}(d-1)}f(z^{\prime},\bar{z}^{\prime})+z^{\prime}\frac{\partial
f(z^{\prime},\bar{z}^{\prime})}{\partial
z^{\prime}}+\bar{z}^{\prime}\frac{\partial
f(z^{\prime},\bar{z}^{\prime})}{\partial\bar{z}^{\prime}},$ (92)

after having performed the following transformation,

$\displaystyle f(z,\bar{z})$
$\displaystyle\rightarrow\frac{1}{\frac{1}{n}-\frac{1}{2}(d-1)}f(z^{\prime},\bar{z}^{\prime})$
(93) $\displaystyle
z^{\prime}=z\sqrt{\bigg{(}\frac{1}{n}-\frac{1}{2}(d-1)\bigg{)}}$
$\displaystyle\bar{z}^{\prime}=\bar{z}\sqrt{\bigg{(}\frac{1}{n}-\frac{1}{2}(d-1)\bigg{)}}.$

The same function already given in equation (87) is the solution here, and the
corresponding eigenvalue $\lambda^{n)}=\lambda$ is

$\displaystyle\lambda^{(n)}=2\bigg{(}\frac{1}{n}-\frac{1}{2}(d-1)\bigg{)}+d,$
(94)

and at $d=4$ when $n\rightarrow\infty$, $\lambda^{(n)}=1$. Take note that
since the Grassmann expansion is nilpotent already at the second order in
expansion parameter $t$, the expansion terminates at that order, therefore it
is unnecessary to go beyond the expansion $n>1$ and $n\rightarrow\infty$ since
the terms are zero. Therefore the Fermionic fields expansion in parameter $t$
is finite, unlike in the Bosonic fields where the expansion is infinite. Now
the perturbed function $V_{p}$ for the Fermionic fields will be,

$\displaystyle V_{P}(z,\bar{z},t)=$
$\displaystyle\bigg{(}\frac{z\bar{z}}{\lambda t}+\ln(\lambda
t)\bigg{)}+g_{f}\sum_{n}e^{\lambda^{(n)}t}f^{(n)}(z,\bar{z})$ (95)
$\displaystyle=$ $\displaystyle\bigg{(}\frac{z\bar{z}}{\lambda t}+\ln(\lambda
t)\bigg{)}+g_{f}\bigg{(}\frac{1}{\lambda
t}f^{(-1)}(z,\bar{z})+\lambda^{(0)}f^{(0)}(z,\bar{z})+\lambda^{(1)}tf^{(1)}(z,\bar{z})\bigg{)}$

where again $f^{(-1)}(z,\bar{z})$ is a constant function,
$f^{(-1)}(z,\bar{z})=const$, and
$f^{(0)}(z,\bar{z})=f^{(1)}(z,\bar{z})=z\bar{z}-\frac{1}{2}$. Then at the
logarithmic fixed point at $t=\frac{z\bar{z}}{\lambda}$, the function $V_{p}$
becomes,

$\displaystyle V_{P}(z,\bar{z})=$
$\displaystyle\big{(}1+\ln(z\bar{z})\big{)}+g_{f}\bigg{(}\frac{1}{z\bar{z}}f^{(-1)}(z,\bar{z})+\lambda^{(0)}f^{(0)}(z,\bar{z})+\frac{\lambda^{(1)}}{\lambda}z\bar{z}f^{(1)}(z,\bar{z})\bigg{)}$
(96) $\displaystyle=$
$\displaystyle\big{(}1+\ln(z\bar{z})\big{)}+g_{f}\bigg{(}\frac{1}{z\bar{z}}f^{(-1)}(z,\bar{z})+\lambda^{(0)}\bigg{(}z\bar{z}-\frac{1}{2}\bigg{)}-\frac{1}{2}\frac{\lambda^{(1)}}{\lambda}z\bar{z}\bigg{)}.$

Notice how different the linearized solution for the Fermionic fields is
different from the Bosonic fields solution in equation (60). First, the
expansion terminates after the first order in expansion parameter $t$ while
for the Bosonic fields the expansion is infinite, and second, the polynomial
function is not a Hermite polynomial which has $n^{th}$ degree expansion as it
was in the case of the Bosonic fields, it is a simple quadratic function
$f(z,\bar{z})$ with no higher degree expansion terms. The truncation in the
expansion is due to the nipotent property of Grassmann variables representing
the Fermionic fields.

Towards this end, in $d=4$, at zeroth order in expansion parameter $t$ we find
the correlation length critical exponent is $\nu=0.33$, and at first order in
expansion parameter $t$ the correlation length critical exponent $\nu=0.5$.
From here the expansion terminates since we are dealing with Grassmann
variables, and therefore there will be no other higher order critical
exponents in expansion parameter $t$. Furthermore, there are no negative
eigenvalues which indicates the absence of correction-to-scaling critical
exponents.

##  5 Discussion

In the paper we have demonstrated that the Polchinski’s ERG flow equation
given in [4], within the LPA limit it is equivalent to the standard elementary
heat equation. The fixed points (steady-state) of this equation are controlled
by a logarithmic function which is the interaction function of the theory, we
therefore conclude that this limit gives rise to a logarithmic interaction
CFT. Contrary to perturbing (linearizing) around the Gaussian fixed point,
here we perturb around the logarithmic fixed point since it is the solution of
our steady-state equation. For the case of the Bosonic fields the perturbed
solution leads to an finite set of Hermite differentials with Hermite
polynomials being the solutions to these differentials. For the Fermionic
fields case we find a finite set of Grassmann differentials analogue to
Hermite differentials, the set is finite because the expansion truncates at
the second order in expansion parameter $t$, since the Grassmann variables are
nilpotent at this order. Then we proceeded to compute the relevant critical
exponents for both Bosonic fields and Fermionic fields cases; For the Bosonic
fields we end up with an infinite set of critical exponents since the values
of the critical exponent are different at each order in expansion parameter
$t$ due to the relevant eigenvalues being different at each order; For the
Fermionic fields we end up with a finite set of critical exponents and the
critical exponents are also different for the zero and first order expansion
in parameter $t$, and the other higher order expansions are nilpotent.
Furthermore, there are no negative eigenvalues for Fermionic fields case, and
therefore the correction-to-scaling critical exponents are absent.

