# When correlations exceed system size: finite-size scaling in free boundary
conditions above the upper critical dimension

Yulian Honchar  $|$ ${}^{\textsf{\footnotesize{\mbox{?},\penalty
1000\mbox{?},\penalty 1000\mbox{?}}}}$ , Bertrand Berche  $|$
${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?}}}}$ , Yurij
Holovatch  $|$ ${}^{\textsf{\footnotesize{\mbox{?},\penalty
1000\mbox{?},\penalty 1000\mbox{?},\penalty 1000\mbox{?}}}}$ , Ralph Kenna
$|$ ${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?}}}}$

###### Abstract

We progress finite-size scaling in systems with free boundary conditions above
their upper critical dimension, where in the thermodynamic limit critical
scaling is described by mean-field theory. Recent works show that the
correlation length is not bound by the system’s physical size, a belief that
long held sway. Instead, two scaling regimes can be observed — at the critical
and pseudo-critical temperatures. We demonstrate that both are manifest for
free boundaries. We use numerical simulations of the $d=5$ Ising model to
analyse the magnetization, susceptibility, magnetization Fourier modes and the
partition function zeros. While some of the response functions hide the dual
finite-size scaling, the precision enabled by the analysis of Lee-Yang zeros
allows this be brought to the fore. In particular, finite-size scaling of
leading zeros at the pseudocritical point confirms recent predictions coming
from correlations exceeding system size. This paper is dedicated to Jaroslav
Ilnytskyi on the occasion of his 60th birthday.

Key words: universality, finite-size scaling, upper critical dimension

###### Abstract

Ми досліджуємо скінченновимірний скейлінґ в системах з вільними граничними
умовами над їх верхньою критичною вимірністю, де в термодинамічній границі
критичний скейлінґ описується середньопольовою теорією. Нещодавні роботи
показують, що кореляційна довжина не обмежується фізичним розміром системи, як
довгий час вважалося раніше. Замість цього спостерігаються два режими
скейлінґу – при критичних та псевдокритичних температурах. Ми демонструємо, що
обидва режими проявляються для вільних границь. Ми використовуємо числові
симуляції моделі Ізінга при $d=5$, щоб проаналізувати намагніченість,
сприйнятливість, магнітні моди Фур’є-перетворень намагнічення та нулі
статистичної суми. Хоча деякі з функцій відгуку приховують подвійний
скінченовимірний скейлінґ, точність, забезпечена аналізом нулів Лі-Янга,
дозволяє вивести це питання на передній план. Зокрема, скінченновимірний
скейлінґ перших нулів у псевдокритичній точці підтверджує нещодавні прогнози,
що виникають з кореляцій, які перевищують розмір системи. Ця стаття присвячена
Ярославу Ільницькому з нагоди його 60-річчя.

Over ten years ago our team published a paper on finite-size scaling (FSS) in
systems in high dimensions which triggered a considerable reaction in the
community [1]. FSS in its standard form was for fifty years expected to fail
in high dimensions and our paper introduced a new form that claimed to rescue
it. However, subtleties exist - one has to carefully differentiate between
boundary conditions and precisely where criticality is measured. A core tenet
of our arguments was that universality, and hence the correct form of FSS
resided at pseudo criticality and not at criticality. At the time, FSS for
free boundary conditions (FBCs) at the critical point was most challenging. At
that point, and in the absence of a more complete theory, results were
unconvincing for they appeared neither to fit with standard FSS nor with the
new regime. One paper inspired by our work was that of Lundow and Markström,
whom we refer to as L & M [2]. The paper focuses entirely on FBCs at the
critical (not pseudocritical) point and uses impressive numerical power to
argue that standard FSS holds there. In the meantime, significant theoretical
developments have occurred in the form of Refs.[4, 5, 3, 6, 7] and we refer
the reader to Ref.[8] for new theory as well as an extended and detailed
review. The new developments come under the broad name of Q-theory, or simply
Q, with the new finite-size scaling prescription called QFSS.

Here we revisit FSS for FBCs at both the critical and pseudocritical points.
We argue that what appeared in L & M as “standard” FSS is in fact a form of
QFSS that falls within Q theory — it is FSS applied to the Gaussian model
instead of to Landau mean-field theory. To differentiate between standard FSS
applied to Landau theory and Q theory applied to the Gaussian fixed point we
revisit the magnetization (which L & M also looked at) as well as Fourier
modes. We also examine Lee-Yang zeros at the pseudocritical and critical
temperatures. In terms of observables used in the study of phase transitions
and critical phenomena, zeros are unique in that they are not moments of the
free energy. Together these provide evidence that validates Q theory for FBCs
in both regimes

The structure of the paper is as follows. In the first section we look at the
background of our research, and discuss previous results on the topic; in
section 2 we present the results of our numerical simulations and the finite-
size scaling of the various observables; section 3 documents the search and
then the scaling of the partition function zeros; results and conclusions are
laid out in the last section. It is our special pleasure to devote this paper
to Jaroslav Ilnytskyi, a colleague and a good friend of ours, whose impressive
work on numerical simulations in condensed matter physics, and in particular
on universality at critical points [9, 10, 11] has inspired us for many years
and continues to do so today.

##  1 Introduction

The subject of phase transitions and critical phenomena is key to unraveling
how matter undergoes transformations [12, 13]. A first approximation of this
is mean-field theory, a subject that is now over 100 years old [14]. Once a
certain space dimension — called the upper critical $d_{\rm c}$ — is
surpassed, the number of interactions is so large that the system’s behavior
is effectively averaged, and mean-field theory holds sway. While valid for
infinite systems this is often also considered to be the case (with suitable
adaption) for finite systems too. However, it was recently exposed that this
is not always the case and one has to take into account the crucial role
played by correlation length. Until recently this was widely believed to scale
no faster than physical size if the system is finite in extent. This is now
known not to be true and correlations can exceed system size. While
significant theoretical progress has been made in verifying this picture for
systems with periodic boundary conditions (PBCs), free boundary conditions
(FBCs) have proved more difficult. In this paper, we explore what happens in
this lesser-known territory using computer simulations. We consider the Ising
model, which has an upper critical dimension $d_{\rm c}=4$, on a five-
dimensional hypercubic lattice.

First, we set the notation. For second-order phase transitions in
ferromagnetic systems, the specific heat, magnetization, susceptibility, and
correlation length near the transition temperature ${{T_{c}}}$, scale as
${{c_{\infty}}}(t)\sim|t|^{-\alpha}$, ${{m_{\infty}}}(t)\sim(-t)^{\beta}$,
${{m_{\infty}}}(h)\sim|h|^{\frac{1}{\delta}}$,
${{\chi_{\infty}}}(t)\sim|t|^{-\gamma}$, ${{\xi_{\infty}}}(t)\sim|t|^{-\nu}$,
where $t$ is a reduced temperature, $h$ is a reduced magnetic field (each is
omitted from functional arguments when it vanishes), and the formula for
${{m_{\infty}}}(t)$ holds only in the broken-symmetry phase where $t<0$. and
the subscript denotes the (infinite) size of the system. The critical
diverging singularities are replaced by finite peaks when the linear extent,
$L$, of the system is finite. These finite peaks are centered around the so-
called pseudocritical points $T_{L}$. The rounding of the susceptibility peak
can be measured by its width at half-height, for example, and is governed by
an exponent $\rho$: $\Delta T_{\rm{rounding}}\sim L^{-\rho}$. The distance of
the pseudocritical temperature $T_{L}$ from the critical one ${{T_{c}}}$ is
described by the shift exponent $\lambda$: $T_{L}-{{T_{c}}}\sim L^{-\lambda}$.
Usually (but not always) [15], the rounding and shifting exponents coincide
and are given by

$\rho=\lambda=\frac{1}{\nu}.$ (1)

Below the critical dimension $d_{c}$ and near to the critical point, the
infinite-volume correlation length $\xi_{\infty}$ can be replaced by the
system size $L$ when the latter is finite. This is sufficient to deliver
appropriate FSS there. For, it for specific heat leads to
$c\sim|t|^{-\alpha}=\xi^{\alpha/\nu}$ being replaced by $L^{\alpha/\nu}$. A
similar applies to other observables so that

$c_{L}\sim L^{\frac{\alpha}{\nu}},\quad m_{L}\sim
L^{-\frac{\beta}{\nu}},\quad\chi_{L}\sim L^{\frac{\gamma}{\nu}}.$ (2)

Eqs. (1) and (2) well describe FSS below the upper critical dimension. When
the dimensionality $d$ exceeds upper criticality $d_{c}$, our current picture
is as follows [8]. The self-interacting term, which is responsible for the
existence of the phase transition by spontaneous symmetry breaking (e.g., the
$\phi^{4}$ term in the field-theoretic equivalent to the Ising-model
Hamiltonian), modifies thermodynamic functions above the critical dimension in
such a way that the above FSS formulae no longer apply. The usual expression
for the hyperscaling relation, $\nu d=2-\alpha$, also fails. These
circumstances can be traced back to the fact that the quartic term is coupled
to a dangerous irrelevant variable [16].

