# Low-temperature behavior of the $O(N)$ models below two dimensions

Andrzej Chlebicki  Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Pawel Jakubczyk
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warsaw, Poland

###### Abstract

We investigate the critical behavior and the nature of the low-temperature
phase of the $O(N)$ models treating the number of field components $N$ and the
dimension $d$ as continuous variables with a focus on the $d\leq 2$ and $N\leq
2$ quadrant of the $(d,N)$ plane. We precisely chart a region of the $(d,N)$
plane where the low-temperature phase is characterized by an algebraic
correlation function decay similar to that of the Kosterlitz-Thouless phase
but with a temperature-independent anomalous dimension $\eta$. We revisit the
Cardy-Hamber analysis leading to a prediction concerning the nonanalytic
behavior of the $O(N)$ models’ critical exponents and emphasize the previously
not broadly appreciated consequences of this approach in $d<2$. In particular,
we discuss how this framework leads to destabilization of the long-range order
in favour of the quasi long-range order in systems with $d<2$ and $N<2$.
Subsequently, within a scheme of the nonperturbative renormalization group we
identify the low-temperature fixed points controlling the quasi long-range
ordered phase and demonstrate a collision between the critical and the low-
temperature fixed points upon approaching the lower critical dimension. We
evaluate the critical exponents $\eta(d,N)$ and $\nu^{-1}(d,N)$ and
demonstrate a very good agreement between the predictions of the Cardy-Hamber
type analysis and the nonperturbative renormalization group in $d<2$.

##  I Introduction

The $O(N)$ models are a cornerstone of modern statistical physics. They have
been employed with a great success to describe phase transitions of a
multitude of universality classes: the liquid-vapor transition for $N=1$, the
helium superfluid transition for $N=2$, the polymer problem for $N=0$ and the
scalar sector of the electroweak standard model for $N=4$ to invoke just a
few. Additionally, their relatively simple formulation has provided a fertile
ground for development and evaluation of new theoretical techniques.

The universality classes of the $O(N)$ models [and the associated critical
exponents] can be parameterized by two variables: $d$ \- the system’s spatial
dimension and $N$. Typically, only the integer pairs of $d$ and $N$ are of
interest from a physical standpoint. However, there is a number of significant
theoretical approaches treating both $d$ and $N$ as continuous variables. The
most notable examples include: the large $N$ expansion, the $4-\epsilon$
expansion and the $2+\epsilon$ expansion. These perturbative renormalization
group (RG) approaches have proven successful both in predicting the values of
the critical exponents as well as development of the general theory of RG [1,
2]. We note, that there has been some effort in engineering systems with non-
integer effective values of $d$ [3, 4, 5] and that recently a rigorous
mathematical description has been developed for systems with non-integer $N$
[6]. The idea of considering non-integer values of $d$ and $N$ was also
implemented in the conformal bootstrap approach [see e.g. [7, 8]].

Depending on $d$ and $N$, the $O(N)$ models can support three very distinct
types of phases: long-range ordered (LRO), disordered and quasi long-range
ordered (QLRO). A LRO phase exhibits a non-vanishing order-parameter
expectation value. A disordered phase, on the other hand, is characterized by
vanishing order-parameter and additionally an exponential decay of the
correlation function $G(r)\propto\exp(-r/\xi)$, where $\xi$ denotes the
correlation length. A QLRO phase, alike a disordered phase, is characterized
by vanishing order-parameter, but the corresponding correlation function
decays as a power-law of the distance $G(r)\propto r^{2-d-\eta}$, where $\eta$
is the anomalous dimension. This correlation structure arises when the RG flow
is controlled by a stable finite-temperature fixed point (FP).

Figure 1: (Color online) Schematic representation of the $(d,N)$ plane in the
vicinity of the Kosterlitz-Thouless point $(2,2)$. The Mermin-Wagner line
[$d=2$, $N>1$] separates the systems that support the long-range order (LRO)
in non-zero temperatures from those that do not. Below two dimensions, for
sufficiently small values of $N$ [$N<N_{c}(d)$] the low-temperature phase is
characterized by QLRO, while for $N>N_{c}(d)$ the system remains disordered
for any non-zero temperature. For $d<2$, the Cardy-Hamber line coincides with
$N_{c}(d)$, while for $d>2$ it is a locus of the hypothetical nonanalyticity
of the critical exponents predicted within the framework of ref. [9], but
contrary to the findings of ref. [10].

Over the years the unusual physics of QLRO phases attracted significant
attention. The most famous example is probably the low-temperature phase of
the Kosterlitz-Thouless universality class $(d,N)=(2,2)$ [11]. This exotic
type of ordering was also found, _inter alia_ , in the random-field [or
random-anisotropy] $O(N)$ models - relevant in the studies of superconductors,
nematic liquid crystals and porous media [12, 13]. For specific values of $d$
and $N$, a weak random-field constitutes a relevant perturbation which
destroys the LRO but allows for the formation of the QLRO sometimes called the
Bragg glass phase [14, 12]. The universal properties of the random-field
$O(N)$ models are related to those of the pure $O(N)$ models in a shifted
dimension due to a property known as dimensional reduction [15, 16, 13]. Fully
grasping the critical behavior of the $O(N)$ models is therefore relevant for
a wide variety of physical contexts. Surprisingly, an old finding that the
$O(N)$ models support a QLRO phase in an extended region of the $(d,N)$ plane
seems relatively unknown and poorly studied [17, 16].

A sketch of the $(d,N)$ plane is presented in fig. 1. The point $(2,2)$ marks
a unique spot in the $(d,N)$ plane, which hosts the exotic Kosterlitz-Thouless
(KT) universality class. The line $d=2$ marks the onset of the $2+\epsilon$
expansion. The Mermin-Wagner theorem states that at finite-temperature
spontaneous breaking of the continuous symmetry can only occur above two
dimensions [18]. Therefore, the $d=2$ Mermin-Wagner line separates the systems
that support LRO in finite temperatures from those that do not. The meaning of
the ‘continuity’ of the symmetry group becomes unclear when we consider non-
integer values of $N<2$, however based on the results from earlier studies
[17, 16] as well as the calculations of the present paper we argue that the
Mermin-Wagner line extends all the way down to $N=1$ [not including the point
$(d,N)=(2,1)$].

The KT theory perturbatively captures the way in which vortices renormalize
the spin-wave stiffness, but remains restricted to a single point in the
$(d,N)$ plane. The $2+\epsilon$ expansion, on the other hand, strictly
describes the spin-wave theory in the vicinity of the KT point $(2,2)$, but
neglects the effects of the topological excitations, which are relevant for
some phase transitions, e.g. at the points $(2,2)$ and $(3,3)$ [see refs. [19,
20] ]. It is natural to ask, how these two approaches can be combined. This
question was first posed by Cardy and Hamber [9]. By combining the RG
equations of KT and the $2+\epsilon$ expansion they predict a collision of
fixed points upon crossing a specific line in the $(d,N)$ plane with important
consequences for the model’s universal properties.

