# Continued functions and Borel-Leroy transformation:
Resummation of six-loop $\epsilon$-expansions from different universality
classes

Venkat Abhignan, R. Sankaranarayanan  Department of Physics, National
Institute of Technology, Tiruchirappali - 620015, India.

###### Abstract

We handle divergent $\epsilon$ expansions in different universality classes
derived from modified Landau–Wilson Hamiltonian. Landau–Wilson Hamiltonian can
cater for describing critical phenomena on a wide range of physical systems
which differ in symmetry conditions and the associated universality class.
Numerically critical parameters are the most interesting physical quantities
which characterize the singular behaviour around the critical point. More
precise estimates are obtained for these critical parameters than previous
predictions from Padé based methods and Borel with conformal mapping
procedure. We use simple methods based on continued functions and Borel-Leroy
transformation to achieve this. These accurate results are helpful in
strengthening existing conclusions in different $\phi^{4}$ models.

###### keywords:

Continued functions , Resummation method , $\epsilon$ expansions

##  1 Introduction

The calculation of recent six-loop [1] and seven-loop renormalization
functions [2] in $O(n)$-symmetric $\phi^{4}$ field theory has lead to
significant improvement in results related to universal critical phenomenon
[3, 4], compared to the 25 years old five-loop functions [5, 6]. It has been
helpful in solving for accurate universal parameters and also better
understand the decade old discrepancy between theoretical predictions and
experimental value in $O(2)$ $\phi^{4}$ model [7]. This was achieved using
Hypergeometric-Meijer resummation to solve for critical exponents [3, 4]. The
issue commonly called as ”$\lambda$-point specific heat experimental anomaly”
was also addressed by using a new blended, continued exponential fraction [8].
We solved for the critical exponents of arbitrary $n$-component field in
different spatial dimensions with the help of other continued functions such
as continued exponential and continued fraction in the $O(n)$-vector model. It
is a scalar field theory which has an infrared fixed point and around its
vicinity the system occupies symmetries of homogeneous, isotropic and dilation
in nature. This makes it a favourable ground to study different physical
systems at phase transitions with similar underlying behaviour but leading to
different universality classes. $O(1)$ $\phi^{4}$ theory can analyse phase
transitions on Ising model, simple fluids and binary mixtures. $O(2)$
$\phi^{4}$ theory deals with superconductivity and superfluid helium-4
transition. Similarly transitions on Heisenberg ferromagnets ($n=3$), some
models of quark-gluon plasma ($n=4$), neutron star matter ($n=10$) and
superfluid helium-3 ($n=18$) can be studied.
Quantities derived from field-theoretical perturbative methods are typically
divergent and resummation procedures are needed to extract meaningful
estimates with high precision. Resummation techniques have been the most
common way to handle perturbation series from field theories. With rapid
development in computational methods the sixth order [1] and seventh order [2]
terms in such series were obtained quite progressively and soon can lead to
more orders. This demands the need for accessible resummation methods which
provide accurate results with only lower order information. Traditionally
resummation techniques such as Padé approximants and Padé-Borel-Leroy
transformation are used for the purpose of obtaining convergence by only using
the lower order information of the perturbation series [9, 10, 11]. Continued
fractions are equivalent to Padé approximants, while they can be also related
to Shanks transformation which are used to accelerate the convergence nature
[12]. The modified orthogonal hypergeometric approximants used in
Hypergeometric-Meijer resummation can also be represented from a particular
form of continued function, namely Gauss’s continued fractions. Such
continuous iterative functions have been used to tackle divergent series in
different ways [13], which is an old idea developed by Yukalov [14, 15]. Here
we implement resummation methods using continued functions and Shanks
transformation on recently derived six-loop $\epsilon$ expansions in different
modified models. Further to improve convergence and remove irregularities we
combine continued functions with Borel-Leroy transformation. Such six-loop
perturbative expansions were derived in $n$-vector model with cubic anisotropy
[9], $O(n)\times O(m)$ spin models [10] and weakly disordered Ising model
[11], which were all solved based on the six-loop expansions from
$O(n)$-symmetric model by Kompaniets et al. [1]. Re-estimation of critical
parameters in these models using different methods with better accuracy are
helpful in removing contradictions and strengthening existing arguments
regarding these theories. Also in all these cases the exact solutions are not
known and the reliability of the extrapolated estimates is only established
when different methods produce comparable results.
The paper is organized as follows: We initially introduce the continued
functions and their implementation through Borel-Leroy transformation in Sec.
2. We then handle the perturbative expansions from $n$-vector model with cubic
anisotropy, $O(n)\times O(m)$ spin models and weakly disordered Ising model in
Sec. 3, 4 and 5 respectively.

