# Model A of critical dynamics: 5-loop $\varepsilon$ expansion study

L. Ts. Adzhemyan  D.A. Evdokimov  M. Hnatič  E. V. Ivanova  M. V. Kompaniets
<EMAIL_ADDRESS>A. Kudlis<EMAIL_ADDRESS>D.V. Zakharov
Saint Petersburg State University, 7/9 Universitetskaya Embankment, St.
Petersburg, 199034 Russia  Bogoliubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, 6 Joliot-Curie, Dubna, Moscow region, 141980,
Russian Federation  Department of Theoretical Physics, SAS, Institute of
Experimental Physics, Watsonova 47, 040 01 Košice, Slovak Republic  Pavol
Jozef Šafárik University in Košice (UPJŠ), Šrobárova 2, 041 80 Košice, Slovak
Republic  New Jersey Institute of Technology, 323 Dr Martin Luther King Jr
Blvd, Newark, NJ 07102, USA  Faculty of Physics, ITMO University, Kronverkskiy
prospekt 49, St. Petersburg 197101, Russia

###### Abstract

We have calculated the five-loop RG expansions of the $n$-component A model of
critical dynamics in dimensions $d=4-\varepsilon$ within the Minimal
Subtraction scheme. This is made possible by using the advanced diagram
reduction method and the Sector Decomposition technique adapted to the
problems of critical dynamics. The $\varepsilon$ expansions for the critical
dynamic exponent $z$ for an arbitrary value of the order parameter dimension
$n$ are derived. Based on these series, the numerical estimates of $z$ for
different universality classes are extracted and compared with the results
obtained within different theoretical and experimental methods.

###### keywords:

renormalization group, critical dynamics, multi-loop calculation, critical
dynamic exponent z, $\varepsilon$ expansion

††journal: Physica A

##  1 Introduction

The observable in the vicinity of critical point physics already for half a
century continues to surprise with its peculiarities. The forgetfulness of the
undergoing second-order phase transition physical systems regarding their
microscopic features allows dividing such systems into so-called universality
classes [1, 2]. The systems united by one universality class should
demonstrate the same critical behavior. Moreover, an elegant feature of the
theory is the fact that each class of universality is determined only by the
most fundamental characteristics: spatial dimensionality, the number of order
parameter components, the range of interaction, as well as the symmetry of the
system above the criticality [3, 4, 5, 6, 7]. However, universality is not a
panacea, for example it does not apply to the value of critical temperature;
it concerns only a limited set of physical observables, where critical
exponents have been the most studied over the years. These, as a rule, non-
integer numbers describe the power-law behavior of the thermodynamic
characteristics near the continuous phase transition. Most of them are related
to the consideration of a static problem. There is, however, an exponent that
requires already to study the critical dynamics of the system. It is the
dynamic critical exponent $z$. The classification of various dynamic models
was suggested in Ref. [8]; it is based on the conservation or not of both the
energy of the system and the order parameter. The purely dissipative
relaxational dynamics of the order parameter corresponds to the case of the
so-called model A, where the critical exponent $z$ describes the critical
slowing down connecting the correlation length $\xi$ and the typical time of
fluctuations $\tau$ as:

$\tau\propto\xi^{z},$ (1)

where $\xi$ and $\tau$ grow indefinitely when approaching the critical point.
This behavior is caused by strong fluctuations that are present in the
vicinity of the phase transition and due to its existence relation (1) does
not allow the system to achieve its equilibrium state. As for numerical
calculations, the struggle is at the level of deviations of tenths and
hundredths from the canonical value $z=2$.

In fact, the A model is the simplest dynamic generalization of the static
$\phi^{4}$ field theory, that allows one to test various techniques being
usual for static consideration of critical dynamics with the least possible
difficulties. The treatment of this model can be realized by using different
Monte Carlo (MC) methods, high-temperature (HT) expansion technique, as well
as using various field theoretical (FT) approaches. Unfortunately, each method
has its drawbacks. This can be either a finite lattice size or poor
convergence of expansions. In this situation, in order to understand the true
picture of the phenomenon, a comprehensive study of the problem using all
available approaches is vitally important. Below, we focus mostly on the
specific results that were obtained within the mentioned methods in the case
of different values of the order parameter dimensionality $n$, as well as the
spatial dimensions $d$ without detailed analysis of reasons of possible
contradictions between them. In this work, we will associate each of the
universality classes or A model for specific $d$ and $n$ with the symbol
$\bm{\mathscr{A}}^{d,n}$ while avoiding mentioning the Ising model or
Heisenberg one in order to exclude possible misunderstanding regarding the
relation with other dynamic models as the J or C models of critical dynamics.
Note that most results are naturally devoted to the one-, two-, and three-
component models A, which physically describe the critical slowing down in
easy-axis, planar and isotropic ferromagnets, respectively. Among them, most
attention of researchers was riveted to the $\bm{\mathscr{A}}^{2,1}$ and
$\bm{\mathscr{A}}^{3,1}$ models as well as to the $\bm{\mathscr{A}}^{3,3}$
one.

Let us begin with MC analysis of lattice models, which can be realized by
means of different options. The system can be prepared both at equilibrium or
out of it. For example, one can suddenly change the temperature of the system
to the critical one and allow it to relax towards equilibrium. In addition,
one can slowly change the external field for a system located in the critical
region. It should also be noted that in practice expression (1) is replaced by
the dependence on the lattice size $L$ instead of the infinite correlation
length as: $\tau\propto L^{z}$. A comprehensive review of different
nonequilibrium universality classes for lattice systems is given in Ref. [9].
For almost forty years, a large number of works presented in which lattice
calculations of the dynamic critical exponent $z$ of the $n$-component model A
in two and three dimensions were performed [10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37]. The corresponding numerical estimates of $z$ are collected in Table 1 and
in Table 2 for the $n=1$ and $n>1$ cases, respectively. Each of the works
deserves a separate consideration, but this is not the purpose of the present
paper. Let us consider only a few of them trying to cover different methods,
for the rest of them we give only numerical estimates of the exponent $z$.

As was said above, a significant number of works were dedicated to the
$\bm{\mathscr{A}}^{3,1}$ model. In Ref. [10], the authors investigated the
equilibrium dynamic critical behaviour of the one-component model on a simple
cubic lattice, where by measuring the decay of the time-displaced correlation
function of magnetization at the critical temperature, they estimated the
critical exponent $z$ as $2.03(4)$. The authors emphasized that such an
estimate agrees with the limitations imposed by the renormalization group
approach: $z-2>0$. In Ref. [12], the authors suddenly quenched the system from
an ordered state at $T=0$ to a critical one. For three dimensions by means of
both Glauber and Metropolis dynamics they found: $z=2.08(3)$. In Ref. [15],
the authors addressed to the so-called damage spreading technique which
consists in observing the dynamics and appropriate simultaneous changing of
boundary spin layers of almost identical systems. Taking into account
logarithmic corrections, the authors extracted the following estimates for $z$
in the $\bm{\mathscr{A}}^{3,1}$ model case: $2.05(4)$. In Ref. [16], by means
of the improved damage spreading method, the authors found the refined
exponents values for 2d and 3d cases as $2.172(6)$ and $2.032(4)$,
respectively. In Ref. [19], in order to accurately determine the critical
behaviour of the relaxation time, the authors introduced the variance-reducing
Monte Carlo algorithm. For the $\bm{\mathscr{A}}^{2,1}$ they found
$2.1665(12)$ for $z$ by means of the single-spin flip Markov dynamics
accompanied by a probability determination according to the heat-bath method.
In other works, the authors analyzed short time dynamics of the system [21].
It turned out that short time behaviour does not depend on the initial
condition. The authors managed to extract the value of the critical dynamic
exponent $z$ from observing relaxation to equilibrium of well-chosen
variables. The exponents were estimated as $2.16(3)$ and $2.075(35)$ for 2d
and 3d cases, respectively. In Ref. [23], the nonequilibrium relaxation was
studied, whereas the initial state is chosen as a completely ordered state at
the critical temperature. By computing the magnetization, energy per bond, and
three-spin correlation, the authors found the following estimate for the
$\bm{\mathscr{A}}^{2,1}$ model: $z=2.169(3)$. Finally, let us pay special
attention to the recent calculations [31] that partly prompted us to implement
the present work. There, the critical exponent $z$ for the
$\bm{\mathscr{A}}^{3,1}$ model was estimated as: $2.0245(15)$. This result was
obtained by considering the so-called improved Blume-Capel model on a simple
cubic lattice that allows authors to eliminate the leading corrections to
scaling. The work also noted that the difference in numerical estimates
obtained within previous MC simulations with the results given by, for
example, FT methods, which is not covered by the error bars, is due to
incorrect data processing, i.e., neglect of such scaling corrections.

