y_{h}^{*}=d\Bigl{(}1-\frac{1}{n}\Bigr{)},\ x_{\phi}^{*}=\frac{d}{n},\
\hbox{{\LGR\textqoppa}}=\frac{d}{2}\Bigl{(}1-\frac{2}{n}\Bigr{)}$ (188)

having used (155) with (121), (122), (123) and extended $x_{\phi}^{*}$ from
Ref. [27] for the $n=4$ case, and where $\tilde{p}$ is given in Eq.(167).
Obviously these scaling forms apply to the Q-sector only (Fourier $Q$-modes).
The G sector is unaffected by DIVs and the finite-size counterparts of
Eqs.(13), (14) and (15) apply there. Following a suggestion by Michael Fisher,
a new exponent ϙ was introduced in Ref. [8] for the Q sector. We like the use
of this exponent, since it is very easy to translate equations from one
universality class to another in terms of $d/d_{\rm uc}$, and also to
generalize from the GFP values (which we recover reverting ϙ to 1), so one can
also rewrite the exponents (188) in the form

$y_{t}^{*}=2\hbox{{\LGR\textqoppa}},\
y_{h}^{*}=\frac{d}{2}+\hbox{{\LGR\textqoppa}},\
x_{\phi}^{*}=\frac{d}{2}-\hbox{{\LGR\textqoppa}},\
\hbox{{\LGR\textqoppa}}=\frac{d}{d_{\rm uc}}.$ (189)

An interesting use of ϙ is in a new form of the hyperscaling relation. Setting
$b=|\tau|^{-1/y_{t}}$ in (185) delivers
$\xi_{\infty}\sim|\tau|^{-{\hbox{\footnotesize{\LGR\textqoppa}}}/y_{t}^{*}}$,
hence $y_{t}^{*}=\hbox{{\LGR\textqoppa}}/\nu$, then in (183) we get
$f_{\infty}\sim|t|^{d/y_{t}^{*}}\sim|\tau|^{2-\alpha}$ which leads to

$\alpha=2-\frac{\nu d}{{\hbox{\footnotesize{\LGR\textqoppa}}}}.$ (190)

This repairs151515Since Josephson’s inequality $\nu d\geq 2-\alpha$ was
introduced in 1967 [88], literature, including textbooks, on statistical
physics, lattice field theory, etc. refer to hyperscaling as “failing” above
the upper critical dimension. This statement should no longer be used in
statistical physics — hyperscaling does not fail because the RG does not fail
above the upper critical dimension. Moreover, the hyperscaling relation should
rather be rewritten properly as (190) and not as (27).  the hyperscaling
relation above $d_{\rm uc}$, and it holds also below $d_{\rm uc}$ where
$\hbox{{\LGR\textqoppa}}=1$. (Obviously $\hbox{{\LGR\textqoppa}}=1$ for the
non-zero or orthogonal Fourier modes in high dimensions too.)

The new exponent already enabled predictions to be more naturally expressed
for models such as the nearest-neighbour Ising model, percolation above its
critical dimension $d_{\rm uc}=6$ and for Long-Range Ising Models (LRIM’ with
various dimensions above $d_{\rm uc}(\sigma)$ (with periodic boundary
conditions) [89, 60]. The extension to general values of $n$ is obvious and we
collect the predictions for arbitrary $d>d_{\rm uc}$ for quantities which have
been discussed above:

$\displaystyle m_{L}\sim
L^{-(\frac{d}{2}-{\hbox{\footnotesize{\LGR\textqoppa}}})},\ \chi_{L}\sim
L^{2{\hbox{\footnotesize{\LGR\textqoppa}}}},\ e_{L}\sim
L^{-(d-2{\hbox{\footnotesize{\LGR\textqoppa}}})},\ c_{L}\sim
L^{4{\hbox{\footnotesize{\LGR\textqoppa}}}-d},$ (191)
$\displaystyle\xi_{L}\sim L^{{\hbox{\footnotesize{\LGR\textqoppa}}}},\
g_{L}(X_{0})\sim L^{-(d-2{\hbox{\footnotesize{\LGR\textqoppa}}})},$ (192)
$\displaystyle t_{L}\sim L^{-2{\hbox{\footnotesize{\LGR\textqoppa}}}},\
\Delta\beta_{L}\sim L^{-2{\hbox{\footnotesize{\LGR\textqoppa}}}},\ |h^{L}|\sim
L^{-(\frac{d}{2}+{\hbox{\footnotesize{\LGR\textqoppa}}})},$ (193)
$\displaystyle h^{\rm LY}_{L}\sim
L^{-(\frac{d}{2}+{\hbox{\footnotesize{\LGR\textqoppa}}})},\ t^{\rm F}_{L}\sim
L^{-2{\hbox{\footnotesize{\LGR\textqoppa}}}}.$ (194)