Following “surprising” analytical [17] and numerical [18] observations of a
superlinear (wrt $L$) correlation length, it was shown in Refs.[4, 20, 19, 3]
that, contrary to earlier expectations, these features of the model are also
affected by such dangerous irrelevant variables, just like the properties
related to free energy [21]. This leads to the introduction of a
pseudocritical exponent ${\hbox{{\LGR\textqoppa}}}=\max(1,d/d_{c})$ for the
FSS of the correlation length, which takes the value $d/d_{c}$ when $d>d_{c}$
[1, 4, 20]. At first sight, the exponent ϙ has a similar status to $\lambda$
and $\rho$, above, in that it most directly refers to finite-size systems as
opposed to the infinite-volume ones which are directly described by $\alpha$,
$\beta$, $\gamma$, $\delta$, and $\nu$. Thus ϙ may be considered a
pseudocritical exponent. The superlinear behavior of the finite-size
correlation length has been verified by Monte Carlo simulations in Refs.[1, 4,
20, 19, 22, 3] and its existence modifies the finite-size scaling relations
for, instead of replacing $\xi_{\infty}$ by $L$, one replaces it by

$\xi_{L}\sim L^{{\hbox{{\LGR\textqoppa}}}},$ (3)

so that

$c_{L}\sim L^{\frac{{\hbox{\footnotesize{\LGR\textqoppa}}}\alpha}{\nu}},\quad
m_{L}\sim
L^{-\frac{{\hbox{\footnotesize{\LGR\textqoppa}}}\beta}{\nu}},\quad\chi_{L}\sim
L^{\frac{{\hbox{\footnotesize{\LGR\textqoppa}}}\gamma}{\nu}}.$ (4)

To distinguish Eq.(2) from Eq.(4), we referred to the latter as QFSS [1] (the
“Q” here and throughout referring to ϙ, a notation suggested by Fisher in
Ref.[20]).

The form $\xi\sim L^{d/4}$ for the correlation length of the short-range
Ising-model (for which $d_{c}=4$) in the analytical study of Ref.[17] and
verified numerically in Ref.[18] was for periodic boundary conditions (PBCs).
That it also applies to free boundary conditions (FBCs) was suggested in
Ref.[1] and verified in Refs.[4, 20, 19, 22, 3]. The existence of ϙ as a new,
universal pseudocritical exponent allows one to extend the hyperscaling
relation to beyond the upper critical dimension. This general form (valid in
all dimensions) is

$\frac{\nu d}{{\hbox{{\LGR\textqoppa}}}}=2-\alpha.$ (5)

In contrast to what was suggested above where ϙ appeared to be a
pseudocritical exponent, this formula links ϙ to critical exponents. This
promotes the new exponent to the level of critical (as opposed to
pseudocritical) exponent. Whether one considers ϙ to be a critical or
pseudocritical exponent is perhaps a personal preference.

To understand the origins of the exponent ϙ, the Fourier modes associated with
high-dimensional systems may be partitioned into two types [3]. Although both
are governed by the Gaussian fixed point, so-called Q-modes are susceptible to
dangerous irrelevant variables while so-called G-modes are not. Here and
throughout, following the terminology used in Ref.[3], “G” refers to the
“pure” Gaussian sector (unmodified by dangerous irrelevant variables) to
distinguish it from the Q-sector [3]. For systems with PBCs only the zero mode
is of the Q-type; all other modes are Gaussian. There, as is common below
$d_{c}$, the rounding exponent $\rho$ has the same magnitude as the shifting
exponent $\lambda$, $\rho=\lambda={{\hbox{{\LGR\textqoppa}}}}/{\nu}$, ensuring
that the Q-scaling window, which is centred on the pseudocritical point,
extends as far as the critical one. In other words, QFSS applies at $T_{L}$
and ${{T_{c}}}$ in the PBC case.

For systems with free boundaries, the partitioning is somewhat more subtle.
The dominance of Q-modes at the pseudocritical point ensures that QFSS governs
finite-size effects there. This is where universality resides above the upper
critical dimension [i.e., the QFSS formulae (4) hold both for PBCs and FBCs
there]. The rounding and shifting exponents are

$\rho=\frac{{\hbox{{\LGR\textqoppa}}}}{\nu},\quad\lambda=\frac{1}{\nu},$ (6)

and, since $\rho>\lambda$, the shifting is bigger than the rounding so that
the QFSS window, centered as it is around the pseudocritical point, does not
extend to the critical point. FSS there is governed by G-type behavior [3].

In this paper, we perform numerical simulations to test the above picture for
FBCs. Again, we expect (as L & M have actually found) G scaling for FSS
results at the critical point and Q scaling at the pseudocritical one. L & M,
however, did not look at the pseudocritical points so we calibrate our efforts
against theirs at the critical one only. The best we can do is $L=50$ which
compares with the impressive $L=160$ of L & M. We also follow L & M when we
look at FSS of the standard observables, namely magnetization and isothermal
susceptibility. Then we go a step further by looking at Fourier modes and
partition function zeros (which L & M did not look at). These lie in the
complex plane of parameters entering the partition function (i.e., external
field or temperature). The notion was developed by Lee and Yang [23, 24], who
studied the partition function as a polynomial in a parameter related to the
external magnetic field. Following a similar idea, Fisher suggested the study
of the zeros for the temperature complex plane [25]. These ideas have been
termed a “fundamental theory of phase transitions” [26]. We discuss these
developments in more detail in section 3, below.

The partition function on a finite lattice is given by

$Z_{L}=\sum_{E,M}p(E,M)e^{-\beta E+hM},$ (7)

where the configurational energy $E$ is conjugate to $\beta=1/kT$ ($k$ being
the Boltzmann factor), the configurational magnetization $M$ is conjugate to
the reduced external field $h=\beta H$ and $p(E,M)$ is the density of states.
From the fundamental theorem of algebra, this can be expressed in terms of the
set of Lee-Yang zeros $\\{z_{j}\\}$ as

$Z_{L}(h)=A(z)\prod_{j}[z-z_{j}(L)],$ (8)

where $z$ is a suitable function of $h$ (e.g., $z=\exp{h}$) and where $A$
denotes a non-vanishing smooth function of its arguments.

Well-established standard FSS, applicable below the upper critical dimension,
gives for the Lee-Yang zeros

$h_{j}(L)\sim L^{-\frac{\Delta}{\nu}},$ (9)

in which

$\Delta=\beta\delta=\beta+\gamma$ (10)

is the gap exponent. These equations are the counterparts of Eq.(2) for the
fundaments of the partition function zeros. A crucial element of Q-theory as
developed in Ref.[1] is the prediction that, for $d>d_{c}$, Lee-Yang zeros
scale in the Q-regime (i.e., at pseudo criticality universally and at
criticality in the PBC case) as

$h_{j}(L)\sim L^{-\frac{{\hbox{\footnotesize{\LGR\textqoppa}}}\Delta}{\nu}}.$
(11)

The aim of this paper is to provide a numerical check of these scaling
formulae. For convenience, we list in Table 1 the various FSS predictions for
the various theoretical possibilities - standard FSS from Landau theory,
Gaussian theory, and QFSS. The first Lee-Yang zeros allow us to access the
G-sector [Eq.(9)] through FBCs at criticality and the Q-sector [Eq.(11)]
through the pseudocritical points.

Quantity | Landau | G | Q
---|---|---|---
$m$ | $L^{-{\beta}/{\nu}}=L^{-1}$ | $L^{-{\beta_{G}}/{\nu}}=L^{-{3}/{2}}$ | $L^{-{\beta_{Q}}/{\nu}}=L^{-{{\hbox{\footnotesize{\LGR\textqoppa}}}\beta}/{\nu}}=L^{-{5}/{4}}$
$\chi$ | $L^{{\gamma}/{\nu}}=L^{2}$ | $L^{{\gamma_{G}}/{\nu}}=L^{2}$ | $L^{{\gamma_{Q}}/{\nu}}=L^{{{\hbox{\footnotesize{\LGR\textqoppa}}}\gamma}/{\nu}}=L^{{5}/{2}}$
$c$ | $L^{{\alpha}/{\nu}}=L^{0}$ | $L^{{\alpha_{G}}/{\nu}}=L^{0}$ | $L^{{\alpha_{Q}}/{\nu}}=L^{{{\hbox{\footnotesize{\LGR\textqoppa}}}\alpha}/{\nu}}=L^{0}$
$h_{1}$ | $L^{{-\Delta}/{\nu}}=L^{-{3}/{2}}$ | $L^{{-\Delta_{G}}/{\nu}}=L^{-{3}/{2}}$ | $L^{{-\Delta_{Q}}/{\nu}}=L^{-{{\hbox{\footnotesize{\LGR\textqoppa}}}\Delta}/{\nu}}=L^{-{15}/{8}}$
Table 1: FSS (Landau and G columns) and QFSS (last column) for the variables
measured in this paper ($d=5$). We expect (and find) QFSS at the
pesudocritical point, standard FSS applied to Gaussian modes at the critical
point, and standard FSS which is of relevance for Landau values not to feature
at all.

We will see that, as emphasized in Ref.[3] for the magnetization, despite
mean-field exponents being accurate in infinite volume, the extension of
Landau theory to finite systems does not manifest physically through standard
FSS. Instead, FSS is either of the G- or Q-types consistent with the theory
developed in Ref.[1, 4, 20, 19, 22, 3].

A significance of this is that pure G-scaling was hitherto considered as
merely a “stepping stone” to matching finite size with the Landau mean-field
exponents. Here we see that the G-sector has direct physical manifestation in
finite volume. Moreover, while FSS and hyperscaling were widely considered
inapplicable above the upper critical dimension, here we again demonstrate
they hold in Q-modified form. Thus the main message of the paper is that
universality holds in high dimensions, but at pseudo criticality rather than
criticality; this also provides a fundamental insight into how criticality at
infinite volume should be approached from finite size.