##  2 Continued functions and Borel-Leroy transformation

The $\epsilon$ expansions in all the models produce quantities of interest in
form of

$Q(\epsilon)\approx\sum^{N}_{i=0}q_{i}\epsilon^{i}$ (1)

in a system with $d=4-\epsilon$ spatial dimensions. As mentioned earlier this
quantity is divergent in nature and hence we convert $Q(\epsilon)$ into
continued exponential (CE),

$Q(\epsilon)\sim
b_{0}\exp(b_{1}\epsilon\exp(b_{2}\epsilon\exp(b_{3}\epsilon\exp(b_{4}\epsilon\exp(b_{5}\epsilon\exp(b_{6}\epsilon\exp(\cdots)))))))$
(2)

or a continued exponential fraction (CEF),

$Q(\epsilon)\sim
c_{0}\exp\left(\frac{1}{1+c_{1}\epsilon\exp\left(\frac{1}{1+c_{2}\epsilon\exp\left(\frac{1}{1+c_{3}\epsilon\exp\left(\frac{1}{1+\cdots}\right)}\right)}\right)}\right)\;\;\;\hbox{for}\;\;\;(\epsilon\rightarrow
0).$ (3)

Coefficients $\\{b_{i}\\}$ and $\\{c_{i}\\}$ can be solved up to arbitary
order $i$ from the perturbative coefficients $\\{q_{i}\\}$ by Taylor expansion
of CE and CEF as shown previously [8]. These affine transformations generally
enlarge the region of convergence for $Q(\epsilon)$ by a different mapping.
And the convergence is obtained by observing the successive approximants

$B_{1}\equiv b_{0}\exp(b_{1}\epsilon),\,B_{2}\equiv
b_{0}\exp(b_{1}\epsilon\exp(b_{2}\epsilon)),\,B_{3}\equiv
b_{0}\exp(b_{1}\epsilon\exp(b_{2}\epsilon\exp(b_{3}\epsilon))),\,\cdots$ (4)

in case of CE and the sequence

$C_{1}\equiv c_{0}\exp\left(\frac{1}{1+c_{1}\epsilon}\right),\,C_{2}\equiv
c_{0}\exp\left(\frac{1}{1+c_{1}\epsilon\exp\left(\frac{1}{1+c_{2}\epsilon}\right)}\right),\\\
C_{3}\equiv
c_{0}\exp\left(\frac{1}{1+c_{1}\epsilon\exp\left(\frac{1}{1+c_{2}\epsilon\exp\left(\frac{1}{1+c_{3}\epsilon}\right)}\right)}\right),\cdots$
(5)

in case of CEF. Such a sequence is further converged with aid of Shanks
transformation and its iteration for any sequence $\\{A_{i}\\}$ such as,

$S(A_{i})=\frac{A_{i+1}A_{i-1}-A_{i}^{2}}{A_{i+1}+A_{i-1}-2A_{i}}\,\,\,\hbox{and}\,\,\,S^{2}(A_{i})=\frac{S(A_{i+1})S(A_{i-1})-S(A_{i}^{2})}{S(A_{i+1})+S(A_{i-1})-2S(A_{i})}.$
(6)

Further the Borel-Leroy (BL) technique we use is derivative of Padé-BL
transformation commonly used [9, 10, 11], where the Padé approximant is
replaced with CE. BL transformation uses a shift parameter $l$ based on which
we can optimize the convergence behaviour. Tuning this parameter $l$ we can
tame the factorial growth of coefficients in quantity $Q(\epsilon)$ and refine
the estimate of $Q(\epsilon)$ to higher accuracy. The BL transformation is
given by

$Q(\epsilon)=\int_{0}^{\infty}e^{-t}t^{l}F(\epsilon
t)dt,\,\,\,F(y)=\sum_{i=0}^{\infty}\frac{q_{i}}{\Gamma(i+l+1)}y^{i}.$ (7)

This transformation which implements sequence of CE is given by

$D_{1}\equiv\int_{0}^{\infty}e^{-t}t^{l}d_{0}\exp(d_{1}\epsilon
t)dt,\,D_{2}\equiv\int_{0}^{\infty}e^{-t}t^{l}d_{0}\exp(d_{1}\epsilon
t\exp(d_{2}\epsilon t))dt,\\\
D_{3}\equiv\int_{0}^{\infty}e^{-t}t^{l}d_{0}\exp(d_{1}\epsilon
t\exp(d_{2}\epsilon t\exp(d_{3}\epsilon t)))dt,\cdots$ (8)

where coefficients $\\{d_{i}\\}$ are obtained similar to $\\{b_{i}\\}$ from
the series $F(y)$. Similar to previous approaches the sequence $\\{D_{i}\\}$
provides the converged estimate which is further accelerated using Shanks
transformation. We observe that implementing CEF with BL transformation does
not lead to significant improvement in the convergence behaviour.
To measure the accuracy of our estimates we calculate error using the relation