In the present study, we first revisit the Cardy-Hamber (CH) scenario with a
focus directed towards the bottom-left quadrant of the $(d,N)$ plane [$d<2$,
$N<2$]. The complementary case of $d\geq 2$ was addressed in our earlier paper
[see ref. [10]]. We begin by summarizing the original CH theory. We then
explain how the CH approach predicts the appearance of the QLRO phase in a
specific region of the $(d,N)$ plane - the conclusion not mentioned in the CH
study, however recognized in Ref. [16]. We subsequently confront these results
with a direct numerical evaluation of the nonperturbative renormalization
group equations. In this framework we calculate the shape of the line
$N_{c}(d)$ bounding the region where the QLRO low-temperature phase exists in
the system’s phase diagram [see fig. 1]. In the vicinity of the KT point, we
find this line laying very close to the CH prediction. We, finally, calculate
the anomalous dimension $\eta$ of the critical and low-temperature fixed
points and the correlation length exponent $\nu$ as a function of $d$ and $N$.

The present paper has the following structure. In section II we readdress the
CH scenario in detail and lay out its previously not appreciated (except Ref.
[16]) consequences. In particular, we show how the fixed point of the
$2+\epsilon$ expansion becomes infrared stable for $d<2$ and $N<2$, leading to
formation of the QLRO in low temperatures. Somewhat surprisingly this fact is
not discussed in the CH paper. In section III we introduce the framework of
the nonperturbative renormalization group and explain the employed
approximations. Section IV contains our results for the shape of the line
$N_{c}(d)$ as well as the critical exponents of the $O(N)$ models below two
dimensions. In section V we summarize our conclusions.

##  II Cardy-Hamber approach

The approach proposed by Cardy and Hamber in [9] relies on the assumption that
the RG equations of the KT theory [21] and the $2+\epsilon$ expansion [22] can
be analytically combined. This way we can create a single set of recursion
relations for two variables: $g$ \- the spinwave interaction coupling and
$y^{2}$ which for $N=d=2$ describes the vortex fugacity and has a somewhat
ambiguous physical interpretation for other values of $d$ and $N$. In specific
contexts parameters analogous to $y^{2}$ may retain their physical meaning in
different dimensions [23, 24].

When investigating phase transitions close to the KT point, we treat
$\epsilon=d-2$, $N-2$ and $y^{2}$ as small quantities of the order no higher
than $O(\epsilon)$. This leads [9] to the equations:

$\displaystyle\dot{g}=-\epsilon g+(N-2)f(g,N)+4\pi^{3}y^{2}+O(\epsilon^{2}),$
(1)
$\displaystyle\dot{y^{2}}=\left(4-\frac{2\pi}{g}\right)y^{2}+O(\epsilon^{2}),$

where $f(g,N)$ is a function which so far has only been studied perturbatively
in powers of $g$. The following reasoning is fairly general and does not
require the knowledge of the exact shape of $f(g,N)$, only the assumption that
$f(g,2)/g$ is a monotonic increasing function for small values of $g$.
However, later on, we are going to exploit the expansion of $f$ to the highest
known order [25]:

$\displaystyle f(g,N)$
$\displaystyle=\frac{g^{2}}{2\pi}+\frac{g^{3}}{4\pi^{2}}+\frac{g^{4}(N+2)}{32\pi^{3}}$
(2)
$\displaystyle-\frac{g^{5}\left(N^{2}-18N\zeta(3)-22N+54\zeta(3)+34\right)}{192\pi^{4}}+O(g^{6}).$

In the analysis of eqs. (1) an essential role is played by the parameter

$\Delta=\epsilon\frac{\pi}{2}-(N-2)f\left(\frac{\pi}{2},N\right)+O(\epsilon^{2}),$
(3)

the sign of which determines the existence and stability of the fixed point
solutions to eq. (1). The condition $\Delta=0$ defines the CH line at which
the distinct families of fixed point solutions collide. As we shall see, for
$\epsilon<0$ this line marks also the lower critical dimension.

Equations (1), admit two families of non-trivial fixed point solutions:

$\displaystyle g_{\text{KT}}=\frac{\pi}{2}+O(\epsilon),\quad
y_{\text{KT}}^{2}=\frac{\Delta}{4\pi^{3}},$ (4)

which can be seen as an extension of the Kosterlitz-Thouless FP to $N\neq 2$,
$d\neq 2$, and

$\displaystyle\epsilon
g_{\text{BZJ}}=(N-2)f(g_{\text{BZJ}},N)+O(\epsilon^{2}),\quad
y_{\text{BZJ}}^{2}=O(\epsilon^{2})$ (5)

identical to the solution of the $2+\epsilon$ expansion of Brézin and Zinn-
Justin [26, 22]. The KT FP solution is physical [$y\in\mathbb{R}$] whenever
$\Delta\geq 0$, i.e. below the CH line [see fig. 1]. The analysis of the RG
eigenvalues shows that this FP, if it exists, has a single unstable direction
and therefore controls the phase transition.

Assuming the form (2) of $f$, the condition (5) can be satisfied by a number
of nontrivial solutions, of which all but one can be discarded as nonphysical
artifacts of the $g$ expansion. From now on, we shall use the term BZJ FP to
describe the solution to $\eqref{bzj_fp}$ with the lowest positive value of
$g_{\text{BZJ}}$, which is the only physical. The BZJ FP exists in the top-
right [$d>2$, $N>2$] and bottom-left [$d<2$, $N<2$] quadrants of the $(d,N)$
plane. In the top-right quadrant, the BZJ FP is critical for $\Delta<0$ and
tricritical for $\Delta>0$. For $\Delta=0$, BZJ and KT FPs coincide and swap
roles, which leads to the hypothetical nonanalyticity of the critical
exponents (predicted by this framework). We discuss this part of the CH
scenario in ref. [10].

A different picture presents itself in the bottom-left quadrant of the $(d,N)$
plane [see fig. 1], which is the main focus of the present study. The CH paper
notes that the KT FP is critical and controls the phase transition when
$\Delta>0$ and that there is no phase transition for $\Delta<0$. However,
Cardy and Hamber also state that only the KT FP is real for $N<2$. It is
nonetheless rather evident from eq. (1), the BZJ FP also exists in this
quadrant, and, as we argue below, controls the low-temperature phase present
in the $\Delta>0$ part of the $(d,N)$ plane below $d=2$. This fact was first
observed in Ref. [16]

For negative values of $\epsilon$, the zero-temperature trivial fixed point
[$g=0$] is unstable, which prevents formation of the long-range order at any
finite temperatures - in accord with the Mermin-Wagner theorem. However, at
the same time, a new infrared stable FP appears. Namely, for $\epsilon<0$ and
$\Delta>0$, the BZJ FP is a stable finite-temperature FP which emerges from
the zero-temperature FP as $\epsilon$ crosses $0$. In this region, the KT FP
controls the transition between the disordered and the QLRO phases.