The MC simulations for the $\bm{\mathscr{A}}^{3,3}$ model are usually
performed on a simple cubic lattice. In Refs. [32, 33], the authors addressed
to the equilibrium dynamic critical behaviour of the autocorrelation function
in order to analyze the dependence of the relaxation time on lattice size and
find the corresponding numerical estimate for the exponent $z$ as $1.96(6)$.
The number was found in good agreement with the result obtained by means of
the above mentioned short time MC simulations $1.976(9)$ in Ref. [34]. These
estimates, however, strongly contradict the renormalization group basis. This
statement was recently supported in [35], where the authors managed to obtain
the following value $2.035(4)$ by analyzing the evolution from various initial
states. In Ref. [37], the authors attacked the problem both in and out of
equilibrium states. There, by computing the autocorrelation time at
equilibrium, they extracted $z=2.033(5)$, while in the out-of-equilibrium
regime, they found the following estimate: $z=2.04(2)$.

As already mentioned, less attention was paid to the rest of the universality
classes. In particular, in Ref. [36], for the $\bm{\mathscr{A}}^{2,2}$ model
the authors used MC simulations to investigate the short time behavior of the
system dynamics starting from both ordered and disordered states. They
estimated the exponent $z$ as $2.04(1)$. In Ref. [38], the authors obtained
the whole set of estimates. Due to the relatively low accuracy, unfortunately,
they all were roughly estimated as $2$.

As for the FT calculations, there is also a vast variety of results [39].
However, from the very beginning, it should be noted that in contrast to
static problems, the degree of elaboration of their dynamic counterparts, due
to the advancement of perturbation theory (PT) to higher orders, leaves much
to be desired. Moreover, as is known, there are a number of different
renormalization group (RG) approaches. Let us review them consequentially.

First, the RG analysis can be performed in fixed spatial dimensionality. The
record-high result here is the four-loop analysis performed in Ref. [40]. For
$\bm{\mathscr{A}}^{2,1}$ and $\bm{\mathscr{A}}^{3,1}$ models for $z$ the
authors obtained $2.093$ and $2.017$, respectively. However, these numbers
were extracted using the most straightforward resummation technique – the
method of Padé approximants. Later, however, these estimates were shifted,
which was made possible by using the advanced resummation strategies [41]. The
improved scheme allowed the authors to extract the following numbers:
$2.0842(39)$ and $2.0237(55)$ for 2d and 3d cases, respectively.

Table 1: Numerical estimates of the dynamic critical exponent $z$ for the $\bm{\mathscr{A}}^{2,1}$ and $\bm{\mathscr{A}}^{3,1}$ models obtained by means of different theoretical and experimental approaches. Abbreviations: ED – equilibrium dynamic critical behaviour; SQtCT – sudden quench to critical temperature; DS – damage spreading; EVm – eigenvalue method; STD – short time dynamics; NED – nonequilibrium dynamics; IBC – improved Blume-Capel model; PR – Padé-resummed expansions; FSD – fixed spatial dimensionality; NP – nonperturbative; wor – without regulators; ri – different regulators. Subscript $f$ denotes "face centred cubic" lattice, while $b$ – body centred cubic one. Method |  Ref. |  Year |  2d |  3d
---|---|---|---|---
MC: ED |  [10] |  1987 |  - |  $2.03(4)$
MC: ED |  [11] |  1991 |  - |  $2.04(3)$
MC: SQtCT |  [12] |  1993 |  $2.21(3)$ |  $2.08(3)$
MC: SQtCT |  [13] |  1993 |  - |  $2.073(16)$
MC: SQtCT |  [14] |  1993 |  2.165(10) |  $2.073(16)$
MC: DS |  [15] |  1994 |  - |  $2.05(4)$
MC: DS |  [16] |  1995 |  $2.172(6)$ |  $2.032(4)$
MC: DS |  [17] |  1995 |  $2.1337(41)$ |  -
MC: DS |  [18] |  1995 |  $2.18(2)$ |  $2.04(1)$
MC: EVm |  [19] |  1996 |  $2.1665(12)$ |  -
MC: STD |  [21] |  1997 |  $2.16(3)$ |  $2.075(35)$
MC: DS |  [22] |  1997 |  $2.166(7)$ |  -
MC: NED |  [23] |  1997 |  $2.169(3)$ |  -
MC: STD |  [24] |  1999 |  - |  $2.042(6)$
MC: SQtCT |  [25] |  2000 |  $2.17$ |  -
MC: SQtCT |  [26] |  2000 |  - |  $2.055(10)$
MC: EVm |  [27] |  2000 |  $2.1667(5)$ |  -
MC: NED |  [28] |  2007 |  $2.16$ |  -
MC: SQtCT |  [29] |  2007 |  $2.165(15)$ |  $2.065(25)_{b}$
MC: SQtCT |  [29] |  2007 |  - |  $2.057(25)_{f}$
MC: IBC |  [30] |  2010 |  - |  $2.020(8)$
MC: IBC |  [31] |  2019 |  - |  $2.0245(15)$
HT: PR |  [42] |  1976 |  $2.125(1)$ |  -
HT: PR |  [43] |  1993 |  $2.183(5)$ |  -
RG: FSD |  [44] |  1995 |  $2.124$ |  $2.022$
RG: FSD |  [40] |  1997 |  $2.093$ |  $2.017$
RG: FSD |  [41] |  2006 |  $2.0842(39)$ |  $2.0237(55)$
RG: NP |  [45] |  2007 |  $2.16(1)$ |  $2.09(4)$
RG: NP |  [46] |  2015 |  - |  $2.025$
RG: NP(wor) |  [47] |  2017 |  $2.28$ |  $2.032$
RG: NP(r1) |  [47] |  2017 |  $2.16$ |  $2.024$
RG: NP(r2) |  [47] |  2017 |  $2.15$ |  $2.024$
RG: NP(r3) |  [47] |  2017 |  $2.14$ |  $2.023$
Exp: |  [48] |  2015 |  - |  $2.06$
Table 2: Numerical estimates of the dynamic critical exponent $z$ for the $\bm{\mathscr{A}}^{2,n}$ and $\bm{\mathscr{A}}^{3,n}$ models obtained by means of different theoretical approaches. Most cases relate to the planar ($n=2$) and Heisenberg ($n=3$) models. Abbreviations: INS – inelastic neutron-scattering. Method | Ref. | Year | Num. of. comp. | 2d | 3d
---|---|---|---|---|---
MC: STD | [36] | 2001 | $n=2$ | $2.04(1)$ |
RG: NP(wor) | [47] | 2019 | $n=2$ | - | $2.029$
RG: NP(r1) | [47] | 2019 | $n=2$ | - | $2.024$
RG: NP(r2) | [47] | 2019 | $n=2$ | - | $2.023$
MC: ED | [32] | 1990 | $n=3$ | - | $1.96(7)$
MC: ED | [33] | 1993 | $n=3$ | - | $1.96(6)$
MC: STD | [34] | 2006 | $n=3$ | - | $1.976(9)$
MC: NED | [35] | 2019 | $n=3$ | - | $2.035(4)$
MC: ED | [37] | 2019 | $n=3$ | - | $2.033(5)$
MC: NED | [37] | 2019 | $n=3$ | - | $2.04(2)$
RG: NP(wor) | [47] | 2019 | $n=3$ | - | $2.025$
RG: NP(r1) | [47] | 2019 | $n=3$ | - | $2.021$
RG: NP(r2) | [47] | 2019 | $n=3$ | - | $2.021$
Exp: ESR | [49] | 1980 | $n=3$ | - | $2.04(7)$
Exp: INS | [50] | 1984 | $n=3$ | - | $2.09(6)$

An alternative approach also applied to the problem is the so-called
nonperturbative or functional renormalization group (NPRG) [45, 46, 47]. In
Ref. [45], the authors obtained the following estimates for the two $2.16(1)$
and three $2.09(4)$ dimensions. A noticeably different number from the latter
for $\bm{\mathscr{A}}^{3,1}$ model was found in Ref. [46]: $2.025$. The
authors of Ref. [47] suggested the whole set of numerical estimates which
correspond to the application of different NPRG regulators and without them.
All numbers have been added to Table 1.

Consider now the field theory approach that revolutionized the critical
behavior theory by its appearance. The Wilson idea to take as a formal small
parameter the difference between the upper critical dimension, above which the
Landau theory is workable, and the physical value of the system spatial
dimension allowed theoreticians to find anomalous deviations of different
physical observables from their canonical values at criticality. Despite the
asymptotic nature of the $\varepsilon$ expansions, even taking into account
the first two or three terms is sufficient to give adequate numerical
estimates for the main mass of physically interesting quantities. In the
present work, the dynamic critical exponent $z$ is obtained within this
formalism. The two-loop result was found almost fifty years ago in Ref. [51].
Ten years later, three-loop results were found, which were record-high for a
long time [52]. The first attempt to calculate four-loop expansions was made
in Ref. [53] with subsequent resummation in Ref. [54]. Only ten years later,
however, the accuracy of four-loop calculations was notably improved [55]
using the new diagram reduction technique. This method can significantly
decrease the number of diagrams and make it possible to carry out calculations
in real time. Thanks to this technique, we are able to calculate the five-loop
contribution to the dynamic critical exponent $z$ within $d=4-\varepsilon$
dimension for an arbitrary number of order parameter components. In such a
high order of PT, the reduction in the number of diagrams turned out to be
crucial: from $1025$ to $201$. The expansion of this length in combination
with the novel resummation approach (we call it the free boundary condition
method, which was developed based on the boundary condition method suggested
by R. Guida and J. Zinn-Justin), makes it possible to extract highly accurate
estimates for $z$ for different values of spatial and spin dimensions. This
paper is an extended version of the recently published letter (ref. on our
future paper), where details of numerical calculations were omitted, and
numerical estimates were given for the most significant case –
$\bm{\mathscr{A}}^{3,1}$ model.