The simplistic results obtained at the LPA limit we find them valuable because
it opens up a new avenue to possible study and understand ERG equations
through using well established tools and theories already being utilized to
solve and understand heat equations. Furthermore these results deserve an in-
depth probing, at this point there are still some open questions on whether
this relation at the LPA limit is something of a mere coincidence or is a
smoking gun to something fundamental. One way to address such questions will
be to move the computation beyond the LPA limit by including higher order
terms in the derivative expansion. However, presently the challenge with
including higher order terms in derivative expansion or momentum expansion has
been seen to result with physical observables being dependent on the choices
of the renormalization scheme [8, 9, 10]. Meaning the theory will be scheme
dependent, and the RG flow equations will depend on the conditions of the
theory, these conditions will include the choices used in the renormalization
scheme to fix the theory. This makes the renormalization of next to leading
order and higher order derivative terms tricky to deal with, and can yield
insensible physical observable results. Since in any renormalization scheme
there are always some choices that one has to make, and these choices are
related to the renormalization conditions that one implements on the theory,
and these choices have to be fixed. In a sensible renormalization scheme, the
physical observables computed from this theory are suppose to be
renormalization scheme independent, independent of the renormalization
conditions. For instance, in perturbation renormalization group for on-shell
renormalization scheme, one of the choices is the propagator has a pole at
mass squared with residue one. Another drawback with this ERG approximation
approach whereby we perform derivative or momentum expansion where at some
point we truncate the expansion is that the expansion is not systematic, one
can’t estimate the error they make when they truncate the expansion, therefore
better systematic truncation approaches still require investigation here.
Including higher order derivative terms is one of the immediate interesting
direction to extend this work. Going forward, we also need to make contact
with experimental results, search for relevant experimental results that can
be used to verify our computational results.

## Acknowledgements

The author would like to thank Robert De Mello Koch for suggesting the project
and providing valuable discussions and comments.

## A Polchinski’s ERG flow equation for Bosonic and Fermionic fields

We used section 2 to formulate Polchincki’s ERG flow equation for both
Fermionic and Bosonic fields, and we have written down two types of
Polchinski’s ERG flow equation, the first one in equation (20) only implements
the coarse graining step of Wilson’s renormalization group flow program, and
the second one in equation (26) implements both the coarse-graining and
rescaling steps of Wilson’s renormalization group flow program. This section
will provide more details regarding the derivations of these equations.

### A.1 Polchinski’s ERG flow equation coarse-graining step

At the start of section 2 and particularly in equation (20) we have shown how
the Polchinski’s ERG flow equation containing both the single Bosonic fields
and the spinless Fermionic fields can be formulated. The detailed derivation
for the formulation is as follows,

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=\int D\psi D\bar{\psi}D\phi
e^{-\frac{1}{2}\int_{p}\phi(p)G_{B}^{-1}(\frac{p}{\Lambda})\phi(-p)-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}J(p)\phi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)-S_{int}[\phi,\bar{\psi},\psi]}.$
(97)

Applying the equation below on the Bosonic fields,

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=$ $\displaystyle\int D\phi
e^{-\int_{p}\frac{1}{2}\phi(p)G^{-1}_{B}(\frac{p}{\Lambda})\phi(-p)+\int_{p}J(p)\phi(-p)-S_{int}^{\Lambda_{0}}[\phi]}$
(98) $\displaystyle=$ $\displaystyle
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{D_{\Lambda}^{\Lambda_{0}}}e^{-S_{int}^{\Lambda_{0}}[\phi]}\bigg{|}_{\phi=G_{B}J(p)},$

where

$\displaystyle D_{\Lambda}^{\Lambda_{0}}=$
$\displaystyle\frac{1}{2}\int_{p}G_{B}\bigg{(}\frac{p}{\Lambda}\bigg{)}\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}$
(99) $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}G_{B}\bigg{(}\frac{p}{\Lambda}\bigg{)}\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)},$

we get

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=$ $\displaystyle\int D\psi
D\bar{\psi}e^{-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)}$
(100) $\displaystyle\times
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}e^{-S_{int}[\phi,\bar{\psi},\psi]}}\bigg{|}_{\phi=G_{B}(\frac{p}{\Lambda})J(p)}$
$\displaystyle=$ $\displaystyle
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}\int D\psi
D\bar{\psi}e^{-\int_{p}\bar{\psi}(p)G_{F}^{-1}(\frac{p}{\Lambda})\psi(-p)+\int_{p}\bar{\eta}(p)\psi(-p)+\int_{p}\eta(p)\bar{\psi}(-p)}$
$\displaystyle\times
e^{\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}e^{-S_{int}[\phi,\bar{\psi},\psi]}}\bigg{|}_{\phi=G_{B}(\frac{p}{\Lambda})J(p)}.$