Above the CH line [see fig. 1], when $\Delta$ becomes negative, the KT FP
ceases to exist. At the same time, the BZJ FP attains a single relevant
direction while its attraction domain shrinks to a subset of the $y^{2}=0$
line. This fixed point does not control any phase transition. This is because,
as we deduce from eq. (1), the RG flows cannot cross between the regions
$y^{2}=0$ and $y^{2}>0$. These two regions differ with respect to the presence
of the vortex-like excitations, which can appear when $y^{2}>0$, but not when
$y^{2}=0$. For this reason, the systems corresponding to those two regions are
subject to very distinct physics. The present study primarily focuses on the
models with $y^{2}>0$. From this perspective BZJ FP acts simply as a repulsive
FP for $\Delta<0$. We, therefore, expect that no phase transition takes place
for $\Delta<0$ and consequently that the condition $\Delta=0$ defines the line
of lower critical dimensions [in the bottom-left quadrant of the $(d,N)$
plane].

The CH scenario for the fixed point collision is illustrated in fig. 2. We
emphasize the contrast between the physical implications of the CH scenario
above and below $d=2$. For $d>2$, the collision takes place between the
critical and the tricritical FPs. The second order phase transition takes
place between the disordered and the LRO phases, and the critical exponents
exhibit (within this approach) nonanalyticities at the line of the collisions
[$\Delta=0$]. For $d<2$, on the other hand, the critical FP collides with the
FP controlling the low-temperature behavior. The phase transitions take place
between the disordered and the QLRO phases, and the line of the collisions
marks the lower critical dimensions $d_{c}(N)$. The above picture, as obtained
for $d<2$, is confirmed with the nonperturbative renormalization group (NPRG)
calculations described in sec. III. Observe, that this remains in contrast to
the case $d>2$ addressed in Ref. [10].

Figure 2: (Color online) Illustration of the Cardy-Hamber scenario for the
fixed point collision. The figure presents the evolution of different fixed
point solutions $g^{*}(\epsilon,N)$ upon varying $\epsilon$ for $N=1.5$ and
$N=2.5$. The six-pointed star denotes a point of FP collision for $N=1.5$, the
five-pointed - for $N=2.5$. The horizontal lines mark the overlapping critical
fixed points $g^{*}(\epsilon,2.5)=g^{*}(\epsilon,1.5)=g_{\text{KT}}=\pi/2$ and
were slightly separated for the purpose of transparency.

##  III Nonperturbative renormalization group

To readdress the presented problem from a different point of view, we employ
the one-particle irreducible variant of the nonperturbative renormalization
group [27]. The central object in the NRPG is a scale-dependent functional
$\Gamma_{k}$ of the order-parameter $\phi$, called the effective average
action. The quantity $\Gamma_{k}[\phi]$ describes the system’s effective free
energy at scale $k$, treating the slow [$q<k$] order-parameter modes
$\phi_{q}$ at the mean-field level while including the fluctuations of the
fast modes [$q>k$]. In this way, it smoothly interpolates from the mean-field
free energy at the ultraviolet scale $k=\Lambda$ to the exact free energy at
the vanishing scale $k\rightarrow 0$. The RG flow of the effective action is
described by the Wetterich equation [27]:

$\partial_{k}\Gamma_{k}=\frac{1}{2}\mathrm{Tr}\left[\partial_{k}R_{k}\left(\Gamma_{k}^{(2)}+R_{k}\right)^{-1}\right],$
(6)

where $\Gamma_{k}^{(2)}$ denotes the second functional derivative of $\Gamma$
and $R_{k}$ is the infrared regulator function. The NPRG has been applied with
remarkable success in a wide variety of contexts ranging from statistical
mechanics and quantum many-body physics to gravity and high-energy physics
[for reviews see e.g. [28, 29, 30, 31] ].

###  III.1 Derivative expansion

The Wetterich equation (6) is not directly soluble. We employ the derivative
expansion (DE) to reduce the functional differential equation (6) to a set of
partial differential equations. The principal idea of this approximation
relies on the expansion of the effective action $\Gamma_{k}$ around the
uniform field configuration in powers of gradient operators acting on the
order-parameter to a specified order. In the present work, we adopt the
$O(\nabla^{2})$ truncation, with $\Gamma_{k}$ parameterized as

$\Gamma_{k}=\int
d^{d}x\left\\{U^{k}(\rho)+\frac{Z_{2}^{k}(\rho)}{2}\left(\nabla\phi\right)^{2}+\frac{\left(Z_{1}^{k}(\rho)-Z_{2}^{k}(\rho)\right)}{4\rho}\left(\nabla\rho\right)^{2}\right\\},$
(7)

including all $O(N)$-symmetric terms up to $\nabla^{2}$. In the effective
action ansatz, the invariant $\rho=\phi^{i}\phi^{i}/2$ was introduced to make
the $O(N)$ symmetry of $\Gamma_{k}$ manifest and simplify the RG flow
equations. Note that no field expansion is imposed here.

Having specified the ansatz, the flow equations are derived in a
straightforward but tedious calculation. The function
$\beta^{k}_{U}(\rho)=k\partial_{k}U^{k}(\rho)$ is obtained by plugging the
ansatz (7) into (6) and evaluating at a uniform field configuration
$\phi^{1}(x)^{2}=2\rho$, $\phi^{i}(x)=0$ for $1<i\leq N$. The derivation of
$\beta^{k}_{Z_{1}}$ and $\beta^{k}_{Z_{2}}$, defined as
$\beta^{k}_{Z_{i}}(\rho)=k\partial_{k}Z^{k}_{i}(\rho)$ first requires us to
differentiate eq. (6) functionally and subsequently expand in momenta:

$\beta_{Z_{i}}=\left.\frac{1}{2d}\left.\Delta_{p}\right|_{p=0}\frac{\delta^{2}\left(\partial_{k}\Gamma_{k}\right)}{\delta\phi^{i}_{p}\delta\phi^{i}_{-p}}\right|_{\mathrm{Unif.}}.$
(8)

Due to the significant length of the flow equations, we relegate their
presentation to the Appendix. The flow equations are fully equivalent to those
analyzed in ref. [10].