$(|S(A_{i+1})-S(A_{i})|+|S(A_{i+1})-S^{2}(A_{i})|)/2,$ (9)

when we take $S^{2}(A_{i})$ as our final estimate for $Q(\epsilon)$ from the
sequence $\\{A_{i}\\}$. This relation which is based on successive
approximations gives the accuracy of our estimate since the value of Shanks
iteration depends on previous iterations. To find the parameter $l$ in BL
transformation, we inspect the space of $l\in[0,50]$ with $\Delta l=0.01$ to
minimize the error given above. We choose the resummation tool empirically
based on observing the convergence nature of coefficients $\\{q_{i}\\}$ and
feasibility of finding parameter $l$.
We have already checked the applicability of CE and CEF on perturbative
$\epsilon$ expansions from $O(n)$-symmetric field theory [8]. To test the
applicability of CE with BL transformation, we implement it to calculate the
correction-to-scaling exponent $\omega$ from $O(n)$-symmetric theory.
Especially in $3d$ system for the $O(4)$ $\phi^{4}$ model we previously used
continued fraction to obtain the estimate $\omega=0.7896(1)$ [8], which is
only compatible with recent calculations from self-consistent resummation
algorithm [16] where $\omega=0.7863(9)$ was obtained. It is also comparable
with estimates from Borel with conformal mapping (BCM) and conformal bootstrap
calculations, $\omega=0.794(9)$ [1] and $\omega=0.817(30)$ [17], respectively.
But there is a mismatch from other recent estimates namely hypergeometric-
Meijer resummation [3], non-perturbative renormalization group [18] and Monte
Carlo simulations (MC) [19] predict as $\omega=0.7519(13)$, $\omega=0.761(12)$
and $\omega=0.765(30)$, respectively. We used continued fraction since the
$\epsilon$ expansion for $\omega$ is of the form $\epsilon Q(\epsilon)$ such
as [3]

$\omega=\epsilon-0.541667\epsilon^{2}+1.15259\epsilon^{3}-3.27193\epsilon^{4}+10.8016\epsilon^{5}-40.5665\epsilon^{6}+166.256\epsilon^{7}.$
(10)

Here we solve for $Q(\epsilon)$ using CE-BL and multiply the estimate with
value of $\epsilon$. In order to obtain critical exponents for three-
dimensional systems we finally equate $\epsilon$ to unity. To introduce the
tuning parameter $l$ we take into consideration Eq. (7) for $Q(\epsilon)$ such
as

$F(\epsilon t)=\frac{1}{\Gamma(l+1)}-\frac{0.541667\epsilon
t}{\Gamma(l+2)}+\frac{1.15259\epsilon^{2}t^{2}}{\Gamma(l+3)}-\frac{3.27193\epsilon^{3}t^{3}}{\Gamma(l+4)}+\frac{10.8016\epsilon^{4}t^{4}}{\Gamma(l+5)}-\frac{40.5665\epsilon^{5}t^{5}}{\Gamma(l+6)}+\frac{166.256\epsilon^{6}t^{6}}{\Gamma(l+7)}.$
(11)

Further by implementing CE as in Eq. (8) we find the sequence for $\epsilon
Q(\epsilon)$ such as

$D_{1}=\epsilon\int_{0}^{\infty}e^{-t}t^{l}\frac{1}{{\Gamma}\left(l+1\right)}\exp\left(-\frac{0.541667\,{\Gamma}\left(l+1\right)}{{\Gamma}\left(l+2\right)}\epsilon
t\right)dt,\\\
D_{2}=\epsilon\int_{0}^{\infty}e^{-t}t^{l}\frac{1}{{\Gamma}\left(l+1\right)}\exp\left(-\frac{0.541667\,{\Gamma}\left(l+1\right)}{{\Gamma}\left(l+2\right)}\epsilon
t\exp\left(1.84615\,{\Gamma}\left(l+2\right)\,{\left(\frac{.00001\,{\Gamma}\left(l+1\right)}{{{\Gamma}\left(l+2\right)}^{2}}-\frac{0.00006}{{\Gamma}\left(l+3\right)}\right)}\epsilon
t\right)\right)dt,\cdots$ (12)

Evaluating these integrals we get approximants $\\{D_{i}\\}$ and further
accelerate their convergence using Shanks transformed sequence
$\\{S(D_{i})\\}$. We take our final estimate for $\omega$ from $S^{2}(D_{4})$
and estimate the error from relation
$(|S(D_{5})-S(D_{4})|+|S(D_{5})-S^{2}(D_{4})|)/2$ as in Eq.(9). By tuning the
parameter $l$ we illustrate the behaviour of estimate in Fig. 1(a) for
$l\in[0.5,1.2]$. We observe that by shifting the parameter $l$ the
nonuniformity in our approximants can be removed and region of precise
estimate can be decided, where the value of estimate is sensistive to $l$.
Further observing the confined region $l\in[1,1.2]$ in Fig. 1(b) we decide
$l=1.097$ to get the precise estimate $\omega=0.79419$ which is most
compatible with BCM prediction. We note here that in BCM procedure it is
necessary to know the asymptotic behaviour of perturbative expansions for
Green’s functions whereas our procedure involves only lower order information.

