# Ralph Kenna’s scaling relations in critical phenomena

Leïla Moueddene 1,2, Arnaldo Donoso 3 and Bertrand Berche 1 Correspondence:
<EMAIL_ADDRESS>0000-0002-4254-807X

(1 Laboratoire de Physique et Chimie Théoriques, CNRS - Université de
Lorraine, UMR 7019, Nancy, France and L4 collaboration, Leipzig-Lorraine-Lviv-
Coventry, Europe
2 Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB,
United Kingdom
3 Department of Experimental Physics, Maynooth University, Maynooth Co.
Kildare, Ireland
)

Abstract: In this note, we revisit the scaling relations among “hatted
critical exponents” which were first derived by Ralph Kenna, Des Johnston and
Wolfhard Janke, and we propose an alternative derivation for some of them. For
the scaling relation involving the behavior of the correlation function, we
will propose an alternative form since we believe that the expression is
erroneous in the work of Ralph and his collaborators.

Keyword: Critical exponents; Logarithmic corrections; Scaling and
renormalization; Scaling laws; Tricritical point; Finite-Size Scaling

## In memory of our friend Ralph Kenna

This paper is dedicated to our friend Ralph Kenna who passed away on Oct. 26th
2023. Ralph was a close collaborator and a very good friend, he was the PhD
co-advisor of one of the authors of the present paper (LM). Renowned
specialist in the study of phase transitions through the partition function
zeros, he developed this formalism in the difficult cases where critical
behaviours are controlled beyond the dominant singularities, by logarithmic
corrections.

The present work was initiated on the occasion of the ComPhys23
workshop111http://www.physik.uni-leipzig.de/ janke/CompPhys23/ organized by W.
Janke in Leipzig, and dedicated to the memory of Ralph. The opening talk was
intended to highlight some of Ralph’s most important contributions to
statistical physics, notably the derivation of scaling laws between the
exponents associated with logarithmic singularities in the vicinity of second-
order phase transitions. Ralph was a well-known physicist in the statistical
physics community and these new scaling laws were very successful in the field
of critical phenomena. It was by revisiting these scaling laws and their
derivation that we noted that one of these relations was incomplete and that
the work undertaken by Ralph and his co-authors around twenty years ago
merited to be completed.

This paper could have been written by Des Johnston or by Wolfhard Janke, the
co-authors of Ralph in this topic (and in many more works).

##  1 Scaling relations and universal combinations of amplitudes, a short
primer

We assume that the reader is familiar with the notion of critical exponents
that describe the singularities of various thermodynamic functions at the
approach of a second-order phase transition. Otherwise, we can suggest to
refer e.g. to the textbook of Kardar [1].

Standard scaling relations among the universal critical exponents are the
following,

$\displaystyle 2\beta+\gamma$ $\displaystyle=$ $\displaystyle 2-\alpha,$ (1)
$\displaystyle\beta(\delta-1)$ $\displaystyle=$ $\displaystyle\gamma,$ (2)
$\displaystyle\nu(2-\eta)$ $\displaystyle=$ $\displaystyle\gamma,$ (3)
$\displaystyle 2-\alpha$ $\displaystyle=$ $\displaystyle\nu d.$ (4)

They are very useful, not only to obtain the values of all 6 fundamental
critical exponents within a universality class from the knowledge of 2 of them
but also because they allow for the definition of other universal quantities,
written as specific combinations of critical amplitudes. Let us show how this
works. For that purpose, we first define the amplitudes as they enter the
expressions of the leading singular behaviours of thermodynamic quantities in
the vicinity of a second-order phase transition:

$\displaystyle\hbox{Specific heat:}\quad C_{\pm}(t,0)\simeq
A_{\pm}|t|^{-\alpha},$ (5) $\displaystyle\hbox{Low temperature
magnetization:}\quad m_{-}(t,0)\simeq B_{-}(-t)^{\beta},$ (6)
$\displaystyle\hbox{Susceptibility:}\quad\chi_{\pm}(t,0)\simeq\Gamma_{\pm}|t|^{-\gamma},$
(7) $\displaystyle\hbox{Critical temperature magnetization:}\quad
m_{c}(0,h)\simeq B_{c}|h|^{1/\delta},$ (8) $\displaystyle\hbox{Correlation
length:}\quad\xi_{\pm}(t,0)\simeq\xi_{0\pm}|t|^{-\nu}.$ (9)

Here, the two arguments of the functions at the l.h.s. are respectively
$t=(T-T_{c})/T_{c}$ and $h=H/T_{c}$, and the indices $\pm$ specify the high
($t>0$) and low ($t<0$) temperature phases, meaning that the field is zero,
and the index $c$ on the contrary implies the field behaviour at the critical
temperature. The symbol $\simeq$ stands for the leading singularity (i.e. the
most singular part, since there could be regular contributions to the
thermodynamic quantities, power law corrections to scaling, and multiplicative
logarithmic corrections, all these being omitted in Eqs. (5)-(9)).

We can also define the singular part of the free energy density in zero field:

$f_{\pm}^{\rm sing}(t,0)\simeq F_{\pm}|t|^{2-\alpha},$ (10)

and, since the specific heat is the second derivative of $f^{\rm sing}$ w.r.t
$t$, $F_{\pm}$ is not independent, since this requires
$A_{\pm}=(1-\alpha)(2-\alpha)F_{\pm}$. The Lee-Yang edge is another quantity
of interest in critical phenomena and we define

$h_{\pm}^{\rm LY}(t)\simeq h_{0\pm}|t|^{\Delta}$ (11)

with the so-called gap exponent $\Delta=\beta+\gamma=\beta\delta$.

Universality is the observation that some quantities only depend on very
general properties, like space dimensionality. The critical exponents are such
universal quantities, but the amplitudes are not, although some combinations
among them have the property of universality. To make it clear, let us write
Widom’s scaling assumption, i.e. the fact that the singular part of the free
energy density is a homogeneous function of the scaling fields,

$f^{\rm
sing}_{\pm}(t,h)=b^{-d}{\mathscr{F}}_{\pm}(\kappa_{t}b^{y_{t}}t,\kappa_{h}b^{y_{h}}h),$
(12)

where ${\mathscr{F}}_{\pm}(x(b),y(b))$ is a universal scaling function of its
arguments, $y_{t}$ and $y_{h}$ are the RG dimensions of the relevant fields
$t$ and $h$, and $\kappa_{t}$ and $\kappa_{h}$ are non universal metric
factors which would differ, say on the square lattice and the triangular
lattice in $2d$. The amplitudes defined above depend on these metric factors,
and this is why they are not universal. E.g. from
$C_{\pm}(t,0)=\frac{\partial^{2}f^{\rm sing}_{\pm}(t,0)}{\partial t^{2}}$,
setting $b=|t|^{-1/y_{t}}$ in the scaling form (12), one obtains

$C_{\pm}(t,0)=\kappa_{t}^{2+(d-2y_{t})/y_{t}}|t|^{(d-2y_{t})/y_{t}}\Bigl{(}\frac{\partial^{2}{\mathscr{F}}_{\pm}(x,0)}{\partial
x^{2}}\Bigr{)}_{x=1}.$ (13)

This identifies the exponent

$\alpha=(2y_{t}-d)/y_{t}$ (14)

and the amplitude

$A_{\pm}=\kappa_{t}^{2-\alpha}\Bigl{(}\frac{\partial^{2}{\mathscr{F}}_{\pm}(x,0)}{\partial
x^{2}}\Bigr{)}_{x=1}.$ (15)

The other exponents are similarly defined in terms of $y_{t}$ and $y_{h}$ by
very famous relations that we do not repeat here and the other amplitudes
depend on the metric factors as $B_{-}\sim\kappa_{h}\kappa_{t}^{\beta}$,
$\Gamma_{\pm}\sim\kappa_{h}^{2}\kappa_{t}^{-\gamma}$,
$B_{c}\sim\kappa_{h}^{1+1/\delta}$.

