,

# Haake-Lewenstein-Wilkens approach to spin-glasses revisited

Maciej Lewenstein<EMAIL_ADDRESS>ICFO - Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  ICREA, Pg. Lluís Companys 23, 08010
Barcelona, Spain  David Cirauqui  ICFO - Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain  Quside Technologies S.L., Carrer d’Esteve Terradas 1,
08860 Castelldefels, Spain  Miguel Ángel García-March  Dpto. de Matemática
Aplicada, Universitat Politècnica de València, Camino de Vera, s/n, 46022
Valencia, Spain  Guillem Guigó i Corominas  ICFO - Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain  Przemysław Grzybowski  Institute of
Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz
University, Umułtowska 85, 61-614 Poznań, Poland  José R.M. Saavedra  Quside
Technologies S.L., Carrer d’Esteve Terradas 1, 08860 Castelldefels, Spain
Martin Wilkens  Universität Potsdam Institut für Physik, Karl-Liebknecht-
Strasse 24/25, 14476 Potsdam-Golm, Germany  Jan Wehr  Department of
Mathematics, The University of Arizona Tucson, AZ 85721-0089 USA

###### Abstract

We revisit the Haake-Lewenstein-Wilkens (HLW) approach to Edwards-Anderson
(EA) model of Ising spin glass [Phys. Rev. Lett. 55, 2606 (1985)]. This
approach consists in evaluation and analysis of the probability distribution
of configurations of two replicas of the system, averaged over quenched
disorder. This probability distribution generates squares of thermal copies of
spin variables from the two copies of the systems, averaged over disorder,
that is the terms that enter the standard definition of the original EA order
parameter, $q_{\rm EA}$. We use saddle point/steepest descent method to
calculate the average of the Gaussian disorder in higher dimensions. This
approximate result suggest that $q_{\rm EA}>0$ at $0<T<T_{c}$ in 3D and 4D.
The case of 2D seems to be a little more subtle, since in the present approach
energy increase for a domain wall competes with boundary/edge effects more
strongly in 2D; still our approach predicts spin glass order at sufficiently
low temperature. We speculate, how these predictions confirm/contradict widely
spread opinions that: i) There exist only one (up to the spin flip) ground
state in EA model in 2D, 3D and 4D; ii) There is (no) spin glass transition in
3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and
Marek Cieplak.

##  I Introduction

Spin glass problem. Spin glasses (SG) have entered solid state and statistical
mechanics in the 1970s, and from the very beginning were considered to be one
of the most outstanding and challenging problems of classical statistical
physics and theory of disordered and complex systems Mezard ; Chuck , not to
mention their quantum version (cf. Sachdev ; Veronica ; Toby  and references
therein). The most important and elaborated models of spin glasses are: the
Edwards-Anderson (EA) model with short range interactions EA , and the
Sherrington-Kirkpatrick (SK) model with infinite range interactions SK .

Sherrington-Kirkpatrick model. The SK model was solved approximately by its
inventors using replica trick and replica symmetric solution of the equations
that ”minimise” the free energy. This solution was clearly physically
incorrect, leading to negative entropy at low temperature. G. Parisi found an
ingenious way to break the replica symmetry in a hierarchical way Parisi .
Parisi’s solution of the SK model turned out to be exact, first as a local
extremum of the free energy Dedominicis , and proven rigorously to be unique
Talagrand . To deepen the understanding of this amazing results it is also
recommended to consult the Ref. Panchenko . Parisi’s solution and Parisi’s
order parameter, interpreted in terms of probability of overlaps between
different frozen configurations of the spin glass is nowadays accepted
commonly. For this achievement, and many others, G. Parisi was awarded the
Nobel Prize in physics in 2021 for ”the discovery of the interplay of disorder
and fluctuations in physical systems from atomic to planetary scales.”

Edwards-Anderson model. In the case of EA model, we are very far from a
rigorous solution. Most of our knowledge is based on numerical simulations on
special purpose classical computers, going back to the 1980s BY ; Ogielski .
It is widely believed that for Ising EA model there is no SG transition at
non-zero temperature in 2D, but there is in 3D and higher dimensions. It is
not clear that Parisi’s picture applies in these low dimensions; an
alternative is provided by the ”droplet model” of Ref. Huse , which predicts
that there exist only one (up to the spin flip) ground state in EA model in
2D, 3D and 4D, but the domain walls, separating the flipped region, from not
flipped one, are complex and might even have fractal dimension. Of course,
there are some rigorous results concerning EA model (cf. Arguin ), but they
are rather weak and very scarce. Thus, the question of the nature of the SG,
as well as many other questions concerning the EA model, is open (cf. Chuck
and references therein). In recent years various aspects have been studied:
ground states in $J=\pm 1$ model Laundry , information theory approach to 3D
EA models people1 , absence of Almeida-Thouless line in 3D SG Katz1 , or
universality in such systems Katz2 , and several other. The goal of this paper
is to look at EA model more than 25 years after the publication of HLW ,
revising the approach developed there.

