On the new universality class in structurally disordered $n$-vector model with
long-range interactions

Dmytro Shapoval, 1,2 Maxym Dudka, 1,2 Yurij Holovatch 1,2,3

1 Institute of Condensed Matter Physics, National Academy of Sciences of
Ukraine, UA – 79011 Lviv, Ukraine

2 ${\mathbb{L}}^{4}$ Collaboration & Doctoral College for the Statistical
Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, Europe

3 Centre for Fluid and Complex Systems, Coventry University, Coventry, CV1
5FB, United Kingdom

We study a stability border of a region where nontrivial critical behaviour of
an $n$-vector model with long-range power-law decaying interactions is induced
by the presence of a structural disorder (e.g. weak quenched dilution). This
border is given by the marginal dimension of the order parameter $n_{c}$
dependent on space dimension, $d$, and a control parameter of the interaction
decay, $\sigma$, below which the model belongs to the new dilution-induced
universality class. Exploiting the Harris criterion and recent field-
theoretical renormalization group results for the pure model with long-range
interactions we get $n_{c}$ as a three loop $\epsilon=2\sigma-d$-expansion. We
provide numerical values for $n_{c}$ applying series resummation methods. Our
results show that not only the Ising systems ($n=1$) can belong to the new
disorder-induced long-range universality class at $d=2$ and $d=3$.

Keywords: Long-range interaction, quenched disorder, renormalization group,
marginal dimension

##  1 Introduction

Year 2022 marks the 110th birth anniversary of Oleksandr (a.k.a. A.S.)
Davydov, an outstanding physicist, known for his seminal contributions in the
fields of solid state theory, nuclear physics and biophysics, for many years
he served as a director of the Bogolyubov Institute for Theoretical Physics in
Kyiv. The authors of this paper who studied physics also from O. Davydov’s
books [1, 2, 3] consider as a great honour to contribute to the Festschrift
prepared on this occasion. In our paper we use the perturbative field
theoretical renormalization approach refined by the resummation of asymptotic
series expansions to study universal features of criticality. Although such
problems were beyond the focus of attention of O. Davydov, the concepts called
for their analysis: Symmetry, Space dimension, Range of interaction belong to
the central ones in physics. In the paper we show how their interplay defines
universal features of one of the key models currently used to understand
quantitatively and to describe qualitatively the critical behaviour in
condensed matter and beyond. Therefore, conceptually the results presented in
this paper are related to those discussed in O. Davydov’s seminal works. For
this reason we have chosen to present these results here.

Since inter-particle forces in various physical, chemical, and biological
systems are often of a long-range nature, models with long-range interaction
attract much attention. They have found their applications in studies of
gravitational, dipolar, cold Coulomb systems, problems in plasma, atomic and
nuclear physics, hydrodynamics and geophysical fluid mechanics (see [4, 5, 6]
and reference therein). Systems with long-range interactions possess
properties that differ from those with short-range interactions. To give an
example, even weak long-range interactions effectively modify the critical
properties and may induce the long-range order in one-dimensional systems [5,
6].

In this paper we will discuss possible changes in the critical behaviour of a
many-particle system caused by mutual effects of long-range interactions and
structural disorder. To this end, we will consider the – now standard –
$n$-vector spin model with the Hamiltonian:

${\cal H}=-\frac{1}{2}\sum_{{\bf x},{\bf x}^{\prime}}J(|{\bf x}-{\bf
x}^{\prime}|)\vec{S}_{\bf x}\vec{S}_{{\bf x}^{\prime}}\,$ (1)

that describes a system of classical $n$-component vectors (’spins’)
$\vec{S}_{\bf x}=(S^{1}_{\bf x},S^{2}_{\bf x},\dots,S^{n}_{\bf x},)$ located
at sites ${\bf x}$ of a $d$-dimensional lattice and interacting via the
distance-dependent potential $J(x)$. The r.h.s of Eq. (1) contains a scalar
product of spins and the sums over ${\bf x},{\bf x^{\prime}}$ span all lattice
sites. Influence of long-range interactions on the critical behavior is
usually exemplified by the power-law decaying interaction:

$J(x)\sim x^{-d-\sigma},$ (2)

where $\sigma>0$ is a control parameter of the interaction decay.