Simple ratios are immediately defined from the fact that the approach to
criticality from above and from below is described by the same exponent for a
given quantity (except for the magnetization, obviously). For example in the
case of the specific heat $C_{+}(|t|)\simeq A_{+}|t|^{-\alpha}$ and
$C_{-}(-|t|)\simeq A_{-}|t|^{-\alpha^{\prime}}$, where
$\alpha^{\prime}=\alpha$. It follows that the metric factors cancel in the
ratio and

$R_{C}(|t|)=\frac{C_{+}(|t|)}{C_{-}(-|t|)}=\frac{A_{+}|t|^{-\alpha}}{A_{-}|t|^{-\alpha^{\prime}}}\to
R_{C}=\frac{A_{+}}{A_{-}}$ (16)

is thus universal. The limit corresponds to the approach to criticality
($|t|\to 0$ here) since the combination $R_{C}(|t|)$ can be temperature-
dependent due to the possible presence of different values for the amplitudes
of the corrections to scaling which has not been taken into account in Eq. (5)
to (9). In the same manner, one defines the universal ratios

$R_{\chi}=\frac{\Gamma_{+}}{\Gamma_{-}},\quad
R_{\xi}=\frac{\xi_{0+}}{\xi_{0-}}.$ (17)

The scaling relations are other examples of relations that allow the
definition of new combinations. For example the ratio $m_{-}^{2}/\chi_{\pm}$
eliminates $\kappa_{h}$, and $\kappa_{t}$ is then eliminated, thanks to Eq.
(1) if we further divide by $C_{\pm}$. There is still an unwanted $|t|^{2}$
dependence that needs to be simplified and for that purpose we consider the
quantity

$R_{\pm}^{(1)}(t)=\frac{m_{-}^{2}(-|t|)}{C_{\pm}(t)\chi_{\pm}(t)|t|^{2}}=\frac{B_{-}^{2}}{C_{\pm}\Gamma_{\pm}}|t|^{2\beta+\gamma+\alpha-2}\to\frac{B_{-}^{2}}{C_{\pm}\Gamma_{\pm}}.$
(18)

Thanks to Eq. (1), the fact that all metric factors cancel out in this latter
quantity makes the combination ${B_{-}^{2}}/{C_{\pm}\Gamma_{\pm}}$ universal.
Proceeding the same way, Eq. (2) and Eq. (3) suggest to contemplate the
expressions

$\displaystyle R_{\pm}^{(2)}(t)$ $\displaystyle=$
$\displaystyle\left.\chi_{\pm}(t)m_{-}^{\delta-1}(-|t|)m_{c}^{-\delta}(h)h\right|_{h=h_{\pm}^{\rm
LY}(|t|)}$ (19)
$\displaystyle=\Gamma_{\pm}B_{-}^{\delta-1}B_{c}^{-\delta}|t|^{-\gamma+\beta(\delta-1)}\to\Gamma_{\pm}B_{-}^{\delta-1}B_{c}^{-\delta},$
$\displaystyle R_{\pm}^{(3)}(t)$ $\displaystyle=$
$\displaystyle\frac{\chi_{\pm}(t)}{\xi_{\pm}^{2-\eta}(t)}=\frac{\Gamma_{\pm}}{\xi_{0\pm}^{2-\eta}}|t|^{-\gamma+\nu(2-\eta)}\to\frac{\Gamma_{\pm}}{\xi_{0\pm}^{2-\eta}},$
(20)

that reach their respective universal values. Eventually, Eq. (4) leads to
consider the following combination

$R_{\pm}^{(4)}(t)=\xi_{\pm}^{d}(t)f_{\pm}^{\rm
sing}(t)=\xi_{0\pm}^{d}F_{\pm}|t|^{-d\nu+2-\alpha}\to\xi_{0\pm}^{d}F_{\pm}$
(21)

as universal also.

##  2 From the universal combinations of amplitudes to scaling laws among
hatted exponents

Having the universal combinations of amplitudes at hand, we consider now the
case where the critical behaviour is described, besides the leading
singularities, by multiplicative logarithmic corrections. This may happen for
example for a system at its upper critical dimension $d_{\rm uc}$, or in the
case of the $2d$ $4-$states Potts model, or the $2d$ disordered Ising model as
well. Many examples can be found in Ref. [3, 4, 2].

Let us first remind the standard definitions of some exponent combinations
which will occur below: $\alpha_{c}=\alpha/\beta\delta$,
$\gamma_{c}=\gamma/\beta\delta$, $\nu_{c}=\nu/\beta\delta$,
$\epsilon_{c}=(1-\alpha)/\beta\delta$. The logarithmic corrections can appear
either in the approach to the critical temperature when the magnetic field is
fixed at zero, or on the other hand right at $T_{c}$, when the magnetic field
approaches zero:

$\displaystyle h=0,\ t\to 0^{\pm},$ (28) $\displaystyle f^{\rm
sing}_{\pm}(t,0)\simeq F_{\pm}|t|^{2-\alpha}(-\ln|t|)^{\hat{\alpha}},$
$\displaystyle m_{-}(t,0)\simeq B_{-}|t|^{\beta}(-\ln|t|)^{\hat{\beta}},$
$\displaystyle
e_{\pm}(t,0)\simeq\frac{A_{\pm}}{(1-\alpha)}|t|^{1-\alpha}(-\ln|t|)^{\hat{\alpha}},$
$\displaystyle\chi_{\pm}(t,0)\simeq\Gamma_{\pm}|t|^{-\gamma}(-\ln|t|)^{\hat{\gamma}},$
$\displaystyle
C_{\pm}(t,0)\simeq{A_{\pm}}|t|^{-\alpha}(-\ln|t|)^{\hat{\alpha}},$
$\displaystyle\xi_{\pm}(t,0)\simeq\xi_{0\pm}|t|^{-\nu}(-\ln|t|)^{\hat{\nu}},$