Before we start with specific calculations, let us review some experimental
results relevant to the problem. There is an extensive list of physical
systems that are described by the model A. For example, the recent results
obtained in Refs. [56, 57] have demonstrated that the phase transition into a
superfluid state is also described by the model A. In order to extract
critical behavior characteristics, one can address the measurement of thermal
transport properties. In particular, special attention of experimentalists is
attracted by thermal diffusivity. It is worth noting that this type of
measurement is challenging because, on the one hand, one needs to create
temperature gradients; on the other, it must be extremely small in order to
preserve the system at criticality. For instance, in Ref. [58], the authors
analyzed the behaviour of strongly anisotropic antiferromagnet FeF2 at
antiferromagnetic paramagnetic transition. The work contains debates on the
applicability of different dynamic models (A or C) to the system. The choice
in favor of the model C is only qualitative, justified only by weak damping of
energy, only with noticeable damping of the order parameter. In Ref. [48], the
magnetoelectric dynamics in the multiferroic chiral antiferromagnet MnWO4 was
analyzed. The observation of critical slowing down of magnetoelectric
fluctuations allowed the authors to conclude about the validity of the
theoretical description of this physical system within the model with
overdamped magnetic 3d-Ising order parameter, i.e., the
$\bm{\mathscr{A}}^{3,1}$ model. The authors of other works [59, 60] studied
the continuous phase transition of the AuAgZn2 alloy by coherent x-ray
scattering. Having quenched the alloy samples, they observed the motion of
interfaces between ordered domains. Using the critical system behaviour, they
also came to the conclusion that the critical dynamics of such a system
corresponds to the A model, although the number for the dynamic critical
exponent is $1.96(11)$. This number indicates that the agreement between
theory and experiments is only qualitative due to the error bar of
experimental numbers. Similar results were obtained in Ref. [61]. In Ref.
[62], different ferromagnets were investigated by means of electron-spin-
resonance and hyperfine interaction. However, the main purpose of the work was
to connect the behavior of the materials with one or other dynamic models. In
particular, the authors aimed to distinguish the physical systems described by
the J model, where the approximate value of $z$ is $2.5$, from systems where
the order parameter is not conserved, i.e., the A model of critical dynamics,
for which the exponent $z$ can be roughly estimated as $2$. There was no talk
of extracting some corrections as tenths or even hundredths to $z=2$. In
Ref.[49], the authors measured the spin-relaxation time for the isotropic
ferromagnet EuO above the critical temperature by means of the zero-field
electron-spin resonance. They estimated the dynamic exponent $z$ as $2.04(7)$.
The authors of different works aimed to analyze spin dynamics of EuS using
inelastic neutron-scattering technique [50]. For the critical exponent $z$
they obtained $2.09(6)$. In addition, some experiments by means of Mössbauer
spectroscopy were carried out on impurity systems, where the primary purpose
was to determine whether the addition of impurities can change the class of
universality or not [63].

The paper is organized as follows. In Sec. 2 the field-theoretical model and
its renormalization are described. Then, in Sec. 3 the features of the diagram
reduction technique are explained. Next, in Sec. 4, the RG expansions for
anomalous dimensions and critical exponents $z$ are found up to
$O(\varepsilon^{5})$ terms. In Sec. 5, the five-loop numerical estimates for
$z$ in case of different $\bm{\mathscr{A}}^{d,n}$ models are presented. At the
end, some conclusion will be drawn.

##  2 Description of the model

The nonrenormalized action of the model A of critical dynamics is defined by a
set of two non-renormalized $n$-component fields
$\phi_{0}\equiv\\{\psi_{0},\,\psi^{\prime}_{0}\\}$ and can be expressed in the
form [7]:

$S_{0}(\phi_{0})=\lambda_{0}\psi_{0}^{\prime}\psi_{0}^{\prime}+\psi_{0}^{\prime}[-\partial_{t}\psi_{0}+\lambda_{0}(\partial^{2}\psi_{0}-m_{0}\psi_{0}-\frac{1}{3!}g_{0}\psi_{0}^{3})]\,,$
(2)

where $\lambda_{0}$ is the Onsager coefficient and $g_{0}$ is the coupling
constant. The model A is multiplicatively renormalizable. The renormalized
action in the spatial dimensionality $d=4-\varepsilon$ reads as follows:

$S_{\text{R}}=Z_{1}\lambda\psi^{\prime}\psi^{\prime}+\psi^{\prime}[-Z_{2}\partial_{t}\psi+\lambda(Z_{3}\partial^{2}\psi-
Z_{4}m\psi-\frac{1}{3!}Z_{5}\mu^{\varepsilon}g\psi^{3})]\,,$ (3)

where the renormalized fields and parameters are expressed in terms of bare
ones as:

$\lambda_{0}=\lambda Z_{\lambda},\quad m_{0}=mZ_{m},\quad
g_{0}=g\mu^{\varepsilon}Z_{g},\quad\psi_{0}=\psi
Z_{\psi},\quad\psi_{0}^{\prime}=\psi^{\prime}Z_{\psi^{\prime}}\,.$ (4)

The renormalization constants $Z_{i}$ are associated with the renormalization
constants from (3) by the following relations:

$\displaystyle Z_{1}=Z_{\lambda}Z_{\psi^{\prime}}^{2}\,,\quad
Z_{2}=Z_{\psi^{\prime}}Z_{\psi}\,,\quad
Z_{3}=Z_{\psi^{\prime}}Z_{\lambda}Z_{\psi}\,,$ (5) $\displaystyle
Z_{4}=Z_{\psi^{\prime}}Z_{\lambda}Z_{m}Z_{\psi}\,,\quad
Z_{5}=Z_{\psi^{\prime}}Z_{\lambda}Z_{g}Z_{\psi}^{3}\,.$

Due to the fact that the renormalization constants $Z_{\psi}$, $Z_{m}$, and
$Z_{g}$ coincide with their static counterparts, the renormalization constants
of the $\phi^{4}$ model [7]

$Z_{\psi}=(Z_{\psi})_{st}\,,\quad Z_{m}=(Z_{m})_{st}\,,\quad
Z_{g}=(Z_{g})_{st}\,,$ (6)

and due to the existence of the relation
$Z_{\psi^{\prime}}Z_{\lambda}=Z_{\psi}$ [7], we can conclude that, in fact, we
are only interested in $Z_{\lambda}$, which is characteristic of this dynamic
model:

$Z_{\lambda}=Z_{1}Z_{\psi^{\prime}}^{-2}=Z_{1}^{-1}Z_{\psi}^{2}=Z_{2}^{-1}Z_{\psi}^{2}\,.$
(7)

For our purposes, it is convenient to calculate it using the renormalization
constant $Z_{1}$, which is determined from the diagrams of the one-irreducible
function
$\Gamma_{\psi^{\prime}\psi^{\prime}}=\langle\psi^{\prime}\psi^{\prime}\rangle_{\text{1-irr}}/(2\lambda)$
at zero external frequency $\omega$ with $m=0$. For the PT expansion of this
function, the charge $u=gS_{d}/(2\pi)^{d}$ is used, where
$S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the area of the $d$-dimensional sphere of
the unit radius. This expansion has the form:

$\Gamma_{\psi^{\prime}\psi^{\prime}}|_{\omega=0,m=0}=Z_{1}(1+u^{2}Z_{g}^{2}(\mu/p)^{2\varepsilon}A^{(2)}+u^{3}Z_{g}^{3}(\mu/p)^{3\varepsilon}A^{(3)}+...)=Z_{1}(1+\sum_{i=2}^{\infty}u^{i}Z_{g}^{i}(\mu/p)^{i\varepsilon}A^{(i)})\,.$
(8)

As the renormalization scheme we use the minimal subtraction one, in which the
counterterms subtract only the polar contributions of $\varepsilon$ from the
diagrams:

$Z_{1}=1+\sum_{n\geq 1}Z_{1}^{(i)}(u)\,\varepsilon^{-i}\,.$ (9)

The coefficients $Z_{1}^{(i)}(u)$ are calculated as series in $u$ based on the
requirements of absence of poles in $\varepsilon$ for
$\Gamma_{\psi^{\prime}\psi^{\prime}}$ from (8). The general theory states that
the coefficients at the highest poles in (8) are expressed in terms of the
coefficients at the first pole in such a way that the possible pole
contributions at logarithms $(\log(\mu/p))^{i}$ should be reduced. This is a
check for calculations of the coefficients $A^{(i)}$ from Eq. (8).