(a) $S^{2}(D_{4})$ vs $l$ for $l\in[0.5,1.2]$

(b) $S^{2}(D_{4})$ vs $l$ for $l\in[1,1.2]$

Figure 1: We plot the estimate of $\omega$ derived from $S^{2}(D_{4})$ vs
shift parameter $l$, with the error bars showing the value of
$(|S(D_{5})-S(D_{4})|+|S(D_{5})-S^{2}(D_{4})|)/2$.

The choices in this CE-BL procedure might be leading to overestimation of
accuracy in the estimates obtained. More stringent test for this procedure
would be to calculate correlation length exponent of well known 3d Ising model
which has a general consensus up to third decimal place, $\nu_{Ising}\approx
0.630$ using different theoretical approaches like renormalization group (RG)
[3, 8], Conformal bootstrap [20] and MC [21]. Using CE-BL resummation of
slowly converging $\epsilon$ expansion [3] we obtain the estimate
$\nu_{Ising}=0.63134$, which assures the accuracy of this procedure to at
least third decimal place.

##  3 $n$-vector model with cubic anisotropy

Critical parameters in a realistic cubic crystal with anisotropy were recently
studied for $n$-vector field [9]. In such model there is a competition of RG
flows between isotropic Heisenberg and anisotropic cubic modes which are in
different regimes of critical behaviour. This leads to stability of fixed
points for this model only in a particular region of order parameter $n$.
Marginal order parameter dimensionality $n_{c}$ determines the stability,
where in case of $n<n_{c}$ the Heisenberg critical behaviour dominates and for
$n>n_{c}$ cubic critical regime is stable. So for $n>n_{c}$, a new class of
universality for critical behaviour of cubic-anisotropy emerges completely
different from the $O(n)$ class (Heisenberg). Hence studying $n_{c}$ is of
physical importance for real ferromagnets with anisotropies to determine which
regime they follow. Initial studies on such models lead to $n_{c}>3$ using RG
method [22] and $n_{c}=3$ using Monte Carlo simulations (MC) [23]. However
later studies based on higher order resummation of RG perturbative series
showed that $n_{c}<3$ [24, 25, 26]. The recent six-loop $\epsilon$ expansion
provides the best prediction as $n_{c}=2.915(3)$ with the Padé-BL technique
[9]. This prediction was derived from six-loop $\epsilon$ expression for
$n_{c}$ which was given as

$n_{c}=4-2\epsilon+2.588476\epsilon^{2}-5.874312\epsilon^{3}+16.82704\epsilon^{4}-56.62195\epsilon^{5}+O(\epsilon^{6})$
(13)

for the physically interesting case of $n=3$. We convert this into CE (Eq. 2)
and calculate its Shanks to get the sequence for $d=3$ as

$B_{1}=2.42612,\,B_{2}=3.35454,\,B_{3}=2.63588,\,B_{4}=3.16593,B_{5}=2.71339$
(14)

and

$S(B_{2})=2.94945,\,S(B_{3})=2.94093,S(B_{4})=2.92181.$ (15)

As we observe the sequence $\\{B_{i}\\}$ is oscillating between the best
prediction given above, while the sequence of Shanks $\\{S(B_{i})\\}$ is
converging monotonically. To accelerate this convergence behaviour we take BL
transformation of CE (Eq. 8) with $l=0.61$ to obtain

$D_{1}=2.58794,\,D_{2}=3.18263,\,D_{3}=2.94130,\,D_{4}=2.91015,D_{5}=2.96437$
(16)

and Shanks of the above sequence is given by

$S(D_{2})=2.96437,\,S(D_{3})=2.91741,S(D_{4})=2.91735.$ (17)

We take the final estimate from $S^{2}(D_{3})$, $n_{c}=2.91735(3)$ which is
more precise compared to previous estimates from 3d RG approaches [24, 25,
26], pseudo-$\epsilon$ expansion [27, 28] and most compatible with the recent
Padé-BL estimate (Table 1). We illustrate the behaviour of shift parameter $l$
versus $S^{2}(D_{3})$ in Fig.2, with the error bars.

(a) $S^{2}(D_{3})$ vs $l$ for $l\in[0.1,8]$

(b) $S^{2}(D_{3})$ vs $l$ for $l\in[0.4,0.8]$

Figure 2: We plot the value of marginal order parameter dimensionality $n_{c}$
derived from CE-BL estimate $S^{2}(D_{3})$ vs shift parameter $l$, with the
error bars showing the value of
$(|S(D_{4})-S(D_{3})|+|S(D_{4})-S^{2}(D_{3})|)/2$.