##  2 Numerical simulations

Lundow and Markström conducted large-scale simulations of the $d=5$ Ising
model with FBCs [2, 27, 28]. The magnetization at $T_{c}$, computed using
space averaging over all sites, was clearly supportive of
${{m_{L}(T_{c})}}\sim L^{-1.5}$. (We note that this result was not determined
through a free FSS fit to $\beta/\nu$ and instead expected values were
imposed.) Table 1 suggests this is indicative of Gaussian (rather than Landau)
FSS: $L^{{{-\beta/\nu}}}=L^{-3/2}$.

Simulations from Ref.[1], conducted by some of the current authors, had
previously yielded different results for the same quantity:
$m_{L}({{T_{c}}})\sim L^{-1.7}$. This result doesn’t support any of the
theoretical suggestions of Table 1. Ref.[1] also looked at the pseudocritical
point and found $m_{L}(T_{L})\sim L^{-1.49}$. Contrary to theory, this aligns
with G-scaling. At the time we suspected that these results — not supportive
of any theory — were a product of the lattices used being not genuinely
representative of five dimensions. In particular, sites at the lattice
boundaries do not have the same number of neighbours as sites in the interior
of the lattice. So, in Ref.[1], the authors took a step to minimize the role
of lower-dimensional sub-manifolds in FBC systems. They computed the “core"
magnetization, averaging only over the central $(L/2)^{d}$ sites. For the
critical point, this yielded results of $m_{L}^{\rm core}(T_{c})\sim
L^{-1.57}$, possibly in line with G-scaling at $d=5$. For the pseudocritical
point, the result was $m_{L}^{\ \\!\rm core}(T_{L})\sim L^{-1.27}$,
potentially aligned with Q-scaling.

There is a clear discrepancy between L & M results in Refs. [2, 27] at $T_{c}$
and those of our team [1] when all sites are used to perform the space
average. The agreement is much better when only core sites are taken into
account in the space average. This suggests that the previous results of Ref.
[1] are strongly affected by crossovers due to the boundary effects, while
numerics in Refs. [2, 27] are more reliable because of the huge sizes reached.

Here we provide numerical simulations of the five-dimensional Ising model with
free boundary conditions. We analyze the scaling of magnetization, isothermal
susceptibility, and magnetization Fourier modes, as well as use the analysis
of the partition function zeros.

In order to investigate the critical behavior of the finite size $d=5$ Ising
model we performed Monte Carlo simulations using the Wolff algorithm [29].
This is a cluster algorithm that updates the lattice by picking a random node
and adding its neighbors to the cluster if they are of the same spin sign with
probability $p=1-e^{-\beta\Delta E}$, where $\Delta E$ is a change of internal
system energy if one spin is flipped [30, 31, 32, 33]. In this way at least
one node is flipped in every simulation step and at zero temperature the whole
lattice is added to the cluster.

We found that starting from an ordered state of the lattice (where all spins
are pointing in the same direction) we complete the thermalization process
much faster than when starting with the disordered state of the lattice (where
all spins are oriented randomly). This effect is more stark for larger lattice
sizes. The reason for this is clear — for cluster algorithms, we need more
adjacent spins of the same sign for the system to change quickly, and the
disordered state does not have enough clusters to flip rapidly. Therefore we
start from ordered states throughout our simulations. For each temperature we
performed $2\times 10^{6}$ iterations, discarding the first 10% of those as
thermalization.

We set boundary conditions to be free, meaning that the nodes on the edges of
the lattice do not have a neighbor in one of the $d$ directions. Instead, they
form a $(d-1)$-dimensional manifold.

###  2.1 Determining pseudo criticality and its finite-size scaling

Besides at the critical temperature itself, we are interested in FSS at the
pseudocritical temperature.

This is shifted away from the actual critical point of the infinite system and
the smaller the system size the more pronounced this effect is. Actually, for
small lattices the system at the critical temperature is effectively
disordered.

Our favored determinant of the pseudocritical point is susceptibility. The
algorithm to locate the pseudocritical temperature $T_{L}$ is illustrated in
Fig. 1. The top part of the figure illustrates the rough values of the
susceptibility for a wide range of temperature values. A peak is clearly
visible and our task is to identify its location $T_{L}$ with some precision.
To identify this pseudocritical peak, then, we first scan a broad range of
temperatures with low-precision simulations to find an approximate region that
contains it for each lattice size. After that, one runs a set of high-
precision simulations narrowed around this temperature. The outcome is
illustrated in the lower figure. Because of the fluctuations, the peak is
difficult to identify manually. So we use a quadratic fit in an attempt to
smoothen the curve so that a local maximum can be more readily identified. At
this point, which we identify as $T_{L}$, we perform another high-precision
simulation and use the data from it for further analysis.

(a)

(b)

Figure 1: The process of estimating the pseudocritical temperature from the
susceptibility for the $d=5$ Ising model on a lattice of size $L=12$. First
(a) we scan greater intervals of temperatures with relatively low precision.
Then we perform this same procedure in more detail for an interval where the
susceptibility is visibly higher. Finally (b) we simulate an interval of
temperatures, where it seems that the peak should be located. with higher
precision. We fit a polynomial through the highest points. This allows us to
select an estimate for the pseudocritical point $T_{L}$ (dashed vertical
line). We perform final simulations at the resulting peak.

To investigate FSS of the pseudocritical point, we first require an estimate
of the critical point itself. We use that ($T_{c}=8.77847{{52}}$) coming from
L & M (2014) [2]. This is the estimate of the critical point that we use
throughout the paper as we also track the behavior of various observables at
the infinite system critical point $T_{c}$.

Figure 2: FSS data of the pseudocritical temperature (blue line). For the G-scaling predicted by theory for FBCs, $t_{L}$ scales as $L^{-2}$ (orange dotted line). This is in marked contrast to $L^{-2.5}$ which comes from Q-scaling (red dashed line) and is relevant for PBCs. Our fit gives $L^{-1.90\pm 0.03}$, which is G-scaling within the error bars. | $t_{L}$ | $m$ | $\chi$ | $m(k)$ | $h_{1}$
---|---|---|---|---|---
MF | $L^{-2}$ | $L^{-1}$ | $L^{2}$ | $L^{-1}$ | $L^{-3}$
G-scaling | $L^{-2}$ | $L^{-1.5}$ | $L^{2}$ | $L^{-1.5}$ | $L^{-3}$
Q-scaling | $L^{\mathbf{-2.5}}$ | $L^{\mathbf{-1.25}}$ | $L^{\mathbf{2.5}}$ | $L^{\mathbf{-1.25}}$ | $L^{\mathbf{-3.75}}$
at $T_{c}$, [2] |  | $L^{-1.5}$ | $L^{2}$ |  |
at $T_{L}$, [1] |  | $L^{-1.39}$ | $L^{2}$ |  | $L^{-3.75}$
at $T_{c}$, [1] |  | $L^{-1.70}$ | $L^{1.71}$ |  |
This paper | $L^{1.90\pm 0.03}$ |  |  |  |
at $T_{L}$, this paper |  | $L^{-{\bf 1.25}\pm 0.09}$ | $L^{1.95\pm 0.02}$ | $L^{-{\bf 1.26}\pm 0.05}$ | $L^{-{\bf 3.81}\pm 0.08}$
at $T_{c}$, this paper |  | $L^{-\bf 1.5}$ | $L^{\bf 2}$ |  | $L^{-{\bf 3.16}\pm 0.48}$
Table 2: FSS behavior for the pseudocritical point, mean magnetization $m$,
isothermal susceptibility $\chi$, magnetization Fourier modes $m(k)$, and the
first Lee-Yang zero $h_{1}$. The first three rows record the theoretical
predictions for mean-field theory (MF), Gaussian theory (G) and QFSS (Q). In
the next three rows, $T_{c}$ and $T_{L}$ refer to the results of numerical
simulations at corresponding temperatures. The last lines record the results
presented in this paper and exponents written in bold font are those for which
there is a reasonable agreement with the theoretical predictions.

In Fig. 2 we present the FSS of the pseudocritical temperatures
$t_{L}=T_{c}-T_{L}$. Since this and all other numerical results of this paper
are obtained for $d=5$, we do not mention this fact explicitly in the
forthcoming figure and table captions. Mean-field or Gaussian FSS would
predict $t_{L}\sim L^{-1/\nu}=L^{-2}$, where $\nu$ is the correlation-lenght
critical exponent. In contrast, Q-scaling predicts $t_{L}\sim
L^{-{\hbox{\footnotesize{\LGR\textqoppa}}}/\nu}=L^{-2{\hbox{\footnotesize{\LGR\textqoppa}}}}=L^{-5/2}$.
The latter was proven for the PBC case, while the former was shown for FBCs in
[34]. We observe $t_{L}\sim L^{-1.90\pm 0.03}$ when we fit over the full range
of data from $L=4$ to $L=50$. This value is much closer to the theoretically
expected G-scaling, $t_{L}\sim L^{-2}$, and further away from the Q-scaling
prediction $t_{L}\sim L^{-2.5}$. It is consistent with observations in the
literature whereby shifting is bigger for FBCs than it is for PBCs. This
result is recorded in Table 2 as are the FSS results for other quantities. As
in the case of pseudocritical points, the table invites comparisons with
available theoretical predictions.