##  3 Diagrammatic technique: reduction scheme

The coefficients $A^{(i)}$ were calculated using the diagram reduction
procedure proposed in [55]. Let us consider it using the five-loop diagrams
with topology ”$e112|33|e44|44||$” as an example (according to the Nickel
notation). There are two different graphs for model A:

$\displaystyle\frac{1}{2}\begin{matrix}\includegraphics[angle={0},width=86.72267pt]{5loop_var1.eps}\end{matrix}+\frac{1}{2}\begin{matrix}\includegraphics[angle={0},width=86.72267pt]{5loop_var2.eps}\end{matrix}$
(10)

Here the lines in the momentum-time ($k,\,t$) representation are matched with
the following expressions:

$\displaystyle\begin{matrix}\includegraphics[angle={0},width=43.36464pt]{1.eps}\end{matrix}=\langle\psi(t_{1})\psi(t_{2})\rangle=\frac{1}{k^{2}}\exp^{-\lambda
k^{2}|t_{1}-t_{2}|}\,,$ (11)
$\displaystyle\begin{matrix}\includegraphics[angle={0},width=43.36464pt]{2.eps}\end{matrix}=\langle\psi(t_{1})\psi^{\prime}(t_{2})\rangle=\theta(t_{1}-t_{2})\exp^{-\lambda
k^{2}(t_{1}-t_{2})}\,.$ (12)

The multiplier in front of the diagram is determined by its symmetry and the
additional factor $1/2$ in the definition of the one-irreducible Green
function
$\Gamma_{\psi^{\prime}\psi^{\prime}}=\langle\psi^{\prime}\psi^{\prime}\rangle_{\text{1-irr}}/2\lambda$.
The diagrams from Eq. (10) are matched with a set of “time versions” which
correspond to all possible time ordering of vertices, taking into account
$\theta$-functions within the lines $\langle\psi\psi^{\prime}\rangle$. In each
variant, the integration over time is easily performed. It turns out that the
result of integration over time is the sum of contributions that already
contain only integration over momentum space. These contributions can also be
represented in the form of diagrams, in the case of diagrams from Eq. (10) the
number of these diagrams is equal to 15. In Ref. [55], a method was proposed
for grouping different time options of diagrams, significantly reducing their
total number and simplifying the integrand expressions. Thus, the sum of the
time versions of the diagrams from Eq. (10) can be reduced to the sum of the
following four unique diagrams:

$\displaystyle\frac{1}{16}\begin{matrix}\includegraphics[angle={0},width=65.04034pt]{5loop_1_1.eps}\end{matrix}+\frac{1}{8}\begin{matrix}\includegraphics[angle={0},width=65.04034pt]{5loop_1_2.eps}\end{matrix}+\frac{1}{4}\begin{matrix}\includegraphics[angle={0},width=65.04034pt]{5loop_1_3.eps}\end{matrix}+\frac{1}{2}\begin{matrix}\includegraphics[angle={0},width=65.04034pt]{5loop_1_4.eps}\end{matrix}$
(13)

In the reduced diagrams from Eq. (13), the lines with momentum $k$ without
strikethrough are associated with the factor $1/k^{2}$, lines with
strikethrough – $1$, and the dashed lines – fractions $1/S$, where $S$ is the
sum of squares of the line momentum that the dashed line crosses. Thus, in the
considered example, the number of integrals necessary for the calculation was
reduced from 15 (the number of time versions) to 4. The factors near the
reduced diagrams from Eq. (13) correspond to the standard symmetry factors
taking into account the additional factor $1/2$ in the definition of the one-
irreducible Green function
$\Gamma_{\psi^{\prime}\psi^{\prime}}=\langle\psi^{\prime}\psi^{\prime}\rangle_{\text{1-irr}}/(2\lambda)$.
In Ref. [55], the authors describe a general way of obtaining the reduced
structures directly from the corresponding static ones. In Table
LABEL:tab:topology, we present all the necessary static diagrams up to the
five-loop approximation, show the Nickel index for each of the static
diagrams, the factors $a(n)$ which allow one to move from the one-component
theory to an arbitrary number of order components, the number of the
corresponding dynamic diagrams, the number of their corresponding time
versions, and the number of diagrams after reduction.

Table 3: Topology of all necessary static diagrams up to 5-loop contributions. Abbreviations: NDD – number of dynamic diagrams; NTV – number of the corresponding time versions; NRD – number of reduced diagrams. Notation for topology of diagrams is chosen in accordance with the Nickel notation. |  |  |  |  |  |
---|---|---|---|---|---|---
Num. | Graph | Topology | $a(n)$ | NDD | NTV | NRD
2 loop
1 |  | e111|e| | $(2+n)/3$ | 1 | 2 | 1
Total: | 1 | 2 | 1
3 loop
1 |  | e112|22|e| | $(8+n)(2+n)/27$ | 1 | 2 | 2
Total: | 1 | 2 | 2
4 loop
1 |  | e112|33|e33|| | $(n^{2}+6n+20)(2+n)/81$ | 2 | 10 | 3
2 |  | e112|e3|333|| | $(2+n)^{2}/9$ | 2 | 10 | 3
3 |  | e123|e23|33|| | $(22+5n)(2+n)/81$ | 3 | 18 | 4
4 |  | e112|23|33|e| | $(22+5n)(2+n)/81$ | 5 | 28 | 7
Total: | 12 | 66 | 17
5 loop
1 |  | e112|33|e44|44|| | $(n^{3}+8n^{2}+24n+48)(2+n)/243$ | 2 | 15 | 4
2 |  | e112|33|444|e4|| | $(8+n)(2+n)^{2}/81$ | 3 | 45 | 9
3 |  | e123|e23|44|44|| | $(2+n)(3n^{2}+22n+56)/243$ | 3 | 30 | 8
4 |  | e112|e3|344|44|| | $(8+n)(2+n)^{2}/81$ | 4 | 30 | 8
5 |  | e123|234|34|4|e| | $(22+5n)(2+n)/81$ | 6 | 90 | 10
6 |  | e112|34|e34|44|| | $(2+n)(3n^{2}+22n+56)/243$ | 9 | 90 | 17
7 |  | e123|e24|34|44|| | $(n^{2}+20n+60)(2+n)/243$ | 12 | 120 | 23
8 |  | e112|34|334|4|e| | $(n^{2}+20n+60)(2+n)/243$ | 21 | 270 | 51
9 |  | e112|23|e4|444|| | $(8+n)(2+n)^{2}/81$ | 7 | 65 | 19
10 |  | e112|23|44|e44|| | $(2+n)(3n^{2}+22n+56)/243$ | 13 | 150 | 29
11 |  | e112|23|34|44|e| | $(n^{2}+20n+60)(2+n)/243$ | 12 | 120 | 23
Total: | 92 | 1025 | 201

All diagrams are calculated using the Sector Decomposition method [64] which
was firstly adapted for critical dynamics in Ref. [55]. The numerically
calculated diagrams are presented in Table LABEL:tab:all_diagrams_exp in A.
The sum of the contributions of the diagrams, taking into account the
combinatorial factors and the coefficients $a(n)$ shown in Table
LABEL:tab:topology, determines the coefficients $A^{(i)}$ from expansion (8).
For clarity, we demonstrate the values of these coefficients in the one-
component case $(n=1)$. They read as follows:

$\displaystyle
A^{(2)}=1/8\,\ln{(4/3)}\,\varepsilon^{-1}+0.0416702(4)+0.0647910(6)\,\varepsilon+0.0446045(9)\,\varepsilon^{2},$
$\displaystyle
A^{(3)}=1/8\,\ln{(4/3)}\,\varepsilon^{-2}+0.1368518(6)\,\varepsilon^{-1}+0.2709195(13)+0.285296(3)\varepsilon,$
$\displaystyle
A^{(4)}=27/64\,\ln{(4/3)}\,\varepsilon^{-3}+0.347358(4)\,\varepsilon^{-2}+0.867242(12)\varepsilon^{-1}+1.29757(3),$
$\displaystyle
A^{(5)}=27/40\,\ln{(4/3)}\,\varepsilon^{-4}+0.77317(4)\,\varepsilon^{-3}+2.41421(13)\,\varepsilon^{-2}+4.8300(4)\,\varepsilon^{-1}.$

Substituting the coefficients found $A^{(i)}$ into Eq. (8) as well as using
the expression known from static consideration:

$\displaystyle
Z_{g}=1+u\frac{8+n}{6\varepsilon}+u^{2}\left(\frac{(8+n)^{2}}{36\varepsilon^{2}}-\frac{14+3n}{24\varepsilon}\right)+$
(14)
$\displaystyle\quad+u^{3}\left(\frac{(8+n)^{3}}{216\varepsilon^{3}}-\frac{7(112+38n+3n^{2})}{432\varepsilon^{2}}+\frac{2960+922n+33n^{2}}{5184\varepsilon}+\frac{\zeta(3)(22+5n)}{54\varepsilon}\right)+O(u^{4}),$

we can find the renormalization constant $Z_{1}$ from the requirement of the
elimination of the poles in Eq. (8). For further calculations, we need only
the coefficient $Z_{1}^{(1)}$ at the first-order pole in $\varepsilon$ in
$Z_{1}$. It can be written in the following form:

$\displaystyle Z_{1}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{n+2}{3}\left[a_{20}u^{2}+(a_{30}+a_{31}\,n)u^{3}+(a_{40}+a_{41}\,n+a_{42}\,n^{2})u^{4}+\right.$
(15)
$\displaystyle\left.+(a_{50}+a_{51}\,n+a_{52}\,n^{2}+a_{53}\,n^{3})u^{5}\right]\,,$

where the coefficients $a_{ij}$ are defined in Table 4.