As we discuss in more details below, in the case of a regular (non-disordered)
lattices, the critical behaviour of the model (1) is governed by the triple of
parameters $(d,n,\sigma)$: depending on their values, the model may manifest a
low-temperature long-range order that emerges as a second-order phase
transition. As long as the model Hamiltonian (1) is formulated in terms of
elementary magnets – ‘spins’, the long-range ordered phase is usually
associated with an emergence of spontaneous magnetization. We will use such
magnetic terminology too, however let us note that the model itself as well as
our discussion without the loss of generality concern much more wide range of
types of ordering [7, 8] in physics and beyond. The transition to the ordered
‘magnetic’ phase is characterised by certain universal (i.e. independent on
specific system details) features. It is said, that it belongs to certain
universality class. Systems, that belong to the same universality class share
the values of critical exponents, amplitude ratios, scaling functions. Our
goal in this paper is to show how these universal features are changed if
instead of a regular lattice structure, one considers the disordered one.
Disorder in the lattice structure may be imposed e.g. by dilution, when a part
of the lattice sites in (1) are not occupied by spins. Such situation mimics
randomness and non-regularities that are so often met in nature and attract
much interest in modern theory of critical phenomena (see e.g. [8] and
references therein.) To quantify our analysis, we will calculate the marginal
dimension $n_{c}(\sigma)$: for given space dimension $d$ it discriminates
between different universality classes. The rest of the paper is organized as
follows. In the next Section we give a short review of the results present so
far, in Section 3 we describe the field-theoretical renormalization group
picture of the critical behaviour for the long-range interacting $n$-vector
model with disorder. The results for $n_{c}$ are given in Section 4 and we
summarize our study in the last Section 5. Some lengthy expressions are given
in the Appendix.

##  2 Review

In this section we briefly review some results relevant for our analysis. To
proceed further, we explain terminology, used throughout the paper. The
$n$-vector model (1) with short-range interactions – since the corresponding
free energy is invariant with respect to rotations in the $n$-dimensional
magnetization space it is also called the $O(n)$-symmetric model – manifests
the second order phase transition for the lattice space dimension $d>d_{lc}$.
The lower critical dimension $d_{lc}=1$ for the discrete (Ising) case $n=1$
whereas $d_{lc}=2$ for $n>1$. Critical exponents and other universal
properties of the short-range $n$-vector model depend on $n$ and $d$ in a non-
trivial way in the region $d_{lc}<d\leq d_{uc}$. It is said that they belong
to the short-range universality class. For $d$ larger than the upper critical
dimension $d_{uc}=4$ the model is governed by the mean-field exponents, see
[9] for more details. Introducing the long-range interaction (2) drastically
changes the picture of the critical behaviour of the $n$-vector model (1).
Calculations performed for the three-dimensional spherical model [10] (it
corresponds to the $n$-vector model at $d=3,n=\infty$) show that for
$\sigma>2$ the critical properties are governed by the short-range critical
exponents, while for $\sigma<2$ one has two regimes depending on the value of
$\sigma$: with mean-field critical exponents and with the $\sigma$-dependent
ones. One-dimensional Ising model ($d=1,n=1$) with interaction (2) was proven
to have phase transition to the long-range-ordered phase at non-zero
temperature [11]. Field-theoretical renormalization group (RG) analysis of the
long-range interacting $n$-vector model gives three universality classes in
dependence on $\sigma$ [12]: (i) the mean-field critical behavior for
$\sigma\leq d/2$, (ii) the short-range universality class for $\sigma\geq 2$
with critical exponents coinciding with those of the model with short-range
interactions, (iii) the long-range universality class for $d/2<\sigma<2$,
where critical exponents depend on $\sigma$. Later it was established that the
actual border between the short-range and the long-range universality classes
lies at $\sigma=2-\eta_{SR}$ [13, 14] rather than at $\sigma=2$, $\eta_{SR}$
is the pair correlation function critical exponent of the short-range model.
Such picture was corroborated by other approaches including non-perturbative
variant of RG (NPRG) [15], Monte Carlo simulations [16] and conformal
bootstrap [17] (for other references and discussion see the review [18]).

