###### Abstract

I start with a review of my personal and scientific interactions with Michael
E. Fisher, who was my post-doc mentor in 1972-1974. I then describe several
recent renormalization group studies, which started during those years, and
still raise some open issues. These include the magnets with dipole-dipole
interactions, the puzzle of the bicritical points and the random field Ising
model.

##  Chapter 0 50 years of correlations with Michael Fisher
and the renormalization group

###  1 My life with Michael E. Fisher

####  1 Personal

I worked on my Ph. D. thesis with the late Yuval Ne’eman at Tel Aviv
University, in parallel to my military service in the Israeli army. Expecting
to be slow relative to the level of activity required in more competitive
topics, I chose to work on the arrows of time, combining recent developments
on the breaking of time reversal symmetry in the decay of the K-meson with the
second law of thermodynamics. I thus studied both high energy physics and
statistical physics, but knew very little about critical phenomena. When I
wrote my thesis (while on duty on the Golan heights in September 1971), I
started looking for a place for my postdoctoral training, and I almost
accepted a position where I would continue my work on irreversible
thermodynamics.

However, two crucial events changed my life. First, the late Joe Imry just
returned from his postdoc at Cornell, and he told me about his good experience
there. Second, the late Pierre Hohenberg visited Tel Aviv, and told us about
Wilson’s renormalization group (RG). Both became close friends and
collaborators for many years. As a result, I wrote to Michael Fisher, and
applied for a postdoc position in his group. In spite of my lack of sufficient
background in phase transitions, Fisher soon offered me a postdoc position. In
particular, he asked me to respond within 2 weeks! Being very precise, he
indicated that I shall have only two duties: do research andattend the weekly
Widom-Fisher seminar. I immediately cancelled my alternative possibilities,
and accepted Fisher’s offer.

My wife, my 2-year old son Ofer (now a high energy physicist at the Weizmann
Institute, studying string theory – Fisher won: he has two sons in physics…)
and I used my Fulbright travel grant111Most of the postdocs at Cornell came
with some external fellowships. and arrived in Ithaca in July 1972, and were
told that we missed the summer, which occurred on the day before our arrival
(Ithaca is cold, and summers are short.) We had already rented an apartment
(for which Michael had kindly loaned us the first rent), and thus we quickly
settled down. During the following two years, we were often invited to the
Fisher home, where we enjoyed the warm hospitality of both Sorrel and Michael.
As on many later occasions, in October 1973 Michael drove me in his car to my
first magnetism conference in Boston, where I talked about our joint work on
dipolar magnets (see below). We returned to Ithaca a few days before my
daughter Tamar was born (in parallel to worrying about the Yom Kippur war in
Israel). Kindly, the Fishers held a party in their home, for exhibiting Tamar
to the local community. I last met Sorrel in 2009, when Michael was awarded an
honorary doctorate at the Weizmann Institute (Fig. 1), and I was very sad to
hear about her death in 2016. It was difficult for Michael to survive without
Sorrel. He reacted emotionally when I sent him the photos in Fig. 1.

Figure 1: Michael and Sorrel Fisher in 2009, when he received an honorary PhD
from the WIS. Left: with David Mukamel, Joe Imry, AA. Right: with Sir Michael
Berry and AA.

After 1974, Michael and I met many times at conferences (e.g., Yeshiva and
Rutgers Statistical Mechanics meetings,222My first Yeshiva meeting, in
December 1972, was a special experience. In particular I met there my life-
long friends and collaborators, the late Dietrich Stauffer and (long live)
Sushanta Dattagupta, and many other players in the field. IUPAP conferences on
Statistical Physics and on Magnetism, Gordon Conferences, and more). He also
visited Tel Aviv in 1980 – on the occasion of receiving the Wolf prize333A
typical Fisher story: The Wolf prize was young. Documents were prepared in
Hebrew and then translated into English. In this translation, “phase
transitions” became “transient phases”, and this appeared on the certificate
given to Michael by Israel’s president at the ceremony. I caught this, and
asked the Wolf people to be more careful in the future. However, when they
asked Michael to return the certificate for correction he refused: “It will
look much more impressive on my office wall as it is”. I do not know how this
ended. (with Wilson and Kadanoff), in 1983 – when he was honored with a
Sackler distinguished visiting scholarship at Tel Aviv
university,444https://ias.tau.ac.il/IAS-Distinguished-Scholars; this was
before Sacklers’ involvement in the opioid crisis became known; since then I
no longer use the Sackler name when I refer to my School and Faculty at TAU.
555 On that visit Michael stayed for a month in an apartment with an
appropriately placed desk and room for playing the guitar – being a
perfectionist in everything, Michael always insisted on proper arrangements in
his rooms and offices., in 1988, when he participated in “Frontiers of
Physics,” Landau Memorial Conference,666“Frontiers of Physics,” Proc. Landau
Memorial Conf., Eds. E. Gotsman, Y. Ne’eman and A. Voronel (Pergamon Press,
Oxford, 1990). and in 1992 – when he was awarded an honorary doctorate from
Tel Aviv University777http://www3.tau.ac.il/honorary$\\#$phd. Michael did not
bring a formal suit for the ceremony, so we rushed downtown to buy one, and I
saw him wear that suit also on later occasions; even when Michael was not in
Tel Aviv, Tel Aviv stayed with him…. As I describe below, we collaborated on
many topics, and I learned a lot from him – on how to use elegant mathematics,
how to write papers, how to fight with referees and with editors (especially
in PRL – both Michael and I refrained for a while from publishing as well as
refereeing there) and how to present talks (with two parallel transparency
projectors!). As other authors in this book describe, I also “suffered” from
his many red comments on my paper drafts, and enjoyed seeing his art of
drawing figures for our papers, both of which taught me a lot. In addition, I
enjoyed participating in Michael’s 60th [1], 70th [2] and 80th [3] birthdays,
see e.g. Figs. 2, 3, which were great reunions of the large group of Michael’s
students and postdocs. Michael also attended my 60th birthday conference in
Eilat, in 2004 888https://www.tau.ac.il/ aharony/public/eilat poster.jpg see
Fig. 4. In 1982, Michael Stephen, Michael Fisher and I joined others to teach
a course on critical phenomena in South Africa [4].999Both kindly drove on the
wrong side of the road, given their prior experience. Also, I visited Michael
in Maryland in 2002. When I arrived at Dulles airport, the person at customs
asked me where I am going, and I told him I am giving a colloquium on quantum
mechanics at the University of Maryland. He immediately asked when we shall
get rid of airports and use teleportation, and I told him this will not happen
in my lifetime. I mentioned this conversation in my colloquium, and Michael
immediately reacted: “This must have been our student”! Note: my talk was
about quantum interference in mesoscopic devices, and at that time Fisher
raised questions about the relevance of quantum mechanics to condensed matter
physics [5], but we did not debate on this. Finally, I must have done
something right: Michael also invited my first doctoral student, the late
Shmuel Fishman, to be a postdoc in his group (first at Cornell and then at
Maryland; Fig. 2).

Figure 2: Fisher and his students/postdocs on his 60h birthday, NAS,
Washington DC 1991. Top: Mustansir Barma, AA, Shmuel Fishman, Joseph Straley,
Pierre Pfeuty, David Nelson, David Jasnow, Ubo Felderhof, Masuo Suzuki,
Michael Barber, Gunduz Caginalp, Arthur Ferdinand. Bottom: Vladimir Privman,
MEF, Lev Mikheev. Figure 3: Card drawn and written by MEF to AA after MEF’s
70th birthday. Note the typical drawings and calligraphy.

Figure 4: AA’s 60th birthday, Eylat, 2004. Top: Fisher lecturing with
transparencies; AA, MEF, Joe Imry, Édouard Brézin and Eytan Domany. Bottom:
MEF with Sir Sam Edwards; Yuval Ne’eman, AA and MEF.

####  2 Scientific

My time at Cornell was unique.101010In a talk celebrating Wilson, David Mermin
called these years ”the Valhalla years of Cornell”. The Wilson-Fisher paper on
the $\epsilon-$expansion [6] had just appeared, and the authors were there to
help with its applications to phase transitions in a variety of physical
systems. That paper, in which they considered the dimensionality
$d=4-\epsilon$ as a continuous variable, allowed simple analytic calculations
(instead of the rather more complicated numerics proposed by Wilson earlier).
This opened the field to many researchers, and led to the flourishing of
critical phenomena ever after.111111Since I came into the field in 1972, I do
not feel competent to review its earlier history, accept for noting that
Wilson and Fisher followed the scaling ideas set earlier by Ben Widom and Leo
Kadanoff, and Widom was also there – open for discussions and help. Fisher
himself reviewed this history in Ref. [7]. Half a century later, this is one
of the main reasons to celebrate Michael Fisher and the RG in this book.

I overlapped at Cornell with many postdocs and students, both in the Baker
laboratory of Chemistry, where Fisher was centered (with Ben Widom, who told
me proudly that he got Fisher there, but was unable to contribute a chapter to
this book), and in the Clark Hall of Physics. The list included Pierre Pfeuty
and Helen Au-Yang, with whom I shared an office, David Jasnow, Howard Tarko,
Alastair Bruce, Michel Droz, Michael Kosterlitz, David Nelson, Eytan Domany,
Paul Gerber, Joe Sak and many more, not to mention Widom’s group (e.g., Jim
Bartis) and Wilson’s group (e.g. H. R. Krishnamurthy). Also, Bob Griffiths was
there on sabbatical. Among others, Pfeuty talked about his work with Elliot on
the Ising model in a transverse field, and Kosterlitz talked about his work
with Thouless on the XY model in two dimensions The atmosphere was friendly,
and Michael encouraged joint discussions and collaborations.121212We also had
the good fortune to read the preprints of chapters from Ashcroft and Mermin’s
“Solid State Physics” (I still have a pile of these heavy typed files), and be
inspired by both authors.

Ken Wilson gave a course on the RG, which he also gave in Princeton (later
published by Wilson and Kogut [8]). Most of us worked on the RG, and often the
course gave us the hints we needed for making progress in our research. Many
visitors came through, and we heard their seminars both on the blackboard in
the Fisher-Widom seminar and in the more conventional physics seminar. I
brought a pile of slides based on my doctoral thesis (transparencies only
began being used then), but did not use them – being forced to use only the
blackboard. Away from this seminar, Michael did use transparencies; he kept
using them all his years, avoiding the more modern use of power point. In
particular, I liked his way of using two projectors in parallel, so that one
could compare theory and experiments, or keep track of definitions. I adopted
this method for many years, until I did switch to power point. Several papers
in this volume also mention Michael’s questions to seminar and conference
speakers. Although sometimes looking sleepy, he usually understood much of the
talks, and posed important questions for which the speakers were not always
prepared. In the following years I tried to imitate Michael also in this
respect, which may have given me the reputation of being “brusque”.

Immediately after my arrival, Michael suggested that I study the dependence of
the critical temperature on the many physical parameters in specific systems.
He later returned to that problem in his work on non-linear fields, and I
return to it in Sec. 2.1 below. However, shortly after that I attended a
physics seminar in Clark Hall, discussing some aspects of dipole-dipole
interactions. Already during the talk I realized that the propagator related
to the two-spin correlation function of such interactions is similar to the
propagator of the photon (which vanishes for longitudinal fluctuations), which
I knew from my field theory courses at Tel Aviv. This propagator projects out
longitudinal correlations. I immediately spoke to Michael about this, and we
agreed that the critical behavior of dipolar systems is a better direction to
follow. Indeed, this project soon ended up with six papers [9, 10, 11, 12, 13,
14], with the first appearing already in March 1973. However, only two of
these were co-authored by Michael! The ‘Heisenberg’ dipolar interactions
involve scalar products like $({\bf S}({\bf R}_{i})\cdot{\bf R}_{ij})$,
involving both the coordinate vector ${\bf R}_{ij}$ and the spin vector ${\bf
S}({\bf R}_{i})$, and therefore we used $n=d=4-\epsilon$, where $n$ is the
number of spin components.131313The equality $n=d$ arises in many physical
systems. Generally, one can extrapolate to the physical limit $n=d=3$ along
many trajectories, e.g., at fixed $n=3$ and expand in $\epsilon=4-d$, expand
in $\epsilon$ along $n=d=4-\epsilon$, at fixed $d=3$ and expand in $1/n$, etc.
Similar questions arise for $n=d=2+\epsilon$ and $n=d=1+\epsilon$. At the end,
all these expansions and extrapolations should approach the same limit at
$n=d=3$. It will be useful to compare all these with existing modern tools.
These scalar products also required angular integrations in continuous $d$
dimensions. This connection between space and spin also led me to generalize
the Wilson-Fisher paper (which had $n=2$ coupled Ising spins) to systems with
cubic symmetry, i.e., coupled $n$ component spins [15, 16], which raised
issues of competing fixed points (FPs), still being investigated today (see
below). After we published these two papers, Michael allowed me to proceed on
my own, taking advantage of Wilson’s course and including Wilson’s direct
diagrammatic way of calculation [11], antiferromagnets [12], anisotropies (in
which $n<d$) [13], $1/n-$expansions (in which $n$ was also treated as a
continuous variable) and more. Still at Cornell, I also wrote papers on
dipolar systems with Alastair Bruce [17]. Needless to say, all my papers from
Cornell acknowledge both Fisher and Wilson for their advice.