Table 4: Numerical values of coefficients from Eq. (15). It is worth noting that the values of the coefficients at higher powers in $n$ ($a_{42}$, $a_{53}$) within each of the corresponding order are known exactly from the analysis of the asymptotics when $n$ tends to infinity [65]. Coef. | Value | Coef. | Value
---|---|---|---
$a_{20}$ | $-0.0359602590565$ | $a_{42}$ | $0.0001671773304$
$a_{30}$ | $0.0105255461549$ | $a_{50}$ | $0.04926(29)$
$a_{31}$ | $0.0013156932693$ | $a_{51}$ | $0.01357(17)$
$a_{40}$ | $-0.0210863(5)$ | $a_{52}$ | $0.000386(16)$
$a_{41}$ | $-0.00399345(12)$ | $a_{53}$ | $-0.0000082188688$

Let us note also that the two- and three-loop contributions to $Z_{1}^{(1)}$
are known analytically:

$\displaystyle Z_{1}^{(1)}$ $\displaystyle=$
$\displaystyle-u^{2}\frac{(2+n)}{24}\ln{(4/3)}-u^{3}\frac{(2+n)(8+n)}{27}$
(16)
$\displaystyle\frac{1}{96}\left(\pi^{2}-8\,\operatorname{Li}_{2}(1/4)+\ln{(4/3)}\left(-6-21\ln{(3})+13\ln{(4)}\right)\right)+O(u^{4})\quad,$

where $\operatorname{Li}_{2}(x)$ is the dilogarithm. These coefficients were
first obtained in the Ref. [65] and [66], respectively.

##  4 Anomalous dimension and critical exponent z

The RG functions are determined by the renormalization constants. In
particular, $Z_{1}$ determines the anomalous dimension $\gamma_{1}$ as:

$\gamma_{1}=\beta(u)\,\partial_{u}\log Z_{1}\,.$ (17)

The expansion for the $\beta$-function is currently known up to the sixth [67]
and seventh [68] loop accuracy. We do not need its explicit form since the
above-mentioned connection between the coefficients at higher poles and
coefficients at the first pole – $Z_{1}^{(1)}$ – together with Eq. (9) allow
us to present $\gamma_{1}$ in a simpler manner:

$\gamma_{1}=-u\partial_{u}Z_{1}^{(1)}.$ (18)

The dynamic critical exponent $z$ is expressed, in turn, in terms of the
$\gamma_{1}^{*}\equiv\gamma_{1}(u_{*})$ function $\gamma_{1}(u)$ at the fixed
point $u_{*}$ as well as the Fisher exponent $\eta$ by means of the following
relation [7]:

$z=2+\gamma_{1}^{*}-\eta\,.$ (19)

The value of the fixed point $u^{*}$ determined by zero of the
$\beta$-function ($\beta(u^{*})=0$), and $\eta$ can be taken with the
necessary accuracy from the $\phi^{4}$ field theory:

$\displaystyle u^{*}$ $\displaystyle=$
$\displaystyle\varepsilon\frac{3}{8+n}+\varepsilon^{2}\frac{9(14+3n)}{(8+n)^{3}}+\varepsilon^{3}\frac{1}{(n+8)^{5}}(-5912.23-2022.99n-175.120n^{2}$
(20) $\displaystyle-$ $\displaystyle
12.375n^{3})+\varepsilon^{4}\frac{1}{(n+8)^{7}}(771918.1+411076.2n+90771.7n^{2}+10753.7n^{3}$
$\displaystyle+$ $\displaystyle 438.956n^{4}-0.3125n^{5}),\,$
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\varepsilon^{2}\frac{2+n}{2(8+n)^{2}}-\varepsilon^{3}\frac{(2+n)(-272-56n+n^{2})}{8(8+n)^{4}}-$
(21) $\displaystyle-$
$\displaystyle\varepsilon^{4}\frac{2+n}{(8+n)^{6}}(1096.74+334.330n+36.9984n^{2}+7.1875n^{3}+0.15625n^{4})+$
$\displaystyle+$
$\displaystyle\varepsilon^{5}\frac{(2+n)}{(8+n)^{8}}(222336.8+117827.7n+25396.1n^{2}+2572.12n^{3}+13.8027n^{4}$
$\displaystyle-$ $\displaystyle 4.58183n^{5}+0.0486946n^{6})\,.$

As a result, the $\varepsilon$ expansion for the dynamic exponent $z$ reads:

$\displaystyle z(\varepsilon,n)=2+$
$\displaystyle+\varepsilon^{2}\frac{(2+n)}{2(8+n)^{2}}(-1+6\ln{(4/3)})+\varepsilon^{3}\frac{(2+n)}{8(8+n)^{4}}(162.462+31.9024n-1.27352n^{2})$
$\displaystyle+\varepsilon^{4}\frac{(2+n)}{32(8+n)^{6}}(-23752.4(16)-6929.0(8)n-770.28(13)n^{2}-170.470(6)n^{3}-4.24424n^{4})$
$\displaystyle+\varepsilon^{5}\frac{(2+n)}{128(8+n)^{8}}\left(1.986(25)10^{7}+1.038(23)10^{7}n+2.15(8)10^{6}n^{2}+1.92(13)10^{5}n^{3}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-3.8(10)10^{3}n^{4}-626(29)n^{5}+7.50053n^{6}\right),$
(22)

where in the one-component case ($n=1$) it is

$\displaystyle
z(\varepsilon)=2+0.01344616\,\varepsilon^{2}+0.01103628\,\varepsilon^{3}-0.0055791(4)\,\varepsilon^{4}+0.01773(31)\,\varepsilon^{5}\,.$
(23)

A more detailed notation of the coefficients entering equation (4) is
presented in B. In the next section we apply different resummation techniques
in order to extract proper numerical estimates for the critical dynamic
exponent $z$ in the case of different values of order parameter
dimensionality.

##  5 Borel resummation

During the last fifty years a large variety of resummation methods were
implemented in order to get relevant numerical estimates for critical
exponents [69, 70, 67]. The most popular among them use the Borel
transformation with consequent analytical continuation by means of conformal
mapping of a special form [71, 72, 68, 73, 70]. This method also has a lot of
realizations. Basically, they differ from each other in the strategy of
choosing resummation parameters, as well as in different ways of using some
additional information about the system behaviour and its observables.

In this work, we address to two specific implementations [67, 74]. Both of
them use the knowledge of asymptotic behavior of RG expansions of the A model
in higher orders. Let us briefly describe how one can take advantage of this
fact. Suppose we have the asymptotic expansion

$A(\varepsilon)=\sum_{k=0}^{\infty}A_{k}\varepsilon^{k},$ (24)

for which only the first $N+1$ coefficients $A_{k}$ are known as well as its
high-order asymptotic behavior (HOA):

$A_{k}\xrightarrow[k\rightarrow\infty]{}c\,k!k^{b_{0}}(-a)^{k}\,.$ (25)

For the model A these parameters were previously calculated in Ref. [75]:
$a=3/(n+8)$ and $b_{0}=3+n/2$. For the truncated part of expansion (24) one
can construct the Borel transformation in the following form:

$\displaystyle
A^{N}(\varepsilon)=\sum\limits_{k=0}^{N}A_{k}\varepsilon^{k}=\int\limits_{0}^{\infty}dt\,e^{-t}t^{b}\sum\limits_{k=0}^{N}B^{b}_{k}\,(\varepsilon
t)^{k}=\int\limits_{0}^{\infty}dt\,e^{-t}t^{b}F^{N}_{b}(\varepsilon t)\,,\quad
B^{b}_{k}=\dfrac{(-1)^{k}A_{k}}{\Gamma(k+b+1)},$ (26)

where the function $F_{b}^{N}(x)$ is called the Borel image. In practice,
within the RG approach any information about observables, say, critical
exponents, can be extracted only from the truncated expansion
$A^{N}(\varepsilon)$. From a formal point of view, however, if the entire
series were known, i.e. the whole function $A(\varepsilon)$, then the
corresponding Borel image $F_{b}(x)\equiv F_{b}^{\infty}(x)$ would possess a
finite radius of convergence ($1/a$), which formally does not allow the
integration in Eq. (26) to be performed. In order to overcome this problem, it
is necessary to find an analytic continuation for the function $F_{b}(x)$
beyond the circle of convergence and here some ambiguity exists. For example,
it can be achieved by analytical continuation of $F_{b}(x)$ via a conformal
mapping of a special form:

$w(x)=\dfrac{\sqrt{1+ax}-1}{\sqrt{1+ax}+1},\quad x(w)=\dfrac{4w}{a(1-w)^{2}}.$
(27)

where the first expression transforms the real semiaxis $x\in(0,\infty)$ into
$w\in(0,1)$ making the integration region within a circle of convergence. Here
we assume that all singularities of the Borel image lie on the cut
$(-\infty,−1/a]$, which are mapped now onto the unit circle $|w|=1$. In terms
of $w$, the expansion $A(\varepsilon)$ can be rewritten as:

$A(\varepsilon)=\int_{0}^{\infty}dte^{-t}t^{b}\left(\frac{\varepsilon
t}{w(\varepsilon t)}\right)^{\lambda}Q(w(\varepsilon
t))=\int_{0}^{\infty}dte^{-t}t^{b}G_{b,{\lambda}}(\varepsilon t).$ (28)

As was said above, we are limited only by a finite number of terms for
$A(\varepsilon)$; in this situation the approximation for the function $Q(x)$
can be taken as:

$Q_{N}(x)=\sum_{n=0}^{N}q_{k}x^{k}\,.$ (29)

The first $N+1$ coefficients $q_{k}$ are determined from the condition of
coincidence of expression (28) reexpanded in terms of $\varepsilon$ up no
$N+1$ terms with the corresponding coefficients $A_{k}$ from (26). The HOA of
these coefficients has the same form in Eq. (25) with the same $b_{0}$ if one
chooses $b=b_{0}+3/2$. The exponent $\lambda$ in Eq. (28) determines the
strong coupling asymptotics:

$A(\varepsilon)\sim\varepsilon^{\lambda},\quad\varepsilon\rightarrow\infty.$
(30)

From this point, the above two methods differ in further modifications
designed to improve the convergence of estimates.

###  5.1 KP17

In Ref. [67], the following idea was suggested. Apart from the two presented
parameters – $b$ and $\lambda$ – the third parameter can be introduced to
improve the results even further. It enters in the following homographic
transformation:

$\varepsilon(\varepsilon^{\prime})\rightarrow\dfrac{\varepsilon^{\prime}}{1+q\varepsilon^{\prime}}\>,\quad\varepsilon^{\prime}(\varepsilon)\rightarrow\dfrac{\varepsilon}{1-q\varepsilon}\;,$
(31)

which allows one to reexpand the original $\varepsilon$ expansion as a series
in $\varepsilon^{\prime}$. Further, all the above performed steps will be
taken with the new $\varepsilon^{\prime}$ expansion. Thus, the final
expression with which the observables can be estimated is:

$A(\varepsilon)\approx
A_{b,\lambda,q}^{N}(\varepsilon)=\int_{0}^{\infty}dte^{-t}t^{b}G_{b,\lambda,q}^{N}\left(\dfrac{\varepsilon
t}{1-q\varepsilon}\right).$ (32)

It is obvious that the estimates obtained by formula (32), in fact depend on
the values of the parameters $b$, $\lambda$, and $q$ and the order of PT
($N$). In Ref. [67], the strategy of their optimal determination was
suggested. In short, they are defined in such a way that in the vicinity of
the selected values the estimate of the physical observable was the least
sensitive to their variation (for details see Ref. [67]).

###  5.2 Free boundary conditions

In Ref. [74], in order to improve the convergence, the authors suggested a
technique called the free boundary condition which is inspired by the
resummation method suggested earlier for calculating static exponents in Ref.
[76]. The authors of [76] proposed to use the known values of critical
exponents in spatial dimensions (further called the boundary dimensions)
different from the one (further called the dimension of consideration) in
which these quantities are desired. However, the information of this kind is
not always known. It is this situation that is observed in the case of the
critical dynamic exponent $z$ for an arbitrary value of the order parameter
dimensionality $n$. In order to overcome this problem, the authors of Ref.
[74] used the trick according to which the value of the critical exponent in
boundary spatial dimension should be considered as a variable quantity, a
certain value of which is chosen based on the best convergence of estimates of
the critical exponent in the considered dimension with growing PT order.
Formally, this procedure can be described as follows. Let us have an initial
series $A(\varepsilon)$, then the first thing we should do is its reexpanding
taking into account the assumed (actually varied) value $A_{\textup{b}}$ in
the boundary spatial dimension (4-$\varepsilon_{\textup{b}}$):

$A(\varepsilon)=A_{\textup{b}}+(\varepsilon_{\textup{b}}-\varepsilon)B_{\textup{b}}(\varepsilon),\quad
B_{\textup{b}}=\sum\limits_{k=0}^{\infty}B_{\textup{b},k}\varepsilon^{k},$
(33)

where by construction the HOAs from (25) for the initial series
$A(\varepsilon)$ and the reexpanded one $B_{\textup{b}}(\varepsilon)$ coincide
with each other.

Figure 1: The behavior of estimates for the critical exponent $z$ for $n=1$
based on the order of PT for different values of $\lambda$.

As noted earlier, in practice we have only a limited number of terms – $N+1$ –
for the series $A(\varepsilon)$, which leads to a limited number (the same
$N+1$) of known terms for the new expansion $B_{\textup{b}}(\varepsilon)$. Now
for this truncated expansion $B_{\textup{b}}^{N}(\varepsilon)$ we repeat the
general steps with the Borel transformation and conformal mapping (28) but
without using the homographic transformation from KP17 (31). Thus, the
following expression can be considered as the final formula for calculation of
the exponent $z$ within the present section:

$A(\varepsilon)\approx
A_{\textup{b},\lambda,b}^{N}(\varepsilon)=A_{\textup{b}}+(\varepsilon_{\textup{b}}-\varepsilon)B_{\textup{b},\lambda,b}^{N}(\varepsilon).$
(34)

Here, in contrast to KP17, the value of $b$ is fixed and chosen according to
the HOA for the A model; only the value of the critical dynamic exponent in
the boundary spatial dimension $z_{\textup{b}}$ as well as the strong coupling
parameter $\lambda$ will be variable.

Figure 2: The behaviour of numerical estimates for critical exponent $z$ based
on the order of PT for different values of $\lambda$ in the case of taking
into account the boundary condition $z_{2d}=2.13$ for $n=1$ (upper left),
$z_{2d}=2.14$ for $n=2$ (upper right), $z_{2d}=2.12$ for $n=5$ (lower left),
and $z_{2d}=2.07$ for $n=10$ (lower right).

For clarity, we have demonstrated the behavior of $z$ with an increase in the
number of loops when the value of $z_{\textup{b}}$ at the boundary dimension
($d=2$) is not fixed (Fig. 1) and taken equal to different numbers for
different values of the order parameter dimension $n$ (Fig. 2). Each line
corresponds to its own value for the $\lambda$ parameter. The dependencies in
Fig. 2 show the trend of convergence of the calculation results to a specific
value as the number of the considered terms increases, similar behavior takes
place in the case of the exactly solvable zero-dimensional model [71]. It
should be also noted that this trend is much more pronounced than in the case
of Fig. 1, where parameter $z_{\textup{b}}$ is not used. As the optimal values
of $\lambda$ and $z_{b}$, one should choose those that provide the best
convergence to a given limit value based on a smaller number of terms of PT
under consideration. It is clear from Fig. 2 that in order to determine these
optimal values, one can require that the slope of the straight line would
decrease in the fastest way with the growth of PT. In order to write down some
formal convergence criterion, we can address to the equation of the straight
line passing through the points of four- ($z^{(4)}$) and five-loop ($z^{(5)}$)
approximations as:

$\displaystyle z(l)=a(\lambda)(l-5)+b(\lambda)\,,$ (35)

where $l$ is number of loops, and $a(\lambda)=z^{(5)}-z^{(4)}$ and
$b(\lambda)=z^{(5)}$. Hence, in order to achieve the highest density of the
straight lines in (35), we demand the least sensitivity of the slope of the
straight line $z(l)$ to the $\lambda$ variation that, in turn, compels the
derivatives $\partial_{\lambda}a$, $\partial_{\lambda}b$, and
$\partial_{\lambda}^{2}b$ to take their minimum values. In this regard, it is
natural to consider as a criterion of convergence the requirement of the
minimum value of the following quantity:

$Q=\sqrt{\left(\partial_{\lambda}a\right)^{2}+\left(\partial_{\lambda}b\right)^{2}+\left(\partial_{\lambda}^{2}b\right)^{2}}.$
(36)