Another factor discussed in this paper is the structural disorder and its
impact on the critical behaviour. The influence of structural inhomogeneity on
the universal properties of physical systems continues to be a hot research
topic both for academic and practical reasons, since almost all materials are
characterized by a certain degree of disorder in their structure. Structural
inhomogeneities in magnetic systems are of different nature, which in turn may
lead to different changes in critical behavior. In the case of strong
structural disorder, randomness is accompanied by frustration and percolation
effects and often leads to absence of the long-range magnetic order. The case
of weak structural disorder is not that obvious. Here we focus specifically on
presence of the weak quenched disorder in lattice structure, which may be
implemented into model (1) via dilution by point-like uncorrelated (or short-
range correlated) quenched non-magnetic inhmogeneities. Its relevance for the
critical behaviour is given by the Harris criterion [19]. The criterion states
that the structural disorder (quenched dilution) leads to a new universality
class of the magnetic phase transition only if the heat capacity critical
exponent of the undiluted (pure) system is positive, $\alpha_{p}>0$, i.e. if
the heat capacity of the pure system diverges at the critical point.
Correspondingly, the disorder is irrelevant if $\alpha_{p}<0$. For the short-
range $n$-vector model at $d=3$ $\alpha_{p}>0$ for $n=1$ (Ising model),
whereas $\alpha_{p}<0$ for $n\geq 2$, therefore due to the Harris criterion
the diluted $n$-vector model at $n\geq 2$ shares the universal properties of
its undiluted counterpart [20, 21]. Unlike the short-range $n$-vector model,
the long-range one manifests the new universality class induced by dilution
also in the region $n\geq 2$ as it was shown in the RG study of Ref. [22]
within two-loop approximation. This result was also corroborated by the low-
temperature RG [23]. However, the estimates of the regions of values
$(d,n,\sigma)$, where the new disordered long-range universality class governs
the critical behaviour were not satisfactory [22]. The perturbative RG
expansions having zero radius of convergence, additional resummation
procedures have to be applied in order to get reliable numerical data on their
basis [7]. Especially it concerns the Ising model ($n=1$), where degeneracy of
the RG equations (similarly as in the short-range case [20, 21]) makes the
expansion parameter to be $\sqrt{\epsilon}$ [24]. Reliable results for the
last case were obtained within a massive renormalization scheme with
resummation of the RG functions [25].

A remarkable feature of the Harris criterion is that it allows to forecast
structural-disorder-induced changes in the universality class of a pure system
without explicit calculation of the RG functions for the diluted one. Indeed,
if the structural disorder changes the universality class only when the heat
capacity of the pure system diverges (i.e. when $\alpha_{p}>0)$, one can use
the condition $\alpha_{p}=0$ as an equation to define parameters
($d,\sigma,n$) that discriminate between different universality classes. For
given space dimension $d$ such equation defines the so-called marginal order-
parameter dimension $n_{c}(\sigma)$. Similar to critical exponents and
critical amplitude ratios, the marginal dimensions are universal quantities,
reachable in experiments and numerical simulations and are the subject of
intensive studies [26, 27, 28, 29, 30, 31]. In the next sections we will
calculate the marginal dimension $n_{c}(\sigma)$ for the diluted long-range
$n$-vector model at space dimensions $d=2$ and $d=3$. This marginal dimension
line in the $(n,\,\sigma)$ parametric plane defines a border of stability
between the pure long-range and the disordered long-range universality
classes. To this end we will make use of the recent three-loop RG results for
the critical exponents of the pure $n$-vector model with long-range power-law
decaying interactions [32].

##  3 Field-theoretical renormalization group description

The progress achieved in quantitative understanding and qualitative
description of critical phenomena to a large extend is due to application of
RG methods [7]. In the field-theoretical RG approach, the critical properties
of the $n$-vector model (1) with the power-law decaying long-range
interactions (2) are described by analysing the effective Hamiltonian [33]:

${\mathcal{H}}=\int
d^{d}x\left\\{\frac{1}{2}\left((\nabla^{\sigma/2}{\vec{\varphi}})^{2}+r_{0}{\vec{\varphi}}^{2}\right)+\frac{u_{0}}{4!}({\vec{\varphi}}^{2})^{2}\right\\},$
(3)

where $\vec{\varphi}=\vec{\varphi}({\bf x})=\\{\varphi_{1}({\bf
x}),\dots,\varphi_{n}({\bf x})\\}$ is an $n$-component field, $u_{0}$ is
unrenormalized coupling, $r_{0}$ defines the temperature distance to the
critical point, and $\nabla^{\sigma/2}$ is a symbolic notation for the
fractional derivative. The last is defined via its action in the momentum
space and leads to the propagator term $q^{\sigma}$ rather than $q^{2}$ as in
the case of short-range interactions. Power counting gives the upper critical
dimension $d_{uc}=2\sigma$, which at $\sigma=2$ coincides with the traditional
$d_{uc}=4$. The effective Hamiltonian (3) is relevant for the case
$0<\sigma<2-\eta_{SR}$. To study crossover to the short-range case $\sigma\to
2$ one should include the traditional term $(\nabla{\vec{\varphi}})^{2}$ into
(3).