To make the fixed point behavior manifest, we follow a standard rescaling
procedure to express the parameterization in a dimensionless form. This is
achieved by multiplying each quantity $X$ by the scale raised to the power of
its canonical dimension $\tilde{X}=Xk^{d_{x}}$. The rescaled variables read:

$\tilde{\rho}=\rho
k^{2-d-\eta_{k}},\quad\tilde{U}^{t}(\tilde{\rho})=U^{k}(\rho)k^{-d},\quad\tilde{Z}^{t}_{1/2}(\tilde{\rho})=Z^{k}_{1/2}(\rho)k^{\eta_{k}}.$
(9)

Above, $\eta_{k}$ denotes the running anomalous dimension defined as a
logarithmic derivative of the scaling factor
$\eta_{k}=-k\partial_{k}\log\left(Z^{k}\right)$. We close the set of RG
equations by defining the rescaling factor
$Z_{2}(\rho_{\eta})/Z_{k}=\tilde{Z}_{2}(\tilde{\rho}_{\eta})\equiv 1$, with an
arbitrary constant $\tilde{\rho}_{\eta}$. The equations are further
transformed into a stationary form by introducing the ‘renormalization time’
$t=\log(k/\Lambda)$ [$\partial_{t}=k\partial_{k}$].

With the flow equations derived, we follow a two-pronged approach to the
described problem. The first avenue relies on a direct integration of the flow
equations to inspect the phase diagram of the $O(N)$ models. As an initial
condition of the RG flow we choose the ‘mexican hat’-type $O(N)$ symmetric
action:

$\Gamma_{\Lambda}[\phi]=\mathcal{S}[\phi]=\int
d^{d}x\left(\frac{1}{2}\left(\nabla\phi\right)^{2}+\frac{u}{8}\left(\phi^{2}-\phi_{0}^{2}\right)^{2}\right),$
(10)

with $\phi_{0}$ tuned to approach the critical domain for $k\rightarrow 0$.
Simultaneously, we follow a complementary path based on solving the fixed
point equation

$\partial_{t}\tilde{\Gamma}^{*}=0.$ (11)

This allows us to extract the values of the critical exponents through
linearization of the flow equations and identify the lower critical dimension
below which the fixed point no longer exists. In our flow equations, $d$ and
$N$ enter as numerical constants. We use that fact in both these approaches to
observe how the critical behavior changes as we traverse the $(d,N)$ plane.

To make the flow equations suitable for the numerical analysis, we introduce
what we now call the functional scheme. In this approach, we represent the
functions $\tilde{V}(\tilde{\rho})=\tilde{U}^{\prime}(\tilde{\rho})$,
$\tilde{Z}_{1}(\tilde{\rho})$ and $\tilde{Z}_{2}(\tilde{\rho})$ on a finite
discrete grid. We approximate the derivatives by finite differences at orders
either $O(h^{4})$ or $O(h^{8})$ depending on the calculation, the momentum
integrals by finite sums using the Gauss-Legendre quadrature and employ the
adaptive $4^{\text{th}}$ order Runge-Kutta-Fehlberg algorithm when integrating
the flow equations. This way we recast the derived set of partial differential
equations into a very large set of algebraic equations that can be solved
numerically. We thoroughly investigated the dependence of our results on the
numerical parameters of the discrete representation. We used a $121$ point
$\tilde{\rho}$-space grid to obtain the presented numerically converged
results.

Throughout the work, we adopt the Wetterich regulator function [28]:

$R_{k}(q^{2})=\alpha
Z_{k}k^{2}\frac{\frac{q^{2}}{k^{2}}}{\exp\left(\frac{q^{2}}{k^{2}}\right)-1},$
(12)

with a variable parameter $\alpha$. In accord with the principle of minimal
sensitivity (PMS) [32, 33, 34], our estimate of a physical quantity
corresponds to the value of $\alpha$ such that the quantity in question is
stationary with respect to $\alpha$.

###  III.2 Low-temperature behavior

When investigating the low-temperature behavior, unlike in the case of the
critical state or the high-temperature behavior, the functional approach
presented above is notoriously difficult to implement. The problem lies in the
structure of the flow equations, which can be depicted as one-loop diagrams
with lines representing ‘dressed’ propagators of the form [scale dependence
suppressed in the notation for clarity]:

$\displaystyle G_{1}^{-1}(\rho,q)=U^{\prime}(\rho)+2\rho
U^{\prime\prime}(\rho)+q^{2}Z_{1}(\rho)+R(q^{2}),$ (13) $\displaystyle
G_{2}^{-1}(\rho,q)=U^{\prime}(\rho)+q^{2}Z_{2}(\rho)+R(q^{2}).$

Both propagators and consequently each term in the $\beta$ functions, suffer
from poles in $q$. At finite scales these poles remain on the imaginary axis,
however as we approach the infrared limit in the low-temperature phase,
$U^{\prime}(0)+R(0)\rightarrow 0^{+}$ taking the poles closer to the real axis
and leading to numerical inaccuracies in the momentum integrals’ evaluation.

To a certain extent, this problem can be dealt with by adopting very precise
procedures for carrying out the momentum integrals. However, the numerical
cost of achieving a given precision diverges rapidly as the pole of the
propagator approaches the real axis. The precision achieved in our numerical
implementation allowed us to capture the low-temperature fixed point inside a
region roughly demarcated by the CH line from the top and the lines $N=1.25$
and $d=1.9$ from the bottom and the right respectively [compare fig. 1]. We
were unable to numerically push the bounds of the region closer to the Mermin-
Wagner line ($d=2$) or the Ising universality class ($N=1$). This is explained
by the fact that the low-temperature behaviour of the models lying on these
two lines is controlled by singular zero-temperature FPs. In the proximity of
these lines, the low-temperature FPs are no longer singular, yet approaching
them, though in principle possible, becomes extremely challenging from the
numerical perspective.

An alternative route for avoiding these problems relies on the expansion of
the parametrizing functions into power-series around the minimum of the local
potential. In the most common approach, the functions $Z_{1}(\rho)$ and
$Z_{2}(\rho)$ are reduced to flowing constants while the local potential is
expanded to a given order - typically such that the quantities of interest
converge to specified precision. This procedure has proven valuable for
studying models with $d\gtrsim 2.5$ as it converges fairly rapidly and greatly
decreases the numerical complexity of the problem. It turns out, however, that
in the case of $O(N)$ models with $N>1$ below dimension $d\lesssim 2.4$, this
type of expansion does not reproduce the expected critical behavior when
carried out beyond the lowest sensible order:

$U(\rho)\approx\frac{u_{2}}{2}\left(\rho-\rho_{0}\right)^{2}.$ (14)

The reasons for the inadequacy of the high-order field expansion in low $d$
remain unclear to us and require future clarifying studies. Henceforth, we
shall refer to the scheme relying on the field expansion as the simplified
scheme [in contrast to the functional scheme described earlier]. The
simplified scheme will serve as a qualitatively accurate tool to outline the
expected results and to describe the parts of the $(d,N)$ plane which we were
not able to access within the functional scheme.

##  IV Results

Our NPRG results firmly support the CH scenario for the fixed point collision
below $d=2$. This is clearly illustrated in fig. 3 showing the evolution of
the anomalous dimensions of the critical and QLRO FPs upon decreasing the
dimension calculated within the simplified scheme. The anomalous dimensions of
the two FPs rise with the decreasing spatial dimension and converge abruptly
when the FPs collide, upon approaching the lower critical dimension. The rest
of the present section provides a detailed description of this phenomenon from
the NPRG perspective with a direct comparison to the predictions of the CH
scenario.