Note that all of the above scaling formulas are readily obtained by replacing
the standard FSS prescription Eq.(10) by the prescription

$Q_{\infty}(t,0)\sim|t|^{\rho}\longrightarrow Q_{L}(t=0,0)\sim
L^{-{\hbox{\footnotesize{\LGR\textqoppa}}}\rho/\nu}.$ (195)

This is what we call Q-scaling and it holds at the pseudocritical point as we
shall see shortly. With Q-scaling to hand, FSS holds above the upper critical
dimension.

Summary of the evolution of the scaling picture above the upper critical
dimension for the $\phi^{4}$ model. The first column presents the correct
results (FSS predictions are for a system with periodic boundary conditions).
In the other columns, we give the (incorrect) results predicted when they are
different. A question mark means that the quantity hasn’t been considered in
the corresponding scenario.  The correct results Landau scaling1 GFP2 Fisher
DIV3 BNPY4 5 thermodynamic limit: $c_{\infty}(\tau,0)\sim|\tau|^{0}$ $\surd$
$\alpha_{\rm\scriptscriptstyle G}=2-\frac{d}{2}$ $\surd$ $\surd$ $\surd$
$m_{\infty}(\tau,0)\sim|\tau|^{1/2}$ $\surd$ $\beta_{\rm\scriptscriptstyle
G}=\frac{d-2}{4}$ $\surd$ $\surd$ $\surd$
$\chi_{\infty}(\tau,0)\sim|\tau|^{-1}$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
$m_{\infty}(0,h)\sim|h|^{1/3}$ $\surd$ $\delta_{\rm\scriptscriptstyle
G}=\frac{d+2}{d-2}$ $\surd$ $\surd$ $\surd$
$\xi_{\infty}(\tau,0)\sim|\tau|^{-1/2}$ $\surd$ $\surd$ $\surd$ ? $\surd$
$g_{\infty}({\bf x},0,0)\sim|{\bf x}|^{-(d-2)}$ $\surd$ $\surd$ $\surd$ ?
$\surd$ FSS: $\xi_{L}(\tau=0,0)\sim L^{d/4}$ $L$ $L$ $L$ $L$ $\surd$
$\chi_{L}(\tau=0,0)\sim L^{d/2}$ $L^{2}$ $L^{2}$ $L^{2}$ $\surd$ $\surd$
$\tau_{L}(\tau=0,0)\sim L^{-d/2}$ $L^{-1/2}$ $L^{-1/2}$ $L^{-1/2}$ $\surd$
$\surd$ $m_{L}(\tau=0,0)\sim L^{-d/4}$ $L^{-1}$ $L^{1-d/2}$ $L^{-1}$ $\surd$
$\surd$ $g_{L}(L/2,\tau=0,0)\sim L^{-d/2}$ $L^{-(d-2)}$ $L^{-(d-2)}$
$L^{-(d-2)}$ ? $\surd$ $h_{L}^{\rm LY}(\tau=0)\sim L^{-3d/4}$ $L^{-3}$
$L^{-(d+2)/2}$ $\surd$ $\tau_{L}^{\rm F}(\tau=0)\sim L^{-d/2}$ $L^{-2}$
$L^{-2}$ $\surd$ 1FSS with Landau exponents,

2Predictions from the RG eigenvalues at the Gaussian Fixed Point,

3Corrections made by the scenario of Fisher,

4Most of the results presented in this column correspond to BNPY’s version of
the scenario of Fisher and are checked in Ref. [20],

5Q scaling of the results presented in the last column are checked e.g. in
Ref. [6].