Then applying equation (98) on the Fermionic fields we get

$\displaystyle e^{-W_{\Lambda}^{\Lambda_{0}}[J]}=$ $\displaystyle
e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}\bigg{[}e^{-\int_{p}\bar{\eta}(p)G_{F}\eta(-p)}e^{-\int_{p}G_{F}\frac{\delta}{\delta\bar{\psi}(p)}\frac{\delta}{\delta\psi(-p)}}e^{-\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}[\phi,\bar{\psi},\psi]}\bigg{]}\bigg{|}_{\phi=G_{B}J,\bar{\psi}=G_{F}\eta,\psi=G_{F}\bar{\eta}}$
(101)

Defining the effective interaction action
$S_{int}[\phi,\bar{\psi},\psi,\Lambda]$ from the functional integral above as,

$\displaystyle
e^{-S_{int}[\phi,\bar{\psi},\psi,\Lambda]}=e^{-\frac{1}{2}\int_{p}J(p)G_{B}(\frac{p}{\Lambda})J(-p)}e^{-\int_{p}\bar{\eta}(p)G_{F}(\frac{p}{\Lambda})\eta(-p)}e^{-\int_{p}G_{F}(\frac{p}{\Lambda})\frac{\delta}{\delta\bar{\psi}(p)}\frac{\delta}{\delta\psi(-p)}}e^{-\frac{1}{2}\int_{p}G_{B}(\frac{p}{\Lambda})\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}}e^{-S_{int}[\phi,\bar{\psi},\psi]},$
(102)

taking the derivative with respect the scale $\Lambda$ we end up with the
following Polchinski’s ERG flow equation for both Bosonic fields and Fermionic
fields,

$\displaystyle-\Lambda\frac{\partial
S_{int}}{\partial\Lambda}=\int_{p}\Lambda\frac{\partial
G_{F}(\frac{p}{\Lambda})}{\partial\Lambda}\bigg{(}\frac{\delta
S_{int}}{\delta\bar{\psi}(p)}\frac{\delta
S_{int}}{\delta\psi(-p)}-\frac{\delta^{2}S_{int}}{\delta\bar{\psi}(p)\delta\psi(-p)}\bigg{)}+\frac{1}{2}\int_{p}\Lambda\frac{\partial
G_{B}(\frac{p}{\Lambda})}{\partial\Lambda}\bigg{(}\frac{\delta
S_{int}}{\delta\phi(p)}\frac{\delta
S_{int}}{\delta\phi(-p)}-\frac{\delta^{2}S_{int}}{\delta\phi(p)\delta\phi(-p)}\bigg{)},$
(103)

which is one of the main equations we were trying to derive in section 2. This
part of the derivation only considers the coarse graining step of Wilson’s
renormalization group program. In the next part we show how the second step of
Wilson’s renormalization group flow is implemented on this equation (103),
this step entails rescaling the Polchinski’s ERG flow equation’s parameters
(variables).

### A.2 Rescaling Polchinski’s ERG flow equation.

At the beginning of section 2 the Polchinski’s ERG flow equation derived
earlier only implements the coarse step, therefore the rescaling step is
required in order to complete Wilson’s renormalization group flow program. We
will show how this step is implemented which will lead to the derivation of
equation (26) shown at the end of section 2. First, the full rescaled
Polchinski’s ERG flow equation derived in [14, 15] for the single Bosonic
fields is as follows,

$\displaystyle\bigg{(}\Lambda\frac{\partial}{\partial\Lambda}+\int\frac{d^{d}p}{(2\pi)^{d}}\phi(p)p\cdot\frac{\partial}{\partial
p}\frac{\delta}{\delta\phi(-p)}+d_{\phi}\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}-d\bigg{)}S[\phi,\Lambda]=\frac{1}{2}\int_{p}\Lambda\frac{\partial
G_{B}(\frac{p}{\Lambda})}{\partial\Lambda}\bigg{(}\frac{\delta
S[\phi,\Lambda]}{\delta\phi(p)}\frac{\delta
S[\phi,\Lambda]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,\Lambda]}{\delta\phi(p)\delta\phi(-p)}\bigg{)}$
(104)

where $d_{\phi}=\frac{1}{2}(d-2+\eta)$, $\eta$ is the anomalous dimension of
the scalar fields, $d$ is the space-time dimension, and take note that
$\int_{p}=\int\frac{d^{d}p}{(2\pi)^{d}}$. Here we will follow similar
arguments used in [14, 15] to apply rescaling on Polchinski’s ERG flow
equation. Dropping the Fermionic fields from the effective action $S_{int}$ in
equation (21), we get the following effective action,

$\displaystyle
S[\phi,\Lambda]=\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}p_{i}}{(2\pi)^{d}}\mathcal{L}_{m,0}(p_{1},\cdots,p_{m},\Lambda)\phi(p_{1})\cdots\phi(p_{m})\delta^{d}(p_{1}+\cdots+p_{m}).$
(105)

where $S[\phi,\Lambda]=S_{int}[\phi,0,0,\Lambda]$. Now, rescaling the momenta
$p$ as,

$\displaystyle p\rightarrow p^{\prime}=sp=(1+t)p$ (106)

where $s=e^{t}$ and $t=\frac{\delta\Lambda}{\Lambda}$ is an infinitesimal
small parameter. The rescaling has consequences on the action
$S=S[\phi,\Lambda]$, it transforms the action $S$ in the following way,

$\displaystyle S\rightarrow S^{\prime}=$ $\displaystyle S+\Delta S$ (107)
$\displaystyle=$ $\displaystyle S+tG_{sc}S$
$\displaystyle\Rightarrow\frac{\partial S}{\partial t}=$ $\displaystyle
G_{sc}S$

where the factor $\Delta S=tG_{sc}S$ and $G_{sc}$ will contain correction
terms to the action $S$ that arise from rescaling. The factor $\Delta
S=tG_{sc}S$ is actually a sum from various factors or terms arising from
inducing changes in the action of equation (105) and in the end these terms
are summed together as $\Delta S=\sum_{i=1}^{4}\Delta S_{i}$. We can unpack
$\Delta S_{1}$ as coming from the differential volume,