In the field-theoretical RG approach, a critical point corresponds to a
reachable and stable fixed point (FP) of the RG transformation. It has been
found [12] that the nontrivial FP determining new long-range universality
class is stable for $d/2<\sigma$. Critical exponents within this universality
class were calculated in $\epsilon=2\sigma-d$-expansion up to order
$\epsilon^{2}$ [12, 33, 34] and up to order $1/n$ [35] in $1/n$-expansion.
Estimates for the critical exponents of three-dimensional systems were
obtained within the massive renormalization approach completed by resummation
in two-loop approximation [36]. The RG results in three-loop approximation
were obtained only recently [32]. Monte Carlo estimates for the critical
exponents in this universality class have been obtained only for the $n=1$
Ising case mainly in one and two space dimensions (for collection of
references see [32]). Only a few Monte Carlo results are available for the
three-dimensional case [37].

The presence of uncorrelated non–magnetic impurities (weak quenched structural
disorder) is usually modeled by fluctuations of the local phase transition
temperature [38]. Introducing $\phi=\phi({\bf x})$ as the field of local
critical temperature fluctuations, one obtains the following effective
Hamiltonian for the structurally disordered system:

${\mathcal{H}}=\int
d^{d}x\left\\{\frac{1}{2}\left((\nabla^{\sigma/2}{\vec{\varphi}})^{2}+\left(r_{0}+\phi\right){\vec{\varphi}}^{2}\right)+\frac{u_{0}}{4!}({\vec{\varphi}}^{2})^{2}\right\\}\,,$
(4)

where the random variable $\phi$ has a Gaussian distribution with zero mean
and a correlator containing the second coupling $v_{0}$:

$\left<\phi({\bf x})\right>=0,\qquad\left<\phi({\bf x})\phi({\bf
x^{\prime}})\right>=v_{0}\delta({\bf x}-\bf{x^{\prime}})\,.$ (5)

The angular brackets $\langle\dots\rangle$ indicate an average over the random
variable $\phi$ distribution.

The RG picture for the disordered long-range model (4) is similar to that for
its short-range analog [20, 21]. In the parametric space of couplings
$(u,\,v)$ the critical properties of the model (4) are governed by four FPs
$(u^{*},\,v^{*})$ in dependence on values $\epsilon=2\sigma-d$ and $n$ [22,
24]: Gaussian FP $(u^{*}=0,\,v^{*}=0)$, unphysical FP
$(u^{*}=0,\,v^{*}\not=0)$, Heisenberg long-range or pure long-range FP
$(u^{*}=0,\,v^{*}\not=0)$ and disordered long-range FP
$(u^{*}\not=0,\,v^{*}\not=0)$. The Gaussian FP is always unstable below
critical $d_{u}=2\sigma$ ($\epsilon>0$) while the unphysical FP is always
stable in this case, however, it is not accessible from initial conditions
appropriate for the model described by (4) and (5). For $\epsilon>0$ and
$n>n_{c}$, the long-range Heisenberg FP is stable and the disordered one is
unstable, while for $n<n_{c}$ the FPs swap their stability. Therefore for
$n>n_{c}$ the universal critical exponents of the diluted model (4) coincide
with those of the model (3). For $n<n_{c}$ model (4) belongs to the new
disorder-induced long-range universality class. The border between these two
regimes is determined by the marginal dimension $n_{c}(d,\sigma)$.

So far, the value of $n_{c}$ for the long-range $n$-vector model was known
within the two-loop approximation [39], with the result

$n_{c}=4-4[\psi(1)-2\psi(\sigma/2)+\psi(\sigma)]\epsilon+O(\epsilon^{2})\,,$
(6)

where $\psi(x)$ is the digamma function. The asymptotic nature of this series
together with its shortness made it difficult to get reliable numerical
estimates on its basis. In the next section we will proceed getting the next
order of the $\epsilon$-expansion and delivering numerical estimates for
$n_{c}(\sigma)$ at certain space dimensions with the help of resummation
procedures.