Figure 3: (Color online) Dependence of the anomalous dimension of both the
critical and the QLRO fixed points on $d$ for a series of values of $N$ within
the simplified scheme. Points denote the anomalous dimension of the QLRO FP
and lines - the critical FP. The stars mark the collision of the FPs at the
lower critical dimension as obtained by the simplified calculation.

###  IV.1 Critical exponents

We calculate the exponents characterizing the critical fixed point within the
functional scheme in a fashion similar to ref. [10]. We begin by finding a
fixed point in a high dimension [$d=3$]. From there we track the evolution of
the critical exponents as we slowly traverse the $(d,N)$ plane along the lines
of constant $N$. In practice, we achieve this by performing small, finite
steps $\delta d$ using the FP solution from the point $(d+\delta d,N)$ as an
initial condition for the FP search at the point $(d,N)$.

The results for the first RG eigenvalue $e_{1}=\nu^{-1}$ of the critical fixed
point [representing also the inverse correlation length exponent] as a
function of $d$ and $N$ are presented in fig. 4. The NPRG results are
juxtaposed with the prediction from the Cardy-Hamber approach. Within both
these approaches, all the curves behave similarly at high dimensions. However,
as we lower the dimension the curve $N=2.5$ departs from the rest.
Interestingly, this distinction arises almost at the same spot as the
nonanalyticity predicted in the CH scenario, but in a smoothed fashion [see
ref. [10] for an extensive discussion].

The degree to which our results for $d<2$ agree with the CH prediction is
somewhat surprising taking into account the low level of expansion employed in
each of the approaches. Most notably the lower critical dimension [where
$e_{1}=0$] is almost identical in the vicinity of $d=2$. However, we also note
some significant quantitative differences between the two approaches. The CH
scenario predicts a square root-like shape of $e_{1}$ close to the lower
critical dimension. This stands in contrast with our results, where $e_{1}$
decays with an exponent of around $0.5$ only for $N=2$, which then falls
rapidly to around $0.3$ for $N=1.8$ and $0.2$ for $N=1.2$.

Figure 4: (Color online) Dependence of the first RG eigenvalue on $\epsilon$
for a series of values of $N$. Points denote our results from the functional
calculation and lines - the predictions from the Cardy-Hamber type analysis.
The red line corresponding to $N=2.5$ features a discontinuity of the first
derivative predicted by the CH type analysis [9], notably absent in the
corresponding fRG calculations [10].

Fig. 5 compares the results for the anomalous dimension of the critical FP
$\eta_{c}$ obtained within the functional scheme with those calculated within
the simplified scheme. In the considered range of values, the data from the
functional scheme shows very weak $N$-dependence of $\eta$ for any dimension
$d$. This is, however, not the case in the simplified scheme, where $\eta$
seems weakly $N$-dependent only in high dimensions. As we lower the dimension,
$\eta_{c}$ starts to depend on $N$ more strongly becoming non-monotonous very
close to the lower critical dimension. This figure visualizes a well-
established fact, that the employed form of the simplified scheme tends to
significantly overestimate the anomalous dimension - in our case by a factor
ranging from $3/2$ in low dimensions to even $2$ close to $d=3$.

Figure 5: (Color online) Dependence of the anomalous dimension of the critical
FP $\eta_{c}$ on $\epsilon$ for a series of values of $N$. Points denote the
results from the simplified scheme and lines - the data from the functional
calculation. The values at the $N$-dependent lower critical dimensions
$\eta_{c}(d_{c}(N))$ are marked by the stars: five-pointed for the functional
calculations and six-pointed for the simplified calculations.

###  IV.2 Low-temperature phase

The fixed point controlling the low-temperature phase can be extracted from
the RG flow. We identify it as the limit of the effective action for vanishing
scale $k\rightarrow 0$ in the flow initiated below the critical temperature.
This means that the search for the low-temperature fixed point, in principle,
should be easier than for the critical one, as it requires no tuning of the
initial condition. However, properly capturing the low-temperature FPs within
the functional scheme is notoriously difficult. One reason for that was laid
out in sec. III.2 and is related to the pole of the propagator. The other
reason lies in the abruptness of the flow in the transient regime in between
the critical and the low-temperature FPs. This abruptness, somewhat
surprisingly, becomes less pronounced the closer to the critical temperature
we get. An example of the RG flow passing close to the critical FP and then
turning to asymptotically approach the low-temperature FP is presented in fig.
6. The figure overlays flows of two quantities: the dimensionless potential
minimum $\tilde{\rho}_{0}$ and the anomalous dimension $\eta$ for
$(d,N)=(1.75,1.3)$ slightly below the critical temperature.

Figure 6: (Color online) Flow of the dimensionless potential minimum
$\tilde{\rho}_{0}$ (left axis, blue circles) and the anomalous dimension
$\eta$ (right axis, orange squares) for $(d,N)=(1.75,1.3)$ slightly below the
critical temperature calculated within the functional scheme. Red vertical
lines roughly demarcate the scales at which the flow is controlled by the
critical and QLRO FPs.

In fig. 7 we juxtapose the functions parametrizing the critical and low-
temperature fixed points for $(d,N)=(1.75,1.3)$. We note two significant
qualitative differences between the two FPs. First and arguably more important
is the much smaller difference between $U^{\prime}(0)$ and $-\alpha$ present
in the QLRO FP leading to difficulties in the numerical treatment. The second
difference is a more complex shape of the $\tilde{Z}_{1}$ function featuring
two maxima - the first around the minimum of the local potential and the
second in significantly lower values of $\tilde{\rho}$. The interpretation of
the structure of the fRG FPs remains somewhat unclear and requires further
investigation. We note that on the qualitative level the QLRO FP is very
similar the KT fixed point [$(d,N)=(2,2)$] recovered in the same approximation
scheme presented in ref. [35] [after a necessary reparametrization].

(a) Critical FP

(b) QLRO FP

Figure 7: (Color online) Fixed point functions for $(d,N)=(1.75,1.3)$. Red
horizontal line marks $-\alpha$ corresponding to the propagator pole. Upon
decreasing $N$ towards $1$, the fixed point potential builds up singularity as
the range of values of $\rho$ where $U^{\prime}(\rho)$ is close to $-\alpha$
increases.