$\displaystyle\int\prod_{i=1}^{m}d^{d}p_{1}\cdots
d^{d}p_{m}=\int\prod_{i=1}^{m}(1+t)^{-md}d^{d}p^{\prime}_{1}\cdots
d^{d}p^{\prime}_{m}$ (108)

expanding to first order in parameter $t$ this induces $\Delta S_{1}$ as,

$\displaystyle\Delta
S_{1}=-t\bigg{(}d\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}\bigg{)}S.$
(109)

The factor $\Delta S_{2}$ is induced from the field rescaling
$\phi(p)=\phi(s^{-1}p^{\prime})$, giving,

$\displaystyle\phi(s^{-1}p^{\prime})=(1+t)^{d-d_{\phi}}\phi(p),$ (110)

to first order in parameter $t$ the inducing factor is,

$\displaystyle\Delta
S_{2}=t\bigg{(}(d-d_{\phi})\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}\bigg{)}S,$
(111)

where $d_{\phi}=\frac{1}{2}(d-2+\eta_{\phi})$ and $\eta_{\phi}$ is the
anomalous dimension of the scalar field $\phi(p)$. The factor $\Delta S_{3}$
is induced from the interaction or vertex function
$\mathcal{L}_{m,0}(p_{1},\cdots,p_{m},\Lambda)=\mathcal{L}_{m,0}(s^{-1}p_{1}^{\prime},\cdots,s^{-1}p_{m}^{\prime};\Lambda)$,
which gives,

$\displaystyle\Delta
S_{3}=-t\bigg{(}\int_{p}\phi_{p}p\cdot\frac{\partial}{\partial
p^{\prime}}\frac{\delta}{\delta\phi_{p}}\bigg{)}S.$ (112)

Lastly, the term $\Delta S_{4}$ is induced from the delta function
$\delta^{d}(p_{1}+\cdots+p_{m})=\delta^{d}(s^{-1}p_{1}^{\prime}+\cdots+s^{-1}p_{m}^{\prime})$,
giving

$\displaystyle\Delta S_{4}=t(dS).$ (113)

Then in the end these induced terms from rescaling are now summed together,

$\displaystyle\Delta S=$ $\displaystyle\Delta S_{1}+\Delta S_{2}+\Delta
S_{3}+\Delta S_{4}$ (114) $\displaystyle=$ $\displaystyle
t\bigg{(}d-\int_{p}\phi_{p}p\cdot\partial_{p^{\prime}}\frac{\delta}{\delta\phi_{p}}-d_{\phi}\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}\bigg{)}S[\phi,\Lambda].$

Allowing the momentum derivative $\partial_{p^{\prime}}$ to act on the
momentum conservation function of $S[\phi,\Lambda]$ we get,

$\displaystyle\Delta
S=-t\bigg{(}\int_{p}\phi_{p}p\cdot\partial_{p}\frac{\delta}{\delta\phi_{p}}+d_{\phi}\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}\bigg{)}S[\phi,\Lambda],$
(115)

then,

$\displaystyle G_{sc}S[\phi,\Lambda]=$
$\displaystyle-\bigg{(}\int_{p}\phi_{p}p\cdot\partial_{p}\frac{\delta}{\delta\phi_{p}}+d_{\phi}\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}\bigg{)}S[\phi,\Lambda]$
(116) $\displaystyle=$
$\displaystyle(d-\Delta_{\partial_{\phi}}-d_{\phi}\Delta_{\phi})S[\phi,\Lambda]$
$\displaystyle\Rightarrow G_{sc}=$
$\displaystyle(d-\Delta_{\partial_{\phi}}-d_{\phi}\Delta_{\phi}),$

where
$\Delta_{\partial_{\phi}}=d+\int_{p}\phi_{p}p\cdot\partial_{p}\frac{\delta}{\delta\phi_{p}}$,
$\Delta_{\phi}=\int_{p}\phi_{p}\frac{\delta}{\delta\phi_{p}}$ and
$d_{\phi}=\frac{1}{2}(d-2+\eta_{\phi})$. Therefore, after applying rescaling
on the single Bosonic fields we get the following Polchinski’s ERG flow
equation,

$\displaystyle\bigg{(}\Lambda\partial_{\Lambda}+d_{\phi}\bigtriangleup_{\phi}+\bigtriangleup_{\partial_{\phi}}-d\bigg{)}S[\phi,\Lambda]=$
$\displaystyle\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,\Lambda]}{\delta\phi(p)}\frac{\delta
S[\phi,\Lambda]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,\Lambda]}{\delta\phi(p)\delta\phi(-p)}\bigg{)}$
(117)
$\displaystyle\bigg{(}\partial_{t}+d_{\phi}\bigtriangleup_{\phi}+\bigtriangleup_{\partial_{\phi}}-d\bigg{)}S[\phi,t]=$
$\displaystyle\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,t]}{\delta\phi(p)}\frac{\delta
S[\phi,t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,t]}{\delta\phi(p)\delta\phi(-p)}\bigg{)},$

where in the last line we used the fact that
$\Lambda\frac{\partial}{\partial\Lambda}=\frac{\partial}{\partial t}$. Now, it
should be easy to infer that the inclusion of Fermionic fields back into the
effective action $S_{int}$ in equation (21) while removing the Bosonic fields
at the same time, then applying rescaling should have the following effect on
the spinless Fermionic fields effective action,

