# Numerical revision of the universal amplitude ratios for the two-dimensional
4-state Potts model

Lev N. Shchur∗,∗∗, Bertrand Berche∗∗ and Paolo Butera∗∗∗
∗ Landau Institute for Theoretical Physics,
Russian Academy of Sciences,
Chernogolovka 142432, Russia

∗∗ Laboratoire de Physique des Matériaux,
Université Henri Poincaré, Nancy 1
BP 239, F-54506 Vandœuvre les Nancy Cedex, France

∗∗∗ Istituto Nazionale di Fisica Nucleare,
Sezione di Milano-Bicocca,
Piazza delle Scienze 3, 20126, Milano, Italia

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###### Abstract

Monte Carlo (MC) simulations and series expansion (SE) data for the energy,
specific heat, magnetization and susceptibility of the ferromagnetic 4-state
Potts model on the square lattice are analyzed in a vicinity of the critical
point in order to estimate universal combinations of critical amplitudes. The
quality of the fits is improved using predictions of the renormalization group
(RG) approach and of conformal invariance, and restricting the data within an
appropriate temperature window.

The RG predictions on the cancelation of the logarithmic corrections in the
universal amplitude ratios are tested. A direct calculation of the effective
ratio of the energy amplitudes using duality relations explicitly demonstrates
this cancelation of logarithms, thus supporting the predictions of RG.

We emphasize the role of corrections and of background terms on the
determnination of the amplitudes. The ratios of the critical amplitudes of the
susceptibilities obtained in our analysis differ significantly from those
predicted theoretically and supported by earlier SE and MC analysis. This
disagreement might signal that the two-kink approximation used in the
analytical estimates is not sufficient to describe with fair accuracy the
amplitudes of the 4-state model.

##  1 Introduction

In a first paper [1] (hereafter denoted as I), we studied the universal
combinations of critical amplitudes of the 3-state Potts model. The present
paper is devoted to a similar analysis in the 4-state case, which is much more
involved due to the presence of logarithmic corrections strongly influencing
the critical behavior.

We analyze numerical data obtained in Monte Carlo (MC) simulations using the
Wolff [2] single-cluster algorithm and also the series expansion (SE) data
available in the literature. In the following, we refer to the data type as MC
and SE data, respectively. For comparison with our own results, we shall also
reconsider the data obtained in MC simulations using the Swendsen-Wang cluster
algorithm [3] by Caselle, Tateo, and Vinci [4] and indicated as CTV data.

Our motivation in using different data sets is to achieve a better control of
the critical behavior, since one may expect, for the three sources, some
differences in the critical region due to the different interplay of the
finite size effects. In addition, one can apply different techniques to the
data analysis: the fits in the case of the MC data and the approximant
technique in the case of the SE data. The consistency of the final results
will increase our confidence. We care so much because the presence of
logarithmic corrections makes the numerical determination of the critical
behavior of the 4-state Potts model a rather delicate task.

To be even safer, two different approaches are used, which were successfully
applied also to the $q=3$ state Potts model in I. First we estimate the
critical amplitudes, which are then used to compute universal ratios. Second,
besides the direct determination of the amplitudes themselves, we estimate
ratios of critical amplitudes, constructing effective ratio functions, and
computing their limiting values at the critical point. This provides a direct
estimate of universal ratios. Analyzing the renormalization group equations,
we have shown in an Appendix (see also [5, 6]) that, in the absence of any
regular background term, the logarithmic corrections cancel in the effective
ratio functions.

The Hamiltonian of the ferromagnetic Potts model [7] reads as

$H=-\sum_{\langle ij\rangle}\delta_{s_{i}s_{j}}\;,$ (1)

where $s_{i}$ is a “spin” variable taking integer values between $0$ and
$q-1$, and the sum is restricted to the nearest neighbor sites $\langle
ij\rangle$ on a lattice of $N$ sites with periodic boundary conditions. The
partition function $Z$ is defined by

$Z=\sum_{conf}{\rm e}^{-\beta H}.$ (2)

with $\beta=1/k_{B}T$ 111According to the usual terminology, the inverse
temperature and the critical exponent of the magnetization are denoted by the
same symbol $\beta$, since there is no risk of confusion in this context., and
$k_{B}$ the Boltzmann constant (fixed to unity). On the square lattice in zero
magnetic field, the model is self-dual. Denoting by $\beta^{*}$ the dual of
the inverse temperature $\beta$, the duality relation

$\left(e^{\beta}-1\right)\left(e^{\beta^{*}}-1\right)=q$ (3)

determines the critical value of the inverse temperature [7]
$\beta_{c}=\ln(1+\sqrt{q})\approx 1.09861$. Dual reduced temperatures $\tau$
and $\tau^{*}$ can be defined by

$\beta=\beta_{c}(1-\tau)\;\;\;{\rm and}\;\;\;\beta^{*}=\beta_{c}(1+\tau^{*}).$
(4)

Close to the critical point, $\tau$ and $\tau^{*}$ coincide through first
order, since
$\tau^{*}=\tau+\frac{\ln(1+\sqrt{q})}{\sqrt{q}}\tau^{2}+O(\tau^{3})$.

The critical amplitudes and the critical exponents describe the singular
behavior of the thermodynamic quantities close to the critical point. For
example the magnetization $M$, the (reduced) susceptibility $\chi$ and the
specific heat $C$ of a spin system in zero external field222In this paper we
only deal with the physical properties in zero magnetic field. behave as
333Note that for simplicity we have dropped for the moment the multiplicative
logarithmic corrections and allowed only for additive corrections.

$\displaystyle M(\tau)$ $\displaystyle=$ $\displaystyle
B(-\tau)^{\beta}\left(1+{\rm corr.\;terms}\right),\ \tau<0$ (5)
$\displaystyle\chi(\tau)_{\pm}$ $\displaystyle=$
$\displaystyle\Gamma_{\pm}|\tau|^{-\gamma}\left(1+{\rm corr.\;terms}\right),$
(6) $\displaystyle C(\tau)_{\pm}$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-\alpha}\left(1+{\rm
corr.\;terms}\right),$ (7)

Here $\tau$ is the reduced temperature $\tau=(T-T_{c})/T$ and the labels $\pm$
refer to the high-temperature (HT) and low-temperature (LT) sides of the
critical temperature $T_{c}$. In addition to the mentioned observables, for
the Potts models with $q>2$ a transverse susceptibility can be defined in the
low-temperature phase444In the following we shall use the notations
$\Gamma_{L}$ or $\Gamma_{T}$ for the longitudinal or transverse susceptibility
amplitudes in the low-temperature phase. When still used, $\Gamma_{-}$ is
identified with $\Gamma_{L}$.

$\chi_{T}(\tau)=\Gamma_{T}(-\tau)^{-\gamma}\left(1+{\rm corr.\;terms}\right).$
(8)

The critical exponents are known exactly for the 2D Potts model [8, 9, 10, 11,
12]:

$x_{\epsilon}=\frac{1+y}{2-y},\quad x_{\sigma}=\frac{1-y^{2}}{4(2-y)},$ (9)

where $y$ is related to the number of states $q$ of the Potts variable by

$\cos\frac{\pi y}{2}=\frac{1}{2}\sqrt{q}.$ (10)

The standard exponents follow from $x_{\epsilon}=(1-\alpha)/\nu$ and
$x_{\sigma}=\beta/\nu$. The central charge of the corresponding conformal
field theory is also simply expressed [11, 12] in terms of $y$

$c=1-\frac{3y^{2}}{2-y}.$ (11)

Analytical estimates of critical amplitude ratios for the $q$-state Potts
models with $q=1$, $2$, $3$, and $4$ were recently obtained by Delfino and
Cardy [13]. They used the exact 2D scattering field theory of Chim and
Zamolodchikov [14] and estimated the ratios using a two-kink approximation for
$1<q\leq 3$. For $3<q\leq 4$, they considered both the two-kink approximation
and the contribution from the bound state. For $q=4$ this approximation leads
to the value $c=0.985$ for the central charge, to be compared to the exact
value $c=1$. Using this approximate value, one can calculate the scaling
dimensions from (11) and (9) obtaining the values $x_{\sigma}=0.117$ and
$x_{\epsilon}=0.577$, to be compared with the exact values $1/8$ and $1/2$
respectively. So, the deviation from the exact values becomes as large as 6
and 15 per cent, emphasizing the difficulty of the $q=4$ case. In the 3-state
case the situation is much better (see I).

Let us recall that the existence of logarithmic corrections to scaling in the
4-state Potts model was pointed out in the pioneering works of Cardy,
Nauenberg and Scalapino [15, 16], where a set of non-linear RG equations was
solved. Their discussion was recently extended by Salas and Sokal [17].

The universal susceptibility amplitude ratio $\Gamma_{+}/\Gamma_{L}$ was
calculated by Delfino and Cardy in [13] for both the 3-state and 4-state Potts
models. Later Delfino, et al. [18] estimated analytically also the ratio of
the transverse to the longitudinal susceptibility amplitudes
$\Gamma_{T}/\Gamma_{L}$. The values obtained in the 4-state case are

$\displaystyle\Gamma_{+}/\Gamma_{L}=4.013,$
$\displaystyle\Gamma_{T}/\Gamma_{L}=0.129.$ (12)

In this latter paper [18], the results of MC simulations were also reported,
but they were considered inconclusive by the authors. More recently Delfino
and Grinza report compatible values in the case of the Ashkin-Teller model,
using the same technique at the same level of approximation [19].

Another contribution to the study of the amplitude ratios in the 2D 4-state
Potts model was reported by Caselle, et al [4]. These authors presented a MC
determination of various amplitudes. In particular, their estimate of the
susceptibility amplitude ratio $\Gamma_{+}/\Gamma_{L}=3.14(70)$ is in
reasonable agreement with the theoretical estimate of Delfino and Cardy, in
spite of a somewhat controversial [18] use of the logarithmic corrections in
the fitting procedure.

Enting and Guttmann [20] also analyzed SE data for the 4-state Potts model and
found

$\displaystyle\Gamma_{+}/\Gamma_{L}=3.5(4),$
$\displaystyle\Gamma_{T}/\Gamma_{L}=0.11(4),$ (13)

results which are compatible with the predictions of [13] and [18]. Their
series analysis does not rely on differential approximants, but, in the hope
to achieve better control of the log-corrections of the $q=4$ case, they
address directly the asymptotic behavior of the series coefficients.

In the present paper we present more accurate MC data supplemented by a
reanalysis of the extended series derived by Enting and Guttmann [20]. We
address the following question: Is it possible to estimate the influence of
the logarithmic corrections on the fit procedure? Is it possible to devise
some procedure in which the role of the logarithmic corrections is properly
taken into account?

In the rest of the paper, we shall be concerned with the following universal
combinations of critical amplitudes

$\frac{A_{+}}{A_{-}},\;\;\frac{\Gamma_{+}}{\Gamma_{L}},\;\;\frac{\Gamma_{T}}{\Gamma_{L}},\;\;R_{C}^{+}=\frac{A_{+}\Gamma_{+}}{B^{2}},\;\;R_{C}^{-}=\frac{A_{-}\Gamma_{L}}{B^{2}}$
(14)

where the last two are a consequence of the scaling relation555We refer the
reader to the review Ref. [21] for a detailed discussion of the universality
of the critical amplitudes ratios. $\alpha=2-2\beta-\gamma$. To the various
critical amplitudes of interest, $A_{\pm}$, $\Gamma_{\pm}$,…, we have
associated appropriately defined “effective amplitudes”, namely temperature-
dependent quantities $A_{\pm}(\tau)$, $\Gamma_{\pm}(\tau),\ldots$, which take
as limiting values, when $\tau\rightarrow 0$, the critical amplitudes
$A_{\pm}$, $\Gamma_{\pm},\dots$. In order to avoid any risk of confusion
between the critical amplitude and the corresponding effective temperature-
dependent amplitude, reference to these temperature-dependent quantities is
always made with their explicit $\tau-$dependence.