In particular, in Ref. [14] I was able to use the Fisher-Wilson approach to
show that the upper critical dimension for the uniaxial dipolar problem
($n=1$) is 3, with logarithmic corrections,[14] confirming the earlier Larkin-
Khmelnitzkii calculations (Larkin then visited Cornell, before he knew
English, and Khmelnitzkii later became a friend) and correcting the simple
Landau theory approach (without the logarithmic corrections) used earlier for
structural phase transitions (about which I learned from Alastair Bruce, who
was a student of Roger Cowley). Guenter Ahlers (whom I also met through
Michael, and later collaborated with [18]) later received at a Gordon
conference (Fig. 5 141414David Nelson reminds me that the conference was
announced in Science magazine as “solids and fruits”, to which Seb Doniach
proposed an improvement: “salads and fruits”.) a prize of 1/3 of a log (of
wood) for confirming these corrections experimentally [19]. Unfortunately,
Guenter was unable to contribute to this book. I return to dipolar systems
below, in Sec. 2.1.

Figure 5: Color version of the original black and white photo of the
participants in the Gordon conference on Dynamics of quantum solids and
fluids: cooperative phenomena in solids and fluids, July 1965. David Nelson,
Bob Birgeneau and AA so far identified Aharony, Ahlers, Als Nielsen, Bak,
Baker, Berker, Birgeneau, Betts, Bishop, Bruce, Chang, Ditzian (now Kadanoff),
Doniach, Emery, Fisher, Grest, Greytak, Griffiths, Hertz, Hohenberg, Imry,
Kadanoff, Kitchens, Kosterlitz, Krishnamurthy, Luther, Martin, Mandelbrot,
Nelson, Nickel, Passell, Pindak, Pynn, Rice, Rohrer, Rudnick, Siggia, Stephen,
van Leeuwen, Widom, Wolf and Wortis. Many of these are also mentioned in the
text. I leave it for the readers to connect names with pictures.

During my years at Cornell, many experimentalists were attracted to visit and
understand the implications of the RG on their work. I later enjoyed very much
collaborations with many of these experimentalists, including, e.g., Ahlers,
Birgeneau, Cowley, Garland, Litster, and Rohrer. One special example was the
visit of Alex Müller, who worked on structural phase transitions and on
mulicritical points in perovskites. This visit resulted in many collaborations
with Alex and with Alastair Bruce [20, 21, 23]. Fortunately, after a visit in
Tel Aviv Alex ignored my enthusiastic recommendation to stay with structural
transitions, and he switched to working on high temperature superconductivity,
for which he was awarded the Nobel prize. Similarly, Heini Rohrer ignored my
advice to stay in magnetism (we collaborated on the random field multicritical
antiferromagnets, see below), and went on to receive the Nobel prize for the
scanning tunneling microscope. Apparently, doing experiments on phase
transitions and talking to theorists are good preparations for Nobel prizes…
Multicritical points were also studied then by Kosterlitz, Nelson, and Fisher
[24], and later by Domany, Mukamel and Fisher [25]. Both of these papers
contain examples of Fisher’s special talent for drawing figures. Both Domany
and Mukamel later became friends and collaborators. I return to this topic
below, in Sec. 2.2.

In May 1973, less than a year after my arrival at Cornell, Mel Green and Jim
Gunton organized a meeting at Temple University, entitled “Renormalization
Group in Critical Phenomena and Quantum Field Theory”. Kindly, Fisher took me
along to that meeting, and gave me a chance to meet all the great people who
were working on that title. I talked on the dipolar systems, and met Paul
Martin (who hosted me at Harvard in the fall of 1974, where I learned about
random systems from Alan Luther and Geoff Grinstein), Shang-keng Ma (who
hosted me at UCSD in the spring of 1975, starting a long collaboration on
random systems), and Bert Halperin and Pierre Hohenberg (who hosted me at Bell
Labs in the summer of 1975, starting a long collaboration on universal
amplitude ratios and starting my long collaboration with Bob Birgeneau). I
also met some of the experimentalists mentioned above, and people like Franz
Wegner, Eberhard Riedel and Édouard Brézin. In the following years, all of
these people worked on the same topics as myself, we corresponded a lot, and
became good friends. The ‘RG community’ of those early years was stimulating
and friendly, and I have been grateful to Michael for helping me become part
of it. Another result of that meeting was that Green invited me to write Ref.
[16], joining a select group of the above list.151515That series of 20 books,
first edited by Cyril Domb (Michael’s mentor at King’s college) and Green and
then by Domb and Joel Lebowitz (so far leader of 124 Yeshiva$\rightarrow$
Rutgers meetings), contains many great reviews on phase transitions and
critical phenomena.

As I mentioned, Wilson also gave the same course at Princeton, where it was
attended by Édouard Brézin and David Wallace. The three of them then used the
RG to derive the universal scaling function of the equation of state [26].
(Brézin and I continued to read and write about each other’s papers for many
years). As soon as we saw that paper, Michael and I realized that similar
universal scaling function should also exist for the two-spin correlation
function. This resulted in the next two joint papers on the
$\epsilon-$expansion of this function [27, 28], and by my own paper on its
$1/n-$expansion [29]. The $\epsilon-$expansion demonstrated for me Michael’s
excellent mathematical insights. In Fourier space, this function depends on
the ‘momentum’ $q$, and the perturbative expansion (in the quartic spin terms)
only contributed to this $q-$dependence at second order, apparently yielding
contributions only at second order in $\epsilon$. However, the short range
correlations were known to have terms which behave like the energy [30],
involving, e.g., the specific heat exponent $\alpha$, which starts at order
$\epsilon$. Michael soon realized that the terms of first order in $\epsilon$
cancel out, so that our expansion of the correlation function fully agreed
with the known scaling forms, at both short and long distances. This work, and
Michael’s insights into the universality of scaling functions, later led me to
work on universal amplitude ratios [31]. In fact, these topics came up again
during Michael’s visits to Tel Aviv, in 1980 and 1983, resulting with two
papers [32, 33] on analytic and non-analytic corrections to
scaling.161616After this paper appeared on the arXiv, Jacques Perk kindly
reminded me that higher order series showed that these papers lacked higher
order log terms (which could not be seen by the short series we had at the
time). [34] In these papers we may have accomplished Michael’s first
assignment to me, back in 1972.

I cannot end this historical introduction without mentioning another
‘interaction’ with Michael. In 1989, he spoke at the Gibbs symposium, and
emphasized the Gibbs rule on the convexity of geometrical surfaces in
thermodynamic surfaces. The summary of that talk [35] ends with the following:
“The striking phase diagrams observed in doped crystals displaying
high-$T_{c}$ superconductivity [12] suggest the intriguing Gibbsian question:
What, if anything, replaces the convexity of the $U(S,V)$ surface when there
is a variable density of frozen or quenched random impurities?” Ref. [12]
refers to a paper [36], in which some of us tried to predict details of the
phase diagrams of the recently discovered high-temperature supercoductors by
Müller and Bednorz (mentioned above). That phase diagram was drawn only
schematically, and some lines apparently did not seem to obey the Gibbs rules.
Michael noticed that, and he would not miss any occasion to tell me that he
keeps watching me, following the biblical rule “spare the rod – spoil the
child” [37]. OK, Michael, I am proud to have been your scientific child!

###  2 Recent developments

The wave of econophysics, sociophysics and the like ‘pushed’ me away from the
‘modern’ statistical physics community. Therefore, my interests concentrated
on random systems, magnetism, quantum mesoscopics and spintronics. However,
many statistical physicists stayed within real physics, and I kept following
the literature on critical phenomena. I often returned to that field, and
wrote on topics which in many respects followed Fisher’s legacy and/or
addressed issues which remained open. Examples include Ref. [38], related to
Fisher’s interest in random systems [39, 35], and Ref. [40], related to
multicritical points [24].

Below I mention a few issues which I think are still open, even after 50
years. Celebrating the 50th anniversary of the $\epsilon-$expansion, it may be
appropriate to remember that this expansion treated the dimensionality of
space as an algebraic abstract variable, without any geometrical picture of a
real system with 3.99 dimensions. One attempt to find such systems came up
when I met Benoit Mandelbrot at Harvard in 1980. Mandelbrot’s fractals are
geometrical constructions with non-integer dimensions, which are not
translationally invariant but in which the ‘mass’ $M$ within a volume of
linear size $L$ scales as $M\sim L^{D}$, identifying $D$ as the (non-integer)
fractal dimensionality. It turned out that real space RG, in the original
spirit of Kadanoff, becomes exact on many fractals, [41], and the ease of such
calculations started a large wave of publications. However, it turned out that
the physics of fractals required much more than just the fractal
dimensionality,[42], and this path of research did not find a geometrical
description of the Wilson-Fisher non-integer $d$. However, fractal models have
turned out to be very useful to model non-translationally invariant physical
systems, e.g., percolation clusters. Hence my long personal involvement with
percolation [43].171717In fact, Michael worked on percolation clusters much
before I met him [44]. He also discussed fractal vesicles in a conference
which I organized with Jens Feder [45].

A major development in recent years was the entrance of field theorists into
the field of critical phenomena, many years after Wilson set the example (not
to mention my own background in high energy). They not only rediscovered many
of the physical systems studied in the 1970’ies, but they also developed and
applied many novel and very accurate methods to study them, including re-
summed high-order $\epsilon-$expansions, [46] fixed-dimensions RG calculations
[47], conformal bootstrap calculations [48], and accurate Monte Carlo
simulations [49]. (To save space, I give only recent references, exemplifying
the large current activity of field theorists on critical phenomena).
Specifically, these calculations identified unambiguously the stable FPs of
the RG, and gave consistent and accurate values for the critical exponents,
both at critical and at multicritical points. The availability of such tools
(and the ‘pushing’ of Slava Rychkov) led me recently back to the puzzle of the
bicritical points [50, 51, 52] discussed in more detail below.181818Assuming
that Michael would have liked this recent work, I tried to communicate it to
him, but it was too late. His E-mail address<EMAIL_ADDRESS>no longer
responded. (The name of this address reflected Michael’s attitude to E-mail,
but he did use it more after he retired. More interestingly, Michael kept
copies of every letter he ever sent or received. I am sure a historian of
science will find many jewels in that archive. I am told that UMD is keeping
it.) Below I also present selected other topics, in which Fisher was
interested already 50 years ago, and demonstrate that even half a century
later, the RG is still an active and exciting field.

####  1 The dipole-dipole interactions

Fifty years after my early papers on the dipolar systems, Refs. [9, 10, 11,
12, 13, 14], sometimes people ask me if it was my father that wrote them.
Well, it was me, and I am still alive and kicking. (The issue may become more
tricky in 30 years, when they will ask my son Ofer the same question…).
Occasionally, some of these results are rediscovered “independently”, and I
send E-mails to those rediscoverers.

In any case, every material with magnetic ions has dipole-dipole interactions,
and therefore it is always necessary to check their relevance to experimental
measurements. Originally, the $\epsilon-$expansion for the $n=d=4-\epsilon$
isotropic dipolar case [9, 10] seemed numerically close to the short-range
Heisenberg case ($2\nu_{dip}=1+9\epsilon/34+\dots$ versus
$2\nu_{iso}=1+\epsilon/4+\dots$). Similar closeness arose for $1<<d$. [13]
However, these numbers should be improved, and I still await extensions to
higher order expansions and other accurate high energy methods, like those
mentioned above. (As Slava Rychkov tells me [53], the dipolar model is not
conformal, hence bootstrap is not applicable. Other methods should still
work.)

As mentioned, the uniaxial Ising case is particularly interesting. The upper
critical dimension is $d_{u}=3$, and one could prove analytically that at this
dimension, e.g., the susceptibility diverges near criticality as
$\chi\sim|t|^{-1}|\log|t||^{1/3}$, where $t$ is the scaling field related to
the temperature distance from the critical point (this is the one third of a
log mentioned above), similar to the behavior of the short range Ising model
at $d=4$. However, the next correction, of order $\log|\log|t||/\log|t|$, has
different amplitudes in these two cases [54]. Note: this dimensionality shift
from $d$ up to $d+1$ dimensions differs from that of the random field shift
from $d$ to $d+2$ dimensions, discussed below [55]; the latter results from
supersymmetry [56], and therefore all the terms in the $\epsilon-$expansions
map from $d$ to $d+2$. Interestingly, unlike at $d<d^{dip}_{u}$, universal
amplitude ratios at $d=d^{dip}_{u}$ have logarithmic corrections, e.g. the
free energy per correlation volume [57]

$\displaystyle\xi^{2}\xi_{\parallel}Ct^{2}/k_{B}=(3/32\pi)|\log|t||,$ (1)

where $C$ is the singular specific heat, $\xi$ is the correlation length,
$\xi_{\parallel}=g_{d}^{1/2}\xi^{2}$, $g_{d}$ measures the strength of the
dipolar interactions and $k_{B}$ is the Boltzmann factor. This implies
logarithmic corrections to the hyperscaling relation, $d\nu=2-\alpha$, which
do not arise at $d<3$, where the free energy per correlation volume is a
universal temperature-independent constant. [31] 191919To show the close
community in those days, the acknowledgements in that paper include G. Ahlers,
J. Als-Nielsen, R. Birgeneau, P. C. Hohenberg, J. M. Kosterlitz, D. R. Nelson
and E. Siggia. Unfortunately the first 3 experimentalists were unable to
contribute to this volume.