In two dimensions, the QLRO FP coincides with the zero-temperature FP. As a
consequence, the dimensionless local potential minimum $\tilde{\rho}_{0}$ [see
eq. (14)], acting as the inverse temperature, takes an infinite value at this
FP. This fact can be used to perform an expansion around
$\tilde{\rho}_{0}=\infty$ similar to that of ref. [36] but extended to
$\epsilon=d-2<0$. With such an expansion we can show that to the first order
in $\epsilon$ the $2+\epsilon$ expansion and the NPRG yield the exact same
value of the anomalous dimension, regardless of the employed infrared
regulator. However, this is no longer true at higher orders. It means that the
simplified scheme is the most accurate exactly in the dimensions where the
functional scheme becomes the most numerically demanding. This offers a unique
opportunity to supplement the functional results with the simplified ones.
Fig. 8 presents the correlation function exponent $d-2+\eta$ of the QLRO fixed
point as a function of $d$ and $N$. In the figure, the combination of
functional and simplified results is compared with the predictions the
$2+\epsilon$ expansion [25]. We note a very strong agreement between the
$2+\epsilon$-expansion and the functional results in all dimensions. The
predictions of the simplified scheme, on the other hand, agree with the
$2+\epsilon$-expansion only in a direct vicinity of $d=2$ and deviate from it
pretty quickly as the dimension decreases.

It is tempting, at this point, to reach out for a comparison with better
understood limiting cases such as the Mermin-Wagner line ($d=2$) and the Ising
universality class ($N=1$). The comparison with the former is straightforward.
The low-temperature FPs of the two-dimensional $O(N)$ models for $2>N>1$ are
characterised by vanishing anomalous dimension. This fact is captured exactly
within the $2+\epsilon$ expansion and as consequence within the derivative
expansion at order $O(\nabla^{2})$. The proper way to compare our results to
the Ising universality class is much more unclear. Due to lack of the
transverse mode (or at least its fraction) leads to a completely different
physics. On a more technical level, one can no longer use $Z_{2}(\rho_{\eta})$
as the order-parameter renormalization factor $Z_{k}$ since $Z_{2}$ has no
physical interpretation for $N=1$. For those reasons, we refrain from
addressing the case of $N=1$ within this study.

Figure 8: (Color online) Dependence of the correlation function exponent of
the QLRO FP on $\epsilon$ for a series of values of $N$. Filled points denote
the results from the functional scheme, empty - from the simplified scheme and
lines - the predictions of the $2+\epsilon$ expansion [ref. [25]]. The stars
mark the lower critical dimensions: five-pointed as obtained in the functional
scheme and six-pointed from the CH scenario. The lines are extended below the
point of the FP collision [where the BZJ FP is no longer infrared stable]
along the prediction $2+\epsilon$ expansion.

Fig. 9 illustrates the collision of the QLRO and the critical FPs by
presenting the evolution of the correlation function exponent $d-2+\eta$ with
the changing dimension. The figure uses the data from the functional
calculation wherever available and supplements it with the data from the
simplified in dimension close to $d=2$. The collision seems very sharp when
compared to fig. 3, where it occur almost smoothly. We note a slight
discrepancy between the lower critical dimension coming from the analyses of
the critical and the QLRO FPs arising due to limited numerical precision.

Figure 9: (Color online) The correlation function exponent $d-2+\eta$ of the
critical and QLRO FPs as functions of $\epsilon$ for a series of values of
$N$. Points denote the exponent of the QLRO FP: filled - from the functional
scheme, empty - from the simplified, and lines denote the exponent of the
critical FP. Stars mark the lower critical dimensions from the analysis of the
critical FPs in the functional scheme.

###  IV.3 Lower critical dimension

We finally examine the line of the lower critical dimension $d_{c}(N)$. We
estimate $d_{c}(N)$ by the lowest dimension at which we are able to extract
the PMS value for the dominant RG eigenvalue in the functional calculation for
a given value of $N$. In fig. 10, we compare our results with the real space
RG data from ref. [17] as well as two distinct estimates springing from the CH
scenario. We recall that the CH line is defined by the equation

$0=\Delta=\epsilon\frac{\pi}{2}-(N-2)f\left(\frac{\pi}{2},N\right)+O(\epsilon^{2}).$
(15)

The first estimate for the CH line follows the original reasoning of Cardy and
Hamber which is based on an additional postulate that $f(g=\pi/2,N=2)=2/\pi$
111This postulate was introduced to satisfy the analytic form for the first RG
eigenvalue conjectured by the authors.. In the alternative estimation, we
employ the perturbative expansion of $f(g,N)$ presented in eq. (2).

Our results fit very nicely into the presented picture, laying very close to
the predictions of refs. [9, 17]. It is instructive to compare the different
estimates of a slope of the CH line at the KT point $(d,N)=(2,2)$. This data
is presented in table 1. Our results lay in between the predictions of other
works. We note at this point that the perturbative CH prediction varies
strongly with the employed order of $g$ expansion and can be not yet
sufficiently converged at the present order.

Figure 10: (Color online) Comparison of different estimates of the shape of the line of the lower critical dimension. Lines denote two different ways of charting the CH line based on the perturbative RG calculations. Points denote results from the present work and from ref. [17]. The NPRG points for $d>2$ mark the position of the CH line estimated within the functional RG scheme in ref. [10], where it was obtained in a form of a crossover.  Calculation | $\left.\frac{\partial N_{c}(d)}{\partial d}\right|_{d=2}$
---|---
CH - original | $\frac{\pi^{2}}{4}\approx 2.5$
CH - perturbative | $3.1$
NPRG | $4.3$
Ref. [17] | $5.8$
Table 1: Different estimates of the slope of the CH line at the KT point
$(d,N)=(2,2)$.

##  V Conclusion

In this paper we addressed the $O(N)$ models for $d<2$ and $N<2$. We
emphasized a consequence of the CH type analysis that below $d=2$ a stable
finite-temperature FP emerges from the zero-temperature FP. The existence of
the low-temperature FP destabilizes the long-range order and leads to
formation of a QLRO phase similar to the KT phase but characterized by a
universal (temperature independent) anomalous dimension $\eta$. We then
demonstrated that the line of collisions of the critical and low-temperature
FPs marks the line of the lower critical dimensions $d_{c}(N)$ of the $O(N)$
models with $1<N<2$ and $d<2$, and bounds the region where the QLRO phase
exists in the model’s phase diagram.

Subsequently, we employed the nonperturbative RG and obtained a picture
supporting the CH scenario for $d<2$ and $N<2$. We verified the existence of
the critical and low-temperature FPs below $d<2$ and demonstrated their
collision by tracking the evolution of their anomalous dimensions. We have
shown a very strong agreement between the predictions of NPRG and the
$2+\epsilon$ expansion for the anomalous dimension of the QLRO FP for all
dimensions between $d=2$ and the lower critical dimension $d_{c}(N)$.

We also examined the line of lower critical dimensions $d_{c}(N)$. We obtained
a shape very similar to the one obtained within the real space RG [17] with
one significant difference. Compared to ref. [17], the slope of our estimate
at the KT point $(d,N)=(2,2)$ is substantially closer to the prediction of the
CH scenario.