$\displaystyle t=0,\ h\to 0^{\pm},$ (35) $\displaystyle f_{c}^{\rm
sing}(0,h)\simeq F_{c}|h|^{1+1/\delta}(-\ln|h|)^{\hat{\delta}_{c}},$
$\displaystyle m_{c}(0,h)\simeq
B_{c}|h|^{1/\delta}(-\ln|h|)^{\hat{\delta}_{c}},$ $\displaystyle
e_{c}(0,h)\simeq E_{c}|h|^{\epsilon_{c}}(-\ln|h|)^{\hat{\epsilon}_{c}},$
$\displaystyle\chi_{c}(0,h)\simeq\Gamma_{c}|h|^{-\gamma_{c}}(-\ln|h|)^{\hat{\delta}_{c}},$
$\displaystyle
C_{c}(0,h)\simeq\frac{A_{c}}{\alpha_{c}}|h|^{-\alpha_{c}}(-\ln|h|)^{\hat{\alpha}_{c}},$
$\displaystyle\xi_{c}(0,h)\simeq\xi_{c}|h|^{-\nu_{c}}(-\ln|h|)^{\hat{\nu}_{c}}.$

We can also define at criticality $t=h=0$ the logarithmic correction of the
correlation function, defining the exponents $\hat{\eta}$ that will play an
essential role in the following of this paper:

$G(0,0,|{\bf r}|)\simeq g_{0}|{\bf r}|^{-(d-2+\eta)}(\ln|{\bf
r}|)^{\hat{\eta}}.$ (36)

We mostly use the notations of Refs. [5, 2], with quantities (amplitudes and
exponents) at the critical temperature defined with the subscript $c$, except
for $\delta$ in (35) which is standard according to the terminology fixed by
Fisher long ago [6].

Ralph Kenna and his co-workers, Des Johnston and Wolfhard Janke, have
established a series of scaling relations [3, 4, 2] among “hatted exponents”,
as Ralph was used to call them. Their approach was based on the zeros of the
partition function, either the Lee-Yang zeros (in complex magnetic field) or
the Fisher zeros (in complex temperature).

Here, we offer an alternative derivation of most of these scaling laws,
probably simpler in its approach. Universality assumes that the previous
ratios of amplitudes are still universal when multiplicative logarithmic
corrections are present, i.e.

$R_{\pm}^{(1)}(t)=\frac{m_{-}^{2}(-|t|)}{C_{\pm}(|t|)\chi_{\pm}(|t|)|t|^{2}}=\frac{B_{-}^{2}}{C_{\pm}\Gamma_{\pm}}(-\ln|t|)^{2\hat{\beta}-\hat{\alpha}-\hat{\gamma}}.$
(37)

The fact that this quantity must tend to ${B_{-}^{2}}/{C_{\pm}\Gamma_{\pm}}$
now demands that

$2\hat{\beta}=\hat{\alpha}+\hat{\gamma}.$ (38)

This is the first of Ralph and coworkers’ scaling relations. Using the same
method, the second ratio easily leads to a second relation,

$\hat{\gamma}+\hat{\beta}(\delta-1)-\delta\hat{\delta}=0.$ (39)

The amplitude of the Lee-Yang edge, $h_{0\pm}$, has a non trivial dependence
with the metric factors,
$h_{0\pm}\sim\kappa_{t}^{\beta\delta}\kappa_{h}^{-1}$. This can be retrieved
from the scaling relation $\Delta=\beta\delta$, and the universality of the
ratio

$R^{(5)}(t)=\frac{m_{c}(h^{\rm
LY}_{\pm}(t))}{m_{-}(t)}=\frac{B_{c}h_{0\pm}^{1/\delta}}{B_{-}}|t|^{\Delta/\delta-\beta}(-\ln|t|)^{\hat{\Delta}/\delta+\hat{\delta}-\hat{\beta}}\to\frac{B_{c}h_{0\pm}^{1/\delta}}{B_{-}}$
(40)

requires that

$\hat{\Delta}=(\hat{\beta}-\hat{\delta})\delta.$ (41)

The scaling relations (38) and (39) were first derived in Ref. [3]. Instead of
(41), Ralph and his co-workers had $\hat{\Delta}=\hat{\beta}-\hat{\gamma}$
which is recovered here using (39) and (41).

In the same paper, they also derived

$\hat{\hbox{\coppa}}=\hat{\nu}+\nu\hat{\alpha}/(2-\alpha).$ (42)

This is an analogue of the hyperscaling relation for logarithmic relations. A
new pseudo-critical exponent [3, 4, 11] appears there, $\hat{\hbox{\coppa}}$,
that describes the finite-size scaling (FSS) of the correlation length,

$\xi_{L}\simeq L(\ln L)^{\hat{\hbox{\footnotesize\coppa}}}.$ (43)

This behaviour is encoded in the scaling hypothesis for the correlation
length, appropriately extended to account for the logarithmic correction:

$\xi_{\pm}(t,h,L^{-1})=b(\ln
b)^{\hat{\hbox{\footnotesize\coppa}}}{\mathscr{X}}(\kappa_{t}b^{y_{t}}(\ln
b)^{\hat{y}_{t}}t,\kappa_{h}b^{y_{h}}(\ln b)^{\hat{y}_{h}}h,bL^{-1}).$ (44)

Like $y_{t}$ (resp. $y_{h}$) is the RG eigenvalue associated with the scaling
field $t$ (resp. $h$), we denote ${\hat{y}}_{t}$ (resp. ${\hat{y}}_{h}$) the
corresponding exponent of the logarithmic correction. For the sake of clarity,
we will later denote the rescaled variables as $x(b)=\kappa_{t}b^{y_{t}}(\ln
b)^{\hat{y}_{t}}t$, $y(b)=\kappa_{h}b^{y_{h}}(\ln b)^{\hat{y}_{h}}h$, and
$z(b)=bL^{-1}$. Eq. (43) follows from the choice $b=L$ at criticality $t=h=0$
in (44). The same scaling form is used in the thermodynamic limit
$L\to\infty$, setting $x=1$. This requires iterations

$\displaystyle b$ $\displaystyle=$
$\displaystyle(\kappa_{t}|t|)^{-1/y_{t}}(\ln b)^{-\hat{y}_{t}/y_{t}}$
$\displaystyle\simeq$
$\displaystyle(\kappa_{t}|t|)^{-1/y_{t}}(-\ln|t|)^{-\hat{y}_{t}/y_{t}}\Bigl{(}1+\frac{\ln(-\ln|t|)}{(-\ln|t|)}+\
\hbox{higher order correction}\Bigr{)}.$

Inserted in the expression of the correlation length leads to leading order to

$\xi_{\pm}(t,0,0)\simeq|t|^{-1/y_{t}}(-\ln|t|)^{\hat{\hbox{\footnotesize\coppa}}-\hat{y}_{t}/y_{t}}{\mathscr{X}}(1,0,0)$
(46)

and requires the usual relation $\nu=1/y_{t}$, but also, to conform to (28),

$\hat{\hbox{\coppa}}=\hat{\nu}+\nu\hat{y}_{t}.$ (47)

We now show that this agrees with Ralph’s scaling relation (42). For that
purpose, we use the compatibility with the extension of the phenomenological
Widom scaling assumption for the free energy density (12) to the presence of
logarithmic corrections, written as far as we know for the first time by Ralph
Kenna in Ref. [15],

$f^{\rm
sing}_{\pm}(t,h,L^{-1})=b^{-d}{\mathscr{F}}_{\pm}(\kappa_{t}b^{y_{t}}(\ln
b)^{\hat{y}_{t}}t,\kappa_{h}b^{y_{h}}(\ln b)^{\hat{y}_{h}}h,bL^{-1}).$ (48)

The second derivative w.r.t. $t$ is the specific heat, and the choice $x=1$ at
$h=L^{-1}=0$ then leads (using $\alpha=(2y_{t}-d)/y_{t}$) to

$C_{\pm}(t,0,0)\simeq|t|^{-\alpha}(-\ln|t|)^{2\hat{y}_{t}-\alpha\hat{y}_{t}}{\mathscr{C}}_{\pm}(1,0,0),$
(49)

(from now on, we always limit (LABEL:eq-highorders) to leading logarithmic
order), hence, from (28),

$\hat{\alpha}=(2-\alpha)\hat{y}_{t}$ (50)

which completes the proof.