The situation is even more interesting when randomness is included. In that
case the expansion in $\epsilon=3-d$ starts at order $\epsilon^{1/2}$, and the
logarithmic corrections for the susceptibility at $d=d_{c}=3$ are replaced by
$\chi\sim|t|^{-1}\exp[(D|\log|t||)^{1/2}]$, with the universal
$D=9/[8\log(4/3)+53]$, different form $D=6/53$ for the short range Ising model
in $d=4$ [58]. I found one reference that claims to have seen a crossover to
this behavior [59], but the evidence there is somewhat indirect.

My recent interest in the dipolar system arose when my colleague at Ben Gurion
University Moshe Schechter told me about his paper on LiHoxY1-xF4
[60].202020This seems to be a relative of LiTbF4, where Ahlers originally
measure the 1/3 log correction [19]. This is a dilute uniaxially anisotropic
quantum dipolar ferromagnet. At $x=1$, an external transverse field $B$ on
this non-random ferromagnet causes a decrease of the critical temperature from
$T_{c}(B=0)\approx 1.53$K all the way down to a zero temperature quantum
transition at a field of $B\approx 4.9$T. $T_{c}$ also decreases with
decreasing $x$ (the Y ion is not magnetic), approaching a spin glass phase at
low $x$. Shechter et al. review various numerical calculations of the
dependence of $T_{c}$ on both $B$ and $x$, including their own – which
emphasized the quantum fluctuations due to off-diagonal spin terms which come
from the dipolar interactions.

Since fluctuations are the “bread and butter” of the RG, I now propose to use
the RG for an alternative approach to this system. As I show, LiHoxY1-xF4
practically contains all the topics studied by RG during the recent decades:
crossover from short range interactions to isotropic dipolar interactions,
then to anisotropic (uniaxial) dipolar interactions, then to random (dilute)
uniaxial Ising behavior (both random exchange and random fields), and finally
adding quantum effects. The order of these crossovers may change, depending on
the relative strengths of the relevant parameters. Below I extend many of my
old papers on some of these topics, and suggest how one can derive the
critical temperature as function of the short/long interaction strengths, the
spin anisotropy, the dilution and the low temperature quantum effects. This
discussion is rather technical, and the unmotivated reader can skip it.

At first, I ignore quantum fluctuations. These can be added later, by adding
another dimension of imaginary times of a width that grows with decreasing
temperature [61, 62, 63, 64]. Classically, we start with the isotropic
Heisenberg short range system. Pure LiTbF4 then requires adding the isotropic
dipole-dipole interactions, with strength $g_{d}$, as treated in Ref. [10],
and the spin anisotropy, with strength $g_{a}$, as treated in Ref. [13]. We
start at $B=0$ and $x=0$. The classical (rescaled) Ginzburg-Landau-Wilson
Hamiltonian of the $n-$component spin model in momentum space is212121Here and
below we denote $\bar{\cal H}=-{\cal H}/(k_{B}T)-\sum_{\bf R}\big{[}|{\bf
S}_{\bf R}|^{2}/2+u[|{\bf S}_{\bf R}|^{4}\big{]}$. [16]

$\displaystyle\bar{\cal H}_{0}(\\{{\bf
S}\\})=-\sum_{\alpha\beta}\Big{[}\frac{1}{2}\int_{\bf
q}U^{\alpha\beta}_{2}({\bf q})S^{\alpha}_{\bf q}S^{\beta}_{-{\bf
q}}+u_{\alpha\beta}\int_{\bf q}\int_{\bf q^{\prime}}\int_{\bf
q^{\prime\prime}}S^{\alpha}_{\bf q}S^{\alpha}_{\bf q^{\prime}}S^{\beta}_{\bf
q^{\prime\prime}}S^{\beta}_{\bf-q-q^{\prime}-q^{\prime\prime}}\Big{]},$ (2)

where the integrals are over a spherical Brillouin zone of radius 1, and [10]

$\displaystyle U^{\alpha\beta}_{2}({\bf q})$
$\displaystyle=[r_{\alpha}+q^{2}]\delta_{\alpha\beta}+g_{d}q^{\alpha}q^{\beta}/q^{2},\
\ {\bf q}\neq 0,$ $\displaystyle=r_{\alpha}\delta_{\alpha\beta}+g_{d}{\cal
D}^{\alpha\beta},\ \ {\bf q}=0,$ (3)

with $r_{\alpha}=(T-T^{MF}_{\alpha})/T^{MF}_{\alpha}$, and ${\cal
D}^{\alpha\beta}$ being the (shape-dependent) demagnetization tensor. The
quartic coefficients $u_{\alpha\beta}$ will be chosen below (for the simplest
isotropic case, $u_{\alpha\beta}\equiv u$.) Our final aim is to find the shift
of the critical temperature $T_{c}$ from its mean field value,
$T^{MF}_{\alpha}$, for the highest ‘initial’ $T^{MF}_{\alpha}$, which we
assume arises for $\alpha=n=\parallel$. For simplicity we ignored in
$U^{\alpha\beta}_{2}$ the terms $q^{\alpha}q^{\beta}$ and $(q^{\alpha})^{2}$
[10, 65]. In pure LiHoF4 there is also a uniaxial anisotropy,
$g_{a}=r_{\perp}-r_{\parallel}$, so that

$\displaystyle r_{n}=r_{\parallel}=r-(1-1/n)g_{a},\ \
r_{\perp}=r_{1}=\dots=r_{n-1}=r+g_{a}/n.$ (4)

Adding a magnetic field will add more shifts in the $r$’s, see below.

Near the isotropic short range FP, the RG iterations for the three main
scaling fields give $t(\ell)=t(0)e^{\ell/\nu},\
g_{d}(\ell)=g_{d}(0)e^{(2-\eta)\ell},\ g_{a}(\ell)e^{\ell\phi/\nu}$. Here, $t$
is the true scaling field for the temperature, which vanishes at the critical
temperature $T_{c}$, and $\phi$ is the crossover exponent for the anisotropy.
[66] Eventually, for $g_{d}(\ell)$ and $g_{a}(\ell)$ sufficiently large, the
system will cross over to the uniaxial dipolar problem, treated in Ref. [14].
At that point, the scaling field $t(\ell)$, associated the with the single
critical temperature parameter $r_{\parallel}$, will be proportional to the
true $T-T_{c}$, and therefore it will yield the temperature shift
$T^{MF}_{c}-T_{c}$, as measured experimentally.

The $\epsilon-$expansion is based on iteratively integrating out of the
partition function the $S_{\alpha}({\bf q})$’s in the spherical shell
$1-\delta\ell<|{\bf q}|<1$ and then rescaling the spin and momenta so that the
momentum cutoff and the coefficient of $q^{2}$ return to 1. [8] At
$d=4-\epsilon$ the recursion relation for $r_{\alpha}$ is [66]

$\displaystyle
dr_{\alpha}/d\ell=2r_{\alpha}+12u_{\alpha\alpha}A^{\alpha\alpha}+4\sum_{\alpha\neq\beta}u_{\alpha\beta}A^{\beta\beta}+\dots,$
(5)

where the dots indicate higher orders in the $u$’s,

$\displaystyle A^{\alpha\beta}=\int^{>}G^{\alpha\beta}({\bf q})/(\delta\ell),$
(6)

the integrals are over a spherical shell in momentum space, with
$1-\delta\ell<|{\bf q}|<1$ and the matrix $G\equiv[U_{2}]^{-1}$ (for ${\bf
q}\neq 0$) is [13]

$\displaystyle G^{\alpha\beta}({\bf
q})=\frac{1}{r_{\alpha}+q^{2}}\Big{[}\delta_{\alpha\beta}-\frac{g_{d}}{q^{2}+g_{d}Q^{2}}\frac{q^{\alpha}q^{\beta}}{r_{\beta}+q^{2}}\Big{]},$
(7)

with $Q^{2}=\sum_{\gamma}(q^{\gamma})^{2}/[r_{\gamma}+q^{2}]$. For $g_{d}=0$
this reduces to $A^{\alpha\beta}=K_{d}\delta_{\alpha\beta}/(r_{\alpha}+1)$,
where $K_{d}$ is the area of a $d-$dimensional sphere. In the dipolar case
$G^{\alpha\alpha}$ depends on the direction of ${\bf q}$, and the integrals,
which depend on $g_{d}$, are more complex [10, 13, 14].

Since $g_{d}(\ell)=g_{d}(0)e^{(2-\eta)\ell}$ is exact [and we ignore the
$\ell-$depedence of $\eta=O(\epsilon^{2})$], we can substitute this
$\ell-$dependence into Eqs. (5) and find $r_{\parallel}(\ell)$ and
$r_{\perp}(\ell)$. Now we can follow the example of Nelson and Domany: [67]
iterate until $r_{\perp}(\ell_{1})=1$, obtain $r_{\parallel}(\ell_{1})$,
integrate the transverse spins out of the partition function and then proceed
with the recursion relations for the uniaxial case with only $S^{\parallel}$.
Note: the propagator at that stage may still have the form

$\displaystyle
G^{\parallel\parallel}=\frac{1}{r_{\parallel}+q^{2}}\Big{[}1-\frac{g_{d}}{q^{2}+g_{d}Q^{2}}\frac{(q^{\parallel})^{2}}{r_{\parallel}+q^{2}}\Big{]},$
(8)

with $Q^{2}=(q^{\parallel})^{2}/[r_{\parallel}+q^{2}]$. It will approach

$\displaystyle
G^{\alpha\beta}=\delta_{\alpha,n}\delta_{\beta,n}\frac{1}{r_{n}+q^{2}+g_{d}\cos^{2}\theta},$
(9)

where $\theta$ is the angle between the unit vector $\hat{\bf q}$ and the
$n-$axis, only when $g_{d}(\ell)\gg 1$. Although some calculations of the
crossover in the presence of both $g_{a}$ and $g_{d}$ was done previously [68,
69], I think the above prescription is more straight forward.

Adding a magnetic field $B$ (in energy units – including the factor
$g\mu_{B}$) on the spin component $S^{1}$, adds a term $--BS^{1}_{q=0}$ to
${\cal H}_{0}$. It is then convenient to replace $S^{1}_{\bf q}\Rightarrow
S^{1}_{\bf q}+M\delta({\bf q})$. This shifts the coefficients in
$U^{\alpha\beta}_{2}$ [17],

$\displaystyle
r_{1}\rightarrow\bar{r}_{1}=r_{\perp}+12uM^{2}=\bar{r}_{\perp}+8uM^{2},$
$\displaystyle r_{\perp}\rightarrow\bar{r}_{\perp}=r_{\perp}+4uM^{2},\ \ {\rm
applies~{}for}\ \beta=2,\dots,n-1,$ $\displaystyle
r_{n}\rightarrow\bar{r}_{\parallel}=r_{\parallel}+4uM^{2},$ (10)

where, for simplicity, we set $u_{\alpha\beta}\equiv u$. The value of $M$ is
determined by requiring that the new coefficient of $S^{1}$ vanishes [17],

$\displaystyle(r_{\perp}+g_{d}{\cal D}^{11}+4uM^{2})M=B,$ (11)

and at small $M$ we can use $M\approx B/(r_{\perp}+g_{d}{\cal D}^{11})$. In
addition, this shift also generates a cubic spin term,
$4uM\sum_{\alpha}S^{1}_{\bf q}S^{\alpha}_{\bf q^{\prime}}S^{\alpha}_{\bf-
q-q^{\prime}}$. [17]

Equations (10) imply

$\displaystyle\bar{r}_{1}>\bar{r}_{\perp}>\bar{r}_{\parallel},$ (12)

implying that $S^{n}=S^{\parallel}$ will be the main order parameter. As we
iterate the recursion relations (5), all the $\bar{r}$’s grow with $\ell$, but
$\bar{r}_{1}(\ell)$ will reach the value 1 first, at $\ell=\ell_{1}$. At that
point the correlations in $S^{\parallel}$ have been eliminated, and we can use
mean field theory to integrate $S^{\parallel}$ out of the partition function,
leaving us with an effective Hamiltonian in the remaining $n-1$ spin
components. We can next iterate until $\ell_{2}$, when
$\bar{r}_{\perp}(\ell_{2})=1$, eliminate the $(n-2)$ transverse components
with $\beta=2,\dots,n-1$ and remain with the single temperature variable
$\bar{r}_{\parallel}(\ell_{2})$. The scaling field for that variable finally
yields

$\displaystyle
T_{c}(B)=T^{MF}_{\parallel}[\bar{r}_{\perp}(\ell_{2})-\bar{r}_{\parallel}(\ell_{2})+8u(\ell_{2})M(\ell_{2})^{2}].$
(13)

The dilution is easily introduced via replicas [70, 71, 72, 58]. Each spin
component $S^{\alpha}$ is replaced by $m$ replicas, $S^{\alpha i},\
i=1,\dots,m$, and at the end one takes the limit $m\rightarrow 0$. In real
space, the Hamiltonian of a specific realization of the occupations,
$\\{p_{\bf R}\\}$, is

$\displaystyle{\cal H}(\\{p_{\bf R}\\},\\{{\bf S}_{\bf R}\\})$
$\displaystyle=-\Big{[}\frac{1}{2}\sum_{{\bf R\neq
R^{\prime}}}\sum_{\alpha\beta}^{n}p_{\bf R}p_{\bf
R^{\prime}}W^{\alpha\beta}({\bf R}-{\bf R^{\prime}})S^{\alpha}_{\bf
R}S^{\beta}_{\bf R^{\prime}}$ $\displaystyle+\sum_{{\bf R}}p_{\bf
R}\big{(}\sum_{\alpha}(r_{\alpha}-r)(S^{\alpha}_{\bf R})^{2}-BS^{1}_{\bf
R}+u|{\bf S}_{\bf R}|^{4}\big{)}\Big{]},$ (14)

where $W^{\alpha\beta}({\bf R}-{\bf R^{\prime}})$ contains all the
interactions and $p_{\bf R}=1$ or $0$ with probabilities $1-x$ or $x$. The
free energy for this realization is $F=-\log Z(\\{p_{\bf R}\\})$, where
$Z(\\{p_{\bf R}\\})={\rm Tr}_{\\{{\bf S}_{\bf R}\\}}\exp[-\beta{\cal
H}(\\{p_{\bf R}\\},\\{{\bf S}_{\bf R}\\})]$.