$\displaystyle\Delta S=$ $\displaystyle
t\bigg{(}\int_{p}d_{\psi\bar{\psi}}\psi(p)\frac{\delta}{\delta\psi(-p)}+\int_{p}d_{\psi\bar{\psi}}\bar{\psi}(p)\frac{\delta}{\delta\bar{\psi}(-p)}+\int_{p}\psi(p)p\cdot\frac{\partial}{\partial
p}\frac{\delta}{\delta\psi(-p)}+\int_{p}\bar{\psi}(p)p^{\cdot}\frac{\partial}{\partial
p}\frac{\delta}{\delta\bar{\psi}(-p)}\bigg{)}S[\psi,\bar{\psi},\Lambda]$ (118)

where $d_{\psi\bar{\psi}}=\frac{1}{2}(d-1+\eta_{\psi})$ and $\eta_{\psi}$ is
the anomalous dimensions of the Fermionic fields. We can continue to write
$\Delta S$ as,

$\displaystyle\Delta S=$
$\displaystyle-t\bigg{(}-d+d_{\psi\bar{\psi}}\Delta_{\psi\bar{\psi}}+\Delta_{\partial_{\psi\bar{\psi}}}\bigg{)}S[\psi,\bar{\psi},\Lambda]$
(119)

where
$\Delta_{\psi\bar{\psi}}=\int_{p}\psi(p)\frac{\delta}{\delta\psi(-p)}+\int_{p}\bar{\psi}(p)\frac{\delta}{\delta\bar{\psi}(-p)}$
and
$\Delta_{\partial_{\psi\bar{\psi}}}=d+\int\frac{d^{d}p}{(2\pi)^{d}}\psi(p)p\cdot\frac{\partial}{\partial
p}\frac{\delta}{\delta\psi(-p)}+\int\frac{d^{d}p}{(2\pi)^{d}}\bar{\psi}(p)p^{\cdot}\frac{\partial}{\partial
p}\frac{\delta}{\delta\bar{\psi}(-p)}$. Then the factor $G_{sc}$ from
rescaling the Fermionic action will be

$\displaystyle
G_{sc}S[\psi,\bar{\psi},\Lambda]=\bigg{(}d-d_{\psi\bar{\psi}}\Delta_{\psi\bar{\psi}}-\Delta_{\partial_{\psi\bar{\psi}}}\bigg{)}S[\psi,\bar{\psi},\Lambda].$
(120)

Therefore, the rescaled Polchinski’s ERG flow equation for the spinless
Fermionic fields is,

$\displaystyle\bigg{(}\partial_{t}-d+d_{\psi\bar{\psi}}\Delta_{\psi\bar{\psi}}+\Delta_{\partial_{\psi\bar{\psi}}}\bigg{)}S[\psi,\bar{\psi},t]=\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\psi(p)}\frac{\delta
S[\psi,\bar{\psi},t]}{\delta\bar{\psi}(-p)}-\frac{\delta^{2}S[\psi,\bar{\psi},t]}{\delta\psi(p)\delta\bar{\psi}(-p)}\bigg{)}.$
(121)

Now, combining the rescaled Polchinski’s ERG flow equations for the single
Bosonic and spinless Fermionic fields together we get the following rescaled
ERG flow equation,

$\displaystyle\big{(}\partial_{t}+d_{\psi\bar{\psi}}\Delta_{\psi\bar{\psi}}+\Delta_{\partial_{\psi\bar{\psi}}}+$
$\displaystyle
d_{\phi}\bigtriangleup_{\phi}+\bigtriangleup_{\partial_{\phi}}-d\big{)}S[\phi,\psi,\bar{\psi},t]=\int_{p}G_{F}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\psi(p)}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\bar{\psi}(-p)}-\frac{\delta^{2}S[\phi,\psi,\bar{\psi},t]}{\delta\psi(p)\delta\bar{\psi}(-p)}\bigg{)}$
(122)
$\displaystyle+\frac{1}{2}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\phi(p)}\frac{\delta
S[\phi,\psi,\bar{\psi},t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,\psi,\bar{\psi},t]}{\delta\phi(p)\delta\phi(-p)}\bigg{)}.$

## B Local Potential Approximation

In section 3 we approximate non-perturbatively the Polchinski’s ERG flow
equation by performing derivative expansions, which is an expansion in powers
of momenta. The expansion is truncated at the LPA by making use of the
Hazenfratz and Hazenfratz projector (operator), which extracts the LPA piece
of the expansion.

### B.1 Bosonic fields Local Potential Approximation (LPA).

The LPA derived in section 3 is the leading term or zeroth order term in the
derivative expansion of the action $S[\phi,\Lambda]$. The action
$S[\phi,\Lambda]$ in momentum space has the following expansion in momentum
$k^{2}$,

$\displaystyle S[\phi,\Lambda]=$
$\displaystyle\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\bigg{(}\mathcal{L}_{m,0}+\frac{k_{1}^{2}+\cdots+k_{m}^{2}}{m(m-1)}\mathcal{L}_{m-2,0}+\mathcal{O}(k^{4})+\cdots\bigg{)}\phi(k_{1})\cdots\phi(k_{m})\delta^{d}(k_{1}+\cdots+k_{m})$
(123) $\displaystyle=$
$\displaystyle\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\bigg{(}V(\phi,\Lambda)+W(\phi,\Lambda)+\mathcal{O}(k^{4})\bigg{)}$

where $V(\phi,\Lambda)=\mathcal{L}_{m,0}(0,\cdots
0;\Lambda)\phi(k_{1})\cdots\phi(k_{m})\delta^{d}(k_{1}+\cdots+k_{m})$ is a
local vertex or interaction term, while the $W(\phi,\Lambda)$ and other higher
order expansion terms are non-local vertex or coupling functions. In position
space this action is written as,