##  3 Solving a disagreement with our friends

In Ref. [2], Ralph Kenna has given a complete account of these, and many more
scaling relations among hatted exponents. This is not our purpose here to be
exhaustive, but rather to show alternative derivations, or to complete what
Ralph and coworkers didn’t do. With this perspective in mind, Eqs. (44), (48)
and an analogous homogeneity form for the correlation function (discussed
later) offers an option to proceed as we show now.

In Ref. [4], two other scaling relations between hatted exponents were
derived:

$\displaystyle\hat{\alpha}=d(\hat{\hbox{\coppa}}-\hat{\nu})\quad\hbox{or}\quad\hat{\alpha}=1+d(\hat{\hbox{\coppa}}-\hat{\nu}),$
(51) $\displaystyle\hat{\eta}=\hat{\gamma}-\hat{\nu}(2-\eta).$ (52)

Concerning the first relation (51), the second formula is valid in such
circumstances where the model has $\alpha=0$ and an impact angle
$\phi\not=\pi/4$ for the Fisher zeros in the complex plane (this is the case
for the pure two-dimensional Ising model). We will not consider this case, but
rather the more general case of the first formula. It can be rederived by
careful use of the ratio 4 (Eq. (21)) in section 1, and even requires to use
of FSS of the correlation length. From
$\xi_{\pm}(t)\simeq|t|^{-\nu}(-\ln|t|)^{\hat{\nu}}$, we first reverse to
$|t|\simeq\xi_{\pm}^{-1/\nu}(t)(\ln\xi_{\pm}(t))^{\hat{\nu}/\nu}$. This
expression is then incorporated into (28) to get

$f^{\rm
sing}_{\pm}(t)\simeq\xi_{\pm}^{-d}(t)(\ln\xi_{\pm}(t))^{d\hat{\nu}+\hat{\alpha}}$
(53)

i.e. a modified version of (21):

$R_{\pm}^{(4bis)}(t)=\frac{\xi_{\pm}^{d}(t)f_{\pm}^{\rm
sing}(t)}{(\ln\xi_{\pm}(t))^{d\hat{\nu}+\hat{\alpha}}}\to\xi_{0\pm}^{d}F_{\pm}.$
(54)

Now, inserting (43) into (53) leads to the FSS behaviour of the free energy
density at criticality,

$f^{\rm sing}_{L}(0)\simeq L^{-d}(\ln
L)^{-d\hat{\hbox{\footnotesize\coppa}}}(\ln L)^{\hat{\alpha}+d\hat{\nu}},$
(55)

and compatibility with (48) at $t=h=0$, $b=L$ then demands

$\hat{\alpha}=d\hat{\hbox{\footnotesize\coppa}}-d\hat{\nu}.$ (56)

which is Kenna and coworkers’ relation.

The same derivation can be done for the magnetic sector, considering the
approach to criticality at $T_{c}$ for $h\to 0$ and yields the scaling
relation

$\hat{\delta}=d\hat{\hbox{\footnotesize\coppa}}-d\hat{\nu}_{c}.$ (57)

Concerning the last Eq. (52), we believe that this relation is incomplete.
Applied to the $4-$state Potts model in two dimensions [7, 8, 9, 10], which
has $\hat{\eta}=-\frac{1}{8}$, $\hat{\gamma}=\frac{3}{4}$,
$\hat{\nu}=\frac{1}{2}$ and $\eta=\frac{1}{4}$, Eq. (52) is fulfilled. We
believe that this is because there $\hat{\hbox{\coppa}}=0$, and that an
additional term $\hat{\hbox{\coppa}}\gamma/\nu$ is missing in the general
case. A test is provided in the case of the Ising model in 4 dimensions which
has $\hat{\hbox{\coppa}}=\frac{1}{4}$ (models at their upper critical
dimensions have $\hat{\hbox{\coppa}}=1/d_{\rm uc}$ [11, 13, 12, 14]). There,
Ralph and his co-authors had anticipated that $\hat{\eta}=0$ for Eq. (52) to
work ($\hat{\gamma}=\frac{1}{3}$, $\hat{\nu}=\frac{1}{6}$ and $\eta=0$ for the
Ising model in 4 dimensions), but according to Luijten [16],
$\hat{\eta}=\frac{1}{2}$ instead, a result that is in contradiction with Eq.
(52).

Let us examine the problem in more detail. In Ref. [4], the authors have
questioned the relation between the correlation function and the square of the
magnetization when the system decorrelates, i.e. for $|{\bf r}|\to\infty$:

$G(t,h,L^{-1}\to\infty,|{\bf r}|\to\infty)\to m^{2}(t,h,L^{-1}\to\infty).$
(58)

On the contrary, we assume that there is no reason why this would not be
valid, so we start from the homogeneity of the (spin-spin) correlation
function with logarithmic corrections as

$G(t,h,L^{-1},|{\bf r}|)=b^{-(d-2+\eta)}(\ln
b)^{\hat{\eta}}G(x(b),y(b),z(b),|{\bf r}|/b),$ (59)

Setting $x=1$, $y=0$ and the thermodynamic limit $z=0$ leads to the following
temperature behaviour when $|{\bf r}|\to\infty$

$G(t,0,0,\infty)\simeq|t|^{\frac{d-2+\eta}{y_{t}}}(-\ln|t|)^{\frac{d-2+\eta}{y_{t}}\hat{y}_{t}+\hat{\eta}}G(1,0,0,\infty)=m^{2}(t,0,0).$
(60)

This requires the usual relation $2\beta=\frac{d-2+\eta}{y_{t}}$ and

$\hat{\eta}=2(\hat{\beta}-\beta\hat{y}_{t}).$ (61)

The two examples given above are test grounds. For the $4-$state Potts model
in two dimensions, we extract immediately $\hat{\eta}=-\frac{1}{8}$ which is
correct. For the $4d$ Ising model on the other hand, we obtain
$\hat{\eta}=\frac{1}{2}$, in agreement with Luijten’s result [16], later
verified numerically in Ref. [17], but we are here in contradiction with the
prediction of Refs. [4, 2].