Further we study the critical exponents for this universality class of cubic
ferromagnets with anisotropy $(n=3)$, which are interesting since they differ
from the Heisenberg class. The exponent $\nu$ which determines the behaviour
of correlation length was given by expression [9]

$\frac{1}{\nu}=2-0.44444\epsilon-0.17513\epsilon^{2}+0.13460\epsilon^{3}-0.34969\epsilon^{4}+0.99461\epsilon^{5}-3.48637\epsilon^{6}+O(\epsilon^{7}).$
(18)

Similarly we convert this into CE for $d=3$ and obtain the oscillating
sequence

$1/B_{1}=0.62442,\,1/B_{2}=0.72262,\,1/B_{3}=0.66793,\,1/B_{4}=0.71540,\,1/B_{5}=0.68772,\\\
1/B_{6}=0.71368.$ (19)

Consecutively the Shanks iterations provide sequence

$S(1/B_{2})=0.68749,\,S(1/B_{3})=0.69334,\,S(1/B_{4})=0.69791,\,S(1/B_{5})=0.70112,$
(20)

and

$S^{2}(1/B_{3})=0.714305,\,S^{2}(1/B_{4})=0.70858.$ (21)

We observe that the final iterated values of Shanks $S^{2}(1/B_{3})$ and
$S^{2}(1/B_{4})$ provide a good neighbourhood for the prediction of $\nu$. To
obtain a more precise prediction, we convert the expression of $\nu$ into CEF
(Eq. 3) whose rapidly convergent sequence is

$1/C_{2}=0.63725,\,1/C_{3}=0.77547,\,1/C_{4}=0.70693,\,1/C_{5}=0.71451,\,1/C_{6}=0.71122,$
(22)

with corresponding Shanks as

$S(1/C_{3})=0.72965,\,S(1/C_{4})=0.71376,\,S(1/C_{5})=0.71222,\,S^{2}(1/C_{4})=0.71205.$
(23)

Our precise CEF estimate for cubic class $\nu=0.71205(85)$ is better than
previous Padé-BL value of $\nu=0.700(8)$ [9]. And it marginally differs from
recent seven-loop precise estimates of Heisenberg class using CEF,
$\nu=0.70787(39)$ [8] and hypergeometric-Meijer resummation, $\nu=0.70906(18)$
[3]. Similarly using the expression for $\gamma$ [9] the critical exponent of
susceptibility, we determine its CE estimate using $S^{2}(B_{4})=1.404(16)$
and CEF estimate using $S^{2}(C_{4})=1.4236(11)$. Similarly to calculate the
correction-to-scaling exponent $\omega$ we take the series which is of the
form $\epsilon Q(\epsilon)$ [9], solve for $Q(\epsilon)$ using CE and CE-BL
and multiply the estimate with value of $\epsilon$. Further we compare them
all with previous most compatible cubic class estimates and also with recent
estimates from Heisenberg class in Table 1. These precise estimates show that
quantitatively the critical exponents of 3d cubic class and Heisenberg class
are numerically close and differ very marginally. This difference may be
refined with help from higher $\epsilon$ expansions.

Table 1: Critical parameters $n_{c}$, $\nu$, $\gamma$, $\omega$ for class of $n=3$ cubic-anisortropy model in three dimensional systems Critical parameter | Continued function estimates | Existing predictions |  | Existing predictions
---
(Heisenberg class)
$n_{c}$ |  | 2.956(26) (CE)
---
2.91735(3) (CE-BL)
|  2.915(3) [9]
---
2.89-2.92 [25]
2.89(4) [26]
2.862(5) [27]
2.86(1) [28]
—
$\nu$ |  | 0.7085(53) (CE)
---
0.71205(85) (CEF)
|  0.700(8) [9]
---
0.706(6) [24]
0.704(4) [26]
|  0.70787(39) (CEF) [8]
---
0.70906(18) [3]
$\gamma$ |  | 1.404(16) (CE)
---
1.4236(11) (CEF)
|  1.368(12) [9]
---
1.419(6) [26]
1.416(4) [27]
|  1.3929(46) (CE) [8]
---
1.385(4) [9]
$\omega$ |  | 0.78417(29) (CE)
---
0.78569(68) (CE-BL)
|  0.799(4) [9]
---
0.7833(54) [24]
0.781(4) [26]
|  0.79083(1) [8]
---
0.794(4) [16]