The quenched average free energy requires averaging of the free energy over
the distribution of the $p_{\bf R}$’s,

$\displaystyle-\beta\overline{F}=\overline{\log
Z}=\overline{\lim_{m\rightarrow 0}[Z^{m}-1]/m}=\overline{\lim_{m\rightarrow
0}[Z_{m}-1]/m},$ (15)

where the overline denotes averaging over the $p_{\bf R}$’s, and

$\displaystyle\overline{Z_{m}}=\overline{{\rm Tr}_{\\{{\bf S}_{\bf
R}\\}}\exp\Big{[}-\beta\sum^{m}_{i=1}{\cal H}(\\{p_{\bf R}\\},\\{{\bf
S}^{i}_{\bf R}\\})\Big{]}}.$ (16)

The exponential on the RHS contains $m$ replicas of the original Hamiltonian,
each with its own spins ${\bf S}^{i}_{\bf R}$’s.

The average of the free energy over the quenched random variables is achieved
using the identity [72, 71]

$\displaystyle\overline{e^{\bar{\cal H}}}=1+\overline{\bar{\cal
H}}+\overline{\bar{\cal H}^{2}}/2+\dots=e^{\bar{\cal H}_{ran}},$ (17)

where

$\displaystyle\bar{\cal H}_{ran}=\overline{\bar{\cal H}}+[\overline{\bar{{\cal
H}^{2}}]_{cum}/2+\dots},$ (18)

and ”cum” means the cumulant of the quenched average. Averaging over cumulants
of products of the $p_{\bf R}$’s without the magnetic field we end up with

$\displaystyle\bar{\cal H}_{ran}=-\sum_{i=1}^{m}\frac{1}{2}\int_{\bf
q}U^{\alpha\beta}_{2}({\bf q})S^{\alpha i}_{{\bf q}}S^{\beta i}_{-{\bf
q}}-\sum_{i,j=1}^{m}(v+u\delta_{ij})\int_{\bf q}\int_{\bf q^{\prime}}\int_{\bf
q^{\prime\prime}}S^{\alpha i}_{\bf q}S^{\alpha i}_{{,\bf q^{\prime}}}S^{\beta
j}_{{\bf q^{\prime\prime}}}S^{\beta j}_{{\bf-q-q^{\prime}-q^{\prime\prime}}},$
(19)

with

$\displaystyle T^{MF}_{\alpha}\rightarrow(1-x$
$\displaystyle)^{2}T^{MF}_{\alpha},\ \ \ g_{a}\rightarrow(1-x)g_{a},\ \ \
r_{d}\rightarrow(1-x)^{2}g_{d},$ $\displaystyle B\rightarrow(1-x)B,\ \ \
u\rightarrow(1-x)u$ (20)

and – to leading order in $x$, $v=x\Delta^{2}$, where $\Delta^{2}$ is the mean
square deviation of the ‘bare’ transition temperature. [72] One can then
follow Ref. [71], derive recursion relations for $u$ and $v$, send $m$ to zero
and follow the 3 steps that led to Eq. (13). Generally, I have shown that
$T_{c}$ is lowered by fluctuations in all three types of spins.

The term with the magnetic field adds a quadratic coupling between the
replicas,

$\displaystyle\overline{\cal H}_{RF}=x(1-x)B^{2}\sum_{\bf
R}\sum_{i,j}S^{1i}_{\bf R}S^{1j}_{\bf R}+\dots,$ (21)

where the dots indicate higher orders. This is the same term generated by the
random field Ising model, see Sec. 2.3, but for the non-ordering spin
component $S^{1}$. This will further lower $T_{c}$, but the details still need
detailed analysis. It should also be noted that, even without randomness, the
off-diagonal terms in ${\cal H}$, e.g. $\sum_{\bf R,R^{\prime}}S^{\alpha}_{\bf
R}S^{\beta}_{\bf R^{\prime}}$, also generate effective-field-like effects,
which may turn the problem onto the random field model (see Sec. 2.3). [73,
74, 75] As discussed in Sec. 2.3 below, this problem is still open, and may be
responsible for the loss of ferromagnetic order at low temperatures i the
preset case.

As stated, quantum fluctuations can be added for low temperatures by adding an
imaginary time axis. At sufficiently low temperatures, these fluctuations
imply mapping of $d$ to $d+1$, 222222This dimensionality shift is based on the
relation $\tau\sim\xi^{z}$, where $\tau$ and $\xi$ are the temporal and
spacial correlation lengths. Here we follow Ref. [62] and adopt $z=1$, as also
found for the quantum one dimensional Ising model in a transverse field [61].
The role of fluctuations will decrease for any $z>0$, e.g., $z=2$ for $n>1$.
thus decreasing all the effects discussed above. This may explain the
flattening of the curves $T_{c}(B,x)$ at low temperatures (observed
experimentally. [60])

So far, we only emphasized the shift in the transition temperature. After
eliminating all the fluctuations, he expected critical behavior should be that
of the uniaxial dipolar system in a transverse field, with (at finite
temperatures) logarithmic corrections to mean field theory. The simulations in
Ref. [60] and the experiments do not yet seem to have reached this limit, so
they probably observe effective exponents in the middle of the crossovers
towards it.

Unfortunately, the editors of this book imposed a deadline, preventing me from
finishing all these calculations. I invite everybody to do that, and – more
importantly – extend these calculations to higher orders in $\epsilon$ or
using other, more accurate methods to solve this RG project. Notably, this
material seems to involve all the outstanding issues in critical phenomena,
and it certainly deserves more research.

####  2 Multicritical points

As mentioned, the modern history of bicritical and tetracritical points, using
the RG, started with Ref. [24], which followed an earlier mean field paper by
Liu and Fisher [76]. The phase diagram of a system with two order parameters,
having $n_{1}$ and $n_{2}$ components, respectively, contains two phases, in
which these order parameters are (separately) non-zero. Fig. 6 exhibits three
possible phase diagrams for the special case of the XXZ antiferromagnet (a
Heisenberg $n=3$ antiferromagnet, with an anisotropy which prefers ordering
along the $Z-$axis.) In that example, the longitudinal magnetic field
$H_{\parallel}$ modifies the spin anisotropy, causing the transition from the
longitudinally ordered phase ($n_{1}=1$) to the ‘flopped’ phase ($n_{2}=2$).
Experimentally and numerically, these phases are often separated by a first-
order “flop” line, which ends at a bicritical point, Fig. 6(a). Alternatively,
the two phases can be separated via second order transition lines by a “mixed”
intermediate phase, and these lines meet the two transition lines between the
ordered phases and the disordered phase at a tetracritical point, Fig. 6(b).
[24, 20] A third possibility, which will be the main topic of the present
subsection, is shown in Fig. 6(c): the bicritical point of Fig. 6(a) is
replaced by a triple point, at which three first order lines meet [25]. In
this case, the transition from the disordered phase into the “mixed” phase on
the “flop” line, along the ‘isotropic’ line $g=0$, is also first order.

Before presenting the detailed RG arguments, here is a summary: At the
multicritical point, and along the line $g=0$ in Fig. 6, the symmetry between
the $n$ spin components is maintained. For a while, it was thought that this
point is described by the isotropic (rotationally invariant) Heisenberg fixed
point, implying a bicritical point. However, many modern accurate techniques
show that for $n=d=3$ this fixed point is slightly unstable, and the RG
trajectories flow slowly either to a biconical fixed point, implying a
tetracritical point, or towards a first order transition, implying a triple
point. Neither the experiments nor the Monte Carlo simulations show either of
these asymptotic predictions, hence the long-time puzzle. Below I present a
novel recent way to calculate the RG trajectories, expanding around the
isotropic fixed point. Since the experiments and simulations show an apparent
bicritical phase diagram, we conclude that they follow the second scenario,
but do not reach the triple point because of the slow flow, which requires a
very large correlation length. However, close to the latter limit the
effective exponents change significantly, and could be measured for checking
the theory.

The nature of this multicritical point in $d=3$ has been a matter of dispute
for many years. The critical properties at this point are determined by the RG
analysis along the line $g=0$ in Fig. 6. This analysis starts with a general
Ginzburg-Landau-Wilson Hamiltonian (in momentum space),

$\displaystyle\overline{\cal H}=-\Big{[}\frac{1}{2}\int_{\bf
q}\sum_{\alpha}\big{(}r_{\alpha}+q^{2}\big{)}S^{\alpha}_{\bf
q}S^{\alpha}_{-{\bf
q}}+\sum_{\alpha\beta\gamma\delta}u_{\alpha\beta\gamma\delta}\int_{\bf
q}\int_{\bf q^{\prime}}\int_{\bf q^{\prime\prime}}S^{\alpha}_{\bf
q}S^{\beta}_{\bf q^{\prime}}S^{\gamma}_{\bf q^{\prime\prime}}S^{\delta}_{\bf-
q-q^{\prime}-q^{\prime\prime}}\Big{]},$ (22)

where the integrals are over a spherical Brillouin zone of radius 1.
Generalizing Eq. (4) to the case of $n_{1}+n_{2}=n$ components, we have

$\displaystyle r_{\alpha}$ $\displaystyle=r_{\parallel}=r-(n_{2}/n)g,\ \ \
1\leq\alpha\leq n_{1};$ $\displaystyle r_{\alpha}$
$\displaystyle=r_{\perp}=r+(n_{1}/n)g,\ \ \ n_{1}+1\leq\alpha\leq n.$ (23)

The quadratic anisotropy $g$ causes the crossover from the multicritical point
to the $n_{1}-$ and $n_{2}-$component critical lines below and above it. [66].
From now on we set $g=0$.

All the previous work started at the Gaussian FP,
$u^{\ast,G}_{\alpha\beta\gamma\delta}=0$, and expanded the free energy in
powers of the quartic coefficients $\\{u\\}$. The recursion relation $\partial
u_{\alpha\beta\gamma\delta}/\partial\ell=\epsilon
u_{\alpha\beta\gamma\delta}-O(u^{2})$ then gave non-trivial FPs, with
$u=O(\epsilon)$. Depending on details, one always found four FPs: the Gaussian
one, the isotropic FP, at $u_{\alpha\beta\gamma\delta}\equiv
u^{\ast,I(n)}=8\pi^{2}\epsilon/(n+8)+O(\epsilon^{2})$, and the decoupled FP,
with $u_{\alpha\beta\gamma\delta}\equiv u^{\ast,I(n_{1})}$ for
$\alpha,\beta,\gamma,\delta\leq n_{1}$ and $u_{\alpha\beta\gamma\delta}\equiv
u^{\ast,I(n_{2})}$ for $\alpha,\beta,\gamma,\delta>n_{1}$. The fourth FP
depends on the details: Kostelitz, Nelson and Fisher [24] found the biconical
FP, and in the cubic case $u_{\alpha\alpha\beta\beta}\equiv
u+v\delta_{\alpha\beta}$ Ref. [15] found the cubic FP. Only one of the four
FP’s was stable (in the RG sense: at criticality ($T=T_{c}$) there is a region
in the $u-$space which flows under the RG iterations to that FP). The
isotropic $n-$component FP was always stable (in the critical space spanned by
the $u$’s) for $n$ below a threshold $n_{c}(d)$. Based on early low order
$\epsilon-$expansions [77], it was thought that $n_{c}(3)>3$, and therefore
the multicritical point is associated with the isotropic FP. However, as
mentioned, higher order expansions (up to $\epsilon^{6}$) and modern re-
summations of these divergent (but Borel summable) series, as well as Monte
Carlo simulations yielded $2.85<n_{c}(3)<3$. Therefore, the isotropic FP is
unstable at $d=3$: As $n$ crosses $n_{c}(d)$, an operator which was irrelevant
at the isotropic FP for $n<n_{c}(d)$ becomes relevant, and the RG trajectory
related to it flows away. If this flow goes to an alternative, stable FP
(biconical, cubic or decoupled), the transition at the multicritical point is
continuous, and is then identified as a tetracritical point (at these points
the minimum of the quartic Hamiltonian below the multicritical point has a
mixed state [76]). Below we return to what happens when the flow does not
reach a stable FP.

Figure 6:  Possible phase-diagrams for the XXZ antiferromagnet in a
longitudinal magnetic field. (a) Bicritical phase diagram. (b) Tetracritical
phase diagram. (c) Diagram with a triple point. Thick lines – first-order
transitions. Thin lines – second-order transitions. The first-order transition
lines between the ordered phases and the disordered paramagnetic phase end at
tricritical points (small empty circles). After Refs. [20, 25].