##  4 Calculation of the marginal dimension

As it has been mentioned above, the marginal dimension $n_{c}$ of a weakly
diluted $n$-vector model with power-law decaying interactions can be obtained
on the base of the critical exponents for the undiluted model. As a
consequence of the Harris criterion, the master equation for determining
$n_{c}$ reads:

${\alpha_{p}}(n_{c},d,\sigma)=0.$ (7)

The treatment of Eq. (7) by means of the field-theoretical RG approach can be
performed in various schemes. Here we will exploit the results of dimensional
regularisation with the minimal subtraction [40] allowing to obtain quantities
of interest by familiar $\epsilon$-expansion with $\epsilon=2\sigma-d$ in our
case. To get $n_{c}(d,\sigma)$ we use the hyperscaling relation
${\alpha_{p}}=2-d{\nu_{p}}$ and $\epsilon$-expansion for the critical exponent
$\nu_{p}$ of the $n$-vector model with long-range interactions, which is known
in the three-loop approximation [32] in the following form:

$\displaystyle{\nu_{p}}^{-1}$
$\displaystyle=\,\sigma\,-\,\frac{(n+2)}{n+8}\,\epsilon\,+\,\frac{(n+2)(7n+20)\alpha_{S,0}}{(n+8)^{3}}\,\epsilon^{2}+\,\frac{(n+2)\epsilon^{3}}{(n+8)^{5}}\bigg{[}-4(5n+22)(7n+20)\alpha_{S,0}^{2}$
$\displaystyle\quad+(n+8)^{2}(7n+20)(\alpha_{S,1}-2\alpha_{D,1}\alpha_{S,0})+(n+8)\Big{(}{-}8(n-1)\alpha_{T}+2(n^{2}+20n+60)\alpha_{U}$
$\displaystyle\quad+2(n^{2}+24n+56)\alpha_{I_{1}}+(5n^{2}+28n+48)\alpha_{I_{2}}+(5n+22)\alpha_{I_{4}}\Big{)}\bigg{]}\,+\,{O}(\epsilon^{4})\,,$
(8)

where $\alpha_{K}$ with $K=\\{S,D,I_{1},I_{2},I_{4},T,U\\}$ are expressed in
terms of the loop integrals (for details see [32]). $\alpha_{S,i}$ and
$\alpha_{D,i}$ are the coefficients at $\epsilon^{i}$ in the
$\epsilon$-expansion series of $\alpha_{S}$ and $\alpha_{D}$. Explicit
expressions for $\alpha_{K}$ are given in the Appendix. We get the following
expression for $n_{c}$:

$n_{c}(d,\sigma)=4+4\alpha_{S,0}\epsilon+(56\alpha_{I_{1}}+40\alpha_{I_{2}}+7\alpha_{I_{4}}-192\alpha_{D,1}\alpha_{S,0}-56\alpha_{S,0}^{2}+96\alpha_{S,1}-4\alpha_{T}+52\alpha_{U})\frac{\epsilon^{2}}{24}+O(\epsilon^{3})\,.$
(9)

Taking into account the explicit expression for $\alpha_{S,0}$ one can check
that up to the first order in $\epsilon$ Eq. (9) coincides with the two-loop
result (6).

Figure 1: Resummed values of $n_{c}(\sigma)$ obtained by the Padé-approximant
[1/1] (dashed lines) and Padé-Borel resummation (solid lines) for $d=2$ (a)
and $d=3$ (b). For the region of values $n$ and $\sigma$ above the lines, the
pure long-range universality class holds, while the new disorder long-range
universality class is induced for the values below the lines. Dots show
results that follow from the interpolation of the NPRG data of Ref. [15]. The
mean-field behaviour holds for $\sigma<d/2$ (regions separated by vertical
dashed lines, coloured online).