It is striking that, while the results obtained for $d<2$ from the NPRG method
in the present work are fully compatible with the predictions of the Cardy-
Hamber approach, they differ rather fundamentally in the case $d>2$ described
in our previous paper [10]. There, we have found no indication of
nonanalyticities of the critical exponents consistent with the CH picture of
the FP collision. Instead, we identified a smooth crossover between two
regions of the $(d,N)$ plane well characterized in the CH paper. On the other
hand, in the present work, we clearly recover a sharp FP collision for $d<2$.
We also observe that the line $N_{c}(d)$ obtained here merges smoothly at the
KT point $(d,N)=(2,2)$ with the line of cross-overs of the critical exponents
presented in ref. [10].

## Acknowledgments

We thank Gilles Tarjus for pointing this problem to us as well as reading the
initial version of the manuscript and providing useful comments. We are
grateful to Bertrand Delamotte for collaboration on related topics and sharing
his experience with us and to Nicolás Wschebor for reading and commenting on
the initial version of the manuscript. We thank Johan Henriksson for useful
correspondence.

We acknowledge funding from the Polish National Science Center via grant
2017/26/E/ST3/00211. AC is grateful for the hospitality of Instituto de
Física, Facultad de Ingeniería, Universidad de la República, Montevideo,
Uruguay, where a substantial part of the paper was completed as part of the
project BPN/BEK/2021/1/00293 funded by Polish National Agency for Academic
Exchange NAWA.

## Appendix

In this section we present the RG equations that were used in this work. To
simplify the expressions we denote the ‘dressed’ propagators as follows:

$\displaystyle G_{1}(\rho)=\left(Z_{1}(\rho)q^{2}+U^{\prime}(\rho)+2\rho
U^{\prime\prime}(\rho)+R(k,\bm{q}^{2})\right)^{-1},$ (16) $\displaystyle
G_{2}(\rho)=\left(Z_{2}(\rho)q^{2}+U^{\prime}(\rho)+R(k,\bm{q}^{2})\right)^{-1},$
(17)

In the equations below we use $\int_{q}$ to denote
$\int\frac{d^{d}q}{\left(2\pi\right)^{d}}$ and we suppress the $k$ dependence
to simplify the notation. The $\beta$ functions employed in the functional
calculation read:

$\displaystyle\beta_{U^{\prime}}(\rho)$
$\displaystyle=k\partial_{k}U^{\prime}(\rho)=-\frac{1}{2}\int_{q}\left(k\partial_{k}R\left(q^{2}\right)\right)\left\\{G_{1}(\rho){}^{2}\left(q^{2}\left(Z_{1}^{\prime}(\rho)\right)+2\rho
U^{(3)}(\rho)+3U^{\prime\prime}(\rho)\right)+(N-1)G_{2}(\rho){}^{2}\left(q^{2}Z_{2}^{\prime}(\rho)+U^{\prime\prime}(\rho)\right)\right\\},$
(18) $\displaystyle\beta_{Z_{1}}(\rho)$
$\displaystyle=k\partial_{k}Z_{1}(\rho)=-\frac{1}{2d}\int_{q}\left(k\partial_{k}R\left(q^{2}\right)\right)\Bigg{\\{}4(N-1)\rho
G_{2}(\rho){}^{4}\left(q^{2}Z_{2}^{\prime}(\rho)+U^{\prime\prime}(\rho)\right)\Bigg{[}-\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+Z_{2}(\rho)\right)\left((d+4)q^{2}Z_{2}^{\prime}(\rho)+dU^{\prime\prime}(\rho)\right)$
$\displaystyle-2q^{2}\left(\partial_{q^{2}}^{2}R\left(q^{2}\right)\right)\left(q^{2}Z_{2}^{\prime}(\rho)+U^{\prime\prime}(\rho)\right)\Bigg{]}+4(N-1)G_{2}(\rho){}^{3}\left(d\left(Z_{1}(\rho)-Z_{2}(\rho)\right)\left(q^{2}Z_{2}^{\prime}(\rho)+U^{\prime\prime}(\rho)\right)+q^{2}\rho
Z_{2}^{\prime}(\rho){}^{2}\right)$ $\displaystyle+4\rho
G_{1}(\rho){}^{4}\left(q^{2}Z_{1}^{\prime}(\rho)+2\rho
U^{(3)}(\rho)+3U^{\prime\prime}(\rho)\right)\Bigg{[}-\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+Z_{1}(\rho)\right)\left((d+4)q^{2}Z_{1}^{\prime}(\rho)+d\left(2\rho
U^{(3)}(\rho)+3U^{\prime\prime}(\rho)\right)\right)$
$\displaystyle-2q^{2}\left(\partial_{q^{2}}^{2}R\left(q^{2}\right)\right)\left(q^{2}Z_{1}^{\prime}(\rho)+2\rho
U^{(3)}(\rho)+3U^{\prime\prime}(\rho)\right)\Bigg{]}+4\rho
G_{1}(\rho){}^{3}Z_{1}^{\prime}(\rho)\left((2d+1)q^{2}Z_{1}^{\prime}(\rho)+4d\rho
U^{(3)}(\rho)+6dU^{\prime\prime}(\rho)\right)$ $\displaystyle-
dG_{1}(\rho){}^{2}\left(Z_{1}^{\prime}(\rho)+2\rho
Z_{1}^{\prime\prime}(\rho)\right)+16(N-1)q^{2}\rho
G_{2}(\rho){}^{5}\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+Z_{2}(\rho)\right){}^{2}\left(q^{2}Z_{2}^{\prime}(\rho)+U^{\prime\prime}(\rho)\right){}^{2}$
$\displaystyle+16q^{2}\rho
G_{1}(\rho){}^{5}\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+Z_{1}(\rho)\right){}^{2}\left(q^{2}Z_{1}^{\prime}(\rho)+2\rho
U^{(3)}(\rho)+3U^{\prime\prime}(\rho)\right){}^{2}$
$\displaystyle-\frac{d(N-1)G_{2}(\rho){}^{2}\left(\rho
Z_{1}^{\prime}(\rho)-Z_{1}(\rho)+Z_{2}(\rho)\right)}{\rho}\Bigg{\\}},$ (19)
$\displaystyle\beta_{Z_{2}}(\rho)$
$\displaystyle=k\partial_{k}Z_{2}(\rho)=-\frac{1}{2d\rho}\int_{q}\left(k\partial_{k}R\left(q^{2}\right)\right)\Bigg{\\{}G_{2}(\rho){}^{2}\Bigg{[}-d\left((N-1)\rho
Z_{2}^{\prime}(\rho)+2\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+2Z_{2}(\rho)\right)$
$\displaystyle-4q^{2}G_{1}(\rho)\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)-\rho
Z_{2}^{\prime}(\rho)+Z_{2}(\rho)\right)\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+\rho
Z_{2}^{\prime}(\rho)+Z_{2}(\rho)\right)-4q^{2}\left(\partial_{q^{2}}^{2}R\left(q^{2}\right)\right)\Bigg{]}$
$\displaystyle+G_{1}(\rho){}^{2}\left(-d\left(2\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+2\rho^{2}Z_{2}^{\prime\prime}(\rho)+5\rho
Z_{2}^{\prime}(\rho)+Z_{1}(\rho)+Z_{2}(\rho)\right)-4q^{2}\left(\partial_{q^{2}}^{2}R\left(q^{2}\right)\right)\right)$
$\displaystyle+G_{2}(\rho)\Bigg{[}G_{1}(\rho)\left(4d\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+\rho
Z_{2}^{\prime}(\rho)+Z_{2}(\rho)\right)+8q^{2}\left(\partial_{q^{2}}^{2}R\left(q^{2}\right)\right)\right)$
$\displaystyle+4q^{2}G_{1}(\rho){}^{2}\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+\rho
Z_{2}^{\prime}(\rho)+Z_{2}(\rho)\right)\left(-\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+\rho
Z_{2}^{\prime}(\rho)-2Z_{1}(\rho)+Z_{2}(\rho)\right)\Bigg{]}$
$\displaystyle+4q^{2}G_{1}(\rho){}^{3}\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+Z_{1}(\rho)\right){}^{2}+4q^{2}G_{2}(\rho){}^{3}\left(\left(\partial_{q^{2}}R\left(q^{2}\right)\right)+Z_{2}(\rho)\right){}^{2}\Bigg{\\}}.$
(20)