Also in this paper, we make use of the duality relation in order to improve
the estimates of the ratios between effective amplitudes measured at dual
temperatures. In the case of the 4-state Potts model, this procedure would
even eliminate all logarithmic corrections from the fit in absence of
background contributions, which unfortunately do exist for most quantities! We
again use the duality relation to estimate the correction-to-scaling
amplitudes in the behavior of the specific heat and of the susceptibility. For
this purpose, we compute ratios also on the duality line, e.g. the
susceptibility effective-amplitude ratio
$\Gamma_{+}(\tau)/\Gamma_{L}(\tau^{*})=\chi_{+}(\beta)/\chi_{L}(\beta^{*})$ as
the ratio of $\chi_{+}(\beta)$, the high-temperature susceptibility at inverse
temperature $\beta$, and of $\chi_{L}(\beta^{*})$, the low-temperature
susceptibility at the dual inverse temperature $\beta^{*}$. Furthermore, we
show analytically that the leading logarithmic corrections cancel on the
duality line for the ratio of the specific-heat amplitudes as extracted from
the energy at dual temperatures.

As a final result of our analysis666The figures given here are an average
between the MC and the SE determinations [5]., we propose estimates of the
susceptibility critical-amplitude ratios $\Gamma_{+}/\Gamma_{L}=6.49(44)$ and
$\Gamma_{T}/\Gamma_{L}=0.154(12)$ which are significantly different both from
the predictions eq.(12) of [13, 18], and from the numerical estimates of [4,
20]. The deviation from the numerical estimates of other authors might be
explained by the complicated logarithmic corrections used to fit the data in
[4, 20]. The difference from the theoretical predictions might be due to the
limited accuracy of the approximation scheme used for $q>3$.

In conclusion, obtaining an accurate (say, within a few per cent) and
generally accepted approximation of the critical amplitude combinations for
the four-state Potts model still remains an open issue both theoretically and
numerically.

##  2 Computational procedures

###  2.1 Monte Carlo simulations

We use the single-cluster Wolff algorithm [2] for studying square lattices of
linear size $L$ with periodic boundary conditions. Starting from an ordered
state, we let the system equilibrate in $10^{5}$ steps measured by the number
of flipped Wolff clusters. The averages are computed over $10^{6}$—$10^{7}$
steps. The random numbers are produced by an exclusive-XOR combination of two
shift-register generators with the taps (9689,471) and (4423,1393), which are
known [22] to be safe for the Wolff algorithm.

The order parameter of a microstate ${\tt M}({\tt t})$ is evaluated during the
simulations as

${\tt M}=\frac{qN_{m}/N-1}{q-1},$ (15)

where $N_{m}$ is the number of sites $i$ with $s_{i}=m$ at the time $\tt t$ of
the simulation [23], and $m\in[0,1,...,(q-1)]$ is the spin value of the
majority of the spins. $N=L^{2}$ is the total number of spins. The thermal
average is denoted $M=\langle{\tt M}\rangle$.

Thus, the longitudinal susceptibility in the low-temperature phase is measured
by the fluctuation of the majority spin orientation

$k_{B}T\;\chi_{L}=\frac{1}{N}(\langle N_{m}^{2}\rangle-\langle
N_{m}\rangle^{2})$ (16)

and the transverse susceptibility is defined in terms of the fluctuations of
the minority of the spins

$k_{B}T\;\chi_{T}=\frac{1}{(q-1)N}\sum_{\mu\neq m}(\langle
N_{\mu}^{2}\rangle-\langle N_{\mu}\rangle^{2}),$ (17)

while in the high-temperature phase $\chi_{+}$ is given by the fluctuations in
all $q$ states,

$k_{B}T\;\chi_{+}=\frac{1}{qN}\sum_{\mu=0}^{q-1}(\langle
N_{\mu}^{2}\rangle-\langle N_{\mu}\rangle^{2}),$ (18)

where $N_{\mu}$ is the number of sites with the spin in the state $\mu$.
Properly allowing for the finite-size effects, this definition of the
susceptibilities is, in both phases, completely consistent with the available
SE data [24].

The internal energy density of a microstate is calculated as

${\tt E}=-\frac{1}{N}\sum_{\langle ij\rangle}\delta_{s_{i}s_{j}}\,$ (19)

its ensemble average is denoted by $E=\langle{\tt E}\rangle$ and the reduced
specific heat per spin measures the energy fluctuations,

$(k_{B}T)^{2}\;C=-\frac{\partial E}{\partial\beta}=\langle{\tt
E}^{2}\rangle-\langle{\tt E}\rangle^{2}.$ (20)

We have simulated the model on square lattices with linear sizes $L=20$, $40$,
$60$, $80$, $100$ and $200$. In each case, we have measured the physical
quantities within a range of reduced temperatures called the “critical window”
and defined as follows. Assuming a proportionality factor of order 1 in the
definition of the correlation length, the relation $L<\xi\propto|\tau|^{-\nu}$
yields the value of the reduced temperature at which the correlation length
becomes comparable with the system size $L$ and thus below which the finite-
size effects are not negligible. This value defines the lower end of the
critical window and avoids finite size effects which would make our analysis
more complex. The upper limit of the critical window is fixed for convenience
at $\tau=0.20-0.25$.

###  2.2 Series expansions

Our MC study of the critical amplitudes will be supplemented by an analysis of
the high-temperature and low-temperature expansions for $q=4$, recently
extended through remarkably high orders by Briggs, Enting and Guttmann [25,
20]. In terms of these series, we can compute the effective critical
amplitudes for the susceptibilities, the specific heat and the magnetization
and extrapolate them by the standard resummation techniques, namely simple
Padé approximants (PA) and differential approximants (DA), properly biased
with the exactly known critical temperatures and critical exponents.

The LT expansions are expressed in terms of the variable $z=\exp(-\beta)$. In
the $q=4$ case, the expansion of the energy extends through $z^{43}$. For the
longitudinal susceptibility the expansion extends through $z^{59}$, and for
the transverse susceptibility through $z^{47}$. In the case of the
magnetization, the expansion extends through $z^{43}$. The HT expansions are
computed in terms of the variable $v=(1-z)/(1+(q-1)z)$. They extend up to
$v^{43}$ in the case of the energy and up to $v^{24}$ for the susceptibility.

It is useful to point out that, for convenience, in Ref. [20] the product of
the susceptibility by the factor $q^{2}/(q-1)$, rather than the susceptibility
itself, is tabulated at HT, because it has integer expansion coefficients. For
the same reason, at LT the magnetization times $q/(q-1)$ is tabulated.
Therefore the appropriate normalization should be restored in order that the
series yield amplitudes consistent with the MC results.

As a general remark on our series analysis, we may point out that the accuracy
of the amplitude estimates given in Ref. I for the $q=3$ case is good due to
the relatively harmless nature of the power-like corrections to scaling, while
in the $q=4$ case the mentioned resummation methods cannot reproduce the
expected logarithmic corrections to scaling and therefore the extrapolations
to the critical point are more uncertain. In this case we have tested also a
somewhat unconventional use of DA’s: in computing the effective amplitudes, we
only retain DA estimates outside some small vicinity of the critical point,
where they appear to be stable and reliable. Finally, we perform the
extrapolations by fitting these data to an asymptotic form which includes
logarithmic corrections. We shall add further comments on the specific
analyses in the next sections.

##  3 Critical amplitudes of 4-state Potts model

###  3.1 Expected temperature-dependence of the observables

In the case of the 4-state Potts model, we have $y=0$ from (10) and the second
thermal exponent [26, 11, 12] $y_{\phi_{2}}=-4y/3(1-y)$ vanishes. Accordingly,
the leading power-behavior of the specific heat (and of other physical
quantities) is modified [15] by a logarithmic factor

${C}(\tau)=\frac{A_{\pm}}{\alpha}|\tau|^{-\alpha}(-\ln|\tau|)^{-1}{\cal
X}_{corr}(-\ln|\tau|)+{\cal Y}_{bt}(|\tau|).$ (21)

The exponent $\alpha$ takes the value $2/3$ and ${\cal X}_{corr}(-\ln|\tau|)$
contains correction terms with powers of $|\tau|$, $-\ln|\tau|$ and
$\ln(-\ln|\tau|)$. It may also contain non-integer power corrections due to
the higher thermal exponents [8, 9, 10, 11, 12] $y_{\phi_{n}}$ or to other
irrelevant fields, as well as power corrections due to the identity field.
${\cal Y}_{bt}(|\tau|)$ contains all regular contributions and is referred to
as the background term.

Extending the pioneering work of Cardy, Nauenberg and Scalapino [15, 16],
Salas and Sokal [17] obtained a set of non-linear RG equations. In the
appendix (see also Refs. [5, 6]), we derive from this set of equations a
closed expression for the leading logarithmic corrections, which is more
suitable to describe the temperature range accessible in a numerical study,
than the asymptotic form given by Salas and Sokal,

${C}(\tau)=\frac{A_{\pm}}{\alpha}|\tau|^{-2/3}(-\ln|\tau|)^{-1}\left[1-\frac{3}{2}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}+O\left(\frac{1}{\ln|\tau|}\right)\right],$
(22)

which is the first term of a slowly convergent expansion of ${\cal
X}_{corr}(-\ln|\tau|)$ in logs. We have observed that the following expansion
(see Appendix) is better behaved in the temperature window near the critical
point accessible by MC and SEs

$\displaystyle{C}(\tau)$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-2/3}{\cal G}^{-1}(-\ln|\tau|),$
(23) $\displaystyle{\cal G}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle(-\ln|\tau|)\times{\cal E}(-\ln|\tau|)\times{\cal
F}(-\ln|\tau|),$ (24) $\displaystyle{\cal E}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)$
(25) $\displaystyle\
\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right),$
$\displaystyle{\cal F}(-\ln|\tau|)$ $\displaystyle\simeq$
$\displaystyle\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{-1}.$
(26)

The function $\cal E(-\ln|\tau|)$ contains the exact form of the leading terms
with universal coefficients predicted by RG. The remaining part is made of log
terms, whose coefficients involve the non-universal dilution field $\psi_{0}$.
The multiplicative function ${\cal F}(-\ln|\tau|)$ mimics, in a given
temperature range close to the critical point, the higher-order terms of this
non-universal part, which is a slowly convergent series in powers of logs
starting with an $O\left(\frac{1}{\ln|\tau|}\right)$ term. The values of
$C_{1}$ and $C_{2}$ in (26) should not be considered as “real” amplitudes of
correction-to-scaling terms, but only as “effective” parameters. A “zeroth-
order” analysis may be performed taking $C_{1}=C_{2}=0$, i.e. ${\cal
F}(-\ln|\tau|)=1.$ A more refined estimate follows from the analysis of the
magnetization, as explained in the section 3.2 and in the Appendix. The
absence of a constant background term is a simplifying feature in the analysis
of the magnetization777The effective amplitudes are constructed by dividing
the corresponding physical quantity by the leading terms (main power
dependence with known logarithmic corrections), and, therefore the background
terms, if present, will be divided by the logarithmic terms. These terms may
seriously complicate the analysis of the limiting behavior of the effective
amplitudes.. The difference in behavior between the two types of expressions
(22) and (23) for the specific heat is illustrated by the fact that, for
example, at a typical value of temperature numerically accessible without
finite-size effects, $\tau=0.10$, we have
$1-\frac{3}{2}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\simeq 0.457$, while
$\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)\times\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\simeq
0.786$ and the two functions are not simply proportional to each other in the
typical range $\tau=0.02-0.20$. Fitting the numerical data with one or the
other choice may thus spoil the outcome for the leading amplitude.