Since the question is of importance, we want to consider it from other
perspectives also. The correlation function is linked to the susceptibility
via the fluctuation-dissipation relation:

$\chi(0,0,L^{-1})=\int_{0}^{L}d^{d}r\ \\!|{\bf r}|^{-(d-2+\eta)}(\ln|{\bf
r}|)^{\hat{\eta}}G(0,0,|{\bf r}|L^{-1},1).$ (62)

Setting $\rho=|{\bf r}|/L$, we have

$\displaystyle\chi(0,0,L^{-1})$ $\displaystyle=$ $\displaystyle L^{2-\eta}(\ln
L)^{\hat{\eta}}\int_{0}^{1}d^{d}\rho\
\\!\rho^{-(d-2+\eta)}\Bigl{(}1+\frac{\ln\rho}{\ln
L}\Bigr{)}^{\hat{\eta}}G(0,0,\rho,1)$ (63) $\displaystyle\simeq$
$\displaystyle L^{2-\eta}(\ln L)^{\hat{\eta}},$

and, since the susceptibility obeys, via the second derivative of (48), to

$\chi(t,h,L^{-1})=\kappa_{h}^{2}b^{-d+2y_{h}}(\ln
b)^{2\hat{y}_{h}}{\mathscr{Y}}(x(b),y(b),z(b)),$ (64)

its FSS compared to (63) demands that $2-\eta=2y_{d}-d=\gamma/\nu$ and

$\hat{\eta}=2\hat{y}_{h}.$ (65)

Again, this confirms $\hat{\eta}=\frac{1}{2}$ for the $4d$ Ising model.

The fluctuation-dissipation theorem has also been used in Refs. [4, 2] in the
form

$\chi_{\infty}(t)\sim\xi_{\infty}^{2-\eta}(t)(\ln\xi_{\infty}(t))^{\hat{\eta}},$
(66)

from where Eq. (52) follows, so there is still some difficulty hidden to solve
our disagreement. Let us set $b=\xi_{\infty}(t)$ in (64):

$\chi_{\infty}(t)=\xi_{\infty}^{-d+2y_{h}}(t)(\ln\xi_{\infty}(t))^{\hat{\eta}}{\mathscr{Y}}(x(\xi_{\infty}(t)),0)$
(67)

where the variable $x$ in the scaling function ${\mathscr{Y}}$ is evaluated at
$\xi_{\infty}(t)$ to give

$x(\xi_{\infty}(t))=\xi_{\infty}(t)^{y_{t}}(\ln\xi_{\infty}(t))^{\hat{y}_{t}}|t|=(-\ln|t|)^{\hat{\hbox{\footnotesize\coppa}}/\nu}.$
(68)

The scaling function must have the behaviour ${\mathscr{Y}}(x,0)\sim
x^{-\gamma}$ when $|t|\to 0$ to recover the temperature singularity of the
susceptibility $\chi_{\infty}(t,0)\sim|t|^{-\gamma}(-\ln|t|)^{\hat{\gamma}}$.
It follows that instead of (66), one has

$\chi_{\infty}(t)\sim\xi_{\infty}^{2-\eta}(t)(\ln\xi_{\infty}(t))^{\hat{\eta}-\gamma\hat{\hbox{\footnotesize\coppa}}/\nu}$
(69)

and instead of Eq. (52), one has a third form for the exponent $\hat{\eta}$:

$\hat{\eta}=\hat{\gamma}-\hat{\nu}(2-\eta)+\gamma\frac{\hat{\hbox{\footnotesize\coppa}}}{\nu},$
(70)

again compatible with $\hat{\eta}=\frac{1}{2}$ for the $4d$ Ising model. This
suggests to use, instead of the ratio $R_{\pm}^{(3)}(t)$, the modified version

$\displaystyle R_{\pm}^{(3{\rm bis})}(t)$ $\displaystyle=$
$\displaystyle\frac{\chi_{\pm}(t)}{\xi_{\pm}^{2-\eta}(t)(\ln\xi_{\pm}(t))^{\hat{\eta}-\gamma\hat{\hbox{\footnotesize\coppa}}/\nu}}$
(71) $\displaystyle=$
$\displaystyle\frac{\Gamma_{\pm}}{\xi_{0\pm}^{2-\eta}}(-\ln|t|)^{\hat{\gamma}-\hat{\eta}-\gamma\hat{\hbox{\footnotesize\coppa}}/\nu-\hat{\nu}(2-\eta)}\to\frac{\Gamma_{\pm}}{\xi_{0\pm}^{2-\eta}}$

which, again, is universal. The 4 standard scaling laws and the corresponding
4 hatted scaling laws are listed in table 1.

Ratio | scaling relation | hatted scaling relation
---|---|---
$\frac{m_{-}^{2}(-|t|)}{C_{\pm}(|t|)\chi_{\pm}(|t|)|t|^{2}}$ | $2\beta+\gamma=2-\alpha$ | $2\hat{\beta}=\hat{\alpha}+\hat{\gamma}$
$\left.\frac{\chi_{\pm}(t)m_{-}^{\delta-1}(-|t|)h}{m_{c}^{\delta}(h)}\right|_{h=h_{\pm}^{\rm LY}(|t|)}$ | $\beta(\delta-1)=\gamma$ | $\hat{\gamma}+\hat{\beta}(\delta-1)-\delta\hat{\delta}=0$
$\frac{\chi_{\pm}(t)}{\xi_{\pm}^{2-\eta}(t)(\ln\xi_{\pm}(t))^{\hat{\eta}-\gamma\hat{\hbox{\footnotesize\coppa}}/\nu}}$ | $\nu(2-\eta)=\gamma$ | $\hat{\gamma}-\hat{\nu}(2-\eta)+\gamma\frac{\hat{\hbox{\footnotesize\coppa}}}{\nu}=\hat{\eta}$
$\xi_{\pm}^{d}(t)f_{\pm}^{\rm sing}(t)$ | $2-\alpha=\nu d$ | $\hat{\alpha}=d\hat{\hbox{\coppa}}-d\hat{\nu}$
Table 1: The main scaling laws and hatted scaling laws and the associated
universal combinations of amplitudes.

So far so good, but the situation is not yet clear, since the case of
percolation in 6 dimensions (its upper critical dimension) is maybe a
counterexample. With $\gamma=1$, $\nu=\frac{1}{2}$, $\eta=0$, and the values
of the logarithmic correction exponents $\hat{\gamma}=\frac{2}{7}$,
$\hat{\nu}=\frac{5}{42}$ and of the pseudo-critical exponent
$\hat{\hbox{\coppa}}=\frac{1}{d_{\rm uc}}=\frac{1}{6}$, using Eq. (70) we
predict $\hat{\eta}=\frac{8}{21}$, while Kenna and coworkers predict
$\hat{\eta}=\frac{1}{21}$ from Eq. (52). This latter result conforms to an
analytic prediction from Ref. [18], but on the other hand, our value is
supported by an FSS prediction by Ruiz-Lorenzo [19], $\chi_{L}L^{-2}\sim(\ln
L)^{8/21}$. This disagreement demands further analysis.

In Ref. [2], Kenna has listed the values of the known hatted critical
exponents for a series of models, and when $\hat{\eta}$ was not known, he has
proposed the expected value from the use of Eq. (52). An interesting model is
missing from the list, the tricritical Ising model, which has logarithmic
corrections at its upper critical dimension $d_{\rm uc}=3$ and has non-zero
$\hat{\hbox{\coppa}}=1/3$. We will now analyse this universality class in more
detail.