The different approaches that we have discussed are collected and compared in
Table 9 for the $\phi^{4}$ model with periodic boundary conditions. In this
table, the first column lists known results in the thermodynamic limit or for
finite-size scaling. The remaining columns correspond to the different
approaches that we have described so far, with the symbol $\surd$ to denote an
agreement and, in cases of disagreement, the prediction made by the
(incorrect) theory considered is given explicitly. The last column is for Q
scaling.

###  10 The case of Free Boundary Conditions

####  1 A mismatch between $T_{L}$ and $T_{c}$ in free boundary conditions

The case of free boundary conditions is more problematic. First the easy part:
the study of FSS properties of physical quantities evaluated at the pseudo-
critical point for FBCs agrees with what we said previously for systems with
PBCs. What is happening right at the critical point on the other hand is still
subject to debate.

Lundow and Markström have studied this problem intensively for the Ising model
in the 5D case, for very large systems (up to $L=160$ in Ref. [90]). They
obtained, for the magnetisation and the susceptibility, the leading behaviours
and corrections to scaling of the form:

$\displaystyle\chi_{L}(T_{c})=0.817\ \\!L^{2}+0.083\ \\!L,$ (196)
$\displaystyle m_{L}(T_{c})=0.230\ \\!L^{-3/2}+1.101\ \\!L^{-5/2}-1.63\
\\!L^{-7/2}.$ (197)

There, the leading and correction exponents are fixed and the coefficients are
free, and the authors concluded in favour of a standard FSS behaviour of the
susceptibility at the critical temperature $\chi_{L}(T_{c})\sim L^{2}$. The
same conclusion, albeit with much lower accuracy, was reached in [6]. There,
by standard it was referred to FSS with Landau exponents, $\chi_{L}\sim
L^{\gamma_{\rm\scriptscriptstyle MFT}/\nu_{\rm\scriptscriptstyle MFT}}$, a
conclusion that we will see is not correct, although $L^{2}$ is (accidentally)
correct.

The case of the magnetisation in (197) is a bit more subtle, because the
leading exponent $-\frac{3}{2}$ is not the standard Landau FSS (the ratio of
MFT exponents would be $-\beta_{\rm\scriptscriptstyle
MFT}/\nu_{\rm\scriptscriptstyle MFT}=-1$ instead).

In order to clarify the situation, Wittmann and Young [81, 91], and then
Flores-Sola et al. [27] considered the behaviour of the Fourier modes in a
finite system with free boundary conditions. Following Ref. [92], a sine-
expansion of the scalar field in the $\phi^{4}$ action is performed with the
boundary conditions $\phi({\bf x})=0$ at the free surfaces.

Wittmann and Young confirmed the scaling at $T_{c}$ for the total
susceptibility (for FBC). They found that the single-mode susceptibility also
obeys a standard FSS behaviour, $\chi_{{\bf k}}\sim L^{2}$ for the modes which
will not acquire a nonzero magnetisation, namely with the smallest wave-vector
with an even wave number $n_{\alpha}$ in the wave expansion
($k_{\alpha}=n_{\alpha}\pi/(L+1)$, $n_{\alpha}=1,2,\dots,L$, $\alpha=1,\dots\
,d$). Wittmann and Young proposed to use $t=T-T_{L}$, with $t=\tau+{\rm
const}\ \\!\times L^{-\lambda}$, as the temperature-like scaling variable and
the starred RG dimensions in the scaling hypothesis for the susceptibility.
They obtain then $\chi_{L}(t)=L^{2y_{h}^{*}-d}{\mathscr{X}}(L^{y_{t}^{*}}t)$
in zero magnetic field and the compatibility with the bulk behaviour in the
thermodynamic limit
$\chi_{\infty}(\tau)\sim|\tau|^{-\gamma_{\rm\scriptscriptstyle MFT}}$ is
recovered by the demand that the asymptotic regime obeys ${\mathscr{X}}(x)\sim
x^{-\gamma_{\rm\scriptscriptstyle MFT}}$ (with $\gamma_{\rm\scriptscriptstyle
MFT}=1$ here) for $x\to\infty$. Thid leads to $\chi_{L}(t)\sim
L^{2y_{h}^{*}-d+y_{t}^{*}}|t|^{-1}$. When $\lambda=d/2$, as is the case in
PBC, this leads at criticality to the correct result
$\chi_{L}(T_{c})=L^{d/2}$. On the other hand if $\lambda=2$ (FBC), this
amounts to $\chi_{L}(T_{c})=L^{2}$, hence the scenario proposed is very
compelling.