Similar expressions of the logarithmic corrections are obtained for the other
physical quantities in Eqs. (79-85) of the Appendix.

Further corrections to scaling may also be present in Eq. (23). They are
discussed in the Appendix and may be of the form $a_{2/3}|\tau|^{2/3}$,
$a_{h_{3}}|\tau|^{\Delta_{h_{3}}}$ or
$a_{\phi_{3}}|\tau|^{\Delta_{\phi_{3}}}$, as well as powers of these terms,
where $\Delta_{h_{3}}=3/4$ and $\Delta_{\phi_{3}}=5/3$. Pure power corrections
and background terms may also be needed. Here we also stress that the
inclusion of a leading correction in $a|\tau|^{2/3}$ and of analytic terms
seems to be necessary according to the papers by Joyce [27, 28], where the
magnetization of a model, expected to belong to the 4-state Potts model
universality class, is shown to have an expression of the form

$M(-|\tau|)=|\tau|^{1/12}(f_{0}(\tau)+|\tau|^{2/3}f_{1}(\tau)+|\tau|^{4/3}f_{2}(\tau))$
(27)

with $f_{i}(\tau)$ analytic functions when $\tau\to 0^{-}$. The correction
exponents are obtained from the table of the conformal scaling dimensions by
Dotsenko and Fateev [11, 12], but not all of them are necessarily present.
However, at least the presence of the exponents $2/3$, and possibly of $4/3$,
seems to be needed in order to account for the numerical results. Caselle et
al. [4] also considered an $a_{2/3}|\tau|^{2/3}$ term to fit the
magnetization. The parameter $a_{h_{3}}$ is a priori possibly needed only for
magnetic quantities, while the corrections in $a_{\phi_{3}}$ will
systematically be dropped in our fits, since they are sub-sub-dominant.

In conclusion, the most general expression that we will consider is the
following:

$\displaystyle{\rm Obs.}(\pm|\tau|)$ $\displaystyle\simeq$ $\displaystyle{\rm
Ampl.}\times|\tau|^{\blacktriangleleft}\times{\cal
G}^{\bigstar}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+$ (28)
$\displaystyle\qquad\qquad+{\rm\ backgr.\ terms},$ $\displaystyle{\rm corr.\
terms}$ $\displaystyle=$ $\displaystyle
a_{2/3}|\tau|^{2/3}+b_{\pm}|\tau|+a_{4/3}|\tau|^{4/3}+\dots,$ (29)
$\displaystyle{\rm backgr.\ terms}$ $\displaystyle=$ $\displaystyle
D_{0}+D_{1}|\tau|+\dots$ (30)

where ${\cal G}(-\ln|\tau|)$ is defined by Eqs. (24-26), while
${\blacktriangleleft}$ and $\bigstar$ stand for exponents which depend on the
observable considered. They are all given in the Appendix.

###  3.2 The magnetization amplitude

The amplitude $B$ of the magnetization is defined by the asymptotic behavior
(see the appendix for details)

$\displaystyle M(-|\tau|)$ $\displaystyle=$ $\displaystyle
B|\tau|^{1/12}(-\ln|\tau|)^{-1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)^{-1}\right.$
(31) $\displaystyle\left.\
\qquad\times\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right){\cal
F}(-\ln|\tau|)\right]^{-1/8}(1+a|\tau|^{2/3}+b|\tau|+\dots).$

We can extract an effective function ${\cal F}_{eff}(-\ln|\tau|)$ which mimics
the real one ${\cal F}(-\ln|\tau|)$ in the convenient temperature range
$|\tau|\simeq 0.01-0.10$. This is done by plotting an effective magnetization
amplitude

$\displaystyle B_{eff}(-|\tau|)$ $\displaystyle=$ $\displaystyle
M(-|\tau|)|\tau|^{-1/12}(-\ln|\tau|)^{1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)\right.$
(32)
$\displaystyle\left.\qquad\qquad\qquad\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right)\right]^{1/8}$

which is then fitted to the expression

$B_{eff}(-|\tau|)=B\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{1/8}\times(1+a|\tau|^{2/3}+b|\tau|+\dots)$
(33)

in which we have also included corrections to scaling. As we have already
noticed, the coefficients appearing in the function ${\cal F}$ are effective
parameters adapted to the temperature window considered, therefore the values
of $C_{1}$ and $C_{2}$ have no special meaning.

In order to analyze the numerical data and to extract the different
coefficients, one needs very accurate data. As an illustration, in Figure 1 we
compare MC and SE data, and MC data from Caselle et al. [4].

Figure 1: The magnetization $M$ in the critical window region. Our MC data are
represented by boxes, the MC data from Ref. [4] by stars and the SE data by a
solid line.

Figure 2: The effective amplitude of the magnetization $M$. Insert: The
magnetization $M$ as function of $|\tau|^{1/12}(-\ln|\tau|)^{-1/8}$. (Our MC
data are represented by open circles, the SE data by boxes and the fit by a
solid line).

The behavior of $B_{eff}(-|\tau|)$ is shown in Fig. 2. In table 1, we present
a selection of our fits of MC data to Eq. (33). The first column of the table
indicates the different choices of the function ${\cal F}(-\ln|\tau|)$. For
each line in the tables, several fits have been tried, varying the number of
points in the interval $|\tau|\in[0.005,0.25]$ (total number of points 50) and
calculating the $\chi^{2}/d.o.f.$ for each fit. A reasonable balance has to be
found between the distance of the points from the critical temperature and
their number. It appeared that limiting the fit window to 20 data points, i.e.
to the interval $[0.005,0.1]$, gives the best confidence level. This choice is
quite satisfactory, since it corresponds to a close vicinity of the critical
point. The criterion that we adopted in order to select a most convincing fit
among all possible fits is the stability of the correction-to-scaling
amplitudes $a$ and $b$ in equation (31) when the temperature window is varied,
typically in the range $|\tau|\in[0,0.06]$ to $|\tau|\in[0,0.30]$. As long as
these numbers have not converged to given values, we cannot pretend that it is
meaningful to include such corrections in the analysis. We have to mention
that stability of the correction amplitudes is never reached if we stop the
correction terms in equation (31) at the leading log, or even at the three
higher log terms (correction function ${\cal E}(-\ln|\tau|)$). Then even the
leading amplitude $B$ is questionable. The effective function ${\cal
F}_{eff}(-\ln|\tau|)$ is essential to reach convergence of all but the two
effective coefficients $C_{1}$ and $C_{2}$ which still strongly depend on the
temperature window.

For a given type of fit defined in the first column, Tables 1 and 2 show the
parameters of the fit minimizing $\chi^{2}$ and the minimization is performed
by varying the width and position of the temperature window.

Once our choice is made for $C_{1}$ and $C_{2}$, the results of the fit for
the amplitude $B$ show a remarkable stability. The entry called fit # 0
corresponds to a “zeroth-order” fit in which the function ${\cal
F}(-\ln|\tau|)$ is taken equal to unity. It is presented for comparison, in
order to emphasize the improvement occurring in the following lines. Fit #1
keeps all correction coefficients, while the lines which follow are obtained
decreasing the number of fit parameters. We favor fits # 2 and #3, where the
coefficient $b$ (the amplitude of the linear term $|\tau|$) is fixed to zero,
since from the first fit (#1) we see that leaving $b$ free leads to a value
close to zero and we consider more reliable a fit with fewer parameters. Fit #
4 corresponds to another extreme choice including no irrelevant correction at
all. It shows that the effective function ${\cal F}(-\ln|\tau|)$ is more
important than the irrelevant corrections to scaling in order to achieve a
stable amplitude $B$ (comparable to the outcomes of fits # 1 to # 3).

Table 1: Various fits of our MC data for the magnetization effective amplitude $M(-|\tau|)|\tau|^{-1/12}(-\ln|\tau|)^{1/8}{\cal E}^{1/8}(-\ln|\tau|)$ to the expression Eq. (33). The stars in the first column indicate our favorite fits (the reasons for this choice are given in the text). Fit # | Amplitude | Correction terms
---|---|---
|  |
| $B$ | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
MC # 0 | $1.1355(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | $-0.41(1)$ | $\phantom{-}0.41(21)$
MC # 1 | $1.1566(5)$ | $-0.740(2)$ | $-0.630(51)$ | $-0.172(4)$ | $-0.018(6)$
MC # $2^{*}$ | $1.1570(1)$ | $-0.757(1)$ | $-0.522(11)$ | $-0.191(2)$ | $\phantom{-}-$
MC # $3^{*}$ | $1.1559(12)$ | $-0.88(5)$ | $\phantom{-}-$ | $-0.21(1)$ | $\phantom{-}-$
MC # 4 | $1.1593(1)$ | $-0.25(2)$ | $-3.04(5)$ | $\phantom{-}-$ | $\phantom{-}-$
Table 2: Same as table 1 for the magnetization data obtained by the SE method. A star in the first column indicates our favorite fit with the coefficients $C_{1}$ and $C_{2}$ (now fixed and shown in bold face in this table and in the forthcoming tables) obtained from the fits of the MC data reported in table 1. Fit # | Amplitude | Correction terms
---|---|---
|  |
| $B$ | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
SE # 0 | $1.1364(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | $-0.435(5)$ | $\phantom{-}0.097(10)$
SE # 1 | $1.1597(1)$ | $-0.637(5)$ | $-1.417(28)$ | $-0.115(2)$ | $\phantom{-}0.013(2)$
SE # $2$ | $1.1589(1)$ | $-0.660(5)$ | $-1.246(25)$ | $-0.126(2)$ | $\phantom{-}-$
SE # $2^{*}$ | $1.1575(1)$ | $-{\bf 0.757}$ | ${\bf-0.522}$ | $-0.194(1)$ | $\phantom{-}-$
SE # $3$ | $1.1583(8)$ | $-0.981(34)$ | $\phantom{-}-$ | $-0.185(10)$ | $\phantom{-}-$
SE # $3^{*}$ | $1.1575(1)$ | ${\bf-0.88}$ | $\phantom{-}-$ | $-0.225(1)$ | $\phantom{-}-$
SE # 4 | $1.1573(1)$ | $\phantom{-}0.164(7)$ | $-4.106(18)$ | $\phantom{-}-$ | $\phantom{-}-$
Table 3: Same as table 1 but using the MC data of Caselle et al. [4] for the magnetization. The fit marked by a star is performed using the same coefficient $C_{1}$ (shown in bold) as in table 1.  Fit # | Amplitude | Correction terms
---|---|---
|  |
| $B$ | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
CTV # 0 | $1.1386(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $-0.518(6)$ | $\phantom{-}0.30(2)$
CTV # 1 | $1.196\pm 1.499$ | $-7.5\pm 163$ | $\phantom{-}19.6\pm 429$ | $-1.09\pm 17.3$ | $1.2\pm 29.2$
CTV # 2 | $1.148(6)$ | $\phantom{-}0.45(56)$ | $-3.6\pm 1.1$ | $-0.17(4)$ | $\phantom{-}-$
CTV # 3 | $1.162(9)$ | $-1.13(38)$ | $\phantom{-}-$ | $-0.14(12)$ | $\phantom{-}-$
CTV # $3^{*}$ | 1.1561(1) | ${\bf-0.88}$ | $\phantom{-}-$ | $-0.215(1)$ | $\phantom{-}-$
CTV # 4 | $1.153(2)$ | $\phantom{-}0.73(31)$ | $-5.5\pm 0.77$ | $\phantom{-}-$ | $\phantom{-}-$

The same procedure is now applied to the SE data. It is a non-conventional
approach to fit SE data, which are usually analyzed by approximant methods,
but the exercise is tempting. We thus apply exactly the same procedure as in
the case of MC data, varying the fitting interval and comparing the values of
the $\chi^{2}/d.o.f.$. The best results are collected in table 2. The
agreement between the results quoted in the two tables is amazing. Not only
the amplitudes, but also the correction coefficients are very close to each
other. In this table (and in the forthcoming tables), we also present the
results of this analysis of SE data with $C_{1}$ and $C_{2}$ fixed to their
best values extracted in table 1 from the MC data. They are indicated again
with a star (SE $\\#2^{*}$ and SE $\\#3^{*}$) and in order to emphasize the
fact that $C_{1}$ and $C_{2}$ in this case are not free parameters, they are
indicated in bold face. A reason for this approach is first to insist on the
consistency of the results and second to privilege a fit with less free
parameters (and in particular no free log term). For the magnetization
amplitude, a compromise between MC and SE data analysis provides our final
estimate

$B=1.157(1).$ (34)

As an additional test, we decided to fit also the MC data of Caselle et al.
[4], obtained using the Swendsen-Wang cluster algorithm, to our functional
expression Eq. (33). Notice that the various coefficients reported in table 3
(not only the critical amplitude, but also the correction terms) are very
close to those reported above in table 1. In particular this fit gives further
support to our choice in favor of fits # 2 and # 3.