$\displaystyle S[\phi,\Lambda]=\int
d^{d}x\bigg{(}V(\phi,\Lambda)+W(\phi,\Lambda)\partial_{\mu}\phi\partial^{\mu}\phi+\mathcal{O}(\partial^{4})\bigg{)}.$
(124)

We can focus on the local potential approximation limit by removing or
neglecting terms from $S[\phi,\Lambda]$ which are non-local or contain
derivatives on them. This can be achieved by making use of the Hasenfratz and
Hasenfratz operator $P(z)$ [5]. Where for example the action of this operator
on a general test functional field $X[\phi]$ acts as follows,

$\displaystyle
P(z)X[\phi]=e^{z\frac{\partial}{\partial\phi(0)}}X[\phi]\bigg{|}_{\phi=0}.$
(125)

Then on the product of test functionals $X_{1}[\phi]X_{2}[\phi]\cdots
X_{m}[\phi]$, the action of the operator $P(z)$ simplifies as follows,

$\displaystyle P(z)\bigg{(}X_{1}[\phi]X_{2}[\phi]\cdots X_{m}[\phi]\bigg{)}=$
$\displaystyle
e^{z\frac{\partial}{\partial\phi(0)}}\bigg{(}X_{1}[\phi]X_{2}[\phi]\cdots
X_{m}[\phi]\bigg{)}$ (126) $\displaystyle=$ $\displaystyle
P(z)X_{1}[\phi]P(z)X_{2}[\phi]\cdots P(z)X_{m}[\phi].$

Now using this simplification, the action of the operator $P(z)$ on
$S[\phi,\Lambda]$ yields the following results,

$\displaystyle P(z)S[\phi,\Lambda]=$ $\displaystyle
e^{z\frac{\partial}{\partial\phi(0)}}S[\phi,\Lambda]\bigg{|}_{\phi=0}$ (127)
$\displaystyle=$ $\displaystyle
e^{z\frac{\partial}{\partial\phi(0)}}\bigg{(}\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\mathcal{L}_{m,0}(k_{1},\cdots,k_{m},\Lambda)\phi(k_{1})\cdots\phi(k_{m})\delta^{d}(\sum_{i}^{m}k_{i})\bigg{)}\bigg{|}_{\phi=0}$
$\displaystyle=$
$\displaystyle\sum_{m}\mathcal{L}_{m,0}(0,0,\cdots,0,\Lambda)z^{m}\delta^{d}(0)$
$\displaystyle=$ $\displaystyle S[z,\Lambda]\delta^{d}(0)$ $\displaystyle=$
$\displaystyle V(z,\Lambda)\delta^{d}(0)$

where
$S[z,\Lambda]=V(z,\Lambda)=\sum_{m}\mathcal{L}_{m,0}(0,0,\cdots,0,\Lambda)z^{m}$.
Therefore, after applying the operator $P(z)$ we see that the fields $\phi$
are replaced by the variable $z$, and the action $S[z,\Lambda]$ contains only
the local terms or local potential terms which are denoted by the function
$V(z,\Lambda)$. Following [5], the delta function $\delta^{d}(0)$ is handled
by working in a finite volume box. Going back to the full rescaled
Polchinski’s ERG flow equation containing only Bosonic fields, which is

$\displaystyle\bigg{(}\partial_{t}+d_{\phi}\triangle_{\phi}+\triangle_{\partial}-d\bigg{)}S[\phi,t]=\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,t]}{\delta\bar{\phi}(p)}\frac{\delta
S[\phi,t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,t]}{\delta\bar{\phi}(p)\delta\phi(-p)}\bigg{)}.$
(128)

First take note that the action of the operator $P(z)$ on the term
$\frac{\delta S[\phi,t]}{\delta\phi}$ yields the following results,

$\displaystyle\int_{p}P(z)\frac{\delta
S[\phi,t]}{\delta\phi(p)}\bigg{|}_{\phi=0}=$
$\displaystyle\int_{p}e^{z\frac{\partial}{\partial\phi(0)}}\frac{\delta}{\delta\phi(p)}\bigg{(}\sum_{m}\int\prod_{i=1}^{m}\frac{d^{d}k_{i}}{(2\pi)^{d}}\mathcal{L}_{m,0}(k_{1},\cdots,k_{m},t)\phi(k_{1})\cdots\phi(k_{m})\delta^{d}(\sum_{i}^{m}k_{i})\bigg{)}\bigg{|}_{\phi=0}$
(129) $\displaystyle=$ $\displaystyle\sum_{m}\mathcal{L}_{m,0}(p,0,\cdots
0;\Lambda)mz^{m-1}\delta^{d}(p)$ $\displaystyle=$ $\displaystyle
V^{\prime}(z,t)\delta^{d}(p),$

where $V(z,t)$ is a local potential and $V^{\prime}(z,t)$ is the derivative of
the function $V(z,t)$ with respect to $z$. From here one can show the
following,