##  4 The tricritical Ising universality class in the Blume-Capel model at the
upper critical dimension

The spin-$1$ Blume-Capel model [20, 21] is a lattice model defined by the
Hamiltonian

$\mathcal{H}=-J\sum_{\langle
i,j\rangle}\sigma_{i}\sigma_{j}+\Delta\sum_{i}\sigma_{i}^{2}-H\sum_{i}\sigma_{i},$
(72)

where the spin variables $\sigma_{i}=-1,0,+1$, $J>0$ denotes the ferromagnetic
exchange interaction among nearest-neighbour sites ($\langle i,j\rangle$
indicates summation over nearest neighbours), and $\Delta$ is the crystal-
field strength that controls the density of vacancies (the $\sigma_{i}=0$
states can be viewed as vacancies in an ordinary $\sigma_{i}=\pm 1$ Ising
model) [22]. For $\Delta=-\infty$, vacancies are suppressed from the partition
function and the Hamiltonian reduces to that of the Ising ferromagnet. At
$\Delta=0$ the second-order transition is in the pure Ising model universality
class. When $\Delta$ increases from $0$, a perturbation theory shows that the
transition temperature decreases along a line which remains of Ising-like
second-order phase transition. On the other hand, in the vicinity of $T=0$ the
transition is first-order and persists first-order at small values of $T$
until it reaches the second-order line. Right at the limit, there is a
tricritical point characterized by specific values of $T_{t}$, $\Delta_{t}$.
Tricriticality corresponds to the $\phi^{6}$ Landau expansion [23] and the
upper critical dimension is thus $d_{\rm uc}=3$, the case that we consider
now.

Figure 1: Typical phase diagram of the Blume-Capel model in the $(T,\Delta)$
plane. The $\lambda$ line is a line of second-order phase transition in the
Ising model universality class that ends at a tricritical point of coordinates
$(T_{t},\Delta_{t})$. The dashed line is a first-order transition line. The
most singular even scaling field at the tricritical point is $g$, a linear
combination of $\tau$ and of $\Delta-\Delta_{t}$.

Using appropriate linear combinations of the physical parameters, determined
by the geometry of the phase diagram and the fact that they have to vanish at
the tricritical point, the even scaling fields are $\tau=(T-T_{t})/T_{t}$,
$g=(\Delta-\Delta_{t})/T_{t}+a\tau$, and the odd scaling field is the usual
magnetic field $h=H/T_{t}$. Lawrie and Sarbach [24] have shown that the free
energy density at the tricritical point in $3d$ reads, in terms of these
scaling fields, as

$f^{\rm sing}_{\rm tri}(\tau,g,h)=b^{-3}{\mathscr{F}}_{\pm}(b^{1}(\ln
b)^{\frac{4}{15}}\tau,b^{2}(\ln b)^{\frac{1}{3}}g(1-a^{-1}b^{-1}(\ln
b)^{-\frac{1}{15}}\tau g),b^{\frac{5}{2}}(\ln b)^{\frac{1}{6}}h),$ (73)

where the non universal metric factors $\kappa_{i}$ have been omitted. Here,
the subscript ‘tri’ indicates that the expression is valid in the vicinity of
the tricritical point where $\tau=g=h=0$. The logarithmic corrections are
explicitly given in this expression.

The notations in Ref. [24] necessitate to be adapted to be consistent with
those that we have used until now. Leaving aside the logarithmic corrections
for a while, let us write

$f^{\rm sing}_{\rm
tri}(\tau,g,h)=b^{-d}{\mathscr{F}}_{\pm}(b^{y_{\tau}}\tau,b^{y_{g}}g,b^{y_{h}}h).$
(74)

The dominant even scaling field is $g$, the usual singularities and critical
exponents are therefore defined by their behaviours w.r.t. $g$ instead of
$\tau$ which brings corrections to scaling due to crossover. This means that
the $t$ and $y_{t}$ of the three previous sections of this paper will now be
replaced by $g$ and $y_{g}$. At $h=0$ and $\tau=0$, setting $b=|g|^{-1/y_{g}}$
leads to $f^{\rm sing}_{\rm tri}(0,g,0)\sim|g|^{d/y_{g}}$. This is compatible
with $\nu=1/y_{g}$. The specific heat measures the total energy fluctuations.
Its most singular part is defined by

$\displaystyle C(\tau,g,h)=\frac{\partial^{2}f^{\rm sing}_{\rm tri}}{\partial
g^{2}}=b^{-d+2y_{g}}{\mathscr{C}}_{\pm}(b^{y_{\tau}}\tau,b^{y_{g}}g,b^{y_{h}}h)\sim|g|^{-\alpha}\quad\hbox{when}\
\tau,h=0$

with $\alpha=\frac{2y_{g}-d}{y_{g}}$. This shows that the exponent $d/y_{g}$
of $f^{\rm sing}_{\rm tri}(0,g,0)$ is thus equal to the usual value $2-\alpha$
and the hyperscaling relation holds.

We can also define less singular exponents w.r.t. the scaling field $\tau$
(with tilde notation), e.g. $C(\tau,0,0)\sim|\tau|^{-\tilde{\alpha}}$. A
similar analysis as above shows that $f^{\rm sing}_{\rm
tri}(\tau,0,0)\sim|\tau|^{d/y_{\tau}}$ with
$d/y_{\tau}=(d/y_{g})\phi=(2-\alpha)\phi$ with $\phi=y_{g}/y_{\tau}$ the
crossover exponent. There is a caveat here since $d/y_{\tau}$ is not equal to
$2-\tilde{\alpha}$ as one can find in the literature [24]. Indeed,
$\tilde{\alpha}=(2y_{g}-d)/y_{\tau}=\alpha\phi$, hence
$2-\tilde{\alpha}=2-\alpha\phi\not=(2-\alpha)\phi$. This is important to
collect correct expressions, and this is done in table 2.

leading even field, $g$ |  subleading even field, $\tau$ |  leading odd field, $h$
---|---|---
$C(0,g,0)\sim|g|^{-\alpha}$ | $\alpha=\frac{2y_{g}-d}{y_{g}}$ | $C(\tau,0,0)\sim|\tau|^{-\tilde{\alpha}}$ | $\tilde{\alpha}=\alpha\phi$ |  |
$m(0,g,0)\sim|g|^{\beta}$ | $\beta=\frac{d-y_{h}}{y_{g}}$ | $m(\tau,0,0)\sim|\tau|^{\tilde{\beta}}$ | $\tilde{\beta}=\beta\phi$ | $m(0,0,h)\sim|h|^{1/\delta}$ | $\delta=\frac{y_{h}}{d-y_{h}}$
$\chi(0,g,0)\sim|g|^{-\gamma}$ | $\gamma=\frac{2y_{h}-d}{y_{g}}$ | $\chi(\tau,0,0)\sim|\tau|^{-\tilde{\gamma}}$ | $\tilde{\gamma}=\gamma\phi$ |  |
Table 2: Leading and subleading singularities for the most common physical
quantities and the definitions of the associated exponents. Here the crossover
exponent is $\phi=y_{g}/y_{\tau}$.