##  4 $O(n)\times O(m)$ spin models

The extension of $O(n)$-symmetric field theory to $O(n)\times O(m)$ symmetry
is helpful in studying frustrated spin systems with noncoplanar and
noncollinear ordering [10]. Such spin structures can be found in physical
systems such as stacked triangular antiferromagnets (STA) and helical magnets
(HM). Such physically interesting systems can be studied in this case for
$n=2$, XY and $n=3$, Heisenberg frustrated antiferromagnets. For $m=2$ the
theory corresponds to system with noncollinear but coplanar ordering (such
that $n\geq m$). Further with $m\geq 3$ we can study the critical behavior of
magnets with noncoplanar ordering. Similar to the previous model based on
stability of fixed points in RG flows, we can deduce the existence of an upper
marginal order parameter value $n^{+}(m,d)$ and lower marginal
dimensionalities $n^{H}(m,d)$, $n^{-}(m,d)$. This demarcation of order
parameter $n$ is useful to differentiate the different regimes of critical
behaviour (where $n^{H}(m,d)<n^{-}(m,d)<n^{+}(m,d)$). For $n>n^{+}(m,d)$ the
system follows second-order continuous phase transition with ”chiral”
universality class and for $n^{-}(m,d)<n<n^{+}(m,d)$ the transition is first-
order. Further for $n<n^{H}(m,d)$ Heisenberg or $O(mn)$-symmetric critical
behaviour dominates and for $n^{H}(m,d)<n<n^{-}(m,d)$ the system either
undergoes first-order transition or ordering happens with simple unfrustrated
sinusoidal spin structure depending on the coupling constant. Finding these
parameters along with critical exponents for such chiral universality class
had given contradictory results in field-theoretical models compared to
experiments [29, 30]. Also the results of different experiments had produced a
wide range of unrelated critical exponents and even the order of phase
transition was found to differ for materials with same symmetry [31, 32, 33,
34, 35, 36, 37, 38, 39].
Hence initially getting precise estimates for upper marginal dimensionality
$n^{+}(m=\\{2,3\\},4-\epsilon)$ is of physical importance to determine the
type of phase transition in real systems. We illustrate the results for
determining $n^{+}(m=\\{2,3\\},4-\epsilon)$ which seem to possess the most
irregular numerical structure compared to other quantities of lower marginal
dimensionalities [10]. The six-loop $\epsilon$ expansion for
$n^{+}(2,4-\epsilon)$ was obtained as [10]

$n^{+}(2,4-\epsilon)=21.798-23.431\epsilon+7.0882\epsilon^{2}-0.0321\epsilon^{3}+4.2650\epsilon^{4}-8.4436\epsilon^{5}+O(\epsilon^{6}).$
(24)

Constructing CE and consecutive Shanks iteration of this expansion for $d=3$
we obtain

$B_{1}=7.440,B_{2}=5.597,B_{3}=5.323,B_{4}=5.596,B_{5}=5.597\\\
\hbox{and}\,\,S(B_{2})=5.275,S(B_{3})=5.450,S(B_{4})=5.597,S^{2}(B_{3})=5.996.$
(25)

Similarly constructing CE-BL ($l=3.69$, Fig 3.(a)) and consecutive Shanks
iteration we obtain

$D_{1}=8.281,D_{2}=6.402,D_{3}=5.911,D_{4}=6.147,D_{5}=6.034\\\
\hbox{and}\,\,S(D_{2})=5.737,S(D_{3})=6.07,S(D_{4})=6.07,S^{2}(D_{3})=6.07.$
(26)

We observe our estimates from CE, $n^{+}(2,3)=5.99(27)$ and CE-BL,
$n^{+}(2,3)=6.07$ are compatible with recent inverse biased Padé-BL prediction
$n^{+}(2,3)=6.1(1)$ and BCM prediction $n^{+}(2,3)=6.0(6)$ [10]. Similarly the
six-loop $\epsilon$ expansion for $n^{+}(3,4-\epsilon)$ was obtained as [10]

$n^{+}(3,4-\epsilon)=32.492-33.719\epsilon+11.100\epsilon^{2}-2.1440\epsilon^{3}+5.2756\epsilon^{4}-8.4830\epsilon^{5}+O(\epsilon^{6}).$
(27)

Constructing CE and consecutively Shanks iteration of this expansion for $d=3$
we obtain

$B_{1}=11.5102,B_{2}=9.267,B_{3}=8.408,B_{4}=9.177,B_{5}=9.109\\\
\hbox{and}\,\,S(B_{2})=7.876,S(B_{3})=8.814,S(B_{4})=9.115,S^{2}(B_{3})=9.257.$
(28)

Constructing CE-BL ($l=5.27$, Fig 3.(b)) and consecutively Shanks iteration we
obtain

$D_{1}=12.437,D_{2}=9.989,D_{3}=9.084,D_{4}=9.532,D_{5}=9.310\\\
\hbox{and}\,\,S(D_{2})=8.554,S(D_{3})=9.384,S(D_{4})=9.384,S^{2}(D_{3})=9.384.$
(29)

We observe our estimates from CE, $n^{+}(3,3)=9.25(22)$ and CE-BL,
$n^{+}(3,3)=9.384$ are compatible with the inverse biased Padé-BL prediction
$n^{+}(3,3)=9.7(1.0)$ and BCM prediction $n^{+}(3,3)=9.3(4)$ [10]. Similarly
using the CE and CE-BL methods we compute estimates of other physically
interesting values of upper marginal order parameter
$n^{+}(m=\\{4,5,6\\},4-\epsilon)$ and lower marginal dimensionalities
$n^{-}(m=\\{2,...,6\\},4-\epsilon)$, $n^{H}(m=\\{2,...,6\\},4-\epsilon)$ [10].
We observe that all our estimates are compatible with Padé-BL and BCM
predictions from six-loop expansion [10], five-loop expansion [40] and
pseudo-$\epsilon$ expansion [40] (Table 2). Further it is interesting to
compute critical exponents of chiral universality class for $m=2$ with $n\geq
6$ since we obtain $n^{+}(2,3)\approx 6$. We compute $\nu$, $\eta$ and
$\gamma$ [10] using CE. CEF or CE-BL could not provide better accuracy. The
series for $\eta$ takes the form $\epsilon^{2}Q(\epsilon)$ which we handle
similar to the way of $\omega$. We compute correction-to-scaling exponents
$\omega_{1}$ and $\omega_{2}$ [10] using CE. $\omega_{1}$ and $\omega_{2}$ are
interesting in physical point of view since they determine the stability of
fixed points in RG flows. We tabulate all the estimates for chiral critical
exponents in Table 3 to compare them with recent predictions from Padé based
resummation of six-loop expansion [10] and five-loop expansion [40]. We
observe that all our estimates have improved precision compared to the Padé-BL
and BCM predictions except for values of $n^{-}(6,3)$ and $\omega_{2}$ for
$n=32$.