Table 5:  Numerical estimates of the critical dynamic exponent $z$ in the case of different spatial dimensions $d$ and values of order parameter dimensionality $n$. The estimates are found by means of the KP17 and FBC resummation procedures of the expansion (4). At each number the set of optimal values of the resummation parameters is presented. The abbreviation WA denotes the weighted average, which is constructed based on the five loop KP17 and FBC results and is taken as the final estimate for each $\bm{\mathscr{A}}^{d,n}$ model. $n$ | Dim. | Free b.c. | KP17 | W. A.
---|---|---|---|---
$-1$ | d=2 | $2.06(2)_{[1.78,0.00038]}$ | $2.08(2)_{[8.0,2.46,0.22]}$ | $2.070(14)$
d=3 | $2.0123(10)_{[1.74,0.00034]}$ | $2.0134(9)_{[8.0,2.46,0.22]}$ | $2.0127(8)$
$0$ | d=2 | $2.11(3)_{[2.48,0.00051]}$ | $2.13(3)_{[8.25,2.62,0.22]}$ | $2.12(2)$
d=3 | $2.020(2)_{[2.48,0.00051]}$ | $2.021(2)_{[8.5,2.42,0.24]}$ | $2.0205(11)$
$1$ | d=2 | $2.13(4)_{[2.49,0.00052]}$ | $2.15(3)_{[9.25,2.42,0.24]}$ | $2.14(2)$
d=3 | $2.023(2)_{[2.49,0.00052]}$ | $2.0239(14)_{[9.25,2.42,0.24]}$ | $2.0236(8)$
$2$ | d=2 | $2.14(4)_{[2.49,0.00047]}$ | $2.16(3)_{[9.5,2.66,0.22]}$ | $2.15(2)$
d=3 | $2.024(3)_{[2.49,0.00047]}$ | $2.0249(13)_{[10.0,2.46,0.22]}$ | $2.0246(10)$
$3$ | d=2 | $2.13(3)_{[2.49,0.00039]22}$ | $2.15(2)_{[10.0,2.68,0.2]}$ | $2.145(15)$
d=3 | $2.024(2)_{[2.49,0.00039]}$ | $2.0247(12)_{[10.75,2.44,0.2]}$ | $2.0244(8)$
$4$ | d=2 | $2.13(3)_{[2.48,0.00030]}$ | $2.13(3)_{[10.5,2.62,0.16]}$ | $2.130(15)$
d=3 | $2.024(2)_{[2.48,0.00030]}$ | $2.0237(10)_{[11.0,2.5,0.16]}$ | $2.0238(7)$
$5$ | d=2 | $2.12(2)_{[2.45,0.00021]}$ | $2.12(2)_{[10.5,2.64,0.12]}$ | $2.120(10)$
d=3 | $2.0223(14)_{[2.45,0.00021]}$ | $2.0225(8)_{[10.75,2.5,0.12]}$ | $2.0224(5)$
$6$ | d=2 | $2.104(12)_{[2.39,0.00013]}$ | $2.107(13)_{[10.5,2.46,0.08]}$ | $2.105(6)$
d=3 | $2.0210(10)_{[2.39,0.00013]}$ | $2.0218(13)_{[2.75,1.0,0.42]}$ | $2.0213(7)$
$7$ | d=2 | $2.094(7)_{[2.31,0.00006]}$ | $2.0946(10)_{[9.75,2.7,0.02]}$ | $2.0945(9)$
d=3 | $2.0197(5)_{[2.31,0.00006]}$ | $2.0202(8)_{[2.5,1.0,0.34]}$ | $2.0199(4)$
$8$ | d=2 | $2.085(3)_{[1.90,0.000008]}$ | $2.085(9)_{[8.25,2.34,0.0]}$ | $2.085(2)$
d=3 | $2.0185(2)_{[1.90,0.000008]}$ | $2.0187(5)_{[2.5,1.2,0.22]}$ | $2.0186(2)$
$9$ | d=2 | $2.077(3)_{[1.48,0.00004]}$ | $2.079(5)_{[2.25,1.32,0.12]}$ | $2.078(2)$
d=3 | $2.0174(2)_{[1.48,0.00004]}$ | $2.0174(2)_{[2.25,1.32,0.12]}$ | $2.01740(10)$
$10$ | d=2 | $2.070(6)_{[1.28,0.00005]}$ | $2.068(3)_{[2.0,1.54,0.0]}$ | $2.069(2)$
d=3 | $2.0163(5)_{[1.28,0.00005]}$ | $2.0162(2)_{[1.5,1.38,0.02]}$ | $2.01622(13)$

Thus, the minimum of $Q$ from (36) will be found in the case of different
spatial dimensions and the number of order parameter components. In the insets
in Fig. 2 we demonstrate the typical behaviour of $Q$ in the vicinity of its
minimal value as a projection to the ($z_{\textup{b}},\lambda$)-surface, which
corresponds to the best convergence of estimates for the 3d when the boundary
dimension is $d=2$ in case of different values of order parameter
dimensionality.

##  6 Numerical results

Figure 3: The behavior of estimates for the critical exponent $z$ for 2d and
3d cases with increasing order parameter dimensionality $n$ calculated on the
basis of the FBC and KP17 methods.

Following two resummation strategies described in the previous section, based
on (4) we extract 2d and 3d numerical estimates for the dynamic critical
exponent $z$ for different $n$. These numbers are presented in Table 5. The
error of each of the estimates, in addition to the internal error of the
particular resummation procedure, takes into account the error of the
coefficients of the 4th and 5th PT orders for the $\varepsilon$ expansion of
the exponent $z$, i.e. the estimates of the critical exponent $z$ were also
calculated for the series with shifted values of the coefficients, and the
obtained "forks" were taken into account. As a final numerical estimates for
$z$ in the case of specific $n$ and d, we have resorted to the weighted
average (WA) in accordance with the procedure described in Ref. [77], where
the inverse error bars were taken as weights. For greater clarity, in Fig. 3
we depicted the trends of the behavior of the critical exponent $z$ with an
increase in the value of the order parameter dimension $n$.

As we can see, all the numbers obtained using the two resummation techniques
are in complete agreement with each other for all values of the order
parameter dimension $n$ taking into account the corresponding error bars. Note
also that in the case of three dimensions, the agreement between each of the
methods is much better than in the case of two dimensions and the amplitude of
errors is lower. This, of course, is not surprising, due to large values of
the expansion parameter ($\varepsilon=2$). In three dimensions FBC gives much
more conservative estimates of errors in contrast to KP17. As we can see in
both dimensions, the maximum value of the exponent is reached in the case of
the two-component A model. Further decrease is estimates can be easily
explained, at least due to the general damping of the expansion coefficients
with an increase in the dimension of the order parameter.

##  7 Summary

Thus, in this work, we have calculated the five-loop RG expansions for the
$n$-component A model of critical dynamics within the Minimal Subtraction
scheme. The possibility of obtaining expansions of this length is dictated by
the fact that the authors managed to decrease the number of actually relevant
diagrams using the diagram reduction technique and to apply the effective
methods for calculating each diagram by means of the Sector Decomposition
technique that had previously been widely used in critical statics. To obtain
the high-precision numerical estimates for the critical exponent $z$, that in
fact has the prime of physical importance, we applied various resummation
techniques for asymptotic series, which were proposed by the authors of this
work. Due to the possibility to change the number of the order parameter
components, as well as to analyze the critical dynamic exponent $z$ in various
spatial dimensions, in this work we have given an exhaustive description of
the behavior of the critical dynamic exponent of the model A which describes a
wide range of physical systems. For example, for the most interesting physical
values of the order parameter dimension ($n=1$), our three-dimensional
estimate ($2.0236(8)$) is in good agreement with the most accurate results
that have been obtained using alternative approaches($2.0245(15)$ [31] and
$2.0237(55)$ [41]).

## Acknowledgment

We would like to thank R.Guida for the helpful discussion and sharing his
notes. The work of A.K. was supported by Grant of the Russian Science
Foundation No 21-72-00108. The work of M.H. was supported by the Ministry of
Education, Science, Research and Sport of the Slovak Republic(VEGA Grant No.
1/0535/21). We are grateful to the Joint Institute for Nuclear Research for
allowing us to use their supercomputer “Govorun”.

## Appendix

##  Appendix A Diagrams for $Z_{1}$ up to five-loop contributions

In Table LABEL:tab:all_diagrams_exp we present numerical values for all
diagrams of the renormalization constant $Z_{1}$ up to five-loop contributions
as part of the Laurent expansion in terms of $\varepsilon$. The coefficients
of expansions are given analytically where it is possible, otherwise numerical
values are suggested.