Until now, no experiments nor simulations observed any sign of a first order
transition at the multicritical point. If indeed the transition is second
order, it must be associated with one of the above stable FP’s, hence the
tetracritical scenario. This conclusion generated Ref. [40], commenting on a
Monte Carlo study [78] of Zhang’s $SO(5)$ model [79] for the high temperature
superconductors, which included the $n_{1}=3-$component antiferromagnet and
the $n_{2}=2-$component superconductor. In that case there exists an exact
proof that the stable FP is the decoupled FP, and my comment wondered why the
$SO(5)$ did not find that, and why they seemed to observe a bicritical point
with exponents associated with the isotropic FP. Obviously, the late Shoucheng
Zhang was not happy with that comment. However, I did end the comment saying:
“All the above statements assume that the initial Hamiltonian is in the region
of attraction of the decoupled FP. Alternatively, one should expect a first
order transition… The apparent experimental observation of isotropic exponents
may still indicate that the initial Hamiltonian is close to the isotropic FP.”
Indeed, the calculation below confirms these last comments: if one has a
finite sample of linear size $L$ then after a finite number of RG iterations
$\ell=\ln L$ the RG flow stays near the isotropic FP, and does not reach the
triple point.

In many earlier papers, when the RG trajectories flowed away from an unstable
point, without reaching any alternative stable FP, it has been generally
expected that the renormalized effective free energy, which can be solved
after all the fluctuations have been removed, has a first order transition. If
the transition at the multicritical point (along the line $g=0$ in Fig. 6)
becomes first order, then (by continuity) the two transition lines from the
disordered phase also become first order in its vicinity, and the
multicritical point then becomes a triple point [25]. These first order
regions then end at tricritical points, Fig. 6(c). However, the experiments
and simulations have not yet exhibited such triple points. Also, there did not
exist accurate estimates on how close to criticality should the triple point
arise. Hence the “puzzle of the bicritical points”, which I studied recently
with my long-time colleague Ora Entin-Wohlman [52], based on calculations with
Andrey Kudlis [50, 51, 80], who was a major player in the re-summed high order
$\epsilon-$expansions [46]. Since these references give full details of our
calculations, I give here only a brief summary.

Generally, the recursion relation near a FP for a parameter $G_{i}$ has the
form

$\displaystyle\partial
g_{i}/\partial\ell=\overline{\beta}_{i}[g_{1},~{}g_{2},\dots].$ (24)

In our case, the $\overline{\beta}-$functions are generated as sixth order
expansions in the $u^{\prime}s$, and people solved them for the FP’s. However,
we need more that just the FP’s. To follow the RG trajectories in the
$u-$space we need the actual variation of $\overline{\beta}_{i}$ on its
$\ell-$dependent arguments, and (like the $\epsilon-$expansion) the series in
these arguments do not converge! For $n=3$ we know that the isotropic and
biconical (or cubic) FP’s are close to each other, and therefore we decided to
construct recursion relations in the vicinity of the isotropic FP, rather than
near the Gaussian FP. Thus, we expanded $\overline{\beta}_{i}$ in powers of
$\delta g_{i}=g_{i}-g^{\ast,I(3)}_{i}$ around the isotropic FP. The
coefficients in these expansions are derivatives of $\overline{\beta}_{i}$
with respect to the various $g_{j}$’s, at the isotropic FP, which are known
series in the $g^{\ast,I(3)}_{i}$. The latter series can be written as series
in $\epsilon$, and can then be resummed to give good estimates of their
accurate values at $d=3$. Close to the isotropic FP it was sufficient to use
only second order expansions in the deviations $\delta g_{i}$’s; the resulting
values of the cubic or of the biconical FP’s were quite accurate.

We now turn to our main task: what happens if we start with a Hamiltonian out
of the region of attraction of the stable (biconical or cubic) FP? Group
theory [81, 82, 83, 49] shows that there exist only three independent
stability exponents near the isotropic $n-$component FP. The quartic terms in
Eq. (22) split into three subgroups of the rotationally invariant $O(n)$
group. For even functions in the spins, the quartic terms can always be
written as

$\displaystyle{\cal H}_{4}=(u^{\ast,I(n)}+p_{0}){\cal P}_{0}+p_{2}{\cal
P}_{2}+p_{4}{\cal P}_{4},$ (25)

where the $p_{i}$’s are scaling fields which flow under the RG iterations,
with (for $d=3$) stability exponents (agreed by all the extrapolations) [48]

$\displaystyle\lambda_{0}\approx-0.78,\ \ \lambda_{2}\approx-0.55,\ \
\lambda_{4}\approx 0.01.$ (26)

The operator ${\cal P}_{0}\equiv|{\bf S}|^{4}$ is the single fully isotropic
quartic term, which characterizes the flow to the isotropic FP. The other
operators break rotational symmetry. The operator ${\cal P}_{2}$ is a linear
combination of operators which ‘prefer’ ordering of one of the competing order
parameters, e.g., ${\cal P}_{2}=|{\bf S}|^{2}\big{(}n_{2}|{\bf
S}_{\parallel}|^{2}-n_{1}|{\bf S}_{\perp}|^{2}\big{)}$. For $n=3$ there are
five such operators. As we said, the multicritical point requires ‘isotropy’,
hence $g=p_{2}=0$. The remaining 9 operators form ${\cal P}_{4}$, which (with
no cubic terms, $v=0$) has the form

$\displaystyle{\cal P}_{4}\equiv\mid{\bf
S}\mid^{4}\Big{[}\frac{n_{1}n_{2}}{(n+2)(n+4)}+x(1-x)-\frac{n_{1}(1-x)+n_{2}x}{n+4}\Big{]},\
\ \ x=\frac{|{\bf S}_{\parallel}|^{2}}{|{\bf S}|^{2}}.$ (27)

It is easy to express the original $u$’s in terms of the three $p$’s, and thus
find the recursion relations for the $p$’s.

Since we set $p_{2}=0$, we are left with the coupled recursion relations for
$p_{0}$ and $p_{4}$. Since $|\lambda_{0}|$ is large,
$p_{0}(\ell)=p_{4}(0)e^{\lambda_{0}\ell}$ decays fast, and can be neglected
after $\ell_{1}$ iterations. Beyond that point, the quadratic recursion
relation for $p_{4}$ can be solved analytically, yielding [50, 80]

$\displaystyle
p_{4}(\ell)=\frac{p_{4}(\ell_{1})e^{\lambda_{4}(\ell-\ell_{1})}}{1+Bp_{4}(\ell_{1})\big{[}e^{\lambda_{4}(\ell-\ell_{1})}-1\big{]}/\lambda_{4}},$
(28)

where the biconical (or cubic) FP is at $p^{\ast,B}=\lambda_{4}/B>0$. Since
$\lambda_{4}>0$ is small, the initial flow is very slow, and the trajectory
stays near the isotropic FP for many iterations. This may explain why Hu’s
Monte Carlo simulations of Zhang’s $O(5)$ model gave an apparent bicritical
diagram with isotropic exponents. For $p_{4}(0)>0$, the trajectory goes to the
biconical (or cubic) FP, but then one should immediately see tetracritical
behavior.

The situation changes for $p_{4}(0)<0$. In that case the trajectory again
remains close to the isotropic FP, but at some point the denominator in (28)
vanishes, and $|p_{4}|$ diverges. Although our quadratic approximation in the
recursion relation may no longer be valid, it is clear that $p_{4}$ varies
strongly before this divergence. In Refs. [50, 51, 52, 80] we show that if the
iterations stop at $\ell_{2}$, when $r(\ell)\sim 1$, then we can use mean
field theory and identify when the transition becomes first order, ending us
with the triple point of Fig. 6(c).

Once we have the $\ell-$dependent $p$’s, we can also substitute them into
other recursion relations, e.g. for $r$ and $g$, and derive the exponents for
the associated scaling fields, $t(\ell)=e^{\ell/\nu(\ell)}t(0)$ and
$g(\ell)=e^{\ell\phi(\ell)/\nu(\ell)}g(0)$. Figure 7 shows these effective
exponents, for the case $n=1+2$ without cubic symmetry, and demonstrates their
fast variation as the bicritical point approaches the triple point.

All these calculations should be extended to higher orders in the $\delta
g_{i}$’s. In our papers we also suggest how one may change the parameters, in
order to approach the triple point and demonstrate Fig. 6(c) by experiments or
simulations.

Figure 7: Dependence of the effective critical exponents $\nu$ and $\phi$ on
the RG flow parameter $\ell$. Different lines correspond to different initial
values $p_{4}(0)$, for a fixed $p_{0}(0)$. In this plot, the trajectories are
shown only at $p_{4}(\ell)>-.8$, where our quadratic approximation is
reasonable. From Ref. [80].

####  3 Random fields Ising model (RFIM)

Quenched random fields exist in many physical systems [84, 85, 86].
Experiments are difficult, due to equilibration problems [87, 88, 89], but
still very desired. Note that some of these experiments study the drastic
effects of the random fields on bicritical phase diagrams (e.g., destroying XY
order below the random field lower critical dimension $d_{l}=4$). This model
was first solved for the $n\rightarrow\infty$ limit (random sources and sinks
of superfluid particles) by Lacour-Gayet and Toulouse [90], who found the new
upper and lower critical dimensions $d_{u}=6,~{}d_{l}=4$. Imry and Ma [91]
then gave qualitative arguments that showed that below a lower critical
dimension $d_{l}$, random magnetic fields cause a breakdown of the system into
domains with opposite spins, hence a destruction of the magnetic long range
order. The size $L$ of the domains is determined by the competition between
the gain in bulk energy ($\pm h_{0}L^{d/2}$, $h_{0}$ is the typical random
field) and the loss in boundary energy ($JL^{d-1}$ for $n=1$ or $JL^{d-2}$ for
$n>1$; $J$ is the exchange energy). They found $d_{l}=4$ for the rotationally
invariant models (with $n\geq 2$) and $d_{l}=2$ for the Ising case ($n=1$). In
both cases, the upper critical dimension was found to be $d_{u}=6$. As
experimental realizations, Imry and Ma suggested antiferromagnetic magnetic
impurities in ferromagnets, or lattice distortions. Following Ref. [84], many
experiments were done with uniform magnetic fields on dilute antiferromagnets.

At dimensions $d>d_{l}=2$, there exists an ordered ferromagnetic phase below
$T_{c}(B)$ for the dilute antiferromagnet in a field $B$, but the Imry-Ma
domains still exist above the transition, with typical size
$L\sim(h_{0}/J)^{2/(d-2)}$, representing metastable states which take a long
time to equilibrate (via avalanches of flopping domains) for large $h_{0}$.
This typical size decreases with $h_{0}$, eventually allowing long rage order.
Indeed, Ref. [92] identified a temperature $T_{i}(B)>T_{c}(B)$, below which
these domains cause slow dynamics. This may explain the difficulties of
experiments to equilibrate [93]. As far as I know, there is not yet a complete
consistent theory to describe this scenario.

Looking at the diagrammatic expansions in the quartic spin terms and in the
random fields, it was shown that each of the leading diagrams in the random
fields model in $d$ dimensions maps exactly onto a similar diagram for the
non-random model in $d-2$ dimensions [55, 94]. It was thus concluded that to
all orders, the $\epsilon-$expansion of the random field model in
$\epsilon=6-d$ is the same as for the non-random model with $\epsilon=4-d$
[55]. This dimensionality shift by 2 was later found to be an exact result of
supersymmetry [56]. Indeed, this dimensionality shift is consistent with the
Imry-Ma argument for $n\geq 2$, where the lower critical dimension without the
random fields is equal to $2$.

However, the rule $d\rightarrow d-2$ does not agree with the Ising case, where
Imry and Ma predicted $d_{l}=2\neq 1+2$ ($1$ is the lower critical dimension
for the Ising model without the random fields). Also, Imbrie [95] and
Brickmont and Kupiainen [96] showed that – unlike the non-random Ising model
at $d=1$ – there is long range order for the RFIM at $d=3~{}(=1+2)$. This
disagreement with dimensionality shift by 2 has been a source of much research
in the last 45 years, suggesting a variety of explanations. Some of these are
reviewed below, but may questions still remain open for future research.

In addition to the problem with the dimensionality shift, there were two other
(related) questions: (1) Do all the random field distributions belong to the
same universality class? and (2) Is the phase transition of the RFIM always
continuous? The literature studied several distributions of the site random
fields, e.g., the Gaussian, $P(h_{i})\sim\exp[-(h_{i}/h_{0})^{2}/2]$ and the
bimodal, $P(h_{i})\sim[\delta(h_{i}+h_{0})+\delta(h_{i}-h_{0})]/2$. Mean field
theory [97, 98] showed that the transition can be continuous at all
temperatures for a distribution with a maximum at $h_{i}=0$, like the Gaussian
one, but it must become of first order at low temperatures (and high random
fields) for distributions with a minimum at $h_{i}=0$, like the bimodal one.