(a) $n^{+}(2,3)$ vs $l$ for $l\in[1,5]$

(b) $n^{+}(3,3)$ vs $l$ for $l\in[4,8]$

Figure 3: We plot the value of upper marginal order parameter $n^{+}$ derived from CE-BL estimate $S^{2}(D_{3})$ vs shift parameter $l$, with the error bars showing the value of $(|S(D_{4})-S(D_{3})|+|S(D_{4})-S^{2}(D_{3})|)/2$. Table 2: Marginal dimensionalities $n^{+}$, $n^{-}$ and $n^{H}$ for class of $O(n)\times O(m)$ symmetric model for physically relevant three dimensional systems. CE-BL estimates are denoted by superscript*. Critical parameter | $m=2$ | $m=3$ | $m=4$ | $m=5$ | $m=6$
---|---|---|---|---|---
$n^{+}(m,3)$ |  | 5.99(27) (CE)
---
6.07* ($l=3.69$)
6.22(12) [40]
6.1(6) [40]
5.96(19) [10]
| 9.25(22) (CE)
---
9.384* ($l=5.27$)
9.9(3) [40]
9.5(5) [40]
9.32(19) [10]
| 12.19(21) (CE)
---
12.363* ($l=6.44$)
13.2(6) [40]
12.7(7) [40]
12.3(3) [10]
| 15.04(22) (CE)
---
15.215* ($l=7.47$)
16.3(1.3) [40]
15.7(1.0) [40]
15.0(3) [10]
| 17.83(23) (CE)
---
18.003* ($l=8.43$)
18.0(5) (BCM) [10]
17.8(3) [10]
$n^{-}(m,3)$ |  | 1.964(3) (CE)
---
1.968(5)* ($l=19.25$)
1.970(3) [10]
| 1.403 (CE)
---
1.406(1)* ($l=6.55$)
1.408(4) [10]
| 1.182(2) (CE)
---
1.182* ($l=21.32$)
1.182(6) [10]
| 1.082(61) (CE)
---
–
1.089(9) [10]
| 1.325(5) (CE)
---
–
1.066(12) [10]
$n^{H}(m,3)$ |  | 1.478(13) (CE)
---
1.459* ($l=0.6$)
1.462(13) [10]
| 0.985(9) (CE)
---
0.972* ($l=0.61$)
0.973(11) [10]
| 0.739(6) (CE)
---
0.729* ($l=0.6$)
0.733(10) [10]
| 0.591(5) (CE)
---
0.583* ($l=0.61$)
0.587(8) [10]
| 0.493(4) (CE)
---
0.486* ($l=0.6$)
0.488(7) [10]
Table 3: Critical exponents $\nu$, $\eta$, $\gamma$, $\omega_{1}$ and $\omega_{2}$ for $m=2$ chiral universality class in physically relevant three dimensional systems with noncollinear but coplanar ordering. Critical exponent | $n=6$ | $n=7$ | $n=8$ | $n=16$ | $n=32$
---|---|---|---|---|---
$\nu$ |  | 0.6516(21) (CE)
---
0.65(2) [10]
| 0.7117(5) (CE)
---
0.71(4) [40]
0.713(8) [10]
| 0.73398 (CE)
---
0.75(4) [40]
0.745(11) [10]
| 0.899(10) (CE)
---
0.89(4) [40]
0.850(16)? [10]
| 0.9368(33) (CE)
---
0.94(2) [40]
0.940(17) [10]
$\eta$ |  | 0.04716(36) (CE)
---
0.047(3) [10]
| 0.04436(69) (CE)
---
0.045(3) [10]
| 0.04144(99) (CE)
---
0.042(2) [10]
| 0.026156(29) (CE)
---
0.0261(7) [10]
| 0.01609 (CE)
---
0.014(3) [10]
$\gamma$ |  | 1.29561 (CE)
---
1.27(3) [10]
| 1.39603(40) (CE)
---
1.39(6) [40]
1.396(14) [10]
| 1.45054 (CE)
---
1.45(6) [40]
1.461(17) [10]
| 1.795(35) (CE)
---
1.75(4) [40]
1.70(5)? [10]
| 1.8565(55) (CE)
---
1.87(4) [40]
1.87(4) [10]
$\omega_{1}$ |  | 0.6516(21) (CE)
---
0.65(2) [10]
| 0.7117(5) (CE)
---
0.71(4) [40]
0.713(8) [10]
| 0.73398 (CE)
---
0.75(4) [40]
0.745(11) [10]
| 0.899(10) (CE)
---
0.89(4) [40]
0.850(16)? [10]
| 0.9368(33) (CE)
---
0.94(2) [40]
0.940(17) [10]
$\omega_{2}$ |  | 0.07327(42) (CE)
---
[0.071,0.167] (BCM) [10]
[0.069,0.171] [10]
| 0.3431(13) (CE)
---
0.33(10) [40]
0.34(2) [10]
| 0.4557(14) (CE)
---
0.45(8) [40]
0.447(15) [10]
| 0.77572(36) (CE)
---
0.77(2) [40]
0.771(6) [10]
| 0.891(27) (CE)
---
0.90(1) [40]
0.904(8) [10]