Formally, the numerical value of $n_{c}$ at fixed $d$ and $\sigma$ can be
calculated from the expansion (9) by using expressions for $\alpha_{K}$ from
the Appendix, recalling that $\epsilon=2\sigma-d$ and substituting the values
of $d$ and $\sigma$. However the $\epsilon$-expansions are known to be
asymptotic at best [7]. Therefore one has to apply special resummation
procedures to restore their convergence in order to get reliable numerical
estimates on their basis. Doing so, we start our analysis by representing
series (9) by means of the diagonal Padé approximant $[1/1](\epsilon)$ [41].
The result for $n_{c}$ as a function of $\sigma$ is shown by dashed lines in
Fig. 1 a,b for fixed $d=2$ and $d=3$, correspondingly. As it was noted in the
previous Section, in the region of $n$ and $\sigma$ above the lines the
critical behaviour of the diluted model is the same as for an undiluted model
with long-range interactions. For the values of $n$ and $\sigma$ below the
lines, the new disorder long-range universality class is induced. It is known
that in the short-range case the Padé approximants of the three-loop expansion
give values of $n_{c}$ that exceed the most accurate estimates [27]. It seems
to be also in the long-range case, since the data of the NPRG approach (shown
by black dots in Fig. 1) are located below the lines calculated via Padé
approximant.111These data were obtained from interpolation of the NPRG results
for critical exponents of the long-range $n$-vector model [15]. To enhance our
estimates we use also a more elaborated Padé-Borel resummation technique [42].
First, to weaken the factorial growth of the expansion coefficients we write
the Borel transform for (9) as:

$\sum_{k=0}^{2}n_{k}\epsilon^{k}\rightarrow\sum_{k=0}^{2}\frac{n_{k}\epsilon^{k}}{k!}\,.$
(10)

An analytical continuation of the Borel transform is achieved by representing
it in the form of the diagonal Padé approximant $[1/1]_{B}(\epsilon)$, where
subscript $B$ is used to distinguish from the Padé approximant of the original
series (9). Finally, the resummed function is obtained via an inverse Borel
transform:

$n_{c}^{res}(\epsilon)=\int_{0}^{\infty}dt{\rm e}^{-t}[1/1]_{B}(\epsilon
t)\,.$ (11)

Results that follow from the Padé-Borel resummation are presented in Fig. 1 by
solid lines for $d=2$ and $d=3$. For $d=3$ the Padé-Borel resummation leads to
lower values of $n_{c}(\sigma)$ as compared to those obtained form the $[1/1]$
Padé approximant. This is a right tendency, as is seen also from comparing our
results to the NPRG data, shown by dots in Fig. 1. As usual with the
perturbative expansions, the accuracy of the results decreases with an
increase of the expansion parameter, in our case it is $\epsilon=2\sigma-d$.
Therefore, our results are less accurate for $d=2$, where the expansion
parameter changes within $0\leq\epsilon\leq 2$ for $1\leq\sigma\leq 2$. But
even then the results of Padé-Borel resummation may serve as reliable
estimates up to the moderate values of $\sigma\simeq 1.5$. A remarkable
feature of the plots presented in Figs. 1 a,b is that for a certain range of
parameters $\sigma$ there are regions in the $n,\sigma$ plane that correspond
to integer values of $n=1,2,3$ and lie below the $n_{c}(\sigma)$ curve. This
means that the new disorder long-range universality class is induced in the
$n$-vector model not only for the Ising ($n=1$) but also for the XY ($n=2$)
and classical Heisenberg ($n=3$) cases.

##  5 Conclusions

In this study we were interested in the question how the critical behaviour of
a many-particle system is changed under the competing influence of two
factors: type of interaction and structural disorder? To this end, we have
considered the archetype model to describe criticality, an $n$-vector spin
model (1), and analysed changes in its critical behaviour provided the
interaction between spins is of a long-range nature (2) and an underlying
lattice structure is disordered. To be more specific, we considered the case
when the weak quenched structural disorder leads to fluctuations in the local
transition temperature (5). The literature available so far [22, 23] reported
that second order phase transition in such a model can belong to the new,
disorder induced long-range universality class. However, such a qualitative
conclusion has to be supported by quantitative estimates of the region of
model parameters where the new universality class can manifest.

To do so, we have calculated the marginal dimension $n_{c}(\sigma)$ of the
structurally-disordered long-range interacting $n$-vector model. For given
space dimension $d$ and interaction decay $\sigma$, the model with the order
parameter component number $n<n_{c}$ belongs to the new disorder-induced long-
range universality class. Based on the recent results for the critical
exponents of the pure long-range $n$-vector model [32], we used the Harris
criterion to calculate $n_{c}(\sigma)$ with the record three-loop accuracy,
Eq. (9). This enabled us to apply familiar resummation techniques to evaluate
numerical values of the marginal dimension, as shown in Figs.1 a,b for space
dimensions $d=2$ and $d=3$. Obtained results serve as a solid argument that
not only the Ising-like ($n=1$) systems, but also systems that are described
by the XY ($n=2$) and Heisenberg ($n=3$) models belong to the new universality
class for the moderate values of $\sigma$ at space dimesions $d=2$ and $d=3$.