The discussion concerning standard FSS with Landau exponents, or Gaussian FP
exponents is nevertheless not settled by the study of the susceptibility alone
and here we will present additional unpublished numerical results [26] for
low-dimensional Ising models with long-range interactions above their
respective upper critical dimensions.

####  2 The three possible scenarios

As we have already noted several times, one cannot disentangle, with the
susceptibility alone, the predictions of standard (or Landau) FSS from those
of the Gaussian Fixed point FSS. The primary aim of Ref. [27] was to test the
discriminating case of the magnetisation. The argument of Wittmann and Young
for the magnetisation would indeed lead to $m_{L}(T_{c})=L^{-\lambda/2}$,
hence $m_{L}(T_{c})=L^{-d/4}$ in PBC and $m_{L}(T_{c})=L^{-1}$ in FBC. The
argument is thus equivalent to Landau scaling, because for FBC,
$\lambda=1/\nu_{\rm\scriptscriptstyle MFT}$.

So we are led to the point where one essentially faces three distinct
hypotheses161616In Ref. [1], a fourth scenario has been considered in which
corrections to scaling could be dominant at small sizes, but this hypothesis
has not been confirmed by the numerics. Therefore we do not discuss this
fourth case further here. among which one has to discriminate (see Table 2).

The three scenarios for the FSS of the susceptibility and the magnetisation at
$T_{c}$. There, $\hbox{{\LGR\textqoppa}}=d/d_{\rm uc}$, $d_{\rm uc}$ and the
${\rm MFT}$ exponents take their usual values and extension to different
universality classes is obvious.  Hypothesis Scaling of $\chi_{L}(T_{c})$
Scaling of $m_{L}(T_{c})$ 1\. Landau (or standard) scaling
$L^{\frac{\gamma_{\rm\scriptscriptstyle MFT}}{\nu_{\rm\scriptscriptstyle
MFT}}}$ $L^{-\frac{\beta_{\rm\scriptscriptstyle
MFT}}{\nu_{\rm\scriptscriptstyle MFT}}}$ 2\. Q scaling
$L^{2{\hbox{\footnotesize{\LGR\textqoppa}}}}$
$L^{{\hbox{\footnotesize{\LGR\textqoppa}}}-d/2}$ 3\. G scaling $L^{2}$
$L^{1-d/2}$

In the case of the $\phi^{4}$ universality class, the options are the
following:

— prediction 1 (Landau scaling) leads to $\chi_{L}(T_{c})\sim L^{2}$ and
$m_{L}(T_{c})\sim L^{-1}$,

— prediction 2 (Q scaling) to $\chi_{L}(T_{c})\sim L^{d/2}$ and
$m_{L}(T_{c})\sim L^{-d/4}$,

— prediction 3 (G scaling) to $\chi_{L}(T_{c})\sim L^{2}$ and
$m_{L}(T_{c})\sim L^{-(d-2)/2}$.

Option 2 is clearly ruled out at $T_{c}$ by the results of Lundow and
Markström in (196) (as well as by the results of Wittmann and Young). Options
1 and 2 are compatible with the results measured for the susceptibility. For
the 5D model, it is easy to discriminate, with the magnetisation, between
option 1 (which predicts the value $-1$), option 2 (prediction $-1.25$) and
option 3 (prediction $-1.5$). Equation (197) is in favour of option 3.

####  3 Towards a Ga ussian Fixed Point scaling at $T_{c}$

In Ref. [27] the magnetisation of the LRIM was investigated, tuning the
parameter $\sigma$ of the interaction decay to control the upper critical
dimension $d/(2\sigma)$. In the thesis of Emilio Flores-Sola [26] results are
reported for various values of $\sigma$ and of $d$. Some of these are
collected in Tables 3, 3 and 3 for the magnetisation, the susceptibility,
correlation length and correlation function at $T_{c}$ and at $T_{L}$. In the
case of the magnetisation (Table 3), the expectation from GFP scaling is an
exponent $-\beta_{\scriptscriptstyle\rm G}/\nu_{\scriptscriptstyle\rm G}$ with
$\beta_{\scriptscriptstyle\rm G}=(d-\sigma)/(2\sigma)$ and
$\nu_{\scriptscriptstyle\rm G}=1/\sigma$, while Q scaling predicts an exponent
$-\hbox{{\LGR\textqoppa}}\beta_{\scriptscriptstyle\rm
MFT}/\nu_{\scriptscriptstyle\rm MFT}=-\hbox{{\LGR\textqoppa}}$.