$\displaystyle\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}P(z)\frac{\delta
S[\phi,t]}{\delta\phi_{p}}\bigg{)}$ $\displaystyle\bigg{(}P(z)\frac{\delta
S[\phi,t]}{\delta\phi_{-p}}\bigg{)}=\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\sum_{m}\mathcal{L}_{m,0}(k_{1},\cdots,k_{m};t)mz^{m-1}\delta^{d}(p)\bigg{)}$
(130)
$\displaystyle\times\bigg{(}\sum_{n}\mathcal{L}_{n,0}(k_{1},\cdots,k_{m};t)nz^{n-1}\delta^{d}(-p)\bigg{)}$
$\displaystyle=$ $\displaystyle
G_{B}^{\prime}(0)\bigg{(}\sum_{m}\mathcal{L}_{m,0}(0,\cdots,0;t)mz^{m-1}\bigg{)}\bigg{(}\sum_{n}\mathcal{L}_{n,0}(0,\cdots,0;t)nz^{n-1}\bigg{)}\delta^{d}(0)$
$\displaystyle=$ $\displaystyle
G_{B}^{\prime}(0)V^{\prime}(z,t)V^{\prime}(z,t)\delta^{d}(0)$ $\displaystyle=$
$\displaystyle G_{B}^{\prime}(0)V^{\prime 2}(z,t)\delta^{d}(0),$

and also,

$\displaystyle
P(z)\bigg{(}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\frac{\delta^{2}S[\phi,t]}{\delta\phi_{p}\delta\phi_{-p}}\bigg{)}=$
$\displaystyle\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\sum_{m}\mathcal{L}_{m,0}(p,-p,0,\cdots,0;t)m(m-1)z^{m-2}\delta^{d}(p-p)$
(131) $\displaystyle=$
$\displaystyle\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\sum_{m}\mathcal{L}_{m,0}(p,-p,0,\cdots,0;t)m(m-1)z^{m-2}\delta^{d}(0)$
$\displaystyle=$
$\displaystyle\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}V^{\prime\prime}(z,t)\delta^{d}(0)$
$\displaystyle=$ $\displaystyle
V^{\prime\prime}(z,t)G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\delta^{d}(0).$

Now when the operator $P(z)$ acts on the entire rescaled Polchinski’s ERG flow
equation as follows,

$\displaystyle
P(z)\bigg{(}\partial_{t}+d_{\phi}\triangle_{\phi}+\triangle_{\partial}-d\bigg{)}S[\phi,t]=P(z)\bigg{(}\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}\bigg{(}\frac{\delta
S[\phi,t]}{\delta\bar{\phi}(p)}\frac{\delta
S[\phi,t]}{\delta\phi(-p)}-\frac{\delta^{2}S[\phi,t]}{\delta\bar{\phi}(p)\delta\phi(-p)}\bigg{)}\bigg{)},$
(132)

plugging the results in equations (127), (129), (130) and (131) we obtain the
following results,

$\displaystyle\bigg{(}\partial_{t}+d_{\phi}\triangle_{\phi}+\triangle_{\partial}-d\bigg{)}V(z,t)=G_{B}^{\prime}(0)V^{\prime
2}(z,t)-V^{\prime\prime}(z,t)\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}$
(133)

where now $d_{\phi}=\frac{d-2}{2}$ and the anomalous dimension $\eta_{\phi}=0$
in the LPA limit since there is no momentum dependence in the zeroth order of
the expansion. Therefore in the end we end up with the following differential
equation,

$\displaystyle\partial_{t}V(z,t)=$
$\displaystyle\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}V^{\prime\prime}(z,t)-G_{B}^{\prime}(0)V^{\prime
2}(z,t)-d_{\phi}zV^{\prime}(z,t)+dV(z,t)$ (134) $\displaystyle\noindent=$
$\displaystyle V^{\prime\prime}(z,t)-V^{\prime
2}(z,t)-d_{\phi}zV^{\prime}(z,t)+dV(z,t),$ (135)

after writing $V(z,t)$ and $z$ as

$\displaystyle V(z,t)\rightarrow$
$\displaystyle\frac{V(z,t)\int_{p}G_{B}^{\prime}(\frac{p}{\Lambda})}{G_{B}^{\prime}(0)},$
(136) $\displaystyle z\rightarrow$ $\displaystyle
z\sqrt{\int_{p}G_{B}^{\prime}\big{(}\frac{p}{\Lambda}\big{)}}.$

### B.2 Set of Hermite differential equations

In section 3 after obtaining the solution in equation (41) which subsequently
at the fixed point the solution is a logarithmic function, perturbing close to
this solution we get an infinite set tower of Hermite differential equations
at each order in expansion parameter $t$. Plugging $V_{p}(z,t)$ into the full
rescaled Polchinski’s differential equation at the LPA limit in equation (43),

$\displaystyle\partial_{t}V_{p}(z,t)=V^{\prime\prime}_{p}(z,t)-V^{\prime
2}_{p}(z,t)-d_{\phi}zV^{\prime}_{p}(z,t)+dV_{p}(z,t),$ (137)

where $V_{p}(z,t)=V(z,t)+g_{b}\sum_{n}f_{n}(z)e^{\lambda_{n}t}$. Expanding and
unpacking the differential equation above we get,