###  2.2 Finite-size scaling of the mean magnetization $m$

Figure 3: Leading behavior and first corrections to the FSS for the
magnetization at the critical temperature $T_{c}$. We plot $m\cdot L^{1.5}$
against $1/L$. Figure 4: Scaling of magnetization at the pseudocritical
temperature $T_{L}$. The theoretical values are presented as orange dotted
(G-scaling), and red dashed (Q-scaling) lines. We see that we can discard a
few smallest lattice sizes to reach the expected scaling behavior.

#### Criticality:

We next analyze the FSS of mean magnetization $m$. Very detailed estimates for
the FSS of the mean magnetization at $T_{c}$ were obtained by L & M in Ref.
[2] with leading behaviors and corrections to scaling given by:

$m_{L}(T_{c})=0.230L^{-3/2}+1.101L^{-5/2}-1.63L^{-7/2}.$ (12)

We performed a similar analysis to compare our numerics with those of L & M.
Although extracted using far smaller lattice sizes, our numerical results
deliver similar outcomes:

$m_{L}(T_{c})=0.215L^{-3/2}+1.514L^{-5/2}-4.280L^{-7/2}.$ (13)

In Fig. 3 we plot corrections to scaling to illustrate the convincing nature
of this fit. This suggests that our results are similar to those of L & M and
$m\sim L^{-3/2}$ as the Gaussian theory predicts.

#### Pseudo criticality:

In Fig. 4 we show the results of the simulations for the mean magnetization
$m$ at the pseudocritical temperatures $T_{L}$ for different values of $L$
ranging from $L=4$ to 50. As was previously suggested, we expect Q-scaling at
$T_{L}$. At $T_{L}$ we obtained $m\sim L^{-1.46\pm 0.02}$ when the power law
fit is performed over all lattice sizes. This is much closer to L & M’s
G-scaling $m\sim L^{-1.5}$ than to the theoretically expected Q-scaling $m\sim
L^{-1.25}$. As explained in Ref. [1], our contention is that the boundary
lattice sites in FBCs may be responsible for this discrepancy between the
expectation and observed FSS; as the surface effects are too strong to show a
properly five-dimensional criticality. To illustrate this we discard the
lowest sizes ($L=4..12$) from the fit, and indeed the FSS reaches the scaling
$m\sim L^{-1.25\pm 0.09}$ (green line in Fig. 6, top). This suggests that for
large enough lattice sizes (and discarding small lattices) the magnetization
for FBCs at pseudo criticality indeed falls within the Q-scaling regime.

###  2.3 Finite-size scaling of isothermal susceptibility $\chi$

Figure 5: Leading behavior and first corrections to the FSS for the
susceptibility at the critical temperature $T_{c}$. For the susceptibility we
plot $\chi\cdot L^{-2}$ against $1/L$.

#### Criticality:

Again, we first compare the results of our simulations at criticality with
those coming from L & M in Ref. [2]. They assume a quadratic dependency of
susceptibility on $L$ and a fit delivers

$\chi_{L}(T_{c})=0.817L^{2}+0.083L.$ (14)

Our equivalent (for far smaller lattices) is

$\chi_{L}(T_{c})=0.285L^{2}+0.030L.$ (15)

The conclusion of Ref. [2] was in favor of standard FSS scaling for the
isothermal susceptibility at the critical temperature $\chi_{L}(T_{c})\sim
L^{2}$, and we can see that, at first sight, that would seem to be the case.

However, as Table 2 illustrates, this could be standard MF scaling or G
scaling. Our contention is that it is G and falls within the theory of Q.

#### Pseudo criticality:

Results for the FSS of isothermal susceptibility at the pseudocritical point
are shown in Fig. 6. At $T_{L}$ we obtained $\chi\sim L^{1.95\pm 0.02}$ using
all simulated lattices, in apparent agreement with G-scaling ($\chi\sim L^{2}$
as opposed to $\chi\sim L^{2.5}$ for Q-scaling). In our fit, data for $L=4$
weighs as much as $L=50$, say. But clearly, $L=4$ is not genuinely
representative of a five-dimensional system. The fact that these data lie on
such a line calls into suspicion $L=50$ data as well. Thus we expect that the
lattice sizes we reach in our simulations are possibly too small to converge
to Q-scaling for susceptibility. Removing a few smaller sizes does not do the
trick as it did for the magnetization.

Figure 6: Scaling of the isothermal susceptibility at the pseudocritical
temperature $T_{L}$. The theoretical values are presented as an orange dotted
line (G-scaling), and a red dashed line (Q-scaling). While for the
magnetization we were able to discard a few smallest lattice sizes to reach
the expected scaling behavior, for susceptibility it is not the case and the
lattice sizes are still too small to show Q-scaling.

###  2.4 Scaling of the magnetization Fourier modes $m\left(k\right)$

Since all observables are expected to be governed at $T_{L}$ by Q-scaling, not
G-scaling, we try other ways to detect FSS at the pseudocritical point for
FBCs. One way to disentangle boundary conditions is through the Fourier
transformation of the magnetization.

As suggested by Flores-Sola et al in Ref. [3] (2016), then, we can check the
influence that the boundary conditions have on the magnetization Fourier modes
and their FSS, since only some modes actually contribute to scaling for FBCs.
In a system with FBCs the Fourier modes are sine waves:

$\psi_{{\bf k}}=\sqrt{2/L}\prod_{\alpha=1}^{d}\sin k_{\alpha}x_{\alpha}$ (16)

with wave vectors $k_{\alpha}=n_{\alpha}\pi/(L+1)$, $n_{\alpha}=1,2,\dots,L$.
Local quantities, like Ising spin $s_{{\bf x}_{i}}$, now read as

$s_{{\bf x}_{i}}=\sum_{{\bf k}}\tilde{s}_{{\bf k}}\psi_{{\bf k}}.$ (17)

However, in order to break the symmetry of the Ising model we should consider
profile $\mu({\bf x})$ of the spin $s_{{\bf x}}$ instead:

$\mu({\bf x})=\left|\left<S({\bf x})\text{sign}(M)\right>_{T}\right|.$ (18)

Here $\left<\dots\right>_{T}$ denotes temperature average. Now Fourier
transform and its inverse counterpart are:

$\displaystyle\mu({\bf x})$ $\displaystyle=$
$\displaystyle\frac{1}{(L+1)^{d}}\sum_{k}\\!\mu({\bf k})\
\prod_{\alpha=1}^{d}\\!\sin(k_{\alpha}\cdot{{x}_{\alpha}}),$ (19)
$\displaystyle\mu({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{x=1}^{N}\\!\mu({\bf x})\
\prod_{\alpha=1}^{d}\\!\sin(k_{\alpha}\cdot{{x}_{\alpha}}),$ (20)

${\bf k}=\pi/(L+1){\bf K}_{L}$, where ${x}_{\alpha}\in[1,L]$ represents the
Cartesian coordinates of a node, ${\bf K}_{L}$ consists of values in the range
$(0,..,L-1)^{d}$.

Figure 7: Fourier modes of the magnetization FSS in the vicinity of
pseudocritical point $T_{L}$. As expected, even $k$ modes vanish, while odd
modes provide accurate scaling.

The mean magnetization is

$\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{1}{L^{d}}\sum_{x}\mu({\bf
x})=\frac{1}{L^{d}}\frac{1}{(L+1)^{d}}\sum_{x}\sum_{k}\mu({\bf
k})\prod_{\alpha}^{d}\\!\sin(k_{\alpha}\cdot{{\bf x}_{\alpha}})$ (21)
$\displaystyle=$
$\displaystyle\frac{1}{L^{d}}\frac{1}{(L+1)^{d}}\sum_{k}\left(\sum_{x}\mu({\bf
x})\prod_{\alpha}^{d}\\!\sin(k_{\alpha}\cdot{{\bf x}_{\alpha}})\right)\cdot
W_{{\bf k}}\,\,,$

where $W_{{\bf k}}=\sum_{{\bf
x}}\prod_{\alpha}^{d}\\!\sin(k_{\alpha}\cdot{{x}_{\alpha}})$. For even modes
${\bf k}$ (i.e. modes for which at least one of the $K_{L}$’s is even), the
weights $W_{{\bf k}}$ are zero. These are called G-modes. Odd modes (i.e.
modes for which all the $K_{L}$’s are odd) are called Q-modes and should
contribute to the scaling. The results provided in Fig. 7 and in Table 2 show
that for our range of lattice sizes it is still not enough for all sites of
the system to represent a five-dimensional lattice. We get $m(k=1)\sim
L^{-1.26\pm 0.05}$, which hints towards the Q-scaling, but for smaller lattice
sizes again we see strong surface effects that shift scaling towards
G-scaling. Without a definite answer, we turn to another technique that is
often used in the theory of criticality: partition function zeros.

##  3 Scaling of the partition function zeros

To summarise so far, although our results are consistent with those of L & M
at $T_{c}$, we have had at best-mixed results at $T_{L}$, which makes it hard
to disambiguate between Q suggested by RG theory and the G scaling are
preferred by numerics. Therefore we seek another way to detect whether we are
in a regime with sufficiently high lattice sizes or not and we turn to
partition function zeros. In our experience, these deliver more reliable
results than those coming from thermodynamic observables. The latter are
moments of the partition function or free energy. As such they are sensitive
to amplitudes which themselves can be variable. E.g., if the amplitude of
correction terms for susceptibility is sufficiently large at $T_{L}$, said
correction terms can dominate and mask the leading FSS behavior coming from
power laws. In such circumstances, it might take extraordinarily large
lattices for leading FSS behavior to take effect. If this happens for FBCs and
not for PBCs, it would lead to the behavior we observed. Partition function
zeros, on the other hand, are not subject to such amplitudes.