To test the issue of the dimensionality shift for the RFIM, in 1993 we
returned to the old and good method of high temperature series [99]. As
reviewed in this book by Singh, Michael Fisher played a crucial role in the
development of this method (see also his list of scientific achievements at
the end of this book). He is also acknowledged in this paper.232323The other
acknowledgement was to Brooks Harris, with whom I have been fortunate to
publish many papers, both on series and on the RG. Series for the
susceptibility were developed for $d=2,3,..,8$. The results confirmed mean
field exponents for $d\geq 6$ (logarithmic corrections could not be seen), and
it was difficult to observe transitions at $d=2$ and $d=3$. It was emphasized
that when the random fields are switched on, one has a crossover from the non-
random to the RFIM behavior (without random exchange, the crossover exponent
is equal to the non-random susceptibility exponent $\gamma$ [100]. Random
exchange causes small corrections). At large random fields and low
temperatures the analysis is erratic, probably hinting at a first order
transition, and this leaves only a intermediate range of fields for extracting
the RFIM exponents. At $d=5$ the result was $\gamma\approx 1.18$, but it was
difficult to give error bars. This value is smaller than the $d=3=5-2$ non-
random $\gamma(d=3)\approx 1.241$, but the uncertainties led the authors to
say that dimensionality shift may hold at $d=5$. Results were not clear at
$d=3$.

To clarify this we produced much better series in 1996 [101]. After stating
that “series expansions have a great advantage over Monte Carlo simulations,
since (like the real experiments) they do not suffer from metastability and
equilibration problems” [see Michael Fisher’s comments on Monte Carlo
simulations in Domany’s paper, this book 242424However, Fisher withdrew that
“slip of the tongue” in 1999 [102].], Ref. [101] presented new high
temperature series for the RFIM in dimensions $d=3,4,5,8$. Results at $d=8$
gave mean-field exponents, as expected for $d_{u}=6$. Results at $d<6$
confirmed the two-exponent (and not three exponent, see below) scenario, but
the numerical values of the exponents and their error bars did not seem
consistent with the $d\rightarrow d-2$ rule. For example, in $d=4,~{}5$ we
found the susceptibility exponent for both field distributions in the ranges
$\gamma=1.45\pm.05,~{}1.13\pm.02$, distinct from the non-random values
$\gamma=1.75,~{}\sim 1.241$ for the non-random Ising model at $d=2,~{}3$. In
addition, the range of temperature for which the series were stable was
shorter for the bimodal distribution, which may have hinted at an approach to
a tricritical point and a first order transition at low temperatures
(presumably including $T=0$). We conclude that dimensionality shift indeed
breaks down somewhere near $d\sim 5$, and that the critical exponents seem to
come from another, supersymmetry breaking, fixed point.

Many theories tried to generalize the $d\rightarrow d-2$ rule in the
hyperscaling relations (e.g., $d\nu=2-\alpha$) by a new rule, $d\rightarrow
d-\theta$, breaking supersymmetry and the dimensionality shift. This procedure
introduced a new independent exponent $\theta$ (presumably related to the
dimensionality of the roughened domain boundaries [103]). These models thus
require three (instead of the usual two) independent critical exponents. Some
of these models, which involve zero temperature fixed points, were reviewed
e.g. in Ref. [101]. The non-random fixed point associated with the transition
as function of $T$ at $T_{c}(0)$, is strongly unstable against the random
fields, and the RG trajectories flow from that fixed point to the RFIM fixed
point, associated with the transition as function of $h_{0}$ at $T=0$, see RHS
of Fig. 8. Bray and Moore [104] use simple arguments to identify the recursion
relations near that fixed point, and argue that it is stable with respect to
the temperature. Therefore, thermal fluctuations are irrelevant, and it is
sufficient to find the ground state as function of the random fields at $T=0$.
Such a scenario would replace the supersymmetric fixed point at low
dimensions. However, it remained to find how the supersymmetric fixed point
turns into this new fixed point. Also, this scenario, which assumes a
continuous transition, cannot be valid for the binary field distributions,
which should yield a first order transition (as predicted by mean field
theory; fluctuations may narrow the range of such a transition, but not
eliminate it completely). As explained in [97], for the binary distributions
there must exist an ustable tricritical point on the critical line of the
RFIM, between the non-random fixed point at $T_{c}(h=0)$ and he one at $T=0$.
This would require a stable RFIM fixed point at a finite temperature between
the non-random fixed point and the tricritical fixed point, as plotted on the
LHS of Fig. 8. This phase diagram differs significantly from that of Bray and
Moore. at temperatures below the tricritical fixed point, the RG trajectories
would have to flow to a zero-temperature fixed point which represents a first
order transition [105], which is presumably different from the Bray-Moore RFIM
fixed point. The apparent universality found in both series and Monte Carlo
must imply that all distributions are described by this finite$-T$ RFIM fixed
point, and not by the zero temperature fixed point. Although some candidates
for this fixed point have been proposed (see below), the full details of such
an RG scenario have not yet been investigated.

Figure 8: Schematic phase diagrams in the temperature-random field $T-h_{0}$
plane for $d=3$. Right: The random field fixed point is at zero temperature.
Left: conjectured diagram for the binary and similar field distribution.
P=paramagnetic (but with Imry-Ma domains), F=ferromagnetic (or any other
ordered Ising phase). NR=non-random, RF=random field, TCP=tricritical point.
$h^{2}_{0}$ is the mean square average of the field distribution. The blue
arrows indicate the renormalization flow at criticality.

Some of the above questions were already addressed in 1997 by Swift et al.
[106]. They used a finite size scaling analysis of the exact ground states of
small RFIM samples in three and four dimensions, and found a difference
between the Gaussian and the bimodal field distributions in four dimensions,
with the latter apparently having a discontinuous jump in the magnetization.
The two distributions generated very different distributions of the
magnetization. Although their analysis was restricted to zero temperature,
they did not rule out a possible RFIM fixed point at a finite temperature. A
similar study at $T=0$ by Sourlas [107], at $d=3$, indicated that the Gaussian
and the bimodal distributions do not belong to the same universality class. He
noticed important differences between individual realizations (which showed
discontinuities) and their averages, but still concluded that all the
distributions give continuous transitions.

The last point raises the important issue of self-averaging. Away from
criticality, the central limit theorem predicts that the width of the
distribution of ay measured quantity decays with system size $L$ as
$(\xi/L)^{d}$, where $\xi$ is the correlation length. However, near a random
(exchange or field) fixed point this width approaches a constant [108]. In
particular, for the RFIM fixed point near six dimensions this width is
proportional to $(h^{2}_{0}u)^{\ast}$ at the random field fixed point, and
independent of $L$, where $h^{2}_{0}$ is the second cumulant of the field
distribution and $u$ is the coefficient of the non-random quadratic spin term
[108]. Since for large $L$ this width becomes finite and independent of $L$,
simulations must be very careful with averages over small numbers of
realizations. Except for a few examples (below), I am not aware of systematic
studies of self-averaging for the general RFIM problem. Such studies might
explain the different results found by different groups.

One exception is the 2002 work of Parisi and Sourlas [109]. They studied the
correlation functions of the three dimensional RFIM, and found that the
correlation length (at finite $L$) is not self-averaging. They blamed this
result on the nonperturbative formation of a bound state in the underlying
replicated field theory. This nonperturbative result may replace the
supersymmetric fixed point at low dimensions, but it is not clear what are the
predictions for the critical behavior.

Assuming the Bray-Moore scenario, of a continuous transition at all $T$, Ref.
[110] performed Monte Carlo simulations at zero temperature. Apparently, such
simulations avoid some of the difficulties in previous simulations. At $d>5$
these simulations agreed with the supersymmetry predictions. However, for
$d<5$ they found deviations from supersymmetry. Like our series results, these
authors also found the same exponents from both distributions. However, they
found the zero temperature transition continuous for all the distributions –
in contrast to the mean field theory for the bimodal distributions. As stated,
such distributions require more research. A possible way out may come from the
effects of random quenched fluctuations on first order transitions. Indeed,
Imry and Wortis [112] presented a domain argument similar to that of Imry and
Ma, to show that randomness (e.g., random interactions or random fields) can
turn a first order transition into a continuous one (the system breaks into
domains of the two competing phases). This argument was supported by various
other arguments [113, 114]. However, all of these arguments still required
some finite temperature below which the argument fails, and there should still
be a first order transition at low temperature. Hopefully, the above story
will stimulate theoreticians (and experimentalists!) to look into this. These
authors also checked self-averaging [111], and found that the widths of the
susceptibility distribution approaches finite $L-$idependent values, as
predicted in Ref. [108]. However, they find different values for different
distributions, which may indicate different universality classes.

The absence of self-averaging implied that the average description may differ
significantly from the behavior of a single realization. Interestingly, Kumar
et al. [115] simulated the zero temperature configurations of RFIM with a
uniform ordering magnetic field at $d=3$, for specific field realizations. The
domain shapes seemed to depend on the field distributions. However, they
measured the dependence of the magnetization $M$ on the uniform field $H$, and
identified the universal (distribution-independent) critical exponents $\beta$
and $\delta$ [$M\sim(T-T_{c})^{\beta}$ at $H=0$, $M\sim H^{1/\delta}$ at
$T=T_{c}$], finding $\beta=.025\pm.005$ and $\delta=72\pm 4$. Remembering that
a first order transition would have $\beta=0,~{}\delta=\infty$, I wonder
whether this allows for the phase diagram on the LHS of Fig. 8, alas for
specific realizations [116]. It is not clear yet how to relate such
realization with the averages used in all the theories.

Apart from the above numerics, there have also been some analytic
calculations, within the field theory ad the Wilsoin-Fisher renormalization
group approach. In 1998, Brézin and De Dominicis [117] noted that the
breakdown of supersymmetry should arise due to additional operators, which
break supersymmetry, to the Parisi-Sourlas Lagrangian [118]. If the
supersymmetric fixed point becomes unstable with respect to such operators,
this would explain the breaking of supersymmetry. However, they found a
singular dependence on the replica number $n$, ending up with a runaway from
the supersymmetric fixed point, without any other, stable, fixed point.

In 2004, Tarjus and Tissier [119] started a long series of papers, in which
they reformulated the functional nonperturbative RG in a superfield formalism,
generalizing the Parisi-Sourlas supersymmetric theory. They concluded that the
failure of dimensionality reduction and standard perturbation theory below
$d\simeq 5.1$ is due to the nonanalytic nature of the zero-temperature ‘cuspy’
fixed point, which is associated with the existence of many metastable states
and the avalanches between them. The cusps appear in the cumulants of the
renormalized field distributions. Near the zero temperature fixed point, the
temperature is a dangerous irrelevant field, related to the appearace of
droplets, which cause rounding of the cusps.

Another exciting result came from a recent theoretical study, which combined
nonperturbative conformal field theory with perturbative RG [120]. These
authors also added interactions which break supersymmetry, but are irrelevant
near the supersymmetic fixed point near $d=6$. Examples include quartic spin
terms, which may become relevat below $d=4$. They found that two such
operators become relevant below $d_{c}\approx 4.2-4.7$, indicating a crossover
to a new, stable, fixed point, at which supersymmetry is broken. As these
authors say, this situation is similar to that discussed in the previous
subsection, where the $n=3$ isotropic fixed point becomes stable at a
dimension slightly below 3, replacing the biconical (or cubic) fixed point
(which are stable at higher dimensions). Since this calculation uses the
expansion near $d=6$ down to the vicinity of $d=4$, it still requires support
from other approximations. In any case, how this new fixed point develops at
lower dimensions, how it relates to the phase diagrams in Fig. 8, and does it
disappear at the Imry-Ma $d_{l}=2$, remain open questions.

In summary, here are some of the remaining open questions:

(a) Which of all the suggested fixed points replaces the supersymmetric fixed
point, and breaks supersymmetry? Are essential singularities important in this
context?

(b) Is that fixed point at zero or at a finite temperature?

(c) If the LHS of Fig. 8 is correct, what are the details of the tricritical
point and of the first-order fixed point?

(d) How does that fixed point relate to the Imry-Ma scenario?

(e) Is that fixed point universal, for all field distributions?

(f) Does that fixed point confirm the breakdown of self-averaging? If so, how
does it relate to observations for single field realizations?

(g) How can the slow dynamics and avalanches of domain flipping be
incorporated into the renormalization group treatment?

(h) All the above was for symmetric field distributions, $P(h)=P(-h)$. There
were claims that non-symmetric distributions, e.g., for a binary mixture in a
porous media, may yield a different behavior, [121] but this issue deserves
further research.

Surely, more work will discover may more questions. There is much more to do
on the RFIM.

###  3 Conclusions

I hope that my story has convinced you that Michael E. Fisher was a great
scientist, a great leader of the statistical physics community, and a great
mentor and teacher. I owe him a lot.

I also hope that I convinced you that both the life of Michael Fisher and the
golden jubilee of the $\epsilon-$expansion deserve a celebration. Although the
life of Fisher ended, the $\epsilon-$expansion and other RG methods are still
very active, as I hope I demonstrated by my few examples. There is much more
to do!

Acknowledgements:

I thank Ora Entin-Wohlman and Andrey Kudlis for collaboration, and Slava
Rychkov and Nikos Fytas for very useful discussions. Eytan Domany and David
Nelson kindly helped checking the history and identifying people in Fig. 2,
Jacques Perk caught many misprints and added some references, and Leo
Radzihovsky corrected some crucial errors in an earlier version. I apologize
to the many people whose references were not included, only due to space and
time constraints.