In the following tables, we will refer to the best two fits by the labels #
$2^{*}$ and # $3^{*}$ (which means that the coefficients $C_{1}$ and/or
$C_{2}$ are fixed to their values indicated in table 1) when fitting other
quantities. Note that the MC data of Caselle et al. [4] are not fitted with
the choice # $2^{*}$, because they do not extend on a range of temperatures
wide enough to apply the corresponding functional expression.

###  3.3 The specific heat and the energy density

We now turn to the study of the specific-heat amplitudes. Figure 3 shows the
effective-amplitude ratio $A_{+}(\tau)/A_{-}(-|\tau|)$ computed at dual and
symmetric temperatures following the same prescription as in the case of the
3-state Potts model studied in paper [1]. The same ratio computed from SE data
is also shown for comparison. The linear fit of the most accurate MC data
(lattice size $L=100$ computed with $10^{7}$ measurement steps) yields for
this ratio the value $0.9999(1)$, which is remarkably close to the exact value
1 derived from duality.

Figure 3: The specific-heat effective-amplitude ratio computed from the energy
ratio
$\frac{(E(\beta)-E_{0})\tau^{\alpha-1}}{(E_{0}-E(\beta^{*}))(\tau^{*})^{\alpha-1}}$
as a function of the reduced temperature $\tau$ on the dual line. The SE data
are represented by boxes, our MC data ($L=100$) by circles. The solid line is
given by Eq. (36). The ratio of the effective amplitudes
$A_{+}(\tau)/A_{-}(-|\tau|)$ is shown for different sizes in the insert (for
lattice linear sizes $L=100$ (open circles), $L=200$ (squares), $L=300$ (up
triangles), and $L=400$ (down triangles). The simulations were performed with
$N_{th}=10^{5}$ thermalization steps and $N_{MC}=10^{6}$ MC steps. The results
of a simulation for $L=100$ with $N_{th}=10^{6}$ and $N_{MC}=10^{7}$, are
represented by closed circles. The SE data are represented by a solid line.
The dotted line represents a linear fit of the closed circles.

The ratio

$\frac{A_{+}(\tau)}{A_{-}(\tau^{*})}=\frac{(E(\beta)-E_{0})\tau^{\alpha-1}}{(E_{0}-E(\beta^{*}))(\tau^{*})^{\alpha-1}},$
(35)

where the constant $E_{0}$ is the value of the energy at the transition
temperature, $E_{0}=E(\beta_{c})=-1-1/\sqrt{q}$, when expanded close to the
transition point leads to

$\displaystyle\frac{A_{+}(\tau)}{A_{-}(\tau^{*})}$ $\displaystyle=$
$\displaystyle 1+\frac{7}{3}\alpha_{q}\tau+O(\tau^{1+\alpha})$ (36)

with
$\alpha_{q}=-E_{0}\beta_{c}e^{-\beta_{c}}=\frac{\ln(1+\sqrt{q})}{\sqrt{q}}\approx
0.5493$. Eq. (36) predicts an asymptotically linear $\tau-$dependence of the
effective-amplitude ratio. This linear dependence is observed in Fig. 3 which
also confirms that the leading logarithmic corrections in the scaling function
${\cal X}_{\it corr}(-\ln(|\tau|))$ asymptotically cancel in the ratio
$A_{+}(\tau)/A_{-}(\tau^{*})$. (Of course this is true also for the same
quantity computed at symmetric temperatures, since $\tau^{*}\approx|\tau|$ for
small values of $\tau$). Figure 3 compares the effective-amplitude ratio
obtained from MC simulation and SE data with Eq. (36). The slope of MC data is
1.25, very close to the predicted value 1.28.

Now, we make the natural conjecture (proven in the Appendix in the absence of
background corrections) that the cancelation of the leading logarithmic
corrections will also occur for the other ratios. For the leading log-
correction, $-\ln|\tau|$, and for the next correction in
$\ln(-\ln|\tau|)/(-\ln|\tau|)$, this can be shown analytically from the RG as
first indicated by Cardy, Nauenberg and Scalapino in Ref. [15] and by Salas
and Sokal in Ref. [17]. Our statement is stronger since it extends also to the
higher order log-terms, such as the next correction in $1/(-\ln|\tau|)$. We
believe that equations (79) to (85) in the appendix are exact, and since all
the log-terms come from Eq. (78), they should cancel in the appropriate ratios
(i.e. when the same powers of the dilution field appear in the numerator and
the denominator. This is always the case when one considers an effective
combination tending to a universal ratio as $\tau\to 0$).

According to a RG analysis (see Appendix A), we may write the energy in the
critical region as

$\displaystyle E_{\pm}(\pm|\tau|)$ $\displaystyle=$ $\displaystyle
E_{0}\pm\frac{A_{\pm}}{\alpha(1-\alpha)\beta_{c}}\frac{|\tau|^{1/3}}{(-\ln|\tau|)}\frac{1+a_{\pm}|\tau|^{2/3}}{{\cal
E}(-\ln|\tau|){\cal F}(-\ln|\tau|)}+D_{1,\pm}|\tau|.$ (37)

In a fixed range of values of the reduced temperature, the “correction
function” ${\cal F}(-\ln|\tau|)$ is now fixed and the only remaining freedom
is to include background terms (coeff. $D$) and possibly additive corrections
to scaling coming from irrelevant scaling fields (coeff. $a$). Therefore, once
the function ${\cal F}(-\ln|\tau|)$ is fixed after our study of the
magnetization, a reasonable fit of the energy data needs only three
parameters888Like in the case of the magnetization, a fourth parameter in
$b_{\pm}|\tau|$ appears to be unnecessary., $A_{\pm}$, $a_{\pm}$, and
$D_{\pm}$. The next step is the fit of the mean $\bar{A}(\tau)$ of the
effective amplitudes,

$\displaystyle\bar{A}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha(1-\alpha)\beta_{c}(E_{+}(\tau)-E_{-}(-|\tau|))\times{\cal
G}(-\ln|\tau|)/|\tau|^{1-\alpha}$ (38) $\displaystyle=$
$\displaystyle\frac{\beta_{c}}{9}(E_{+}(\tau)-E_{-}(-|\tau|))\times{\cal
G}(-\ln|\tau|)/|\tau|^{1/3}.$

We thus fit the MC data to the expression

$\bar{A}(\tau)=A(1+a|\tau|^{2/3})+D_{1}|\tau|\times{\cal
G}(-\ln|\tau|)/|\tau|^{1/3}.$ (39)

The log corrections, which indeed cancel in the singular part, unfortunately
reappear in the background term, albeit suppressed by the power
$|\tau|^{2/3}$.

Table 4: Fits of the energy difference (MC data computed at dual temperatures) to the expression $\bar{A}(\tau)\simeq{\rm Ampl.}\times(1+{\rm corr.\ terms})+{\rm backgrd.\ terms}\times|\tau|^{-1/3}{\cal G}(-\ln|\tau|)$.  Fit # | Amplitude | correction terms | background term
---|---|---|---
|  |  |
| Ampl. | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
MC # $2^{*}$ | $1.338(3)$ | ${\bf-0.757}$ | ${\bf-0.522}$ | $-4.98(8)$ | $\phantom{-}0.920(13)$
MC # $3^{*}$ | $1.316(10)$ | ${\bf-0.88}$ | $\phantom{-}-$ | $-4.88(38)$ | $\phantom{-}0.899(62)$

Figure 4: The effective amplitude $\bar{A}(\tau)$ computed from our MC data
using Eq. (38) (open circles) and from the fit to Eq. (39) (solid line).

Figure 5: The energy differences $\Delta E_{+}$ and $\Delta E_{-}$, calculated
from our MC data for lattice size $L=100$ (circles) and the fitted expressions
(solid lines).

We note that $\bar{A}(\tau)$ is constructed in Eq. (38) using the values of
energy density computed at symmetric temperatures. The same quantity
constructed from the energy densities at dual temperatures (with
$E_{-}(\tau^{*})$ instead of $E_{-}(-|\tau|)$ in Eq. (38)) can also be studied
and provides better results. In table 4, we show our results (the best fit is
obtained with the choices of $C_{i}$’s coefficients labeled $2^{*}$ in tables
1 and 2). Again, the agreement between the two fits is quite good, but this
time it is more trivial since the same data set is fitted. Taking the average
of the parameters from the two fits, we conclude

$A=1.327(12),$ (40)

and $a\simeq-4.93(23)$, and $D_{1}\simeq 0.910(38)$. Padé approximants of SE
data for the specific heat provide $A=1.35(1)$.

The expansion including corrections to scaling and background terms for the
specific heat follows from the expressions of the energy density. There is
some disagreement between our amplitude
$\frac{A}{\alpha(1-\alpha)\beta_{c}}\simeq 5.922(40)-6.021(13)$ and the result
reported by Caselle et al. [4], $6A\simeq 7.80(36)$999Notice that Caselle et
al. use a different definition of the energy.. We have to notice that the
amplitudes are very sensitive to the expression used for the fits. Our choice
of effective amplitude in Eq. (38) is supported by the quite regular behavior
shown in figure 4, and also by the natural choice of the fitting expression
(39). The comparison between the MC data and the resulting fit is shown in
figure 5. We agree on this point with Enting and Guttmann [20] who emphasized
that their estimates depend critically on the form assumed for the logarithmic
sub-dominant terms, and on the further assumption that the other sub-dominant
terms, including powers of logarithms, powers of logarithms of logarithms
etc., can all be neglected.