## References

  * Amit and Martin-Mayor [2005] D. J. Amit and V. Martin-Mayor,  _F. Theory, Renorm. Group, Crit. Phenom. Graphs to Comput. Third Ed._ (World Scientific Publishing Co., 2005) pp. 1–544.
  * Zinn-Justin [2010] J. Zinn-Justin,  _Phase Transitions Renorm. Gr._ (Oxford University Press, 2010) pp. 1–464.
  * Boada  _et al._ [2012] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Phys. Rev. Lett. 108, 133001 (2012), arXiv:1112.1019 .
  * Boada  _et al._ [2015] O. Boada, A. Celi, J. Rodriguez-Laguna, J. I. Latorre, and M. Lewenstein, New J. Phys. 17, 045007 (2015), arXiv:1409.4770 .
  * Łebek and Jakubczyk [2020] M. Łebek and P. Jakubczyk, Phys. Rev. A 102, 013324 (2020).
  * Binder and Rychkov [2020] D. J. Binder and S. Rychkov, J. High Energy Phys. 2020, 117 (2020), arXiv:1911.07895 .
  * El-Showk  _et al._ [2014] S. El-Showk, M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Phys. Rev. Lett. 112, 141601 (2014).
  * Henriksson [2022] J. Henriksson, “The critical o(n) cft: Methods and conformal data,”  (2022).
  * Cardy and Hamber [1980] J. L. Cardy and H. W. Hamber, Phys. Rev. Lett. 45, 1217 (1980).
  * Chlebicki and Jakubczyk [2021] A. Z. Chlebicki and P. M. Jakubczyk, SciPost Phys. 10 (2021), 10.21468/SCIPOSTPHYS.10.6.134, arXiv:2012.00782 .
  * Kosterlitz and Thouless [1973] J. M. Kosterlitz and D. J. Thouless, J. Phys. C Solid State Phys. 6, 1181 (1973).
  * Feldman [2001] D. E. Feldman, Int. J. Mod. Phys. B 15, 2945 (2001).
  * Tarjus and Tissier [2020] G. Tarjus and M. Tissier, The European Physical Journal B 93, 50 (2020).
  * Giamarchi and Le Doussal [1994] T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. 72, 1530 (1994).
  * Parisi and Sourlas [1979] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979).
  * Tissier and Tarjus [2006] M. Tissier and G. Tarjus, Phys. Rev. Lett. 96, 12 (2006).
  * Lau and Dasgupta [1987] M. H. Lau and C. Dasgupta, Phys. Rev. B 35, 329 (1987).
  * Mermin and Wagner [1966] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * Kamal and Murthy [1993] M. Kamal and G. Murthy, Phys. Rev. Lett. 71, 1911 (1993).
  * Motrunich and Vishwanath [2004] O. I. Motrunich and A. Vishwanath, Phys. Rev. B - Condens. Matter Mater. Phys. 70 (2004), 10.1103/PhysRevB.70.075104, arXiv:0311222 [cond-mat] .
  * José  _et al._ [1977] J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
  * Brézin and Zinn-Justin [1976] E. Brézin and J. Zinn-Justin, Phys. Rev. Lett. 36, 691 (1976).
  * Herbut [1998] I. F. Herbut, Phys. Rev. B 58, 971 (1998).
  * Herbut [2000] I. F. Herbut, Phys. Rev. B 61, 14723 (2000).
  * Bernreuther and Wegner [1986] W. Bernreuther and F. J. Wegner, Phys. Rev. Lett. 57, 1383 (1986).
  * Polyakov [1975] A. M. Polyakov, Phys. Lett. B 59, 79 (1975).
  * Wetterich [1993] C. Wetterich, Phys. Lett. B 301, 90 (1993), arXiv:1710.05815 .
  * Berges  _et al._ [2002] J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, 223 (2002), arXiv:0005122 [hep-ph] .
  * Kopietz, Peter; Bartosch, Lorenz; Schütz [2010] F. Kopietz, Peter; Bartosch, Lorenz; Schütz,  _Phase Transitions Renorm. Gr._ (Springer Berlin, Heidelberg, Heidelberg, 2010) p. 380.
  * Gies [2012] H. Gies, in  _Renorm. Gr. Eff. F. Theory Approaches to Many-Body Syst._, Vol. 852, edited by A. Schwenk and J. Polonyi (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012) pp. 287–348, arXiv:0611146 [hep-ph] .
  * Dupuis  _et al._ [2020] N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, arXiv (2020), arXiv:2006.04853 .
  * Balog  _et al._ [2020] I. Balog, G. De Polsi, M. Tissier, and N. Wschebor, Phys. Rev. E 101 (2020), 10.1103/PHYSREVE.101.062146, arXiv:2004.02521 .
  * Canet  _et al._ [2003] L. Canet, B. Delamotte, D. Mouhanna, and J. Vidal, Phys. Rev. B - Condens. Matter Mater. Phys. 68, 64421 (2003).
  * De Polsi  _et al._ [2020] G. De Polsi, I. Balog, M. Tissier, and N. Wschebor, Phys. Rev. E 101, 1 (2020), arXiv:2001.07525 .
  * Jakubczyk  _et al._ [2014] P. Jakubczyk, N. Dupuis, and B. Delamotte, Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 90, 62105 (2014), arXiv:1409.1374 .
  * Gräter and Wetterich [1995] M. Gräter and C. Wetterich, Phys. Rev. Lett. 75, 378 (1995).
  * Note [1] This postulate was introduced to satisfy the analytic form for the first RG eigenvalue conjectured by the authors.