Table 6: Numerical values of all diagrams for renormalization constant $Z_{1}$ up to the five loops in $\varepsilon$. The way of description of diagram topology is chosen in accordance with Nickel notation. N. | Topology | Numerical results
---|---|---
2 loop
1 | e111|e| | $1/8\ln{(4/3)}\varepsilon^{-1}+0.041670207(4)+0.0647910(6)\varepsilon+0.0446045(9)\varepsilon$
3 loop
1 | e112|22|e| | $1/4\ln{(4/3)}\varepsilon^{-2}+0.1368518(6)\varepsilon^{-1}+0.27091943(4)+0.285296(3)\varepsilon$
4 loop
1 | e112|33|e33|| | $9/64\ln{(4/3)}\varepsilon^{-3}+0.1070793(11)\varepsilon^{-2}+0.256086(3)\varepsilon^{-1}+0.358251(6)$
2 | e112|e3|333|| | $-0.000852(3)\varepsilon^{-2}+-0.005031(10)\varepsilon^{-1}+-0.017487(15)$
3 | e123|e23|33|| | $3/32\ln{(4/3)}\varepsilon^{-}3+0.0848713(9)\varepsilon^{-2}+0.216134(3)\varepsilon^{-1}+0.377071(5)$
4 | e112|23|33|e| | $3/16\ln{(4/3)}\varepsilon^{-3}+0.1562589(12)\varepsilon^{-2}+0.400053(4)\varepsilon^{-1}+0.579729(8)$
5 loop
1 | e112|33|e44|44|| | $3/40\,\ln{(4/3)}\,\varepsilon^{-4}+0.073163(4)\varepsilon^{-3}+0.204788(16)\varepsilon^{-2}+0.35477(5)\varepsilon^{-1}$
2 | e112|33|444|e4|| | $-0.000666(12)\varepsilon^{-3}-0.00451(4)\varepsilon^{-2}-0.01797(12)\varepsilon^{-1}$
3 | e123|e23|44|44|| | $1/40\,\ln{(4/3)}\,\varepsilon^{-4}+0.031582(3)\varepsilon^{-3}+0.098616(9)\varepsilon^{-2}$
|  | $+0.23615(3)\varepsilon^{-1}$
4 | e112|e3|344|44|| | $-0.00133(3)\varepsilon^{-3}-0.01160(9)\varepsilon^{-2}-0.0524(3)\varepsilon^{-1}$
5 | e123|234|34|4|e| | $0.0172897(9)\varepsilon^{-2}+0.076055(4)\varepsilon^{-1}$
6 | e112|34|e34|44|| | $1/10\,\ln{(4/3)}\,\varepsilon^{-4}+0.111931(5)\varepsilon^{-3}+0.33142(2)\varepsilon^{-2}+0.67233(6)\varepsilon^{-1}$
7 | e123|e24|34|44|| | $1/20\,\ln{(4/3)}\,\varepsilon^{-4}+0.070358(4)\varepsilon^{-3}+0.23961(2)\varepsilon^{-2}+0.61230(6)\varepsilon^{-1}$
8 | e112|34|334|4|e| | $3/20\,\ln{(4/3)}\,\varepsilon^{-4}+0.182275(8)\varepsilon^{-3}+0.58732(3)\varepsilon^{-2}+1.21353(9)\varepsilon^{-1}$
9 | e112|23|e4|444|| | $-0.00196(2)\varepsilon^{-3}-0.01250(7)\varepsilon^{-2}-0.0491(3)\varepsilon^{-1}$
10 | e112|23|44|e44|| | $7/40\,\ln{(4/3)}\,\varepsilon^{-4}+0.188694(8)\varepsilon^{-3}+0.57368(3)\varepsilon^{-2}+1.03326(9)\varepsilon^{-1}$
11 | e112|23|34|44|e| | $1/10\,\ln{(4/3)}\,\varepsilon^{-4}+0.119119(6)\varepsilon^{-3}+0.39009(2)\varepsilon^{-2}+0.75114(8)\varepsilon^{-1}$

##  Appendix B Five-loop $\varepsilon$ expansion for critical exponent $z$

In this Appendix we present the five-loop $\varepsilon$ expansion for the
critical dynamic exponent $z$ for an arbitrary value of the order parameter
dimension $n$. In terms of the coefficients $b_{ij}$ from Table
LABEL:tab:coefficients_b the expansion can be written as:

$\displaystyle z(\varepsilon,n)=2$ $\displaystyle+$
$\displaystyle\varepsilon^{2}\frac{(2+n)}{2(8+n)^{2}}(b_{20})+\varepsilon^{3}\frac{(2+n)}{8(8+n)^{4}}(b_{30}+b_{31}n+b_{32}n^{2})$
(37) $\displaystyle+$
$\displaystyle\varepsilon^{4}\frac{(2+n)}{32(8+n)^{6}}(b_{40}+b_{41}n+b_{42}n^{2}+b_{43}n^{3}+b_{44}n^{4})+$
$\displaystyle+$
$\displaystyle\varepsilon^{5}\frac{(2+n)}{128(8+n)^{8}}\left(b_{50}+b_{51}n+b_{52}n^{2}+b_{53}n^{3}+b_{54}n^{4}+b_{55}n^{5}+b_{56}n^{6}\right).\qquad$

Table 7: Analytical values of coefficients $b_{ij}$ entering Eq. (37) in terms of variables $a_{ij}$ from Table 4. Coef. | Value
---|---
$b_{20}$ | $-1+6\ln{(4/3)}$
$b_{30}$ | $-272-16128\,a20-13824\,a_{30}$
$b_{31}$ | $-56-3456\,a_{20}-1728\,a_{30}-13824\,a_{31}$
$b_{32}$ | $1-1728\,a_{31}$
$b_{40}$ | $-46144-2227200\,a_{20}-6967296\,a_{30}-3538944\,a_{40}+67584\zeta(3)+3244032\,a_{20}\zeta(3)$
$b_{41}$ | $-17920-918528\,a_{20}-2363904\,a_{30}-6967296\,a_{31}-884736\,a_{40}-3538944\,a_{41}$
| $+23808\zeta(3)+1142784\,a_{20}\zeta(3)$
$b_{42}$ | $-1124-83328\,a_{20}-186624\,a_{30}-2363904\,a_{31}-55296\,a_{40}-884736\,a_{41}$
| $-3538944\,a_{42}+1920\zeta(3)+92160\,a_{20}\zeta(3)$
$b_{43}$ | $230+6336\,a_{20}-186624\,a_{31}-55296\,a_{41}-884736\,a_{42}$
$b_{44}$ | $5-55296\,a_{42}$
$b_{50}$ | $-5655552-214384640\,a_{20}-1547403264\,a_{30}-2378170368\,a_{40}-849346560\,a_{50}$
|
$+(720896\pi^{4})/5+(23068672\,a_{20}\pi^{4})/5+21921792\zeta(3)+1436024832\,a_{20}\zeta(3)$
| $-60948480\zeta(5)+1401421824\,a_{30}\zeta(3)-2925527040\,a_{20}\zeta(5)$
$b_{51}$ | $-2912768-127770624\,a_{20}-841052160\,a_{30}-1547403264\,a_{31}-1104150528\,a_{40}$
| $-2378170368\,a_{41}-849346560\,a_{51}+(434176\pi^{4})/5$
|
$+(13893632\,a_{20}\pi^{4})/5+11530240\zeta(3)+793313280\,a_{20}\zeta(3)+1401421824\,a_{31}\zeta(3)$
|
$-33259520\zeta(5)-1596456960\,a_{20}\zeta(5)-318504960\,a_{50}+668860416\,a_{30}\zeta(3)$
$b_{52}$ | $-(787544/3)-22469632\,a_{20}-143824896\,a_{30}-841052160\,a_{31}-164560896\,a_{40}$
| $-1104150528\,a_{41}-39813120\,a_{50}-318504960\,a_{51}-849346560\,a_{52}$
|
$+(95232\pi^{4})/5+(3047424\,a_{20}\pi^{4})/5+1244160\zeta(3)+111329280\,a_{20}\zeta(3)$
|
$+101523456\,a_{30}\zeta(3)+668860416\,a_{31}\zeta(3)-6113280\zeta(5)-293437440\,a_{20}\zeta(5)$
| $-2378170368\,a_{42}$
$b_{53}$ | $121472+1018880\,a_{20}-5121792\,a_{30}-143824896\,a_{31}-7962624\,a_{40}$
|
$-164560896\,a_{41}-1104150528\,a_{42}-1658880\,a_{50}-39813120\,a_{51}-318504960\,a_{52}$
| $-849346560\,a_{53}+(9088\pi^{4})/5-137984\zeta(3)-1867776a_{20}\zeta(3)$
|
$+4976640\,a_{30}\zeta(3)+101523456\,a_{31}\zeta(3)-445440\zeta(5)+(290816\,a_{20}\pi^{4})/5$
| $-21381120\,a_{20}\zeta(5)$
$b_{54}$ | $27620+569856\,a_{20}+342144\,a_{30}-5121792\,a_{31}-7962624\,a_{41}$
|
$-164560896\,a_{42}-1658880\,a_{51}-318504960\,a_{53}+64\pi^{4}+2048\,a_{20}\pi^{4}$
|
$-20800\zeta(3)-774144a_{20}\zeta(3)+4976640a_{31}\zeta(3)-10240\zeta(5)-39813120\,a_{52}$
| $-491520a_{20}\zeta(5)$
$b_{55}$ | $2798/3+640\,a_{20}+342144\,a_{31}-7962624\,a_{42}-1658880\,a_{52}$
| $-39813120\,a_{53}-288\zeta(3)$
$b_{56}$ | $13-1658880a_{53}-16\zeta(3)$

Using the coefficients from Table LABEL:tab:coefficients_b we come to the
equations (4) for $z$ for an arbitrary value of $n$.

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