###  3.4 Susceptibilities amplitudes

####  3.4.1 High temperature susceptibility amplitude

We proceed along the same lines as for the other physical quantities and fit
the high-temperature susceptibility to the expression

$\chi_{+}(\tau)=\Gamma_{+}\tau^{-7/6}{\cal
G}^{3/4}(-\ln\tau)(1+a_{+}\tau^{2/3}+b_{+}\tau)+D_{+}.$ (41)

Figure 6: The effective amplitude of the high-temperature susceptibility $\chi_{+}(\tau)$. We have shown SE data for $\tau\leq 0.05$, our MC data for $\tau>0.05$ and a fit to Eq. (41) (dotted line). Table 5: Fits of the high-temperature susceptibility to the expression $\chi_{+}(\tau)\simeq{\rm Ampl.}\times\tau^{-7/6}\times{\cal G}^{3/4}(-\ln\tau)\times(1+{\rm corr.\ terms})+{\rm backgr.\ terms}$. A star in the first column indicates that the coefficients $C_{i}$’s are those deduced from the MC fits of table 1. fit # | amplitude | correction terms | backgr.
---|---|---|---
|  |  |
| $\Gamma_{+}$ | $\propto\frac{1}{-\ln\tau}$ | $\propto\frac{\ln(-\ln\tau)}{(-\ln\tau)^{2}}$ | $\quad\propto\tau^{2/3}$ | $\quad\propto\tau^{0}$
MC # $2^{*}$ | $0.03144(15)$ | $\bf-0.757$ | $\bf-0.522$ | $\phantom{-}0.561(60)$ | $-0.053(17)$
MC # $3^{*}$ | $0.03178(30)$ | $\bf-0.88$ | $\phantom{-}-$ | $\phantom{-}0.53(23)$ | $\phantom{-}0.052(120)$
CTV # $3^{*}$ | $0.03051(29)$ | ${\bf-0.88}$ | $\phantom{-}-$ | $\phantom{-}1.48(34)$ | $-0.45(24)$
SE # $2^{*}$ | $0.03041(1)$ | ${\bf-0.757}$ | $\bf-0.522$ | $\phantom{-}1.30(1)$ | $-0.362(9)$
SE # $3^{*}$ | $0.03039(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $\phantom{-}1.67(1)$ | $-0.59(2)$

We can easily obtain the amplitude $\Gamma_{+}$ observing that a single
constant as a background term $D_{+}$ is sufficient for the fit (this will
also be the case at low temperature). The effective amplitude

$\Gamma_{eff}(\tau)=\chi_{+}(\tau)\tau^{7/6}{\cal G}^{-3/4}(-\ln\tau)$

is represented in Fig. 6 and table 5 collects the coefficients determined by
the fits. We finally obtain the high-temperature susceptibility amplitude

$\Gamma_{+}=0.0310(7).$ (42)

The value which follows from differential approximants to SE data, although
less accurate, is consistent with it, $\Gamma_{+}=0.033(2)$.

####  3.4.2 Low temperature susceptibilities amplitudes

The behavior of the longitudinal susceptibility in the low-temperature phase
is less easy to analyze [5]. We use the expression

$\chi_{L}(-|\tau|)=\Gamma_{L}|\tau|^{-7/6}{\cal
G}(-\ln|\tau|)^{3/4}(1+a_{L}|\tau|^{2/3}+b_{L}|\tau|)+D_{L}.$ (43)

and the various coefficients are collected in table 6. For the transverse
susceptibility, the same procedure leads to the amplitudes also listed in the
table 6.

One may note that the values of the transverse susceptibility amplitude are
more stable than those of the longitudinal amplitude, while the estimates of
the corrections to scaling are less scattered in the latter case. Our final
estimates are

$\Gamma_{L}=0.00478(24)$ (44)

and

$\Gamma_{T}=0.00074(2).$ (45)

DA analysis of SE data gives approximately $\Gamma_{L}=0.005(1)$.

Table 6: Fits of the low-temperature longitudinal susceptibility to the expression $\chi_{L}(-|\tau|)\simeq{\rm Ampl.}\times|\tau|^{-7/6}\times{\cal G}^{3/4}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+{\rm backgr.\ terms}$ and of the low-temperature transverse susceptibility to the expression $\chi_{T}(-|\tau|)\simeq{\rm Ampl.}\times|\tau|^{-7/6}\times{\cal G}^{3/4}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+{\rm backgr.\ terms}$. A star in the first column indicates our favorite fit with the coefficients deduced from the MC fits of table 1. fit # | amplitude | correction terms | backgr.
---|---|---|---
|  |  |
| $\Gamma_{L}$ | $\propto\frac{1}{-\ln\tau}$ | $\propto\frac{\ln(-\ln\tau)}{(-\ln\tau)^{2}}$ | $\quad\propto\tau^{2/3}$ | $\quad\propto\tau^{0}$
MC # $2^{*}$ | $0.00454(2)$ | $\bf-0.757$ | $\bf-0.522$ | $-2.83(3)$ | $\phantom{-}0.050(2)$
MC # $3^{*}$ | $0.00484(3)$ | $\bf-0.88$ | $\phantom{-}-$ | $-3.73(14)$ | $\phantom{-}0.13(1)$
CTV # $3^{*}$ | $0.00494(3)$ | $\bf-0.88$ | $\phantom{-}-$ | $-4.35(15)$ | $\phantom{-}0.210(19)$
SE # $2^{*}$ | $0.00483(1)$ | $\bf-0.757$ | $\bf-0.522$ | $-3.77(3)$ | $\phantom{-}0.116(3)$
SE # $3^{*}$ | $0.00493(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $-4.18(5)$ | $\phantom{-}0.178(6)$
| $\Gamma_{T}$ | $\propto\frac{1}{-\ln\tau}$ | $\propto\frac{\ln(-\ln\tau)}{(-\ln\tau)^{2}}$ | $\quad\propto\tau^{2/3}$ | $\quad\propto\tau^{0}$
MC # $2^{*}$ | $0.00076(1)$ | $\bf-0.757$ | $\bf-0.522$ | $-0.805(34)$ | $-0.0028(2)$
MC # $3^{*}$ | $0.00073(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $-0.25(13)$ | $-0.0050(14)$
SE # $2^{*}$ | $0.00073(1)$ | $\bf-0.757$ | $\bf-0.522$ | $-0.577(14)$ | $-0.00373(15)$
SE # $3^{*}$ | $0.00073(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $-0.369(15)$ | $-0.00457(16)$

###  3.5 Universal amplitude ratio $R_{c}^{-}$.

Figure 7: The functions ${R}^{-}_{C}(-|\tau|)$ (open symbols) and
${R}^{-}_{C^{*}}(-\tau)$ (closed symbols) which approaches the universal ratio
$R_{C}^{-}$ as $\tau\to 0^{-}$. Our MC data are represented by circles and the
CTV data by stars.

We use the available MC data for $C$, $M$, and $\chi$ to estimate the
universal amplitude ratio $R_{C}^{-}$ in the low-temperature phase. To this
purpose, we form the function (compare with Eq. (69))

$R_{C}^{-}(-|\tau|)=\alpha\tau^{2}\frac{C(-|\tau|)\chi_{L}(-|\tau|)}{M^{2}(-|\tau|)}$
(46)

which is an estimator of the universal amplitude ratio in the limit
$|\tau|\rightarrow 0$. As discussed in the Appendix, we expect that all sets
of logarithmic corrections cancel in this ratio. Figure 7 shows with open
symbols the combination from Eq. (46) for two sets of MC data, those of
Caselle et al. (CTV) and our simulations. We may fit these data with
correction-to-scaling terms starting from $|\tau|^{2/3}$ or, assuming in plain
analogy with the energy ratio that such corrections cancel, with terms
starting with $|\tau|$. In table 7 we include these fits for our MC data set
varying the temperature window and the number of correction terms.

Table 7: Estimates of the critical amplitude ratio $R_{C}^{-}$ from our MC data using different fits and varying the temperature window. $\tau-$window | amplitude | correction terms
---|---|---
|  |
| $R_{C}^{-}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{4/3}$ | $\quad\propto|\tau|^{5/3}$
$0.01-0.29$ | 0.00651(3) | $-3.04(12)$ | $-1.24(39)$ | $\phantom{-}4.18(32)$ | $\phantom{-}-$
| 0.00685(3) | $-4.60(3)$ | $\phantom{-}3.84(5)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00619(1) | $\phantom{-}-$ | $-16.13(11)$ | $\phantom{-}29.25(37)$ | $-14.34(31)$
| 0.00591(2) | $\phantom{-}-$ | $-11.01(10)$ | $\phantom{-}12.13(16)$ | $\phantom{-}-$
$0.01-0.10$ | 0.00628(7) | $-1.54(58)$ | $-6.78\pm 2.36$ | $\phantom{-}9.41\pm 2.53$ | $\phantom{-}-$
| 0.00654(3) | $-3.66(9)$ | $-1.99(18)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00627(3) | $-2.70(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00618(3) | $\phantom{-}-$ | $-15.90\pm 1.26$ | $\phantom{-}28.64\pm 5.51$ | $-14.09\pm 6.18$
| 0.00611(1) | $\phantom{-}-$ | $-13.05(17)$ | $\phantom{-}16.10(35)$ | $\phantom{-}-$
$0.01-0.046$ | 0.00642(6) | $-3.09(33)$ | $\phantom{-}0.62(80)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00637(1) | $-2.83(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00616(4) | $\phantom{-}-$ | $-14.20(94)$ | $\phantom{-}18.99\pm 2.44$ | $\phantom{-}-$
| 0.00588(4) | $\phantom{-}-$ | $-6.92(18)$ | $\phantom{-}-$ | $\phantom{-}-$

A similar analysis was also performed for the data from Caselle et al (not
shown here). The results of both analysis are consistent.

More traditionally, we may evaluate the ratio $R_{C}^{-}$ using the estimated
values of the amplitudes reported in sections 3.2, 3.3 and 3.4 (see Eq. (14)).
The results will be presented later, but anticipating the forthcoming
analysis, we can quote as a reliable estimate $R_{C}^{-}\simeq 0.0052\pm
0.0002$ (approximants of SE data lead to $0.0050(2)$). One can see from the
table 7 that the ratio obtained from the effective function Eq. (46) is
systematically larger. The difference may be explained by the fact that
fitting the effective ratio function Eq. (46) without any logarithmic
correction, we assume that the background corrections for the longitudinal
susceptibility $\chi_{L}$ and for the specific heat $C$ are small in the
critical temperature window. While this is indeed the case, these background
terms are not negligibly small and their presence leads to systematic
deviations of the estimates of $R_{C}^{-}$ presented in the table 7. The ratio
(46) may be written as
$\alpha\tau^{2}(C_{s}+C_{bt})(\chi_{s}+\chi_{bt})/M^{2}$, where $C_{s}$ and
$\chi_{s}$ are the singular parts of the specific heat and of the
susceptibility and $C_{bt}$ and $\chi_{bt}$ are the corresponding background
nonsingular terms. Eq. (46) may be rewritten as
$\alpha\tau^{2}C_{s}\chi_{s}(1+C_{bt}/C_{s})(1+\chi_{bt}/\chi_{s})/M^{2}$.
Thus, the background terms $C_{bt}$ and $\chi_{bt}$ contribute when divided by
the singular terms (or in other words multiplied by factors $\tau^{2/3}{\cal
G}$ and $\tau^{7/6}{\cal G}^{-3/4}$, respectively). Clearly, in the critical
region, the first factor has the dominant contribution. This “large” term may
be eliminated completely if we form a quantity equivalent to Eq. (46) from the
energy difference $E_{-}(|\tau|)-E_{0}$ instead of the specific heat,

$R^{-}_{C^{*}}(-|\tau|)=\alpha(\alpha-1)\beta_{c}\tau\frac{(E_{-}(-|\tau|)-E_{0})\chi_{L}(-|\tau|)}{M^{2}(-|\tau|)},$
(47)

which is shown in the Figure 7 with closed symbols. The extrapolation at
$\tau\to 0^{-}$ is obviously different. The results of the fit of MC data to
Eq. (47) are given in table 8. The outcome for the universal combination
$R_{C}^{-}$ is now fully consistent with the value $0.0052(2)$ and supports
our idea that the specific heat background term spoils the behavior of the
estimator (46).