## Acknowledgements

We thank the Editors, Yurij Naidyuk, Larissa Brizhik, and Oleksandr Kovalev,
for their invitation to contribute to the Festschrift in memory of Oleksandr
Davydov. This work was supported in part by the grant of the National Academy
of Sciences of Ukraine for research laboratories/groups of young scientists No
07/01-2022(4)) (D.S.). M.D. thanks members of LPTMC for their hospitality
during his academic visit in the Sorbonne University, where the part of this
work was done.

## Appendix

In this appendix, we list expressions for $\alpha_{K}$ as they were given in
Ref. [32]

$\displaystyle\alpha_{D}\,$ $\displaystyle=$
$\displaystyle\,1+\frac{\epsilon}{2}\big{[}\psi(1)-\psi(\tfrac{d}{2})\big{]}+\frac{\epsilon^{2}}{8}\left[\left(\psi(1)-\psi(\tfrac{d}{2})\right)^{2}+\psi_{1}(1)-\psi_{1}(\tfrac{d}{2})\right]\,,$
(12) $\displaystyle\alpha_{S}\,$ $\displaystyle=$
$\displaystyle\,2\psi(\tfrac{d}{4})-\psi(\tfrac{d}{2})-\psi(1)+\frac{\epsilon}{4}\Big{[}\left[2\psi(\tfrac{d}{4})-\psi(\tfrac{d}{2})-\psi(1)\right]\left[3\psi(1)-5\psi(\tfrac{d}{2})+2\psi(\tfrac{d}{4})\right]$
(14)
$\displaystyle+3\psi_{1}(1)+4\psi_{1}(\tfrac{d}{4})-7\psi_{1}(\tfrac{d}{2})-4J_{0}(\tfrac{d}{4})\Big{]}\,,$
$\displaystyle\alpha_{U}$ $\displaystyle=$
$\displaystyle\alpha_{I_{2}}=\,-\psi_{1}(1)-\psi_{1}(\tfrac{d}{4})+2\psi_{1}(\tfrac{d}{2})+J_{0}(\tfrac{d}{4})\,,$
(15) $\displaystyle\alpha_{T}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{2}\Big{[}2\psi(\tfrac{d}{4})-\psi(\tfrac{d}{2})-\psi(1)\Big{]}^{2}+\frac{1}{2}\psi_{1}(1)+\psi_{1}(\tfrac{d}{4})-\frac{3}{2}\psi_{1}(\tfrac{d}{2})-\,J_{0}(\tfrac{d}{4})\,,$
(16) $\displaystyle\alpha_{I_{1}}\,$ $\displaystyle=$
$\displaystyle\,\frac{3}{2}\left[2\psi(\tfrac{d}{4})-\psi(\tfrac{d}{2})-\psi(1)\right]^{2}+\frac{1}{2}\psi_{1}(1)-\frac{1}{2}\psi_{1}(\tfrac{d}{2})\,,$
(17) $\displaystyle\alpha_{I_{4}}\,$ $\displaystyle=$ $\displaystyle
6\,\frac{\,\Gamma(1+\tfrac{d}{4})^{3}\Gamma(-\tfrac{d}{4})}{\,\Gamma(\tfrac{d}{2})}\;\Big{[}\psi_{1}(1)-\psi_{1}(\tfrac{d}{4})\Big{]}\,.$
(18)

In the above expressions $\psi_{i}$ are the polygamma functions of order $i$,
while $J_{0}$ is the following sum:

$J_{0}(\tfrac{d}{4})=\frac{1}{\Gamma(\tfrac{d}{4})^{2}}\sum_{n\geq
1}\frac{\Gamma(n+\tfrac{d}{2})\Gamma(n+\tfrac{d}{4})^{2}}{n(n!)\Gamma(\tfrac{d}{2}+2n)}\Big{[}2\psi(n+1)-\psi(n)-2\psi(n+\tfrac{d}{4})-\psi(n+\tfrac{d}{2})+2\psi(\tfrac{d}{2}+2n)\Big{]}\,.$
(19)

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