This being said, we can now incorporate the logarithmic corrections in Eq.
(74) to get

$f^{\rm sing}_{\rm tri}(\tau,g,h)=b^{-d}{\mathscr{F}}_{\pm}(b^{y_{\tau}}(\ln
b)^{{\hat{y}}_{\tau}}\tau,b^{y_{g}}(\ln b)^{{\hat{y}}_{g}}g,b^{y_{h}}(\ln
b)^{{\hat{y}}_{h}}h),$ (76)

and the comparison with Eq. (73) simply leads to

$\displaystyle y_{g}=2,\quad y_{h}=\frac{5}{2},\quad y_{\tau}=1,\quad$ (77)
$\displaystyle\hat{y}_{g}=\frac{1}{3},\quad\hat{y}_{h}=\frac{1}{6},\quad\hat{y}_{\tau}=\frac{4}{15},\quad\hat{\hbox{\coppa}}=\frac{1}{3}.$
(78)

| leading exponent | logarithmic correction exponent
---|---|---
Quantity |  | IM4D | Tri. IM | Perco. |  | IM4D | Tri. IM | Perco.
$C(t,0)$ | $\alpha=\frac{2y_{t}-d}{y_{t}}$ | $0$ | $\frac{1}{2}$ | $-1$ | $\hat{\alpha}=(2-\alpha)\hat{y}_{t}$ | $\frac{1}{3}$ | $\frac{1}{2}$ | $\frac{2}{7}$
$m_{-}(t,0)$ | $\beta=\frac{d-y_{h}}{y_{t}}$ | $\frac{1}{2}$ | $\frac{1}{4}$ | $1$ | $\hat{\beta}=\beta\hat{y}_{t}+\hat{y}_{h}$ | $\frac{1}{3}$ | $\frac{1}{4}$ | $\frac{2}{7}$
$\chi(t,0)$ | $\gamma=\frac{2y_{h}-d}{y_{t}}$ | $1$ | $1$ | $1$ | $\hat{\gamma}=2\hat{y}_{h}-\gamma\hat{y}_{t}$ | $\frac{1}{3}$ | $0$ | $\frac{2}{7}$
$m_{c}(0,h)$ | $\frac{1}{\delta}=\frac{d-y_{h}}{y_{h}}$ | $\frac{1}{3}$ | $\frac{1}{5}$ | $\frac{1}{2}$ | $\hat{\delta}=\frac{1}{\delta}\hat{y}_{h}+\hat{y}_{h}$ | $\frac{1}{3}$ | $\frac{1}{5}$ | $\frac{2}{7}$
$h_{\rm LY}(t,0)$ | $\Delta=\frac{y_{h}}{y_{t}}$ | $\frac{3}{2}$ | $\frac{5}{4}$ | $2$ | $\hat{\Delta}=\Delta\hat{y}_{t}-\hat{y}_{h}$ | $0$ | $\frac{1}{4}$ | $0$
$\xi(t,0)$ | $\nu=\frac{1}{y_{t}}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\hat{\nu}=\hat{\hbox{\coppa}}-\nu\hat{y}_{t}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{5}{42}$
$G(0,0,|{\bf r}|)$ | $\eta=d-2y_{h}+2$ | $0$ | $0$ | $0$ | $\hat{\eta}=2\hat{y}_{h}$ | $\frac{1}{2}$ | $\frac{1}{3}$ | $\frac{8}{21}$
Table 3: Leading and logarithmic correction exponents for the most common
physical quantities.

We can now deduce the values of the standard critical exponents and the
associated logarithmic corrections exponents. They are listed in Table 3 for
three universality classes which all have non-zero values of the pseudo-
critical exponent $\hat{\hbox{\coppa}}$, the Ising model in four dimensions,
the tricritical Ising model in three dimensions, and the problem of
percolation in six dimensions.

In this table, the first six lines are not controversial. The seventh line
presents the correlation function correction exponent $\hat{\eta}$ which
follows from our scaling law, in any of the forms given in Eqs. (61), (65) or
(70). These three expressions are mutually consistent, but they differ from
Eq. (52) used by Kenna and his co-authors. This latter formula would
respectively predict for the three universality classes the values $0$,
$-\frac{1}{3}$ and $\frac{1}{21}$. We have seen that the first value, $0$, is
falsified in the $4d$ IM case by Refs. [16, 17], but the results of Ref. [18]
invalidates our third value $\frac{8}{21}$, the last entry of in table 3 while
Ref. [19] on the contrary supports this value.

The case of the tricritical Ising model in three dimensions appears crucial
and we have to provide numerical results in support of our result. The
numerical computation of the correlation function is known to be a very
delicate problem and we will approach the value of the exponent $\hat{\eta}$
differently, using FSS. Another delicate aspect is the also well-known fact
that extracting logarithmic corrections in the vicinity of a critical point
can be extremely difficult [25]. Recently, it was found that very accurate
results can be obtained numerically in the Blume-Capel model with relatively
small system sizes [26] via the analysis of the zeros of the partition
function, and in particular the Lee-Yang zeros [27, 28]. The Lee-Yang zeros
are connected to the susceptibility [29] via

$\chi(g,0,L^{-1})\simeq L^{-d}\sum_{j=1}^{L^{d}}h_{j}^{-2}(g,0,L^{-1}),$ (79)

where $j$ labels the zeros in the upper half complex plane are indexed in
order of increasing distance from the critical point. The sum is dominated by
the lowest zero, the Lee-Yang edge $h^{\rm LY}$, and at the tricritical point,
the FSS of the susceptibility is thus linked to that of $h^{\rm LY}$:

$\chi(0,0,L^{-1})\simeq L^{-d}h^{\rm LY}(0,0,L^{-1})^{-2}\simeq L^{2-\eta}(\ln
L)^{\hat{\eta}}.$ (80)

In the presence of logarithmic corrections to the scaling form of the Lee-Yang
edge obeys

$h^{\rm LY}(g,h)=b^{-y_{h}}(\ln
b)^{-\hat{y}_{h}}{\mathscr{H}}_{\pm}(b^{y_{\tau}}(\ln
b)^{{\hat{y}}_{\tau}}\tau,b^{y_{g}}(\ln b)^{{\hat{y}}_{g}}g,b^{y_{h}}(\ln
b)^{{\hat{y}}_{h}}h)$ (81)

compatible with the behaviour in terms of the thermal scaling field $g$, as it
can be shown using the scaling laws of Table 3, $h^{\rm
LY}(g,0)\sim|g|^{\Delta}(-\ln|g|)^{\hat{\Delta}}$. If one sits exactly at the
tricritical point, $\tau=g=h=0$, we can extract the FSS behaviour of the zeros
by setting $b=L$,

$h^{\rm LY}(0,0,L^{-1})\simeq L^{-y_{h}}(\ln L)^{-\hat{y}_{h}}$ (82)

and it follows that we expect

$h^{\rm LY}(0,0,L^{-1})\simeq L^{(\eta-2-d)/2}(\ln L)^{-\hat{\eta}/2}$ (83)

which agrees with Eq. (80).