Table 8: Estimates of the critical amplitude ratio $R_{C^{*}}^{-}$ (Expr. (47)) from our MC data using different fits and varying the temperature window. $\tau-$window | amplitude | correction terms
---|---|---
|  |
| $R_{C}^{-}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{4/3}$ | $\quad\propto|\tau|^{5/3}$
$0.01-0.29$ | 0.00551(9) | $-2.53(5)$ | $\phantom{-}0.57(15)$ | $\phantom{-}1.11(12)$ | $\phantom{-}-$
| 0.00558(3) | $-2.97(3)$ | $\phantom{-}1.95(2)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00527(8) | $\phantom{-}-$ | $-11.44(16)$ | $\phantom{-}20.98(51)$ | $-11.25(42)$
| 0.00508(1) | $\phantom{-}-$ | $-7.29(9)$ | $\phantom{-}7.38(14)$ | $\phantom{-}-$
$0.01-0.10$ | 0.00558(4) | $-3.28(36)$ | $\phantom{-}3.66\pm 1.45$ | $\phantom{-}2.23\pm 1.55$ | $\phantom{-}-$
| 0.00553(1) | $-2.77(5)$ | $\phantom{-}1.58(9)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00535(2) | $-1.98(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00538(2) | $\phantom{-}-$ | $-16.11(99)$ | $\phantom{-}40.39\pm 4.30$ | $31.97\pm 4.81$
| 0.00525(2) | $\phantom{-}-$ | $-9.55(24)$ | $\phantom{-}11.82(50)$ | $\phantom{-}-$
$0.01-0.046$ | 0.00558(3) | $-3.09(20)$ | $\phantom{-}2.34(48)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00543(2) | $-2.11(3)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00535(3) | $\phantom{-}-$ | $-12.40(74)$ | $\phantom{-}18.95\pm 1.99$ | $\phantom{-}-$
| 0.00511(4) | $\phantom{-}-$ | $-5.04(19)$ | $\phantom{-}-$ | $\phantom{-}-$
Table 9: Estimates of the critical amplitude ratio $R_{C^{*}}^{-}$ from Caselle et al MC data using different fits and varying the temperature window. $\tau-$window | amplitude | correction terms
---|---|---
|  |
| $R_{C}^{-}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{4/3}$
$0.0058-0.029$ | 0.00548(2) | $-2.59(17)$ | $1.27(48)$ | $\phantom{-}-$
| 0.00543(1) | $-2.14(2)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00535(1) | $\phantom{-}-$ | $-13.77(66)$ | $\phantom{-}21.79\pm 1.98$
| 0.00521(2) | $\phantom{-}-$ | $-5.93(23)$ | $\phantom{-}-$

Again, a similar analysis of CTV data (see table 9) leads to fully consistent
results.

##  4 Discussion

Our final goal is the determination of some universal combinations of
amplitudes. This can be done either directly from the values of the amplitudes
listed in the various tables of this paper, or also by extrapolating the
effective ratios to $\tau=0$. Let us start with an estimate obtained by the
second method, and let us concentrate on the most controversial amplitude
ratios, those of the susceptibilities. As we have just shown in the section on
$R_{C}^{-}$, this method may lead to systematic deviations if the background
terms are not handled with care. Later on we shall discuss the ratios of the
amplitudes listed in the previous tables.

The ratio $\Gamma_{+}/\Gamma_{L}$ can be estimated from the ratio of the SE of
the susceptibility $\chi_{+}$ at high-temperature and of the longitudinal
susceptibility $\chi_{L}$ in the low-temperature phase. We have again two
options to form this ratio, either from quantities computed at temperatures
symmetric with respect to the critical temperature $T_{c}\pm\tau$ or at
inverse temperatures $\beta$ and $\beta^{*}$ related by the duality relation
(3).

Figure 8: The ratio of the effective amplitudes
$\Gamma_{+}(\tau)/\Gamma_{L}(-|\tau|)$ obtained from SE data at symmetric
temperatures (boxes) and dual temperatures (circles) together with the
corresponding linear fits (solid lines).

Figure 8 shows both the ratios $\Gamma_{+}(\tau)/\Gamma_{L}(-|\tau|)$ and
$\Gamma_{+}(\tau)/\Gamma_{L}(\tau^{*})$, while the straight lines are drawn as
a guide for the eye. It would be naive to take the linear fit too seriously,
otherwise one should conclude that the background (non-singular) contribution
to the ratio is negligible. The value of the universal amplitude ratio
$\Gamma_{+}/\Gamma_{L}$ obtained from the SE data is approximately
$\Gamma_{+}/\Gamma_{L}\simeq 6.16(1)$ and $6.30(1)$, when using respectively
the fits MC $\\#2^{*}$ and MC $\\#3^{*}$. We have also analyzed the effective-
amplitude ratio from MC data obtained by dividing the high-temperature reduced
susceptibility by the longitudinal reduced susceptibility computed at
temperatures related by the duality relation. Neglecting the constant
background terms in the susceptibilities eliminates all logarithms and makes
the fit quite simple, leading to a ratio in the range
$\Gamma_{+}/\Gamma_{L}\simeq 6.30-6.60$. On the other hand, if we keep in the
fit the background terms, the logs reappear and we are lead to
$\Gamma_{+}/\Gamma_{L}\simeq 6.0-6.1$.

Thus we get values which are quite different from the analytical prediction
$\Gamma_{+}/\Gamma_{L}=4.013$ of Delfino and Cardy [13], as well as from the
value $3.5(4)$ estimated by Enting and Guttmann from an analysis of the SE
data for the susceptibility in both phases, and from the value $3.14(70)$
estimated by Caselle et al. [4].

Let us now estimate the effective-amplitude ratio
$\Gamma_{T}(-|\tau|)/\Gamma_{L}(-|\tau|)$. This ratio, shown in figure 9, has
been computed both by MC simulation, for various lattice sizes, and from SE
data. Due to the non-singular correction terms, its behavior is far from being
linear in $\tau$ as was the ratio $\chi_{+}(|\tau|)/\chi_{L}(-|\tau|)$
(compare with figure 8). A possible explanation is that there might be some
symmetry in the correction-to-scaling amplitudes occurring in the asymptotic
expansion of $\chi_{+}(|\tau|)$ and $\chi_{L}(-|\tau|)$, but not of
$\chi_{T}(-|\tau|)$, which introduces here a stronger background term still
containing the logs. Following again the same procedure, we arrive at
estimates close to $0.146$ when neglecting all logarithmic corrections, while
we get $0.152-0.153$ when allowing for these corrections and from the analysis
of SE data. Finally we obtain $\Gamma_{T}/\Gamma_{L}=0.151(3)$ and $0.148(3)$
from the fits MC $\\#2^{*}$ and $3^{*}$. All these values differ from the
analytical prediction $\Gamma_{T}/\Gamma_{L}=0.129$ of Delfino et al. [18] and
from the value $0.11(4)$ of Ref. [20].

Figure 9: 4-state Potts model. The ratio of the effective amplitudes
$\Gamma_{T}(-|\tau|)/\Gamma_{L}(-|\tau|)$ for the 4-state Potts model on
square lattices of linear sizes $L=20$ (boxes), $L=40$ (up triangles), $L=60$
(down triangles), $L=80$ (diamonds), $L=100$ (stars) computed with
$N_{MC}=10^{5}$ MC steps, and $L=100$ (closed circles) computed with
$N_{MC}=10^{6}$. The solid line represents the SE data.

Let us now extract what we believe are more reliable estimates for the
universal combinations of amplitudes by a direct evaluation of the ratios of
the numbers presented in this paper and collected again in table 10. The
universal combinations are presented in table 11, together with the
corresponding results available in the literature. Averaging our different
results, we quote the following final estimates:

$\displaystyle\Gamma_{+}/\Gamma_{L}$ $\displaystyle=$ $\displaystyle 6.49\pm
0.44,$ (48) $\displaystyle\Gamma_{T}/\Gamma_{L}$ $\displaystyle=$
$\displaystyle 0.154\pm 0.012,$ (49) $\displaystyle R_{C}^{+}$
$\displaystyle=$ $\displaystyle 0.0338\pm 0.0009,$ (50) $\displaystyle
R_{C}^{-}$ $\displaystyle=$ $\displaystyle 0.0052\pm 0.0002.$ (51)

These results clearly confirm the above-mentioned limits of effective ratios.

Table 10: Critical amplitudes and correction coefficients for the 4-state Potts model. They are written in the following format ${\rm Obs.}(\pm|\tau|)\simeq{\rm Ampl.}\times|\tau|^{\blacktriangleleft}\times{\cal G}^{\bigstar}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+{\rm backg.\ terms}.$ The results of the MC analysis of Ref.[4] are compiled together with our results obtained by combining the MC and series expansion (SE) data analysis.  observable | amplitude |  correction terms, $\times{\cal G}^{\bigstar}(-\ln|\tau|)$ | background terms | source
---|---|---|---|---
|  |  |  |
|  | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{0}$ | $\quad\propto|\tau|$ |
$E_{\pm}(\tau)$ | $1.338(3)/\alpha(1-\alpha)\beta_{c}$ | $-4.98(8)$ | $\phantom{-}\alpha_{4}$ | $\phantom{--}E_{0}$ | $\phantom{-}3.77(6)$ |  this paper MC# $2^{*}$
| $1.316(9)/\alpha(1-\alpha)\beta_{c}$ | $-4.88(38)$ | $\phantom{-}\alpha_{4}$ | $\phantom{--}E_{0}$ | $\phantom{-}3.68(30)$ |  this paper MC# $3^{*}$
$\chi_{+}(\tau)$ | $\Gamma_{+}=0.0223(14)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}0.05(14)$ | $\phantom{-}-$ | [4]
| $\Gamma_{+}=0.031(5)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
| $\Gamma_{+}=0.03144(15)$ | $\phantom{-}0.561(60)$ | $\phantom{-}-$ | $-0.053(17)$ | $\phantom{-}-$ |  this paper MC# $2^{*}$
| $\Gamma_{+}=0.03178(30)$ | $\phantom{-}0.53(23)$ | $\phantom{-}-$ | $\phantom{-}0.052(120)$ | $\phantom{-}-$ |  this paper MC# $3^{*}$
| $\Gamma_{+}=0.03041(1)$ | $\phantom{-}1.30(1)$ | $\phantom{-}-$ | $-0.362(9)$ | $\phantom{-}-$ |  this paper SE# $2^{*}$
| $\Gamma_{+}=0.03039(1)$ | $\phantom{-}1.67(1)$ | $\phantom{-}-$ | $-0.59(2)$ | $\phantom{-}-$ |  this paper SE# $3^{*}$
$\chi_{L}(-|\tau|)$ | $\Gamma_{L}=0.00711(10)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}0.02(1)$ | $\phantom{-}-$ | [4]
| $\Gamma_{L}=0.0088(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
| $\Gamma_{L}=0.00454(2)$ | $-2.83(3)$ | $\phantom{-}-$ | $\phantom{-}0.050(2)$ | $\phantom{-}-$ |  this paper MC# $2^{*}$
| $\Gamma_{L}=0.00484(3)$ | $-3.73(14)$ | $\phantom{-}-$ | $\phantom{-}0.13(1)$ | $\phantom{-}-$ |  this paper MC# $3^{*}$
| $\Gamma_{L}=0.00483(1)$ | $-3.77(3)$ | $\phantom{-}-$ | $\phantom{-}0.116(3)$ | $\phantom{-}-$ |  this paper SE# $2^{*}$
| $\Gamma_{L}=0.00493(1)$ | $-4.18(5)$ | $\phantom{-}-$ | $\phantom{-}0.178(6)$ | $\phantom{-}-$ |  this paper SE# $3^{*}$
$\chi_{T}(\tau)$ | $\Gamma_{T}=0.0010(3)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
| $\Gamma_{T}=0.00076(1)$ | $-0.805(34)$ | $\phantom{-}-$ | $-0.0028(2)$ | $\phantom{-}-$ |  this paper MC# $2^{*}$
| $\Gamma_{T}=0.00073(1)$ | $-0.25(13)$ | $\phantom{-}-$ | $-0.0050(14)$ | $\phantom{-}-$ |  this paper MC# $3^{*}$
| $\Gamma_{T}=0.00073(1)$ | $-0.577(14)$ | $\phantom{-}-$ | $-0.00373(15)$ | $\phantom{-}-$ |  this paper SE# $2^{*}$
| $\Gamma_{T}=0.00073(1)$ | $-0.369(15)$ | $\phantom{-}-$ | $-0.00457(16)$ | $\phantom{-}-$ |  this paper SE# $3^{*}$
$M(-|\tau|)$ | $B=1.1621(11)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}0.05(14)$ | $\phantom{-}-$ | [4]
| $B=1.1570(1)$ | $-0.191(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ |  this paper MC# $2^{*}$
| $B=1.1559(12)$ | $-0.210(10)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ |  this paper MC# $3^{*}$
| $B=1.1575(1)$ | $-0.194(1)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ |  this paper SE# $2^{*}$
| $B=1.1575(1)$ | $-0.225(1)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ |  this paper SE# $3^{*}$
Table 11: Universal combinations of the critical amplitudes in the 4-state Potts model.  $A_{+}/A_{-}$ | $\Gamma_{+}/\Gamma_{L}$ | $\Gamma_{T}/\Gamma_{L}$ | $R_{C}^{+}$ | $R_{C}^{-}$ | source
---|---|---|---|---|---
$1.$ | $4.013$ | $0.129$ | $0.0204$ | $0.00508$ | [13, 18]
$\phantom{-}-$ | $4.02$ | $0.129$ | $\phantom{-}-$ | $\phantom{-}-$ | [19]
$\phantom{-}-$ | $3.14(70)$ | $\phantom{-}-$ | $0.021(5)$ | $0.0068(9)$ | [4]
$\phantom{-}-$ | $3.5(4)$ | $0.11(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
$1.000(5)$ | $6.93(6)$ | $0.1674(30)$ | $0.03452(25)$ | $0.00499(3)$ |  this paper MC# $2^{*}$
$1.000(13)$ | $6.57(10)$ | $0.1508(26)$ | $0.03439(63)$ | $0.00524(9)$ |  this paper MC# $3^{*}$
$\phantom{-}-$ | $6.30(1)$ | $0.1511(24)$ | $0.03336(9)$ | $0.00530(3)$ |  this paper SE# $2^{*}$
$\phantom{-}-$ | $6.16(1)$ | $0.1481(23)$ | $0.03279(24)$ | $0.00532(5)$ |  this paper SE# $3^{*}$