For any finite system, the partition function can be written as a polynomial
in terms of fugacities and the partition function zeros are thus zeros in the
complex plane of the appropriate variables. Analyzing phase transitions via
the behavior of the zeros when approaching the critical point was first
suggested by Tsung-Dao Lee and Chen-Ning Yang for complex fields [23, 24] and
later generalized for complex temperatures by Michael Fisher [25, 26]. In what
follows below, we will study the properties of the Lee-Yang zeros. For most of
the models though, the zeros and their critical properties can be only
estimated by the extensive use of computer simulations [35]. Analytically,
solving the partition function for zeros in the complex plane is required. In
particular, as the system size increases, Lee-Yang zeros exhibit a scaling
behavior, accumulating near critical points in the thermodynamic limit. This
scaling pattern allows for the extraction of critical exponents and the
classification of phase transitions into universality classes. Understanding
the scaling properties of Lee-Yang zeros provides valuable insights into
critical behavior and phase transitions, enabling the characterization of
macroscopic properties and underlying physical phenomena.

(a)  (b)

(c)  (d)

Figure 8: Lines of zeros ${\rm Im}\ \\!Z(h^{R},h^{I})=0$ (black curves) and
${\rm Re\ \\!}Z(h^{R},h^{I})=0$ (yellow curves) at the pseudocritical
temperature $T_{L}$ for lattice sizes $L=12,14,16,18$ (a,b,c,d respectively).
Crossings of different color lines give the position of Lee-Yang zeros. All
zeros are located on the imaginary axis, in agreement with the unit circle
theorem.

Let us write down the definition of the partition function in the complex
$h=h^{R}+ih^{I}$ plane at $\beta=\beta_{c}$ in more details:

$Z(\beta_{c},h^{R}+ih^{I})=\sum_{\\{s\\}}e^{-\beta_{c}\sum_{i,j}S_{i}S_{j}+\beta_{c}h^{R}\sum_{i}S_{i}}\Big{(}\cos(\beta_{c}h^{I}\sum_{i}S_{i})+i\sin(\beta_{c}h^{I}\sum_{i}S_{i})\Big{)}\,,$
(22)

where the first sum spans all spin configurations. This can be conveniently
rewritten as:

$Z(\beta_{c},h^{R}+ih^{I})=Z(\beta_{c},h^{R})\Big{(}\Big{\langle}\cos(\beta_{c}h^{I}\sum_{i}S_{i})\Big{\rangle}_{\beta_{c},h^{R}}+i\Big{\langle}\sin(\beta_{c}h^{I}\sum_{i}S_{i})\Big{\rangle}_{\beta_{c},h^{R}}\Big{)}\,,$
(23)

here the averaging $\left<\dots\right>_{\beta_{c},h^{R}}$ is performed at the
fixed values of temperature $\beta_{c}$ and magnetic field $h^{R}$.

With the data of numerical simulations to hand, to determine the points where
the partition function becomes zero, it is necessary for both the real and
imaginary parts of the partition function to be zero, so we look at the values
of trigonometric functions averages
$\langle\cos(\beta_{c}h^{I}\sum_{i}S_{i})\rangle_{\beta_{c},h^{R}}$ and
$\langle\sin(\beta_{c}h^{I}\sum_{i}S_{i})\rangle_{\beta_{c},h^{R}}$ by
plotting the contours in the complex magnetic field plane along which these
function separately vanish [36, 37], and then, more precise methods are
implemented around the intersection of the said contours [38]. The simulations
at the complex $h$ values are not needed as we use the histogram reweighting
technique around the real simulations at $h=0$. Since the probability of
states “up" and “down" are the same for the Ising model, we artificially
improve our statistics by not only using the simulation data but also applying
random signs to the magnetization of each state with the probability $50\%$.
This way our statistics include more possible states and improve drastically.

Results presented in Fig.8 at $T_{L}$ show the position of several first Lee-
Yang zeros for different lattice sizes. The zeros follow a unit circle
theorem, which in our coordinates means that all zeros should lay on the
imaginary axis. This is what we observe for a few first zeros, though it is
obscured later by the extrapolation from the simulation point.

Nonetheless, we can be sure that our analysis of the first zeros is valid. To
further show this, we present a FSS of the first Lee-Yang zero in Figs.9,10.
The FSS of first Lee-Yang zero is governed by [39]:

$h_{1}\sim L^{-\frac{\Delta}{\nu}}=L^{-\frac{\beta+\gamma}{\nu}}.$ (24)

Q and G scaling predict $h_{1}\sim L^{3.75}$ and $L^{3}$, respectively.

Figure 9: FSS of the first Lee-Yang zero for sizes $L=10,..,25$ at $T_{c}$.
Expected outcome: G-scaling.

#### Criticality:

Fig.9 shows the FSS of the first Lee-Yang zero at $T_{c}$, where we get
$h_{1}\sim L^{-3.16\pm 0.48}$, this might hint at G-scaling according to our
other exponents at $T_{c}$.

#### Pseudo criticality:

For the first zero $h_{1}$ we see that for $T_{L}$ our fit when we discard the
smallest lattice sizes, $h_{1}\sim L^{-3.81\pm 0.08}\,$, is well compared to
Q-scaling and far from G-scaling: $h^{(Q)}_{1}\sim L^{-3.75}$ versus
$h^{(G)}_{1}\sim L^{-3}$, cf. Eq.(24).

Furthermore, from Eq.(8), we may write the susceptibility as

$\chi\propto\frac{1}{L^{d}}\sum_{j}{\frac{1}{(z-z_{j}(L))^{2}}}\,,$ (25)

where $z_{j}(L)$ is the $j$th zero for a system of size $L$. Assuming that the
first zeros dominate this gives that

$\chi\sim L^{-d}h_{1}^{-2}\sim L^{-2d\Delta/\nu}\,.$ (26)

I.e., we can provide an estimate for the susceptibility in this way. We find
the contribution of the first zero to the susceptibility,
$L^{-d}h_{1}^{-2}\sim L^{2.62\pm 0.10}$ which is far closer to Q
($\chi_{L}\sim L^{5/2}$) than to G ($\chi_{L}\sim L^{2}$). This means that, at
least from the Lee-Yang-zeros analysis, even the susceptibility is supportive
of Q for pseudo criticality and free boundaries.

Figure 10: FSS of the first Lee-Yang zero $h_{1}$ at $T_{L}$. Expected
outcome: Q-scaling. We observe that for smaller sizes the results have
corrections while for larger lattice sizes the scaling comes in line with
Q-scaling.

##  4 Conclusions

Mean-field theory holds sway for infinite volume systems above the upper
critical dimension. Standard FSS and standard hyperscaling both fail there.
Instead, Q theory governs finite-size systems in high dimensions. This
accounts for different types of boundary conditions and whether or not one
sits at the critical or pseudocritical point. Universality resides at the
pseudocritical point where standard FSS is replaced by QFSS. FSS at the
critical point depends on boundaries. For periodic boundaries, QFSS again
applies. For free boundaries standard FSS applied to Gaussian or G modes – not
Landau MFT – holds (although Landau MFT is supported by simulations in the
case of percolation with FBC at the percolation threshold [7]). Here we have
used Monte Carlo simulations of the $d=5$ Ising model to verify this picture
for free boundaries.

In this paper, we looked at the FSS of the coordinates of the pseudocritical
temperature $T_{L}$, mean magnetization $m$, isothermal susceptibility and
other observables. For the scaling of the pseudocritical point, we obtained
$t_{L}\sim L^{-2}$, as predicted before Q theory in Ref. [34]. For mean
magnetization at the critical temperature $T_{c}$ we observe G-scaling, while
at the pseudocritical temperature $T_{L}$ the small lattice sizes distort the
scaling picture, though discarding a few smallest lattice sizes gives us
Q-scaling. For the isothermal susceptibility the lattice sizes in question are
too small to present a proper five-dimensional scaling, and the scaling is
Gaussian at both $T_{c}$ and $T_{L}$. Furthermore, the magnetization Fourier
modes were studied, since for FBCs only odd magnetization Fourier modes
contribute to Q-scaling. We observed that even for small lattice sizes odd
modes manifest Q-scaling at the $T_{L}$.

Finally, we observed that expressing susceptibility in terms of Lee-Yang zeros
delivers a very strong signal of Q rather than G (or standard Landau) scaling.
To conclude, then, all of the observables measured are supportive of the Q
picture and, although small lattices cast dubious results, twelve years after
our original paper [1], the overall holistic picture is supportive of Q.

## Acknowledgements

We thank the Editors of this issue, Viktoria Blavatska, Taras Patsahan and
Orest Pizio for the invitation to submit this paper to the Festschrift on the
occasion of the 60th anniversary of Jaroslav Ilnytskyi. Doing so we send our
best wishes to Slavko, as friends call him, and express our admiration for his
scientific – and many other – activities, some of which we had the privilege
to participate in. We are grateful for the regular fruitful discussions among
the participants of the $\mathbb{L}^{4}$ Collaboration and Doctoral College
for the Statistical Physics of Complex Systems, especially to Leïla Moueddene,
Andy Manapany, and Marjana Krasnytska. This work was supported in part by the
National Academy of Sciences of Ukraine, project KPKBK 6541030(YuH & YuH). The
numerical simulations were performed at Coventry University’s HPC. Our work
would not be possible without the tireless and noble work of the Armed Forces
of Ukraine.