##  5 Randomly diluted Ising model

Randomly diluted Ising model (RIM) for systems with ”frozen” impurities have
different universality class of critical exponents derived from that of pure
Ising model [41, 42]. Studying this model using RG approach is interesting
since other disordered systems such as randomly site-diluted Ising model [43,
44], randomly bond-diluted Ising model [45, 46] and $\pm J$ Ising model [47,
48] fall under the same category of RIM universality class. Experimentally the
critical behaviours are studied in crystalline mixture of two compounds such
as ordering in an anisotropic uniaxial antiferromagnet of $FeF_{2}$ or
$MnF_{2}$ with $ZnF_{2}$ as impurity.
Initial RG studies on RIM model has led to conclusion that critical exponents
have to be in the powers of $\sqrt{\epsilon}$ instead of $\epsilon$ [49, 50].
However the most recent studies show that resummation of critical exponents in
$\sqrt{\epsilon}$ expansion do not give reliable estimates in comparison with
predictions from other theoretical approaches [51, 11]. For instance the
recent six-loop $\sqrt{\epsilon}$ expansions derived for $\nu$ and $\gamma$
are in the form of [11]

$\nu_{\sqrt{\epsilon}}=0.5+0.0841158\epsilon^{1/2}-0.016632\epsilon+0.0477535\epsilon^{3/2}+0.272584\epsilon^{2}+0.223298\epsilon^{5/2}+O(\epsilon^{3})$
(30)

and

$\gamma_{\sqrt{\epsilon}}=1+0.168232\epsilon^{1/2}-0.0285471\epsilon+0.0788288\epsilon^{3/2}+0.564505\epsilon^{2}+0.440615\epsilon^{5/2}+O(\epsilon^{3}).$
(31)

We construct CEF sequences such as in Eq. (5) for variable $t$ with a change
in variable $\epsilon\rightarrow\epsilon t^{2}$ for $d=3$ and finally equate
$t$ to unity. We get CEF estimates $\nu_{\sqrt{\epsilon}}=0.57956(16)$ and
$\gamma_{\sqrt{\epsilon}}=1.16108(19)$ which are more precise than the recent
Padé-BL predictions $\nu_{\sqrt{\epsilon}}=0.577(31)$ and
$\gamma_{\sqrt{\epsilon}}=1.172(55)$ [11]. However we come to the same
conclusion that these values of critical exponents obtained from resummation
of $\sqrt{\epsilon}$ expansions are in contradiction from theoretical and
experimental results, further confirming that these series are not resummable.
One can further try resummation of the RG functions directly through continued
functions instead of $\sqrt{\epsilon}$ expansions to obtain critical exponents
as tried previously [51, 11].

##  6 Conclusion

Simple methods using continued functions were utilised to derive precise
critical parameters and critical exponents from six-loop RG perturbative
expansions of $n$-vector model with cubic anisotropy, $O(n)\times O(m)$ spin
models and weakly disordered Ising model. This is useful in better defining
the nature of continuous phase transitions in their corresponding physical
systems. To summarise there is an interesting quote in the recent work of
Kompaniets and Panzer regarding resummation methods [1] : In work on
resummation, there is always an undeclared parameter: the number of methods
tried and rejected before the paper was written, which was followed by their
comment ”We stopped counting”. Taking into consideration the essence of this
problem, we have devised resummation methods with ease of use and less
computations, which provide estimates with better accuracy than Padé based
methods. Also Padé methods are typically riddled with spurious poles, where a
thorough inspection is required to remove them when finding a reliable
estimate. However in our approach one has to further rigorously study which
strategic method using CE, CEF or CE-BL provides best accuracy and reliable
convergence based on the numerical structure of the perturbation series. Also
in all our cases we have handled only physically relevant systems with
$d=3(\epsilon=1$), further trying to study $d=2(\epsilon=2)$ systems can cause
trivial problems to emerge from the structure of continued functions we have
considered. Such problems can perhaps be removed by implementing long-order
asymptotic behaviour of the coefficients in the perturbation series if
available.

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