As we said, this form can be checked with good accuracy at the price of
relatively light Monte Carlo simulations. The coordinates of the tricritical
point of the Blume-Capel model in $3d$ are found in the literature [30],
$T_{t}\simeq 1.4182$, $\Delta_{t}\simeq 2.84479(30)$, but the value of
$\Delta_{t}$ does not seem to be as accurate as that of the temperature and
for example Zierenberg et al [31] report instead $\Delta_{t}\simeq 2.8446(3)$.
Let us first analyse this problem ourselves. In Figure 2, we report the FSS of
the magnetization at $T_{t}\simeq 1.4182$ for several values of $\Delta$
ranging from 2.8440 to 2.8448. The magnetization is expected to follow the FSS
behaviour

$m(0,0,L^{-1})\sim L^{-d+y_{h}}(\ln L)^{\hat{y}_{h}}$ (84)

with $d-y_{h}=\frac{1}{2}$ and $\hat{y}_{h}=\frac{1}{6}$. The data points are
fitted as $m(0,0,L^{-1})L^{\frac{1}{2}}=a(\ln L)^{b}$ with $a,b$ free
parameters. The closest to the expected result (the black dashed line) is at
$\Delta=2.8442$ where we get $b=0.155\pm 0.007$. We will thus consider this
value of $\Delta$ as our estimate for the coordinate of the tricritical point
$\Delta_{t}$.

Figure 2: FSS of the magnetization for the tricritical Ising model (Blume-
Capel model at its tricritical temperature) in $3d$ at $T_{t}=1.4182$ and
various values of the crystal field parameter $\Delta$ for sizes ranging from
$L=12$ to $22$. The best fit is for $\Delta=2.8442$ ($\chi^{2}/{\rm
dof}=30.40/4=7.6$ at $\Delta=2.8440$, $\chi 2/{\rm dof}=5.39/4=1.35$ at
$\Delta=2.8442$, $\chi 2/{\rm dof}=28.36/4=7.09$ at $\Delta=2.8446$ and $\chi
2/{\rm dof}=76.74/4=19.18$ at $\Delta=2.8448$).  Figure 3: FSS of the Lee-Yang
edge for the tricritical Ising model (Blume-Capel model at its tricritical
temperature) in $3d$ at $T_{t}=1.4182$ and various values of $\Delta$ for
sizes ranging from $L=12$ to $22$.

The analysis of the Lee-Yang edge is presented in figure 3 with a larger
choice of values of $\Delta$ and, again, the best fit is at
$\Delta_{t}=2.8442$ where the estimate of $\hat{y}_{h}$ is now slightly larger
at $0.172\pm 0.008$.

The reader could still question the sensitivity of the value of $\hat{y}_{h}$
with the choice of tricritical temperature $T_{t}$. Indeed, when one looks at
the FSS of the tricritical magnetization, for example, the effective exponent
of the log term is either positive and close to the expected value or can
differ from the expectation and even be negative, depending on the values of
the crystal field $\Delta$ (see Figure 2). It makes sense to ask whether the
role of $T_{t}$ may also have a significative influence. We believe that the
results presented in this work are reliable and to support the consistency of
the numerical data, we show in figure 4 that slight variations of $T$ change
the regime from the pure 3d Ising model at $T=1.4197$, for which
$y_{h}={2.4815(15)}$ [32] is expected, to first-order at $T=1.4070$ where an
effective FSS $y_{h}=d$ is expected [33]. $T=1.4182$ safely recovers
$y_{h}^{\rm Tri}=2.5$ to a very good accuracy and confirms the tricritical
value of $T_{t}\simeq 1.4182$. Note that the transition line in the phase
diagram in the vicinity of the tricritical point is almost at a fixed value of
$\Delta$, this is why the three regimes are found at the same crystal field
value of $\Delta=2.8442$.

Figure 4: FSS of the Lee-Yang edge in the vicinity of the tricritical point of
the Blume-Capel model in $3d$, at $T_{t}=1.4070$, 1.4182 and 1.4197 and
$\Delta=2.8442$ to show the first-order, the tricritical, and the ordinary
second-order regimes from the values of the corresponding RG dimensions
$y_{h}$.

##  5 Conclusions

The numerical results obtained for the tricritical Ising model universality
class in $3d$ confirm the prediction that $\hat{y}_{h}=\frac{1}{6}$, hence the
prediction $\hat{\eta}=\frac{1}{3}$, while the scaling law of Kenna and co-
workers would have given $-\frac{1}{3}$ instead.

In Ref. [2], Ralph Kenna concluded his review with a table collecting the sets
of critical exponents and hatted critical exponents for various models and
predicting those which were still unknown from the use of the newly discovered
scaling laws, and in particular Eq. (52) that we scrutinize and propose to
replace by Eq. (70) or any of the equivalent forms that we have derived.

In the list, the $O(n)$ model with long-range interactions [34, 35] was
predicted to have $\hat{\eta}=0$. We propose instead $\hat{\eta}=\frac{1}{2}$,
following from $\hat{y}_{t}=(4-n)/[2(n+8)]$, $\hat{y}_{h}=\frac{1}{4}$.

The Lee-Yang edge in $6d$ [19] was predictedto have $\hat{\eta}=\frac{1}{9}$.
We rather have $\hat{y}_{t}=-\frac{2}{9}$ and $\hat{y}_{h}=\frac{2}{9}$, hence
$\hat{\eta}=\frac{4}{9}$.

For lattice animals in $8d$ [19], Kenna predicted $\hat{\eta}=\frac{1}{9}$ and
we have $\hat{y}_{t}=\hat{y}_{h}=\frac{2}{9}$ and $\hat{\eta}=\frac{4}{9}$.

The case of scale-free networks [36, 37, 38] is particular in the sense that
Ralph Kenna did not make any prediction for $\hat{\eta}$, because some
exponents were missing. From those which are known, we can deduce that
$\hat{y}_{t}=-\frac{1}{2}$ and $\hat{y}_{h}=-\frac{1}{4}$ and we deduce thus
$\hat{\eta}=-\frac{1}{2}$ which is a new prediction.

Eventually, we believe that the $n$-colour Ashkin-Teller model in $2d$ is
still under question since the exponents collected by Shalaev and Jug [39] do
not satisfy the “standard” scaling laws, e.g. the values reported do not obey
$\hat{\alpha}+\hat{\gamma}=2\hat{\beta}$.

To finish this paper, we would like to say that the scaling laws discovered by
Ralph Kenna and his co-workers are invaluable because they make it possible to
establish (or falsify) the consistency of the results obtained for various
models. The case of the $n$-colour Ashkin-Teller model in $2d$ is such an
example where it seems that there are still some inconsistencies to solve.
Although we happened to contradict one of these scaling laws, we admire the
piece of work done in Refs. [2, 3, 4] where we recognize Ralph’s footprint.

Acknowledgements: We are thankful to Wolfhard Janke for very interesting
discussions. We thank the Collège Doctoral 02-07 Statistical Physics of
Complex Systems (University of Leipzig-Université de Lorraine) and the L4
collaboration Leipzig - Lviv - Lorraine - Coventry. LM thanks Coventry
University and the Collège Doctoral for financial support.

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