$\displaystyle\partial_{t}V(z,t)+g_{b}\sum_{n}\lambda_{n}f_{n}(z)e^{\lambda_{n}t}=\bigg{(}\frac{\partial^{2}V(z,t)}{\partial
z\partial z}+g_{b}\sum_{n}\frac{\partial^{2}f_{n}(z)}{\partial z\partial
z}e^{\lambda_{n}t}\bigg{)}$ (138)
$\displaystyle-\bigg{(}\bigg{(}\frac{\partial V(z,t)}{\partial
z}\bigg{)}^{2}+2g_{b}\frac{\partial V(z,t)}{\partial z}\sum_{n}\frac{\partial
f_{n}(z)}{\partial z}e^{\lambda_{n}t}+g^{2}_{b}\sum_{m,n}\frac{\partial
f_{n}(z)}{\partial z}\frac{\partial f_{m}(z)}{\partial
z}e^{(\lambda_{n}+\lambda_{m})t}\bigg{)}$ $\displaystyle-
d_{\phi}\bigg{(}z\frac{\partial V(z,t)}{\partial
z}+zg_{b}\sum_{n}\frac{\partial f_{n}(z)}{\partial
z}e^{\lambda_{n}t}\bigg{)}+d\bigg{(}V(z,t)+g_{b}\sum_{n}f_{n}(z)e^{\lambda_{n}t}\bigg{)}$
$\displaystyle\underbrace{\partial_{t}V(z,t)-\frac{\partial^{2}V(z,t)}{\partial
z\partial z}+\bigg{(}\frac{\partial V(z,t)}{\partial
z}\bigg{)}^{2}}_{=0,\text{ from homogeneous
equation}}=-g_{b}\sum_{n}\lambda_{n}f_{n}(z)e^{\lambda_{n}t}+g_{b}\sum_{n}\frac{\partial^{2}f_{n}(z)}{\partial
z\partial z}e^{\lambda_{n}t}$ $\displaystyle-\bigg{(}2g_{b}\frac{\partial
V(z,t)}{\partial z}\sum_{n}\frac{\partial f_{n}(z)}{\partial
z}e^{\lambda_{n}t}+g^{2}_{b}\sum_{m,n}\frac{\partial f_{n}(z)}{\partial
z}\frac{\partial f_{m}(z)}{\partial z}e^{(\lambda_{n}+\lambda_{m})t}\bigg{)}$
$\displaystyle-d_{\phi}\bigg{(}z\frac{\partial V(z,t)}{\partial
z}+\epsilon\sum_{n}\frac{f_{n}(z)}{\partial
z}e^{\lambda_{n}t}\bigg{)}+d\bigg{(}V(z,t)+g_{b}\sum_{n}f_{n}(z)e^{\lambda_{n}t}\bigg{)}$
$\displaystyle
0=-g_{b}\sum_{n}\lambda_{n}f_{n}(z)e^{\lambda_{n}t}+g_{b}\sum_{n}\frac{\partial^{2}f_{n}(z)}{\partial
z\partial z}e^{\lambda_{n}t}-\bigg{(}2g_{b}\frac{\partial V(z,t)}{\partial
z}\sum_{n}\frac{\partial f_{n}(z)}{\partial
z}e^{\lambda_{n}t}+g^{2}_{b}\sum_{m,n}\frac{\partial f_{n}(z)}{\partial
z}\frac{\partial f_{m}(z)}{\partial z}e^{(\lambda_{n}+\lambda_{m})t}\bigg{)}$
$\displaystyle-d_{\phi}\bigg{(}z\frac{\partial V(z,t)}{\partial
z}+g_{b}z\sum_{n}\frac{f_{n}(z)}{\partial
z}e^{\lambda_{n}t}\bigg{)}+d\bigg{(}V(z,t)+g_{b}\sum_{n}f_{n}(z)e^{\lambda_{n}t}\bigg{)},$

then at order $\mathcal{O}(g_{b})$ we get the following differential equation,

$\displaystyle e^{\lambda t}\frac{\partial^{2}f(z)}{\partial z\partial
z}=e^{\lambda t}\bigg{(}(\lambda-d)f(z)+d_{\phi}z\frac{\partial f(z)}{\partial
z}+2\frac{\partial V(z,t)}{\partial z}\frac{\partial f(z)}{\partial
z}\bigg{)},$ (139)

and expanding the last term of the RHS as follows,

$\displaystyle e^{\lambda t}\frac{\partial V(z,t)}{\partial z}\frac{\partial
f(z)}{\partial z}=$ $\displaystyle-e^{\lambda
t}\frac{\partial\log(U(z,t))}{\partial z}$ (140) $\displaystyle=$
$\displaystyle-e^{\lambda t}\frac{1}{U}\frac{\partial U}{\partial
z}\frac{\partial f(z)}{\partial z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{z}{\lambda t}e^{\lambda t}\frac{\partial
f(z)}{\partial z}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{z}{\lambda
t}\bigg{(}1+\lambda t+\frac{\lambda^{2}t^{2}}{2!}+\cdots\bigg{)}\frac{\partial
f(z)}{\partial z}$ $\displaystyle=$
$\displaystyle\frac{z}{2}\bigg{(}\frac{1}{\lambda
t}+\frac{\lambda}{\lambda}+\frac{\lambda^{2}t}{2!\lambda}+\cdots\bigg{)}\frac{\partial
f(z)}{\partial z}$ $\displaystyle=$
$\displaystyle\frac{z}{2}\bigg{(}\frac{1}{\lambda t}+1+\frac{\lambda
t}{2!}+\cdots\bigg{)}\frac{\partial f(z)}{\partial z},$

then collecting the zeroth order ($\mathcal{O}(t^{0})$) terms in expansion
parameter $t$ from equation (139) we get the following differential equation
below,

$\displaystyle\frac{\partial^{2}f(z)}{\partial z\partial z}=$
$\displaystyle(\lambda-d)f(z)+d_{\phi}z\frac{\partial f(z)}{\partial
z}+2\frac{\lambda}{\lambda}\frac{z}{2}\frac{\partial f(z)}{\partial z}$ (141)
$\displaystyle=$
$\displaystyle(\lambda-d)f(z)+\big{(}d_{\phi}+1\big{)}z\frac{\partial
f(z)}{\partial z}$ $\displaystyle=$
$\displaystyle(\lambda-d)f(z)+c_{\phi\lambda}z\frac{\partial f(z)}{\partial
z},$

where $c_{\phi\lambda}=d_{\phi}+1$.

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