FSS of the magnetisation (upper panel) and the susceptibility (lower panel)
for the LRIM in $d=1$, 2 with FBCs at $T_{c}$ and at $T_{L}$. The results at
the critical temperature support G scaling ($\frac{d-\sigma}{2}$, resp.
$\sigma$) while those at the pseudo-critical temperature support Q scaling
($d/4$, resp. $d/2$).  $m_{L}(T_{c})$ $\frac{d-\sigma}{2}$ $m_{L}(T_{L})$
$\frac{\hbox{{\LGR\textqoppa}}\beta_{\scriptscriptstyle\rm
MFT}}{\nu_{\scriptscriptstyle\rm
MFT}}=\frac{d}{4}$$\frac{\hbox{{\LGR\textqoppa}}\beta_{\scriptscriptstyle\rm
MFT}}{\nu_{\scriptscriptstyle\rm MFT}}$ $d=1$ $\sigma=0.1$ $L^{-0.450(4)}$
0.45 $L^{-0.233(4)}$ 0.25 $\sigma=0.2$ $L^{-0.401(3)}$ 0.40 $L^{-0.230(4)}$
0.25 $d=2$ $\sigma=0.1$ $L^{-0.949(1)}$ 0.95 $L^{-0.501(2)}$ 0.50 $\sigma=0.2$
$L^{-0.897(1)}$ 0.90 $L^{-0.494(1)}$ 0.50 $\chi_{L}(T_{c})$ $\sigma$
$\chi_{L}(T_{L})$ $\frac{\hbox{{\LGR\textqoppa}}\gamma_{\scriptscriptstyle\rm
MFT}}{\nu_{\scriptscriptstyle\rm
MFT}}=\frac{d}{2}$$\frac{\hbox{{\LGR\textqoppa}}\beta_{\scriptscriptstyle\rm
MFT}}{\nu_{\scriptscriptstyle\rm MFT}}$ $d=1$ $\sigma=0.1$ $L^{0.099(1)}$ 0.1
$L^{0.522(3)}$ 0.5 $\sigma=0.2$ $L^{0.200(1)}$ 0.2 $L^{0.525(5)}$ 0.5 $d=2$
$\sigma=0.1$ $L^{0.094(2)}$ 0.1 $L^{0.985(2)}$ 1. $\sigma=0.2$ $L^{0.198(2)}$
0.2 $L^{0.994(2)}$ 1.

FSS of the correlation length for the LRIM in $d=1$, 2 with FBCs at $T_{c}$
and at $T_{L}$. The results at the critical temperature support G scaling
($1$) while those at the pseudo-critical temperature are more in favour of Q
scaling ($d/(2\sigma)$).  $\xi_{L}(T_{c})$ $1$ $\xi_{L}(T_{L})$
$\hbox{{\LGR\textqoppa}}=\frac{d}{2\sigma}$$\frac{\hbox{{\LGR\textqoppa}}\beta_{\scriptscriptstyle\rm
MFT}}{\nu_{\scriptscriptstyle\rm MFT}}$ $d=1$ $\sigma=0.1$ $L^{1.01(3)}$ 1
$L^{4.03(7)}$ 5 $\sigma=0.2$ $L^{1.03(2)}$ 1 $L^{2.21(4)}$ 2.5 $d=2$
$\sigma=0.1$ $L^{1.07(7)}$ 1 $L^{7.48(4)}$ 10 $\sigma=0.2$ $L^{0.95(6)}$ 1
$L^{3.97(4)}$ 5