## References

  * [1] Current problems in statistical mechanics, ed. by E. Domany and D. Jasnow, Physica A 177, Issues 1-3 (1991).
  * [2] J. Stat. Phys. 110, Special issue in honor of Michael E. Fisher’s 70th birthday, (edited by Joel Lebowitz), Numbers 3-6 (March 2003).
  * [3] 106th Rutgers Statistical Physics Conference, Dec. 2011: https://cmsr.rutgers.edu/docman-lister/cmsr/events-cmsr/ statistical-mechanics-conference/106/1070-smm106-finalprogram/file
  * [4] Critical Phenomena, Proc. Stellenbosch, S. A. 1982, F. J. W. Hahne, ed., (Springer, Berlin, 1983).
  * [5] Michael E. Fisher, Condensed Matter Physics: Does Quantum Mechanics Matter?, Excursions in the Land of Statistical Physics, pp. 207-253 (World Scientific 2016). Reprinted from “Niels Bohr: Physics and the World,” Eds. H. Feshbach, T. Matsui and A. Oleson (Harwood Academic Publisher, Chur, 1988) pp. 65–115.
  * [6] Kenneth G. Wilson and Michael E. Fisher, Critical Exponents in 3.99 Dimensions, Phys. Rev. Lett. 28, 240 (1972).
  * [7] Michael E. Fisher, Renormalization group theory: Its basis and formulation in statistical physics, Rev. Mod. Phys. 70, 653 (1998). See Also, Michael E Fisher, Renormalization group theory, the epsilon expansion and Ken Wilson as I knew him, Int. Jn Mod. Phys. B 29, 1530006 (2015). Selected reviews by Fisher appeared in Excursions in the land of statistical physics (World Scientific, 2017); see also Cyril Domb’s paper there.
  * [8] Kenneth G. Wilson and J. Kogut, The renormalization group and the $\epsilon$ expansion, Phys. Repts. 12, 75 (1974).
  * [9] M. E. Fisher and A. Aharony, Dipolar interactions at ferromagnetic critical points, Phys. Rev. Lett. 30, 559 (1973).
  * [10] A. Aharony and M. E. Fisher, Critical behavior of magnets with dipolar interactions, I. Renormalization group near four dimensions, Phys. Rev. B 8, 3323 (1973).
  * [11] A. Aharony, Critical behavior of magnets with dipolar interactions, II. Feynman graph expansion for ferromagnets near 4 dimensions, Phys. Rev. B 8, 3342 (1973).
  * [12] A. Aharony, Critical behavior of magnets with dipolar interactions, III. Antiferromagnets, Phys. Rev. B 8, 3349(1973).
  * [13] A. Aharony, Critical behavior of magnets with dipolar interactions, IV. Anisotropy, Phys. Rev. B 8, 3358 (1973).
  * [14] A. Aharony, Critical behavior of magnets with dipolar interactions, V. Uniaxial magnets in d dimensions, Phys. Rev. B 8, 3363 (1973). Erratum Phys Rev. B 9, 3946 (1974).
  * [15] A. Aharony, Critical behavior of anisotropic cubic systems, Phys. Rev. 8, 4270 (1973).
  * [16] A. Aharony, Dependence of universal critical behavior on symmetry and range of interaction, in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds., Vol. 6 (Academic Press, NY, 1976), pp. 357-424.
  * [17] E. g., A. Aharony and A. D. Bruce, Equation of state and scaling relations for isotropic ferromagnets with dipolar interactions, Phys. Rev. B 10, 2973 (1974).
  * [18] A. Aharony and G. Ahlers, Universal ratios among corrections to scaling amplitudes and effective critical exponents, Phys. Rev. Lett. 44, 782 (1980).
  * [19] Guenter Ahlers, Avinoam Kornblit, and H. J. Guggenheim, Logarithmic Corrections to the Landau Specific Heat near the Curie Temperature of the Dipolar Ising Ferromagnet LiTbF4, Phys. Rev. Lett. 34, 1227 (1975).
  * [20] E.g., A. D. Bruce and A. Aharony, Coupled order parameters, symmetry breaking irrelevant scaling fields and tetracritical points, Phys. Rev. B 11, 478 (1975).
  * [21] A. Aharony, K. A. Müller and W. Berlinger, Trigonal-to-tetragonal transition in stressed SrTiO3 : A realization of the 3-state Potts model, Phys. Rev. Lett. 38, 33 (1977). This paper was directly triggered by Ref. [22].
  * [22] David Mukamel, Michael E. Fisher, and Eytan Domany, Magnetization of Cubic Ferromagnets and the Three-Component Potts Model, Phys. Rev. Lett. 37, 565 (1976).
  * [23] The following 2 papers, describing theory and experiments, appeared back to back: A. Aharony and A. D. Bruce, Lifshitz point, critical and tricritical behavior in anisotropically-stressed Perovskites, Phys. Rev. Lett. 42, 462 (1979); J. Y. Buzaré, J. C. Fayet, W. Berlinger, and K. A. Müller, Tricritical Behavior in Uniaxially Stressed RbCaF3, Phys. Rev. Lett. 42, 465 (1979).
  * [24] J. M. Kosterlitz, D. R. Nelson and M. E. Fisher, Bicritical and tetracritical points in anisotropic antiferromagnetic systems, Phys. Rev. B 13, 412 (1976).
  * [25] E. Domany, D. Mukamel, and M. E. Fisher, Destruction of first-order transitions by symmetry-breaking fields, Phys. Rev. B 15, 5432 (1977).
  * [26] E. Brézin, D. J. Wallace, and Kenneth G. Wilson, Feynman-Graph Expansion for the Equation of State near the Critical Point (Ising-like Case), Phys. Rev. Lett. 29, 591 (1972).
  * [27] M. E. Fisher and A. Aharony, Scaling function for critical scattering, Phys. Rev. Lett. 31, 1238 (1973).
  * [28] M. E. Fisher and A. Aharony, Scaling function for two point correlations, I. Epsilon expansion near 4 dimensions, Phys. Rev. B 10, 2818 (1974).
  * [29] A. Aharony, Scaling function for two-point correlations. II. Expansion to order $1/n$, Phys. Rev. B 10, 2834 (1974). Interestingly, these results were rediscovered independently by Andrey Katanin, in 2021! After our correspondence, which followed the ArXiv version of his paper, he found a misprint in my paper, and the corrected results appeared: A. Katanin, Nonanalytic momentum dependence of spin susceptibility for Heisenberg magnets in the paramagnetic phase and its effect on critical exponents, Phys. Rev. B 103, 054415 (2021).
  * [30] M. E. Fisher and J. S. Langer, Resistive anomalies at magnetic critical points, Phys. Rev. Lett. 20, 665 (1968).
  * [31] V. Privman, P. C. Hohenberg, and A. Aharony, Universal critical-point amplitude relations, in Phase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, eds., Vol. 14 (Academic, NY, 1991), pp. 1-134, 364-367.
  * [32] A. Aharony and M. E. Fisher, Universality in analytic corrections to scaling for planar Ising models, Phys. Rev. Lett. 45, 679 (1980).
  * [33] A. Aharony and M. E. Fisher, Nonlinear scaling fields and corrections to scaling near criticality, Phys. Rev. B 27, 4394 (1983).
  * [34] Y. Chan, A.J. Guttmann, B.G. Nickel and J.H.H. Perk, The Ising Susceptibility Scaling Function, J. Stat. Phys. 145 549 (2011) and references therein.
  * [35] Michael E. Fisher, Phases and Phase Diagrams – Gibbs’ Legacy Today, Physica A 163, 15 (1990).
  * [36] A. Aharony, R. J. Birgeneau, A. Coniglio, M. A. Kastner, and H. E. Stanley, Magnetic phases and magnetic pairing in doped La2CuO4, Phys. Rev. Lett. 60, 1330 (1988).
  * [37] Proverbs 13, 20.
  * [38] A. Aharony and A. B. Harris, Absence of self-averaging and universal fluctuations in random systems near critical points, Phys. Rev. Lett. 77, 3700 (1996).
  * [39] P. B. Weichman and M.E.Fisher, Helium in Vycor, constrained randomness and the Harris criterion, Phys. Rev. B 34, 7652 (1986); R .R. P. Singh and M. E.Fisher, Random coupling crossover in Ising ferromagnets, Phys. Rev. B 37, 1980 (1988); R. P. Singh and Michael E. Fisher, Disordered systems which escape the bound $\nu\geq 2/d$, Phys. Rev. Lett. 60, 548 (1988).
  * [40] A. Aharony, Comment on “Bicritical and tetracritical phenomena and scaling properties of the SO(5) theory”, Phys. Rev. Lett. 88, 059703 (2002); A. Aharony, Old and new results on multicritical points, in Ref. [2], dedicated to Fisher’s 70th birthday, p. 659.
  * [41] Yuval Gefen, Benoit B. Mandelbrot and Amnon Aharony, Critical Phenomena on Fractal Lattices, Phys. Rev. Lett. 45, 855 (1980).
  * [42] Yuval Gefen, Amnon Aharony, Benoit B. Mandelbrot and Scott Kirkpatrick, Solvable Fractal Family, and Its Possible Relation to the Backbone at Percolation, Phys. Rev. Lett. 47, 1771 (1981).
  * [43] D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor and Francis, London (1992); revised 2nd edition (1994).252525 This book was written while Saddam Hussain sent missiles from Iraq to Israel. The universities were closed, and I had time to work in the bomb shelter. However, no journal editor allowed to acknowledge Hussain for helping my research – so here it is. At that time Michael wrote to me about his experiences during the German Blitz bombing of London, and invited me and my family to come and stay in safe Maryland. We did not come, and stayed alive, but remain grateful for the kind intention.
  * [44] M. E. Fisher, Critical probabilities for cluster size and percolation problems, J Math. Phys. 2, 620 (1960).
  * [45] M. E Fisher, Fractal and nonfractal shapes in two-dimensional vesicles, Physica D 38, 112 (1989); Proc. Conf. “Fractals in Physics,” 1-4 October 1989, Vence, France, Eds. A. Aharony and J. Feder (North Holland Publ. Co., 1989).
  * [46] E.g. L. T. Adzhemyan, E. V. Ivanova, M. V. Kompaniets, A. Kudlis, and A. I. Sokolov, Six-loop $\epsilon$ expansion study of three-dimensional $n-$vector model with cubic anisotropy, Nucl. Phys. B 940, 332 (2019) and references therein. See also chapter by Adzhemyan et al. in this book.
  * [47] E.g., J. M. Carmona, A. Pelissato, and E. Vicari, $N-$component Ginzburg-Landau Hamiltonians with cubic anisotropy: A six-loop study, Phys. Rev. B 61, 15136 (2000) and references therein.
  * [48] E.g., D. Poland, S. Rychkov, and A. Vichi,The conformal bootstrap: Theory, numerical techniques, and applications, Rev. Mod. Phys. 91, 015002 (2019); S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Bootstrapping Heisenberg magnets and their cubic anisotropy, Phys. Rev. D 104, 105013 (2021) and references therein. See also chapter by Poland and Simmons-Duffin in this book.
  * [49] E.g., M. Hasenbusch and E. Vicari, Anisotropic perturbations in three-dimensional $O(N)-$symmetric vector models, Phys. Rev. B 84, 125136 (2011); Martin Hasenbusch, Cubic fixed point in three dimensions: Monte Carlo simulations of the $\varphi^{4}$ model on the simple cubic lattice, Phys. Rev. B 107, 024409 (2023) and references therein.
  * [50] A. Aharony, O. Entin-Wohlman and A. Kudlis, Different critical behaviors in perovskites with a structural phase transition from cubic-to-trigonal and cubic-to-tetragonal symmetry, Phys. Rev. B 105, 104101 (2022).
  * [51] A. Aharony, O. Entin-Wohlman and A. Kudlis, Bi- and tetracritical phase diagrams in three dimensions, Low Temperature Physics 48, 483 (2022) [Fiz. Niz. Temp. (Kharkhov) 48, 542 (2022)].
  * [52] A. Aharony and O. Entin-Wohlman, The puzzle of bicriticality in the XXZ antiferromagnet, Phys. Rev. B 106, 094424 (2022).
  * [53] Slava Rychkov, Numerical conformal bootstrap: targets, Talk at the Simons Collaboration on the Nonperturbative Bootstrap Annual Meeting, Simons Foundation, New York, November 8-9 (2018). Recording is available: https://www.simonsfoundation.org/event/simons-collaboration-on-the-nonperturbative-bootstrap-annual-meeting-2018/ Also, Aleix Gimenez-Grau, Yu Nakayama, Slava Rychkov, unpublished.
  * [54] E. Brézin and J. Zinn-Justin, Critical behavior of uniaxial systems with strong dipolar interactions, Phys. Rev. B 13, 251 (1976).
  * [55] Amnon Aharony, Yoseph Imry, and Shang-keng Ma, Lowering of Dimensionality in Phase Transitions with Random Fields, Phys. Rev. Lett. 37, 1364 (1976).
  * [56] G. Parisi and N. Sourlas, Random Magnetic Fields, Supersymmetry, and Negative Dimensions, Phys. Rev. Lett. 43, 744 (1979).
  * [57] A. Aharony and B. I. Halperin, Exact relations among amplitudes at critical points of marginal dimensionality, Phys. Rev. Lett. 35, 1308 (1975).
  * [58] A. Aharony, New singularities in the critical behavior of random Ising models at marginal dimensionalities, Phys. Rev. B 13, 2092 (1976).