_Note added in proof._ One of the Authors of this paper, Ralph Kenna passed
away when this paper was in the course of editorial processing. This will be
probably the first paper we have to finalize without him, recalling again and
again our common life.

## References

  * [1] Berche B., Kenna R., Walter J.-C., Hyperscaling above the upper critical dimension, Nuclear Physics B, 2012, 865(1), 115, doi:10.1016/j.nuclphysb.2012.07.021
  * [2] Lundow P., Markström K., Finite size scaling of the 5d Ising model with free boundary conditions, Nuclear Physics B, 2014, 889, 249, doi:10.1016/j.nuclphysb.2014.10.011
  * [3] Flores-Sola E., Berche B., Kenna R., Weigel M., Role of Fourier modes in finite- size scaling above the upper critical dimension, Phys. Rev. Lett., 2016, 116, 115701, doi:10.1103/PhysRevLett.116.115701
  * [4] Kenna R., Berche B., A new critical exponent ’coppa’ and its logarithmic counterpart ’hat coppa’. Condensed Matter Physics, 2013, 16(2), 23601, doi:10.5488/CMP.16.23601
  * [5] Kenna R., Berche B., Universal finite-size scaling for percolation theory in high dimensions, Journal of Physics A: Mathematical and Theoretical, 2017, 50(23), 235001, doi:10.1088/1751-8121/aa6bd5
  * [6] Berche B., Chatelain C., Dhall C., Kenna R., Low R., Walter J.-C., Extended Scaling in High Dimensions, Journal of Statistical Mechanics: Theory and Experiment, 2008, P11010, doi:10.1088/1742-5468/2008/11/P11010
  * [7] Ellis T., Kenna R., Berche B., The fifty-year quest for universality in percolation theory in high dimensions, Condensed Matter Physics, 2023, 26, No. 3, 33606 doi:10.5488/CMP.26.33606
  * [8] Berche B., Ellis T., Holovatch Yu., Kenna R., Phase transitions above the upper critical dimension. SciPost Phys. Lect. Notes, 2022, 60, doi:10.21468/SciPostPhysLectNotes.60
  * [9] Ivaneyko D., Berche B., Holovatch Yu., Ilnytskyi J., On the universality class of the 3d Ising model with long-range-correlated disorder. Physica A, 2008, 387, 4497–4512, doi:10.1016/j.physa.2008.03.034
  * [10] Ilnytskyi J.M., Holovatch Yu., How does the scaling for the polymer chain in the dissipative particle dynamics hold? Condens. Matter Phys., 2007, 10, 539-552,doi:10.5488/CMP.10.4.539
  * [11] Ivaneyko D., Ilnytskyi J., Berche B., Holovatch Yu., Local and cluster critical dynamics of the 3d random-site Ising model. Physica A, 2006, 370, 163-178, doi:10.1016/j.physa.2006.03.010
  * [12] Chaikin P.M., Lubensky T.C., Principles of Condensed Matter Physics, Cambridge University Press, 2000.
  * [13] Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory, Yu.Holovatch (Ed.), vols.1–7, World Scientific, Singapore, 2004–2023.
  * [14] Berche B., Henkel M., Kenna R., Critical Phenomena: 150 Years Since Cagniard de la Tour, Journal of Physical Studies, 2009, 13, 3201, (arxiv 0905.1886)
  * [15] Janke W., Kenna R., Finite-size scaling and corrections in the Ising model with Brascamp-Kunz boundary conditions, Phys. Rev. B, 2002, 65(6),064110, doi:10.1103/PhysRevB.65.064110
  * [16] Privman V., Fisher M.E., Finite-size scaling of the correlation length above the upper critical dimension in the five-dimensional Ising model, J. Stat. Phys., 1983, 33, 385, doi:10.1007/BF01009803
  * [17] Brezin E., Zinn-Justin J., Finite size effects in phase transitions, Nuclear Physics B, 1985, 257, 867, doi:10.1016/0550-3213(85)90379-7
  * [18] Jones J.L., Young A.P., Finite-size scaling of the correlation length above the upper critical dimension in the five-dimensional Ising model, Phys. Rev. B, 2005, 71, 174438, doi:10.1103/PhysRevB.71.174438
  * [19] Flores-Sola E., Berche B., Kenna R., Weigel M., Finite-size scaling above the upper critical dimension in Ising models with long-range interactions, Eur. Phys. J. B, 2015, 88, 28, doi:10.1140/epjb/e2014-50683-1
  * [20] Kenna R., Berche B., Fisher’s Scaling Relation above the Upper Critical Dimension, Eur. Phys. Lett., 2014, 105, 26005, doi:10.1209/0295-5075/105/26005
  * [21] Binder K., Nauenberg M., Privman V., Young A.P., Finite-size tests of hyperscaling, Phys. Rev. B, 1985, 31, 1498, doi:10.1103/PhysRevB.31.1498
  * [22] Kenna R., Berche B., _Scaling and Finite-Size Scaling above the Upper Critical Dimension_ , in “Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory” Vol. 4, Yu. Holovatch (Ed.), World Scientific, Singapore, 2015, pp 1-54.
  * [23] Yang C.N., Lee T.D., Statistical theory of equations of state and phase transitions. I. Theory of condensation, Physical Review, 1952, 87, 404, doi:10.1103/PhysRev.87.404
  * [24] Lee T.D., Yang C.N., Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Physical Review, 1952, 87(3), 410, doi:10.1103/PhysRev.87.410
  * [25] Fisher M.E., The nature of critical points, In ed. Britten, W.E., Lectures in theoretical physics, v. 7C, p. 1–159. University of Colorado Press, Boulder, Colorado, USA, 1965.
  * [26] Wu F.Y., Professor C.N. Yang and statistical mechanics, International Journal of Modern Physics B, 2008, 22(12), 1899, doi:10.1142/S0217979208039198
  * [27] Lundow P.H., Markström K., Non-vanishing boundary effects and quasi-first-order phase transitions in high dimensional Ising models, Nucl. Phys. B, 2011, 845, 120–139, doi:10.1016/j.nuclphysb.2010.12.002
  * [28] Lundow P., Boundary effects on finite-size scaling for the 5-dimensional Ising model, Nuclear Physics B, 2021, 967, 115422, doi:10.1016/j.nuclphysb.2021.115422
  * [29] Wolff U., Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett, 1989, 62, 361, doi:10.1103/PhysRevLett.62.361
  * [30] Kasteleyn P.W., Fortuin C.M., Phase transitions in lattice systems with random local properties, Journal of the Physical Society of Japan Supplement, 1969, 26, 11;
  * [31] Fortuin C.M., Kasteleyn P.W., On the random-cluster model: I. Introduction and relation to other models, Physica, 1972, 57(4), 536, doi:10.1016/0031-8914(72)90045-6
  * [32] Ferrenberg A.M., Swendsen R.H., New Monte Carlo technique for studying phase transitions, Physical Review Letters, 1988, 61, 2635, doi:10.1103/PhysRevLett.61.2635
  * [33] Ferrenberg A.M., Swendsen R.H., Optimized Monte Carlo data analysis, Physical Review Letters, 1989, 63, 1195, doi:10.1103/PhysRevLett.63.1195
  * [34] Rudnick J., Gaspari G., Privman V., Effect of boundary conditions on the critical behavior of a finite high-dimensional Ising model, Phys. Rev. B, 1985, 32, 7594, doi:10.1103/PhysRevB.32.7594
  * [35] Bena I., Droz M., Lipowski A., Statistical mechanics of equilibrium and nonequilibrium phase transitions: the Yang–Lee formalism, International Journal of Modern Physics B, 2005, 19(29), 4269, doi:10.1142/S0217979205032759
  * [36] Krasnytska M., Berche B., Holovatch Yu., Kenna R., Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks, Journal of Physics A: Mathematical and Theoretical, 2016, 49(13), 135001, doi:10.1088/1751-8113/49/13/135001
  * [37] Krasnytska M., Berche B., Holovatch Yu., Kenna R., Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks, Europhys. Lett., 2015, 111, 60009, doi:10.1209/0295-5075/111/60009
  * [38] Press W.H. et. al., Numerical Recipes 3rd Edition, The Art of Scientific Computing, Cambridge University Press (2007).
  * [39] Itzykson C., Pearson R., Zuber J., Distribution of zeros in Ising and gauge models, Nuclear Physics B, 1983, 220(4), 415, doi:10.1016/0550-3213(83)90499-6

Коли кореляції перевищують розмір системи: скінченновимірний скейлінґ за
вільних граничних умов над верхньою критичною вимірністю
Ю.Гончар${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?},\penalty
1000\mbox{?}}}}$, Б.Берш${}^{\textsf{\footnotesize{\mbox{?},\penalty
1000\mbox{?}}}}$, Ю.Головач${}^{\textsf{\footnotesize{\mbox{?},\penalty
1000\mbox{?},\penalty 1000\mbox{?},\penalty 1000\mbox{?}}}}$,
Р.Кенна${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?}}}}$

${}^{\textsf{\footnotesize 1}}$ Інститут фізики конденсованих систем НАН
України, Львів, 79011, Україна ${}^{\textsf{\footnotesize 2}}$ Центр плинних і
складних систем, Університет Ковентрі, Ковентрі, CV1 5FB, Велика Британія
${}^{\textsf{\footnotesize 3}}$ Лабораторія теоретичної фізики та хімії,
Університет Лотарингії, CNRS, Нансі, Франція ${}^{\textsf{\footnotesize 4}}$
Співпраця $\mathbb{L}^{4}$ і Коледж докторантів ‘Статистична фізика складних
систем’, Ляйпціґ-Лотарингія-Львів-Ковентрі, Європа ${}^{\textsf{\footnotesize
5}}$ Центр науки про складність, Відень, 1080, Австрія