FSS of the correlation function at ${\bf x}|=L/2$ for the LRIM in $d=1$, 2
with FBCs at $T_{c}$ and at $T_{L}$. The results at the critical temperature
are more in favour of G scaling (${d-\sigma}$) while those at the pseudo-
critical temperature support Q scaling ($d/2$).  $g_{L}(T_{c},\frac{L}{2})$
$d-2+\eta_{\scriptscriptstyle\rm G}$ $g_{L}(T_{L},\frac{L}{2})$
$d-2+\eta_{\hbox{\footnotesize{\LGR\textqoppa}}}$$\frac{\hbox{{\LGR\textqoppa}}\beta_{\scriptscriptstyle\rm
MFT}}{\nu_{\scriptscriptstyle\rm MFT}}$ $d=1$ $\sigma=0.1$ $L^{-0.86(6)}$ 0.9
$L^{-0.487(5)}$ 0.5 $\sigma=0.2$ $L^{-0.83(6)}$ 0.8 $L^{-0.483(6)}$ 0.5 $d=2$
$\sigma=0.1$ $L^{-2.07(9)}$ 1.9 $L^{-0.954(3)}$ 1 $\sigma=0.2$ $L^{-1.70(9)}$
1.8 $L^{-0.974(3)}$ 1

Other quantities (temperature shift, correlation length, correlation function)
are also reported in Ref. [26] (and partially collected here) and, although
perfectible, all numerical results at $T_{c}$ support the option 3 above (and
also confirm Q scaling, i.e. option 2, at $T_{L}$). A recent work[93] has
revisited FSS in the 5D Ising model (with short-range interactions) with FBCs
and also confirms the present results.

The picture now is the following for FBCs: _Either_ we study physical
quantities which are related to the Q modes (i.e. modes that have a non-zero
projection on the average magnetisation) or those related to the G modes
(modes that do not contribute to the average magnetisation). In the first
case, which is expected for FBC at $T_{L}$, the DIV has to be taken into
account and Q scaling rules (option 2 in Table 2). In the second case, which
holds for FBC at $T_{c}$, $u$ is not dangerous and the physics is controlled
by the GFP (option 3 in Table 2). The exponents there are Gaussian exponents
collected in Table 4 and not Landau exponents.

###  11 Conclusion

We have reached a point where we believe that ϙ and
$\hat{\hbox{{\LGR\textqoppa}}}$ are necessary ingredients of FSS at and above
the upper critical dimension. They allow for a natural, and simple, extension
of the scaling hypothesis that governs the behaviour of thermodynamic averages
and correlations in systems having a non-zero finite-size order parameter.
This is the case at the pseudo-critical points $T_{L}$, and also at $T_{c}$
for systems with PBCs. There, the scaling hypothesis for the singular part of
the free energy density, for the correlation length and for the order
parameter correlation function can be written as follows

$\displaystyle d>d_{\rm uc}$ $\displaystyle
f(\tau,h,L^{-1})=b^{-d}f(b^{2{\hbox{\footnotesize{\LGR\textqoppa}}}}\tau,b^{d/2+{\hbox{\footnotesize{\LGR\textqoppa}}}}h,bL^{-1}),$
$\displaystyle\xi(\tau,h,L^{-1})=b^{{\hbox{\footnotesize{\LGR\textqoppa}}}}\xi(b^{2{\hbox{\footnotesize{\LGR\textqoppa}}}}\tau,b^{d/2+{\hbox{\footnotesize{\LGR\textqoppa}}}}h,bL^{-1}),$
$\displaystyle g(\tau,h,L^{-1},{\bf
x})=b^{-(d-2{\hbox{\footnotesize{\LGR\textqoppa}}})}g(b^{2{\hbox{\footnotesize{\LGR\textqoppa}}}}\tau,b^{d/2+{\hbox{\footnotesize{\LGR\textqoppa}}}}h,bL^{-1},b^{-1}{\bf
x}),$ $\displaystyle d=d_{\rm uc}$ $\displaystyle f_{\rm
uc}(\tau,h,L^{-1})=b^{-d_{\rm uc}}f(b^{2}(\ln b)^{\hat{y}_{t}}\tau,b^{d_{\rm
uc}/2+1}(\ln b)^{\hat{y}_{h}}h,bL^{-1}),$ $\displaystyle\xi_{\rm
uc}(\tau,h,L^{-1})=b\ \\!(\ln
b)^{\hat{\hbox{\footnotesize{\LGR\textqoppa}}}}\xi(b^{2}(\ln
b)^{\hat{y}_{t}}\tau,b^{d_{\rm uc}/2+1}(\ln b)^{\hat{y}_{h}}h,bL^{-1}),$
$\displaystyle g_{\rm uc}(\tau,h,L^{-1},{\bf x})=b^{-(d-2)}g(b^{2}(\ln
b)^{\hat{y}_{t}}\tau,b^{d_{\rm uc}/2+1}(\ln
b)^{\hat{y}_{h}}h,bL^{-1},b^{-1}{\bf x})$

and the scaling of other quantities follows from appropriate derivatives wrt
the scaling fields.