The main outcome of this work are the surprisingly high values of the ratios
$\Gamma_{+}/\Gamma_{L}$, $\Gamma_{T}/\Gamma_{L}$ and $R_{C}^{+}$ (and the low
value for $R_{C}^{-}$), significantly deviating from the predictions of
Delfino and Cardy. We emphasize that our results are also supported by a
direct extrapolation of effective-amplitude ratios for which most corrections
to scaling disappear. We believe that our fitting procedure is reliable, and
since the disagreement with the theoretical calculations can hardly be
resolved, we suspect that it might be attributed to the approximations made in
Ref. [13] in order to predict the susceptibility ratios. Even more puzzling is
the fact that Delfino and Cardy argue in favor of a higher robustness of their
results for $\Gamma_{T}/\Gamma_{-}$ than for $\Gamma_{+}/\Gamma_{-}$, while
the disagreement is indisputable in both cases. Indeed, in the conclusion of
their paper, and in a footnote, Delfino et al [18] (p.533) explain that their
results are sensitive to the relative normalization of the order- and
disorder-operator form-factors, which could be the origin of some troubles for
$q=3$ and $4$ for the ratios $\Gamma_{+}/\Gamma_{L}$ and $R_{C}$ only.

As a final argument in favor of our results, we may mention a work of W. Janke
and one of us (LNS) on the amplitude ratios in the Baxter-Wu model (expected
to belong to the 4-state Potts model universality class), leading to the
estimate $\Gamma_{+}/\Gamma_{-}\simeq 6.9$ and $R_{C}^{-}\approx 0.005$ [30].
This result, obtained from an analysis of MC data shows a similar discrepancy
with Delfino and Cardy’s result and suggests that further analysis might still
be necessary.

##  5 Acknowledgements

Discussions with A. Zamolodchikov, V. Dotsenko, V. Plechko, W. Janke and M.
Henkel, and a correspondence with J. Cardy and J. Salas were very helpful.

LNS is grateful to the Statistical Physics group of the University Henri
Poincaré Nancy 1 for the kind hospitality. Both LNS and PB thank the
Theoretical group of the University Milano–Bicocca for hospitality and
support. Financial support from the Laboratoire Européen Associé “Physique
Théorique et Matière Condensée”, a common research program between the Landau
Institute, the Ecole Normale Supérieure de Paris, Paris Sud University and the
Russian Foundation for Basic Research is also gratefully acknowledged.

##  Appendix A Solution of non-linear RG equations and cancelation of
logarithmic corrections in effective-amplitude ratios

For the 4-state Potts model, the non-linear RG equation for the relevant
thermal and magnetic fields $\phi$ and $h$, with the corresponding RG
eigenvalues $y_{\phi}$ and $y_{h}$, and the marginal dilution field $\psi$ are
given by

$\displaystyle\frac{d\phi}{d\ln b}$ $\displaystyle=$
$\displaystyle(y_{\phi}+y_{\phi\psi}\psi)\phi,$ (52)
$\displaystyle\frac{dh}{d\ln b}$ $\displaystyle=$
$\displaystyle(y_{h}+y_{h\psi}\psi)h,$ (53) $\displaystyle\frac{d\psi}{d\ln
b}$ $\displaystyle=$ $\displaystyle g(\psi)$ (54)

where $b$ is the length rescaling factor and $l=\ln b$. The function $g(\psi)$
may be Taylor expanded,
$g(\psi)=y_{\psi^{2}}\psi^{2}(1+\frac{y_{\psi^{3}}}{y_{\psi^{2}}}\psi+\dots)$.
Accounting for marginality of the dilution field, there is no linear term. The
first term has been considered by Nauenberg and Scalapino, and later by Cardy,
Nauenberg and Scalapino. The second term was introduced by Salas and Sokal. In
this appendix, we slightly change the notations of Salas and Sokal, keeping
however the notation $y_{ij}$ for all coupling coefficients between the
scaling fields $i$ and $j$. These parameters take the values
$y_{\phi\psi}=3/(4\pi)$, $y_{h\psi}=1/(16\pi)$, $y_{\psi^{2}}=1/\pi$ and
$y_{\psi^{3}}=-1/(2\pi^{2})$, while the relevant scaling dimensions are
$y_{\phi}=\nu^{-1}=3/2$ and $y_{h}=15/8$.

The fixed point is at $\phi=h=0$. Starting from initial conditions $\phi_{0}$,
$h_{0}$, the relevant fields grow exponentially with $l$. The field $\phi$ is
analytically related to the temperature, so the temperature behavior follows
from the renormalization flow from $\phi_{0}\sim|\tau|$ up to some $\phi=O(1)$
outside the critical region. Notice also that the marginal field $\psi$
remains of order $O(\psi_{0})$ and $\psi_{0}$ is negative, $\psi_{0}=O(-1)$.
In zero magnetic field, under a change of length scale, the singular part of
the free energy density transforms as

$f(\psi_{0},\phi_{0})=e^{-Dl}f(\psi,\phi),$ (55)

where $D=2$ is the space dimension. Solving Eq. (52) leads to
$\ln(\phi/\phi_{0})=y_{\phi}l+y_{\phi\psi}\int\psi dl$ where the last integral
is obtained from Eq. (54) rewritten as $\int_{0}^{l}\psi
dl=\frac{1}{y_{\psi^{2}}}\ln({\psi}/{\psi_{0}})+\frac{1}{y_{\psi^{2}}}\ln
G(\psi_{0},\psi)$. Note that $G(\psi_{0},\psi)$ takes the value 1 in Cardy,
Nauenberg and Scalapino and the value
$\frac{y_{\psi^{2}}+y_{\psi^{3}}\psi_{0}}{y_{\psi^{2}}+y_{\psi^{3}}\psi}$ in
Salas and Sokal. Since this term appears always in the same combination, we
write $z=\psi_{0}/\psi$, $\bar{z}=\frac{z}{G(\psi_{0},\psi)}$ and in the same
way we set $x=\phi_{0}/\phi$. One thus obtains

$l=-\frac{1}{y_{\phi}}\ln
x+\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln\bar{z}.$ (56)

At the critical temperature $\phi=0$, the $l-$dependence on the magnetic field
obeys a similar expression and one is led to the equality

$l=-\nu\ln x+\mu\ln\bar{z}=-\nu_{h}\ln y+\mu_{h}\ln\bar{z},$ (57)

where $y=h_{0}/h$ and for brevity we will denote $\nu=1/y_{\phi}=\frac{2}{3}$,
$\mu=\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}=\frac{1}{2}$,
$\nu_{h}=1/y_{h}=\frac{8}{15}$ and
$\mu_{h}=\frac{y_{h\psi}}{y_{h}y_{\psi^{2}}}=\frac{1}{30}$. We can thus deduce
the following behavior for the free energy density in zero magnetic field in
terms of the thermal and dilution fields, or at the critical temperature in
terms of magnetic and dilution fields

$\displaystyle f(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle x^{D\nu}\
\\!\bar{z}^{-D\mu}f(\phi,\psi)$ (58) $\displaystyle f(h_{0},\psi_{0})$
$\displaystyle=$ $\displaystyle y^{D\nu_{h}}\ \\!\bar{z}^{-D\mu_{h}}f(h,\psi)$
(59)

The other thermodynamic properties follow by derivation with respect to the
scaling fields, e.g.
$E(\phi_{0},\psi_{0})=\frac{\partial}{\partial\phi_{0}}f(\psi_{0},\phi_{0})$,
or

$\displaystyle E(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\frac{\partial\phi}{\partial\phi_{0}}E(\phi,\psi),$ (60) $\displaystyle
C(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\left(\frac{\partial\phi}{\partial\phi_{0}}\right)^{2}C(\phi,\psi),$
(61) $\displaystyle M(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\frac{\partial h}{\partial h_{0}}M(\phi,\psi),$ (62)
$\displaystyle\chi(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\left(\frac{\partial h}{\partial h_{0}}\right)^{2}\chi(\phi,\psi).$
(63)

The derivatives $\frac{\partial\phi}{\partial\phi_{0}}$ and $\frac{\partial
h}{\partial h_{0}}$ will thus determine the scaling behavior of all the
thermodynamic quantities. The first one is obvious,
$\frac{\partial\phi}{\partial\phi_{0}}=x^{-1}$ and for the second one,
$\frac{\partial h}{\partial h_{0}}=y^{-1}$, we express $y$ in terms of $x$ and
$z$ using eq. (57) 101010It follows that
$y=x^{y_{h}/y_{\phi}}\bar{z}^{y_{h\psi}/y_{\psi^{2}}-y_{h}y_{\phi\psi}/y_{\phi}y_{\psi^{2}}}$..
Altogether, introducing the notations $\lambda=y_{h}/y_{\phi}=\frac{5}{4}$ and
$\kappa=y_{h}y_{\phi\psi}/y_{\phi}y_{\psi^{2}}-y_{h\psi}/y_{\psi^{2}}=\frac{7}{8}$
one has

$\displaystyle f(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle x^{D\nu}\
\\!\bar{z}^{-D\mu}f(\phi,\psi)$ (64) $\displaystyle E(\phi_{0},\psi_{0})$
$\displaystyle=$ $\displaystyle x^{D\nu-1}\ \\!\bar{z}^{-D\mu}E(\phi,\psi)$
(65) $\displaystyle C(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
x^{D\nu-2}\ \\!\bar{z}^{-D\mu}C(\phi,\psi)$ (66) $\displaystyle
M(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle x^{D\nu-\lambda}\
\\!\bar{z}^{-D\mu+\kappa}M(\phi,\psi)$ (67)
$\displaystyle\chi(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
x^{D\nu-2\lambda}\ \\!\bar{z}^{-D\mu+2\kappa}\chi(\phi,\psi)$ (68)

What appears extremely useful in these expressions is that when defining
appropriate effective ratios111111i.e. effective ratios which eventually tend
to universal limits when $\tau\to 0$, the dependence on the quantity $\bar{z}$
cancels, due to the scaling relations among the critical exponents. This
quantity $\bar{z}$ is precisely the only one where the log terms are hidden,
and thus we may infer that not only the leading log terms, but all the log
terms hidden in the dependence on the marginal dilution field disappear in the
conveniently defined effective ratios. For example in an effective ratio like

$R_{C}(x)=x^{2}\frac{C(x,z)\chi(x,z)}{M^{2}(x,z)},$ (69)

all corrections to scaling coming from the variable $z$ disappear.