#  Solving Simulation Systematics in and with AI/ML
Bridging the systematic gap between simulated and real data

Brett Viren  Dept. of Physics, Brookhaven National Laboratory  Corresponding
author<EMAIL_ADDRESS>Jin Huang  Dept. of Physics, Brookhaven National
Laboratory  Yi Huang  Computer Science, Brookhaven National Laboratory
Meifeng Lin  Computer Science, Brookhaven National Laboratory  Yihui Ren
Computer Science, Brookhaven National Laboratory  Kazuhiro Terao  SLAC
National Accelerator Laboratory  Dmitrii Torbunov  Dept. of Physics,
Brookhaven National Laboratory  Haiwang Yu  Dept. of Physics, Brookhaven
National Laboratory  Computer Science, Brookhaven National Laboratory

###### Abstract

Training an AI/ML system on simulated data while using that system to infer on
data from real detectors introduces a systematic error which is difficult to
estimate and in many analyses is simply not confronted. It is crucial to
minimize and to quantitatively estimate the uncertainties in such analysis and
do so with a precision and accuracy that matches those that AI/ML techniques
bring. Here we highlight the need to confront this class of systematic error,
discuss conventional ways to estimate it and describe ways to quantify and to
minimize the uncertainty using methods which are themselves based on the power
of AI/ML. We also describe methods to introduce a simulation into an AI/ML
network to allow for training of its semantically meaningful parameters. This
whitepaper is a contribution to the Computational Frontier of Snowmass21.

##  1 Introduction

A common AI/ML application in HEP and elsewhere adopts a strategy like:

  1. 1.

Train a network on samples from a simulated detector data domain.

  2. 2.

Apply the trained network to samples from a real detector data domain.

  3. 3.

???

  4. 4.

Profit.111With apologies to South Park.

Of course, our simulations are always wrong at some level. Great pains are
taken to minimize this wrongness, but the result can only be as accurate as
the precision of our validation methods. We are best to keep in mind that the
two data domains (simulated and real) remain distinct. When we fail to fill in
the “???” with systematic error analyses of a precision commensurate with that
of the primary analyses then we make what is called here a “domain-equivalency
mistake”. With so much of our mainline analyses exploiting the high precision
of artificial-intelligence and machine-learning (AI/ML) we must also evaluate
our systematic errors at an equal level. Failing that, we do not know how
wrong we truly are.

Here, we discuss select ways to supply step three. First we describe some
conventional approaches to reducing the difference between the two domains.
Then we look at a special case where learned, generative AI/ML techniques are
employed, give some problems with their standard validation methods and
possible ways to address these problems. Next we introduce a new concept being
developed at Brookhaven National Lab for bridging the two data domains. And
finally describe new techniques being explored at SLAC National Lab and others
for integrating simulation to AI/ML training. Though these problem are general
to many fields and applications but we discuss them in the context of recent
developments and applications toward DUNE and other liquid argon time
projection chamber (LArTPC) detectors.

##  2 Transforms for closer equivalence

Prior to the rise of machine learning, we used our “natural intelligence” to
understand our detectors and minimize the errors made by its simulation. Of
course, we still do this. We make our simulation produce data as similar to
real detector data as it is feasible to do so. We typically strike a balance
between effort and quality so the fidelity is “good enough” for a given
purpose. As the high-precision of AI/ML is applied to analyses, the fidelity
of the simulation must be raised to an equivalent level.

Some features of the real detector data are a “nuisance”. They are
particularly difficult to model with required precision yet are also are not
related to the physics information we seek to extract from our detectors. For
example, real LArTPC detectors typically have some forms of coherent noise
which greatly confound the extraction of signal and which are also impractical
to model with high precision. As a practical and effective work around, we
apply transformations to both the simulated and the real data which attempt to
minimize the prevalence of “nuisance” features while leaving unchanged those
due to signals we wish to extract.

In the LArTPC arena, we have collected expertise in to a variety of such
transforms. We have developed software “noise filters” targeting a set of
noise features that are specific to certain detectors or general across all.
Transforms that attack coherent noise include explicit “signal protection”
guards so they retain certain features which can not be readily distinguished
as being from signal. We also apply crucial “signal processing” to amplify
signal by deconvolving detector response. As a side effect, this reduces both
the general high frequency thermal noise from electronics as well as low
frequency noise that would otherwise be amplified by the deconvolution. The
capability of these transforms to bring the simulated and real data domains
closer has been well validated [1],[2].

However, these validations are rough in nature compared to the precision of
AI/ML. Evaluation is in terms of metrics formed as some average over the
features of samples from each data domain. The averaging hides fine features
which can nonetheless still be noticed by AI/ML networks. The “domain-
equivalency mistake” is reduced by applying this manner of validation but how
much error remains and how to measure it is still a concern.

##  3 Generative network evaluations

Generative adversarial networks (GANs) are class of AI/ML networks with many
practical benefits. The GAN architecture consists of two parts. A generator
part is trained to produce samples from an underlying but “ersatz”
distribution such they are indistinguishable from those drawn from a reference
distribution. Simultaneously, a discriminator part is trained to distinguish
generated samples from reference samples. We consider the “ersatz” and
reference distributions distinct and samples from the latter are typically
provided by a detailed, conventional simulation.

In a scientific data context, GANs are often used to implement a “fast
simulation” and in section 4 we introduce another way to exploit them. The
goal in this case is to replace a computationally expensive conventional
simulation with a faster GAN network.

However, such a replacement introduces the potential for another domain-
equivalency mistake. The network may not fully learn the reference
distribution due to use of a limited size training set and/or non-optimal
network architectures. We must keep in mind that the “ersatz” distribution is
thus twice-removed from the one sampled by the real detector. Next, we
describe ways to get a handle on these errors.

###  3.1 Network based metrics

A GAN as a fast simulation is still useful if we can measure the quality of
samples it produces. When the samples are in the form of 2D pixel images
measures such as the Fréchet [3] (FID) and/or kernel [4] (KID) _inception
distance_ (together xID) are commonly applied. Both measure a certain vector
distance between two sets of samples. The vectors are formed from high-level
features extracted from the 2D pixel images with a help of the Inception
network. An important detail for here is that the features (labels) and the
data set used to train the network are fixed by the metric definitions. The
training uses a large, standardized dataset called ImageNet. It includes
images of cats, cars and cardigans but certainly not any images from our
detectors.

Thus, an image vector derived from a sample, be it real, simulated or ersatz,
represents the result of a projection from the actual distribution domain to a
domain spanned by the set of internal features associated to the ImageNet
labels. For example, one element of the vector may represent in some abstract
way a “%-catness” of an image. Applying xID as metrics for a GAN trained in
scientific domains is like asking how much “catness” does the GAN preserve in
the detector data. It is an absurd, if adorable, metric to judge scientific
worthiness of a detector simulation.

Furthermore, xID metrics are themselves “metric free” (in the sense of a
metric tensor). We can not assign any absolute interpretation to its distance
values across the latent space. Furthermore, xID posses some known biases
which confound attempts to compare the reported values of the “same” metric
across different applications. While they are a practical tool rank different
GANs trained on and compared with identical data, they are not useful for
providing a quantitative systematic uncertainty measure.

The manual human effort put into the labeling of ImageNet images was certainly
nontrivial. The similar kind of effort to label millions of detector events
would exhaust even the most diligent pool of grad students. Happily, with
simulated data our “labels” may be our “Monte Carlo truth” values. We may
generate labeled images in massive quantities with little human effort. These
images can then replace the ImageNet training set and a variant of the xID
metrics can be constructed which is tailored to a given data domain. By
construction such a metric will not be subject to the projection problem
suffered with the xID metrics. Though, we must recognize that like the GAN
output itself, such metrics are still “twice removed” from real data. We may
have escaped comparing scientific data in terms of “%-catness” but we still
would make comparisons in terms of “%-simulation-domain-featureness” which we
then use as a measure of what is really “%-real-domain-featureness”.

Nonetheless, the LArTPC GAN-using community may come together to develop
domain-specific xID-like metrics in this manner. This would require strong
coordination to standardize simulation configuration to generate training
samples, the labels to use, what post-processing to apply, how to train
Inception (or perhaps another network), distribution of pre-trained models and
standardizing their application. Different GANs target different data domains
(raw ADC, signal processed, output of reconstruction) and thus some variety
such metrics need construction. Studies would also be needed to understand how
these metrics differ when applied to data representing different LArTPC
detector designs.

Another approach is to form a comparison metric by replacing the label-trained
network with an auto-encoder (AE) network trained on domain-specific data
samples. The values of nodes forming the central bottleneck layer of the AE
network can form the vectors used in a distance measure. A major benefit of
this approach is that it requires no labeling and thus can be trained on
samples from all relevant domains (eg, real, simulated and ersatz).

Furthermore, some AE techniques allow for the latent vector space to be made
smooth so that interpolating between different the points corresponding to two
samples produces meaningfully similar decoded samples. Ideally we may find
some means by which we can understand a geometric metric tensor of this latent
space so that we may form a distance with some semantic and quantitative
meaning. In section 5.3 we will suggest one way that may provide this.