Hyperscaling relation among critical exponents and “hatted” critical exponents
of the logarithmic corrections take the form

$\displaystyle\alpha=2-\frac{\nu d}{\hbox{{\LGR\textqoppa}}},$ (200)
$\displaystyle\hat{\nu}=\hat{\hbox{{\LGR\textqoppa}}}-\frac{\nu\hat{\alpha}}{2-\alpha}.$
(201)

They are valid above, below, and at $d_{\rm uc}$ for the first one (with
$\hbox{{\LGR\textqoppa}}=1$ below and $\hbox{{\LGR\textqoppa}}=d/d_{\rm uc}$
above $d_{\rm uc}$), and the second one is valid at $d_{\rm uc}$.

In FBCs, a special situation arises due to the rounding which does not scale
like the shift of the pseudo-critical temperature, and, as a consequence, the
pseudo-critical temperature is somehow pushed far below the critical
temperature. It follows that at the critical temperature, there is no room for
the order parameter to develop in the free energy mode expansion (it develops
at the pseudo-critical point). The non-zero order parameter (this would be the
zero-mode in the PBC case), which is responsible for the contamination of the
scaling by the dangerous irrelevant variable, has no influence there. It
follows that the correct homogeneity assumption that describes the singular
behaviour in the very vicinity of the critical temperature $\tau\to 0$ is as
above except that ϙ stays stuck to 1 and the physical quantities are
controlled by the Gaussian fixed point. As far as we know, this is the only
instance (FBC, $T=T_{c}$) in which ratios of the Gaussian exponents are
measured in FSS and not ratios of MFT exponents. In a similar manner, one may
expect that the logarithmic corrections in the Q sector which are due to the
scaling field $u$ becoming marginal at $d_{\rm uc}$, do not show up in the G
sector.

### Acknowledgments

We would like to thank Yu. Holovatch for his constant support, and for
accepting a delay of about a year in the delivery of the present chapter!

This may not be a common practice, but here I would like also to thank two of
my collaborators, two PhD students who are working with me, and were under the
co-supervision of Ralph. Andy Manapany and Leïla Moueddene. They are wonderful
people with whom we can share opinions and views on the world. I think that
maybe I opened their minds to some aspects, and I am sure that they opened
mine. They knew Ralph, they appreciated him a lot, and we share this.

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## Index

  * anomalous dimension §1, §8
  * coppa §1, §1, §2, §3
    * hat §3
  * correction
    * logarithmic §1, §2, §2
    * to finite-size scaling §8
  * critical
    * amplitude §4, §1
    * dimension
      * upper §1, §1, §2, §9
    * exponent §1, §1, §1, §2, §5
      * hatted §2
      * starred §3
    * point §1
  * dangerous irrelevant variable §4, §6, §3, §9
  * exponent
    * critical §1, §1, §1, §2, §5
    * hatted §2
    * rounding §3
    * shift §3, §3
    * starred §3
  * finite-size scaling §1, §1, §1, §2, §1, §2, §2
  * Fisher, M.E. §1, §1, §1, §3
  * Fourier modes §7
  * Gaussian
    * fixed point §3, §4, §5
    * model §4
  * Ginzburg-Landau
    * theory §4
  * homogeneity §2, §1
    * assumption §1
      * with logarithms §2
  * hyperscaling relation §1, §1, §1, §11, §1
  * logarithmic
    * correction §1, §2
    * exponent §1
  * mean-field theory §4
  * Palestine Chapter 0
  * pseudocritical point §1, §3
  * scaling relations §1, §3
  * singularities §1
  * Ukraine Chapter 0
  * zeros
    * Fisher §3, §2
    * Lee-Yang §3, §1
    * of the partition function §3