Now we proceed by iterations of $l=-\nu\ln x+\mu\ln\bar{z}$ and
$\bar{z}=z\frac{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi}{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi_{0}}$.
The asymptotic solution of Eq. (54) is121212When $y_{\psi^{3}}=0$ the
asymptotic solution of Eq. (54) is simply
$\frac{\psi_{0}}{1-\psi_{0}y_{\psi^{2}}l}$. Thus one can try the ansatz
$\psi=\frac{\psi_{0}}{1-\psi_{0}y_{\psi^{2}}l}(1+X(l))$ with a small
correction $X(l)$ to solve asymptotically the full Eq. (54). Keeping only
terms of order $O(l^{-3})$ at most, we are led to the following expression
$X^{\prime}(l)=-(1/l)X(l)+y_{\psi^{3}}/((y_{\psi^{2}})^{2}l^{2})$ where we use
$X(l)=Y(l)/l$ to eventually obtain $Y(l)=(y_{\psi^{3}}/(y_{\psi^{2}})^{2})\ln
l$.

$\frac{\psi}{\psi_{0}}=\frac{1}{1-\psi_{0}y_{\psi^{2}}l}\left(1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}}\frac{\ln
l}{l}+O(1/l)\right).$ (70)

We get for the variable $z$

$\displaystyle
z=\frac{1-\psi_{0}y_{\psi^{2}}l}{1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}}\frac{\ln
l}{l}}\simeq-\psi_{0}y_{\psi^{2}}\nu(-\ln|\tau|)\frac{1+\frac{\mu}{\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}+O\left(\frac{1}{-\ln|\tau|}\right)}{1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}+O\left(\frac{1}{-\ln|\tau|}\right)}.$
(71)

Similarly, one has the combination

$\frac{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi}{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi_{0}}\simeq\frac{1}{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi_{0}}\left(1-\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{1}{(-\ln|\tau|)}+O\left(\frac{1}{(-\ln|\tau|)^{2}}\right)\right)$
(72)

and eventually one gets for the full correction-to-scaling variable the heavy
expression

$\displaystyle\bar{z}$ $\displaystyle=$ $\displaystyle{\rm
const}\times(-\ln|\tau|)\frac{1+\frac{\mu}{\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}{1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}\times\left(1-\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{1}{(-\ln|\tau|)}\right)\times{F}(-\ln|\tau|)$
(73) $\displaystyle=$ $\displaystyle{\rm
const}\times(-\ln|\tau|)\frac{1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}{1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}\times\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right)\times{\cal
F}(-\ln|\tau|)$

where ${\cal F}(-\ln|\tau|)$ is a function of the variable $(-\ln|\tau|)$ only
where also appears the non-universal constant $\psi_{0}$. Using Eq. (67), we
deduce the behavior of the magnetization for example

$\displaystyle M(-|\tau|)$ $\displaystyle=$ $\displaystyle
B|\tau|^{1/12}(-\ln|\tau|)^{-1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)\right.$
(74) $\displaystyle\left.\
\qquad\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{-\ln|\tau|}\right){\cal
F}(-\ln|\tau|)\right]^{-1/8}.$

Since all these log expressions are “lazy functions”, it is unsafe to expand
such terms, e.g.
$\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\simeq
1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}$, since the correction term is
not small enough in the accessible temperature range $|\tau|\simeq 0.05-0.10$.
We can thus only extract an effective function ${\cal F}_{eff}(-\ln|\tau|)$
which mimics the real one ${\cal F}(-\ln|\tau|)$ in the convenient temperature
range. This is done through a plot of an effective-magnetization amplitude

$\displaystyle B_{eff}(-|\tau|)$ $\displaystyle=$ $\displaystyle
M(-|\tau|)|\tau|^{-1/12}(-\ln|\tau|)^{1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)\right.$
(75)
$\displaystyle\left.\qquad\qquad\qquad\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{-\ln|\tau|}\right)\right]^{1/8}$

which is found to behave as

$B_{eff}(-|\tau|)=B\left(1-\frac{C_{1}}{-\ln|\tau|}-\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{1/8}$
(76)

from which one deduces that the function ${\cal F}(-\ln|\tau|)$ takes the
approximate expression

${\cal
F}(-\ln|\tau|)\simeq\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{-1}.$
(77)

What is remarkable is the stability of the fit to Eq. (76). We obtain (see
table 1) $C_{1}=-0.757$ and $C_{2}=-0.522$ which yields an amplitude
$B=1.1570(1)$. It is also possible to try a simpler choice, fixing $C_{1}=0$
and approximating the whole series by the $C_{2}-$term only. We then find the
value $C_{2}^{\prime}=-0.88$ and this leads to a very close magnetization-
amplitude $B=1.1559(2)$.

For the following, we group all the terms coming from the variable $\bar{z}$
into a single function ${\cal G}(-\ln|\tau|)=(-\ln|\tau|)\times{\cal
E}(-\ln|\tau|)\times{\cal F}(-\ln|\tau|)$ where

$\displaystyle{\cal E}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)$
(78)
$\displaystyle\qquad\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{-\ln|\tau|}\right)$

in terms of which the singular parts of the physical quantities take a very
compact form,

$\displaystyle f(\tau)$ $\displaystyle=$ $\displaystyle
F_{\pm}|\tau|^{4/3}{\cal G}^{-1}(-\ln|\tau|)$ (79) $\displaystyle M(-|\tau|)$
$\displaystyle=$ $\displaystyle B|\tau|^{1/12}{\cal G}^{-1/8}(-\ln|\tau|)$
(80) $\displaystyle\chi_{+}(\pm|\tau|)$ $\displaystyle=$
$\displaystyle\Gamma_{+}|\tau|^{-7/6}{\cal G}^{3/4}(-\ln|\tau|)$ (81)
$\displaystyle\chi_{L}(-|\tau|)$ $\displaystyle=$
$\displaystyle\Gamma_{L}|\tau|^{-7/6}{\cal G}^{3/4}(-\ln|\tau|)$ (82)
$\displaystyle\chi_{T}(-|\tau|)$ $\displaystyle=$
$\displaystyle\Gamma_{T}|\tau|^{-7/6}{\cal G}^{3/4}(-\ln|\tau|)$ (83)
$\displaystyle E_{\pm}(\pm|\tau|)$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha(\alpha-1)}|\tau|^{1/3}{\cal
G}^{-1}(-\ln|\tau|)$ (84) $\displaystyle C_{\pm}(\pm|\tau|)$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-2/3}{\cal G}^{-1}(-\ln|\tau|).$
(85)

The function ${\cal E}$ is known exactly while the function ${\cal F}$ needs
to be fitted to the numerical data. In the same range of values of the reduced
temperature, the “correction function” ${\cal F}(-\ln|\tau|)$ is now fixed and
the only remaining freedom is to include background terms and possibly
additive corrections to scaling coming from irrelevant scaling fields 131313To
introduce corrections to scaling, let us consider the case of an irrelevant
scaling field, let say $g$, coupled to the temperature field through
$\frac{d\phi}{dl}=y_{\phi}\phi+y_{\phi\psi}\phi\psi+y_{\phi g}\phi
g\quad\hbox{and}\quad\frac{dg}{dl}=y_{g}g$ ($\Delta>0$ above plays the rôle of
$-y_{g}/y_{\phi}$, and is thus linked to the corresponding negative RG
eigenvalue $y_{g}$). Solving for $g$ gives $g=g_{0}e^{y_{g}l}$ (the irrelevant
scaling field decays exponentially when one approaches the fixed point).
Solving for $\psi$ gives $\psi=\frac{\psi_{0}}{1-\psi_{0}y_{\psi^{2}}l}$, and
for $\phi$,
$l=\frac{1}{y_{\phi}}\ln(\phi/\phi_{0})-\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln(\psi/\psi_{0})+\frac{y_{\phi
g}}{y_{\phi}y_{g}}g_{0}(e^{y_{g}l}-1).$ Iteration now leads to
$l=-\frac{1}{y_{\phi}}\ln|\tau|+\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln(-\ln|\tau|)+\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln\frac{|\psi_{0}|y_{\psi^{2}}}{y_{\phi}}+\frac{y_{\phi
g}g_{0}}{y_{\phi}|y_{g}|}(1-|\tau|^{|y_{g}|/y_{\phi}})$ and thus a free energy
density including the additive correction term  $f\simeq e^{-Dl}={\rm
const}\times|\tau|^{D/y_{\phi}}(-\ln|\tau|)^{-Dy_{\phi\psi}/y_{\phi}y_{\psi^{2}}}\left(1+\frac{Dy_{\phi
g}g_{0}}{y_{\phi}|y_{g}|}|\tau|^{|y_{g}|/y_{\phi}}\right).$ In our case, the
$\ln|\tau|$ terms are due to the first scaling field (marginal) through the
complicated variable $\bar{z}$ and other correction terms could be added, e.g.
$f(\tau)=F_{\pm}|\tau|^{4/3}(-\ln|\tau|)^{-1}{\cal E}^{-1}(-\ln|\tau|){\cal
F}^{-1}(-\ln|\tau|)(1+D|\tau|^{\frac{2}{3}|y_{g}|}).$ . Among the additive
correction terms, we may have those coming from the thermal sector
$\Delta_{\phi_{n}}=-\nu y_{\phi_{n}}$, where the RG eigenvalues are
$y_{\phi_{n}}=D-\frac{1}{2}n^{2}$. The first dimension $y_{\phi_{1}}=y_{\phi}$
is the temperature RG eigenvalue, the next one, $y_{\phi_{2}}$, vanishes and
is responsible for the appearance of the logarithmic corrections, so the first
irrelevant correction to scaling in the thermal sector comes from
$\Delta_{\phi_{3}}=-\nu y_{\phi_{3}}=5/3$. One can also imagine a coupling of
the magnetic sector to irrelevant scaling fields. The magnetic scaling
dimensions $x_{\sigma_{n}}$ lead to RG eigenvalues
$y_{h_{n}}=D-x_{\sigma_{n}}$. The first dimension $y_{h_{1}}=y_{h}$ is the
magnetic field RG eigenvalue. The second one is still relevant,
$y_{h_{2}}=7/8$, and it could lead, if admissible by symmetry, to corrections
generically governed by the difference of relevant eigenvalues
$(y_{h_{1}}-y_{h_{2}})/y_{\phi}=2/3$. The next contribution comes from
$y_{h_{3}}=-9/8$ and leads to a Wegner correction-to-scaling exponent [29]
$\Delta_{h_{3}}=-\nu y_{h_{3}}=3/4$. Eventually, spatial inhomogeneities of
primary fields (higher order derivatives) bring the extra possibility of
integer correction exponents $y_{n}=-n$ in the conformal tower of the
identity. The first one of these irrelevant terms corresponds to a Wegner
exponent $\Delta_{1}=-\nu(-1)=2/3$ and it is always present. We may thus
possibly include the following corrections: $|\tau|^{2/3}$, $|\tau|^{3/4}$,
$|\tau|^{4/3}$, $|\tau|^{5/3}$, …, the first and third ones being always
present, while the other corrections depend on the symmetry properties of the
observables.

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