###  3.2 Invariant metrics

When we apply GAN to scientific data we can exploit another validation
strategy. We may form metrics which are sensitive to certain conserved
quantities, symmetries or correlations which are expected to be invariant over
the features of any given sample. These are given to us by the physics of the
detector operation and of fundamental particles and their interactions.

For example, raw waveforms read out from capacitive-coupled sense electrodes
in LArTPC (“induction channels”) should integrate over long times to a value
nearly zero (after baseline subtraction). Or, at high level after
reconstructing GAN-generated data, particle momentum must be conserved and
invariant mass peaks must be recovered at expected energies and Bragg peaks
should appear with proper characteristics. Where a GAN is involved in
generating or where other networks may provide categorizing, we may validate
their capability against these types of constraints.

Such metrics pose a conundrum. We may choose to bake them into the loss
function in order to get a better trained network. But then, such a network
will, by construction, produce results which are valid according to the
metrics.

##  4 Domain isomorphism

A novel approach to attacking the “domain-equivalency mistake” is being
investigated as part of a lab-directed research and development project at
Brookhaven National Lab. We are developing a method allowing a sequence of
attacks on systematic errors. First is to reduce the domain difference, then
to quantitatively measure remaining difference and finally to provide a
mechanism to propagate that remaining systematic difference through any
subsequent processing and analysis.

The work goes under the name Large-Scale Scientific Simulation Systematic GAN
(LS4GAN). It uses GAN-style networks to form a two-way isomorphic map that
transforms samples (eg images) from one domain to another (eg “real” to
“simulated” and vice versa). The network architecture and training is such
that information specific to an individual sample is retained by the mapping
while information that is general to the source domain is replaced by
information general to the target domain.

For example, given a simulated detector readout of a neutrino interaction
event, the features from the modeled noise and detector response would be
replaced with equivalent features from the real detector. Meanwhile, the
information which is specific to the particular neutrino interaction in the
event will be retained. The simulation can be thought of as keeping its
intended nature but being made “more real”.

An existing proof of principle is demonstrated by CycleGAN [5],[6]. The LS4GAN
effort has also produced an initial application of our variant network UVCGAN
[7] on standard data photographic image sets. The LS4GAN effort is now working
to apply this technique to LArTPC data which confront novel challenges due to
sample size and the fact we may not rely on the human bias of “artistic
judgment” of the quality of results.

Once those challenges are overcome, we may exploit this mapping in a few ways.
First, we may compare the output of the simulated-to-real mapping against its
simulated input on a sample-by-sample, pixel-by-pixel basis. For example, a
simple pixel-wise difference image may be formed and it is expected to
highlight certain features which would otherwise be invisible in the
conventional validation techniques such as described in section 2. Given
substantial domain expertise we may reasonably expect such high fidelity
comparisons will point us to improvements we may make to the simulation and
thus bring its subsequent output yet closer to reality.

After some number of iterations on the above process we expect to reach a
point of diminishing returns. We may then retrain LS4GAN a final time and use
the nominal simulation output and the more realistic simulated-to-real mapping
to create two data sets of near identical sample pairs. The event-by-event
difference holds a representation of whatever residual systemic difference
exists between the simulated and the real data domains.

As these two datasets differ slightly in their information content but not at
all in their format, each may be individually sent down a common
reconstruction and analysis chain, be it based on conventional or AI/ML
methods or some mixture. The results of processing each set may then be
compared to form a quantitative estimate of the systematic uncertainty which
due to our use of an imperfect simulation.

This will provide us a quantitative measure of the domain-equivalency mistake
but it must be recognized it still neglects some residual uncertainty. The
simulated-to-real mapping is itself imperfect and subject to systematic
differences driven by, for example, the statistical sampling of simulated and
real data that is made in defining a training set. So, the resulting
systematic uncertainty is itself uncertain. The value of this residual unknown
systematic uncertainly can arguably be estimated as being of some lower order
compared to the systematic uncertainty measure the technique provides. Arguing
that becomes a detailed problem for the particular experiment applying this
method.

##  5 Trainable simulation

A trained neural network encodes some learned information into an abstract
representation over a number of parameters. Any given parameter in that set
has no identifiable meaning. Here we describe techniques to develop models
with semantically meaningful parameters that can be efficiently trained in the
same manner as are neural networks. Two techniques are described and both
provide computationally efficient methods to calculate the gradient over the
learned parameters.

###  5.1 Backpropagation

We may “simply” develop (or reimplement) a detailed simulation with code that
uses a software library that provides the “auto gradient” feature. This
feature allows the gradient of every function return value with respect to its
arguments to be computed with no explicit code provided by the developer.
Perhaps the most known examples of libraries providing this feature include
PyTorch and JAX for Python while implementations for C++ applications also
exist. Given a means to compute gradients we may apply gradient descent
optimizations such as are commonly used to train neural networks. In fact, the
resulting code is structured as a network of computation nodes in a very
similar manner to that of neural networks differing only in terms of
connectivity and functions. When the auto gradient library provides support
for GPUs, as many do, we gain the side benefit of hardware acceleration.

###  5.2 Forward gradient

Recent work [8] has introduced a method to efficiently calculate an estimate
of a gradient in the context of a forward pass of the simulation. It is
stochastic in nature which is perhaps particularly acceptable to calculate
gradients through a simulation which is also stochastic. In principle, this
method may be applied to our “usual” C++ simulations running on CPU though
this will likely prove prohibitively costly and still requires a
reimplementation of our simulations to run on GPU. More work is needed to see
if this is a viable alternative to the auto-gradient based approach.

###  5.3 Usage

A broad class architectures have been collected under the general umbrella of
“simulation-based inference” [9]. In addition, a trainable simulation can
provide a potentially powerful extension to the LS4GAN approach. Instead of
passing minted samples to LS4GAN we may directly integrate a trainable
simulation network to the corresponding input nodes of the architecture.
Training will then optimize over the semantically meaningful parameters of the
simulation network. Resulting changes in these parameters can be interpreted
as a measure of their systematic errors.

Such a scheme is potentially very powerful but needs to be tempered by the
fact that the semantic parameters of the simulation represent a particular
information basis that may not fully span the domain of the real data.
Certainly, this basis is more relevant than the one based on labels of cats,
cars and cardigans of xID but a qualitatively similar domain-equivalency
mistake will confound the results. For example, optimizing against real data
with sources of noise that are not modeled in the trainable simulation will
cause systematic shifts to be attributed to the unrelated parameters.

We may also combine this conceptual thread with that of an AE-based metric in
an attempt to provide some semantic mapping into the latent vector space. For
example, the gradient the latent vector with respect to the trainable
simulation parameters may give semantic meaning to the latent space. Metrics
formed on that latent space can then be assigned semantically meaningful
uncertainties.

##  6 Summary

We have highlighted a core systematic problem in applying analyses which
utilize high-precision AI/ML networks that are trained on simulation and
applied to real detector data which we name the “domain-equivalency mistake”.
We may reduce this mistake by applying data transforms based on domain
expertise. We suggest to develop more appropriate variants of existing metrics
used by current GAN researchers as well as novel ones in order to provide more
meaningful ways to compare the quality of a generated samples against those in
a reference set. A developing AI/ML-based method, LS4GAN, is described and
expected to reduce, measure and propagate systematic differences between two
data domains with high precision. Finally, we describe trainable simulations
which bring a promise to provide semantic meaning to measures of systematic
error estimates based on training between the simulated and real domains.

## References

  * [1] MicroBooNE collaboration  “Ionization Electron Signal Processing in Single Phase LAr TPCs I: Algorithm Description and Quantitative Evaluation with MicroBooNE Simulation”  In _JINST_ 13, 2018, pp. P07006  arXiv:1802.08709 [hep-ex]
  * [2] “Ionization Electron Signal Processing in Single Phase LAr TPCs II: Data/Simulation Comparison and Performance in MicroBooNE”  In _JINST_ 13, 2018, pp. P07007  arXiv:1804.02583 [hep-ex]
  * [3] Martin Heusel et al.  “GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium”  In _Advances in Neural Information Processing Systems_ 30 Curran Associates, Inc., 2017  URL: https://proceedings.neurips.cc/paper/2017/file/8a1d694707eb0fefe65871369074926d-Paper.pdf
  * [4] Mikołaj Bińkowski, Danica J Sutherland, Michael Arbel and Arthur Gretton  “Demystifying MMD GANs”  In _arXiv preprint arXiv:1801.01401_ , 2018
  * [5] Jun-Yan Zhu, Taesung Park, Phillip Isola and Alexei A Efros  “Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks”  In _Computer Vision (ICCV), 2017 IEEE International Conference on_ , 2017
  * [6] Jun-Yan Zhu, Taesung Park, Phillip Isola and Alexei A. Efros  “Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks”, 2020  arXiv:1703.10593 [cs.CV]
  * [7] Dmitrii Torbunov et al.  “UVCGAN: UNet Vision Transformer cycle-consistent GAN for unpaired image-to-image translation”, 2022  arXiv:2203.02557 [cs.CV]
  * [8] Atılım Güneş Baydin et al.  “Gradients without Backpropagation”, 2022  arXiv:2202.08587 [cs.LG]
  * [9] Kyle Cranmer, Johann Brehmer and Gilles Louppe  “The frontier of simulation-based inference”  In _Proceedings of the National Academy of Sciences_ 117.48, 2020, pp. 30055–30062  DOI: 10.1073/pnas.1912789117