  * [59] A. Ferdinand, A.-C. Probst, A. Michels, R. Birringer and S. N. Kaul, Critical behaviour of nanocrystalline gadolinium: evidence for random uniaxial dipolar universality class, J. Phys.: Condens. Matter 26, 056003 (2014).
  * [60] Tomer Dollberg, Juan Carlos Andresen, and Moshe Schechter, Effect of intrinsic quantum fluctuations on the phase diagram of anisotropic dipolar magnets, Phys. Rev. B 105, L180413 (2022).
  * [61] R. J. Elliott, P. Pfeuty and C. Wood, Ising model with a transverse field, Phys. Rev. Lett. 25, 443 (1970).
  * [62] M. Suzuki, Relationship between $d-$dimensioal quantal spin systems and $(d+1)-$dimensional Ising systems, Prog. Theor. Phys. 56, 1454 (1976).
  * [63] Subir Sachdev, Quantum Phase Transitions, Cambridge University Press (2011).
  * [64] E.g., David Andelman and Amnon Aharony, Critical behavior with axially correlated random bonds, Phys. Rev. B 31, 4305 (1985) and references therein.
  * [65] Cubic symmetry is reflected by terms like $hq^{2}_{\alpha}\delta_{\alpha\beta}$ in Eq. (3). [10] In parallel to our work, Thomas Nattermann sat isolated in East Germany, and impressively found indepedently many of our dipolar results (produced in the supporting Cornell atmosphere). In particular, he concentrated on these cubic terms: e.g., T. Nattermann and S. Trimper, Critical behaviour and cubic anisotropy, J. Phys. A: Math. Gen., 8, 2000 (1975). Here I ignore these terms for simplicity. They (as well as the terms with $q^{\alpha}q^{\beta}$ also seem to be irrelevant, but only at order $\epsilon^{2}$. In those years Nattermann and I corresponded a lot by snail mail (this was before the electronic mail!), and met many times after he escaped from East Germany and joined the university of Köln in West Germany.
  * [66] M. E. Fisher and P. Pfeuty, Critical Behavior of the Anisotropic $n-$Vector Model, Phys. Rev. B 6, 1889 (1972).
  * [67] David R. Nelson and Eytan Domany, Equations of state for bicritical points. I. Calculations in the disordered phase, Phys. Rev. B 13, 236 (1976).
  * [68] K. Reid, Y. Millev, M. Fähnle and H. Kronmüller, Phase transitions in ferromagnets with dipolar interactions and uniaxial anisotropy, Phys. Rev. B 51, 15229 (1995).
  * [69] S.N. Kaul, Critical Behaviour of Heisenberg Ferromagnets with Dipolar Interactions and Uniaxial Anisotropy, Lect. Notes Phys. 678, 11 (2005).
  * [70] S. Grinstein and A. Luther, Application of the reormalization group to phase transitions in disordered systems, Phys. Rev. B 13, 1329 (1976).
  * [71] A. Aharony, Critical properties of random and constrained dipolar magnets, Phys. Rev. B 12, 1049 (1975).
  * [72] A. Aharony, Y. Imry and S. Ma, Comments on the critical behavior of random systems, Phys. Rev. B 13, 466 (1976). This reference averaged over the local transition temperature, while here we average over the site occupation.
  * [73] A. Aharony, Absence of ferromagnetic long range order in random isotropic dipolar magnets and in similar systems, Solid State Comm. 28, 667 (1978).
  * [74] M. Schechter, LiHoxY1-xF4 as a random-field Ising ferromagnet, Phys Rev. B 77, 020401(R) (2008).
  * [75] D. M. Silevitch, D. Bitko, J. Brooke, S. Ghosh, G. Aeppli and T. F. Rosenbaum, A ferromagnet in a continuously tunable random field, Nature 448, 567– (2007).
  * [76] K S. Liu and M. E. Fisher, Ouantum lattice gas and the existence of a supersolid, J. Low Temp. Phys. 10, 655 (1973).
  * [77] I. J. Ketley and D. J. Wallace, A modified epsilon expansion for a Hamiltonian with cubic point-group symmetry, J. Phys. A: Math. Nucl. Gen. 6 1667 (1973).
  * [78] Xiao Hu, Bicritical and Tetracritical Phenomena and Scaling Properties of the SO(5) Theory, Phys. Rev. Lett. 87, 057004 (2001).
  * [79] E. Demler, W. Hanke, and S.-C. Zhang, $SO(5)$ theory of antiferromagnetism and superconductivity, Rev. Mod. Phys. 76, 909 (2004).
  * [80] A. Kudlis, A. Aharony and O. Entin-Wohlman, Effective exponents near bicritical points, ArXiv:2304.08265. To appear in EPJ special edition on Non-Equilibrium Quantum Physics, Many Body Systems, and Foundations of Quantum Mechanics.
  * [81] F. J. Wegner, Critical Exponents in Isotropic Spin Systems, Phys. Rev. B 6, 1891 (1972).
  * [82] A. Codello, M. Safari, G. P. Vacca, and O. Zanusso, Critical models with $n\leqq 4$ scalars in $d=4-\epsilon$, Phys. Rev. D 102, 065017 (2020) and references therein.
  * [83] P. Calabrese, A. Pelissetto, and E. Vicari, Multicritical phenomena in $O(n_{1})\bigoplus O(n_{2})-$symmetric theories, Phys. Rev. B 67, 054505 (2003).
  * [84] S. Fishman and A. Aharony, Random field effects in disordered anisotropic antiferromagnets, J. Phys. C: Solid State Phys. 12, L729 (1979).
  * [85] J. L. Cardy, Random-field effects in site disordered Ising antiferromagnets, Phys. Rev. B 29, 505 (1984).
  * [86] P. G. de Gennes, Liquid-liquid demixing inside a rigid network. Qualitative features, J. Phys. Chem. 88, 6469 (1984).
  * [87] H. Rohrer, A. Aharony and S. Fishman, Critical and multicritical properties of random antiferromagnets, J. Mag. Mag. Mat. 15-18, 396 (1980).
  * [88] E.g., J. P. Hill, T. R. Thurston, R. W. Erwin, M. J. Ramstad, and R. J. Birgeneau, Transition to long-range order in the three-dimensional random-field Ising model, Phys. Rev. Lett. 66, 3281 (1991).
  * [89] R. A. Cowley, A. Aharony, R. J. Birgeneau, R. A. Pelcovits, G. Shirane and T. R. Thurston, The bicritical phase diagram of two-dimensional antiferromagnets with and without random fields, Zeitschrift für Physik B Condensed Matter 93, 5 (1993); R. J. Birgeneau, A. Aharony, R. A. Cowley, J. P. Hill, R. A. Pelcovits, G. Shirane and T. R. Thurston, Effects of random fields on bicritical phase diagrams in two and three dimensions, in Ref. [2], dedicated to Fisher’s 70th birthday, p. 58.
  * [90] P. Lacour-Gayet et G. Toulouse, Ideal Bose Einstein condensation and disorder effects, J. Phys. France 35, 425 (1974).
  * [91] Y. Imry and S.-k. Ma, Random-fields instability of the ordered state of continuous symmetry, Phys. Rev. Lett. 35, 1399 (1975).
  * [92] U. Nowak et al., Dyamic state model for exchage bias. I. Thory, Phys. Rev. B 66, 014430 (2002).
  * [93] R. J. Birgeeau, Random fields ad phase transitions in model magnetic systems, J. Mag. Mag. Mat. 177-178, 1 (1998).
  * [94] See also P. A. Young, On the lowering of dimensionality in phase trasitions with random fields, J Physics C 10, L257 (1977).
  * [95] John Z. Imbrie, Lower Critical Dimension of the Random-Field Ising Model, Phys. Rev. Lett. 53, 1747 (1984).
  * [96] J. Brickmont and A. Kupiainen, Lower critical dimension for the random-field Ising model, Phys. Rev. Lett. 59, 1829 (1987).
  * [97] A. Aharony, Tricritical points in systems with random fields, Phys. Rev. B 18, 3318 (1978).
  * [98] D. Andelman, First- and second-order phase trasitions with random fields at low temperatues, Phys. Rev. B 27, 3079 (1983).
  * [99] Yonathan Shapir and Amnon Aharony, High-temperature series and exact relations for the Ising model in a random field, J. Phys. C: Solid State Phys. 15, 1361 (1982).
  * [100] Y. Shapir and A. Aharony, Rigorous relations for the cross-over behaviour due to random perturbations, J. Phys. C: Solid State Phys. 14, L905 (1981).
  * [101] Misha Gofman, Joan Adler, Amnon Aharony, A. B. Harris, and Moshe Schwartz, Evidence for two exponent scaling in the random field Ising model, Phys. Rev. Lett. 71, 1569 (1993); Critical behavior of the random-field Ising model, Phys. Rev. B 53, 6362 (1996).
  * [102] M. E. Fisher, Some views from forty years as a statistical mechanician, Proc. STATPHYS20, Physica A 263, 554 (1999).
  * [103] e.g., G. Grinstein and S.-k. Ma, Roughening and lower critical dimension in the random-field Ising model, Phys. Rev. Lett. 49, 685 (1982); D. S. Fisher, Interface fluctuations in disordered systems: $5-\epsilon$ expansion and failure of dimensional reduction, Phys. Rev. Lett. 56, 1964 (1986).
  * [104] A. J. Bray and M. A. Moore, Scaling theory of the random-field Ising model, J. Phys. C: Solid State Phys. 18, L927 (1985).
  * [105] M. E. Fisher and A. N. Berker, Scaling for first-order transitions in thermodynamic and finite systems, Phys. Rev. B 26, 2507 (1982).
  * [106] M. R. Swift, A. J. Bray, A. Marita, M. Cieplak and J. R. Baavar, Scaling of the random-field Ising model at zero temperature, EuroPhys. Lett. 38, 273 (1997).
  * [107] N. Sourlas, Universality in random systems: the case of the 3D random field Ising model, Computer Phys. Comm. 121-122, 183 (1999).
  * [108] A. Aharony and A. B. Harris, Absece of self-averagig and uiversal fluctuatios i random systems near critical poits, Phys. Rev. Lett. 77, 3700 (1996).
  * [109] G. Parisi and N. Sourlas, Scale invariance in disordered systems: The example of the random-field Ising model, Phys. Rev. Lett. 89, 257204 (2002).
  * [110] Nikolaos G. Fytas and Víctor Martín-Mayor, Universality in the Three-Dimensional Random-Field Ising Model, Phys. Rev. Lett. 110, 227201 (2013); Nikolaos G. Fytas, Víctor Martín-Mayor, Marco Picco, and Nicolas Sourlas, Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions, Phys. Rev. Lett. 116, 227201 (2016); Nikolaos G. Fytas, Víctor Martín-Mayor, Giorgio Parisi, Marco Picco, and Nicolas Sourlas, Evidence for Supersymmetry in the Random-Field Ising Model at $D=5$, Phys. Rev. Lett. 122, 240603 (2019).
  * [111] Nikolaos G. Fytas and Víctor Martín-Mayor, Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions, Phys. Rev. E 93, 063308 (2016).
  * [112] Y. Imry and M. Wortis, Influence of quenched impurities on first-order phase transitions, Phys. Rev. B 19, 3580 (1979).
  * [113] M. Aizenman and J. Wehr, Rounding of first-order phase trasitions in systems with quenched disorder, Phys. Rev. Lett. 62, 2503 (1989).
  * [114] K. Hui and A. N. Berker, Random-field mechanism in random-bond multicritical systems, Phys. Rev. Lett. 62, 2507 (1989).
  * [115] M. Kumar, V. Banerjee and S. Puri, Random field Ising model in a uniform magnetic field: Ground states, pinned clusters and scaling laws, Eur. Phys. J. E 40, 96 (2017).
  * [116] Similar apparent first order trasitions for sigle realizations were also indicated by L. Hernández ad H. Ceva, Wang-Landau study of the critical behavior of the bimodal 3D random field Ising model, Physica A 387, 2793 (2008) and by N. G. Fytas and A. Malakis, Phase diagram of the 3D bimodal random-field Ising model, Eur. Phys. J. B 61, 111 (2008).
  * [117] E. Brézin and C. De Dominicis, New phenomena in the random field Ising model, Europhys. Lett. 44, 13 (1998).
  * [118] In fact, such a possibility was already mentioned by G. Parisi, An introduction to the statistical mechanics of amorphous systems, in Recent advances in field theory and statistical mechanics, ed. J.-B. Zuber and R. Stora (Elsevier, 1984).
  * [119] G. Tarjus and M. Tissier, Random-field Ising model and $O(N)$ models: theoretical description through the functional renormalization group, Eur. Phys. J. B 93, 50 (2020) and references therein.
  * [120] Apratim Kaviraj, Slava Rychkov, and Emilio Trevisani, Parisi-Sourlas Supersymmetry in Random Field Models, Phys. Rev. Lett. 129, 045701 (2022); Random field Ising model and Parisi-Sourlas supersymmetry I. Supersymmetric CFT, JHEP 04, 090 (2020); Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group, arXiv:2009.10087; S. Rychkov, Four lectures on the random field Ising model, Parisi-Sourlas supersymmetry and dimensional reduction, arXiv:2303.09654. These lecture notes contain many relevant references.
  * [121] A. Maritan et al., Orderig and phase trasitions in random-field Ising model, Phys. Rev. Lett. 67, 1821 (1991).