HLW approach In 1984 Universität Essen GHS initiated the extremely successful
Sonderforschung Bereich ”Unordnung und große Fluktuazionen” with several
neighbouring centres. Fritz was a speaker of this initiative for the next 12
years. He convinced Maciej Lewenstein (his summer-time postdoc) and his new
PhD student Martin Wilkens to study short-range spin glasses. They formulated
a new approach to this problem, based on idea of studying disorder-averaged
probability distribution for configurations of two replicas/copies of the
system HLW . The idea of the HLW approach is as follows. We consider two
replicas/copies of the system and evaluate the joint probability distribution
of configurations averaged over the disorder:

$\displaystyle P(\sigma,\sigma^{\prime})$ $\displaystyle=$
$\displaystyle\left\langle\left\langle\exp\\{-\beta\left[H(\sigma,\\{K_{ij}\\}_{\langle
ij\rangle})\right.\right.\right.$ $\displaystyle+$
$\displaystyle\left.\left.\left.H(\sigma^{\prime},\\{K_{ij}\\}_{\langle
ij\rangle)})\right]\\}/Z(K)^{2}\right\rangle\right\rangle,$

where $H(\sigma,\\{K_{ij}\\}_{\langle
ij\rangle})=-\sum_{<ij>}K_{ij}\sigma_{i}\sigma_{j}$, $<ij>$ denotes nearest
neighbors, $\langle\langle\cdot\rangle\rangle$ denotes average over disorder,
and $Z(K)$ is the partition function calculated for a given configuration of
the quenched disorder variables $K_{ij}$. We assume that $K_{ij}$ are iidrv’s
(independent identically distributed random variables) with a Gaussian
distribution, $P(K)=\exp(-K^{2}/2\Delta^{2})/\sqrt{2\pi\Delta^{2}}$ or a
binary distribution, $P(K=\pm\Delta)=1/2$. Note that both distributions are
even, that is, invariant under the change of sign of $K_{ij}$. The idea is to
absorb the sign of $\sigma_{i}\sigma_{j}$ into $K_{ij}\to
K_{ij}\sigma_{i}\sigma_{j}$, and introduce the spin overlap variables
$\tau_{i}=\sigma_{i}\sigma_{i}^{\prime}$. We obtain the effective probability
distribution for $\tau$’s

$P(\tau)=2^{N}\left\langle\left\langle\exp{[\beta\sum_{\langle
ij\rangle}K_{ij}(1+\tau_{i}\tau_{j})]}/Z(K)^{2}\right\rangle\right\rangle.$
(1)

Here the number of relevant variables is reduced as we summed over dummy
variables. Note that magnetic order for $\tau$’s implies the non-zero EA order
parameter $q_{\rm EA}$ and vice versa,

$\langle\sum_{i}\tau_{i}\rangle_{T}/N=\langle\langle\sum_{i}\langle\sigma_{i}\sigma_{i}^{\prime}\rangle_{T}/N\rangle\rangle=\sum_{i}\langle\langle\,\langle\sigma_{i}\rangle_{T}^{2}\rangle\rangle=q_{\rm
EA}.$

We term $\langle\cdot\rangle$ or $\langle\cdot\rangle_{T}$ the thermal average
over possible configurations. Denoting $\alpha=\beta\Delta$, with $\Delta$ the
parameter characterizing the probability distributions for the disorder, HLW
used a convenient high temperature expansion to calculate (1) up to 12 order
in the expansion parameters $\alpha^{2}/(1+\alpha^{2})$ for the Gaussian, and
$\tanh^{2}(\alpha)/(1+\tanh^{2}(\alpha))$ for binary case. In effect, they
calculated

$P(\tau)=\exp[-H_{\rm eff}(\alpha,\tau)]/Z_{\rm eff},$ (2)

where effective Hamiltonian contained nearest neighbors couplings $-K_{1}$,
next nearest neighbors couplings $K_{2}$, and elementary plaquette terms,
$K_{3}$. The coefficients of these terms were explicit functions of
temperature ($\alpha$ in the notation of the present paper). The critical
surface separating ferromagnetic from paramagnetic region was estimated then
using (optimized) real space renormalization group approach. It turned out
that in 2D the $H_{\rm eff}$ never enters the ferromagnetic region, in 4D it
enters the ferromagnetic region for sure, and in 3D the situation was not
clear, suggesting that $H_{\rm eff}$ touches the critical region in a
quadratic manner. That would imply that the critical exponents of the spin
glass model are two times bigger than those of the standard Ising model, in
agreement with the best numerical simulation available at that time.

HLW followers The paper by HLW did not found too many followers, but some very
prominent are worth mentioning. Indeed, R. Swendsen with collaborators
published two papers on HLW method in Phys. Rev. B in the end of 1980s. In the
first one by J.-S. Wang and R.H. Swendsen Swendsen1 , the authors studied
Monte Carlo renormalization-group of Ising spin glasses. Application of this
approach to the $\pm J$ Ising spin glass showed clear differences between 2D,
3D, and 4D models. The data were consistent with a zero-temperature transition
in two dimensions, and non-zero temperature transitions in three and four
dimensions. In another paper Swendsen2  Monte Carlo and high-temperature-
expansion calculations of a spin-glass effective Hamiltonian were performed.
The authors studied the quenched random-coupling spin-glass problem from the
point of view of a nonrandom effective Hamiltonian, by Monte Carlo and high-
temperature-expansion methods. It was found that the high-temperature series
of the spin-glass effective Hamiltonian diverges below the ferromagnetic
transition temperature. The Monte Carlo approach does give reliable results at
low temperatures. The results were compared with the HLW picture of spin-glass
phase transitions.

Present work. In this paper we revise HLW approach. The idea is to estimate
$P(\tau)$, performing saddle point/steepest descent approximation in
calculating the Gaussian average of the disorder, which should be correct in
the limit $\alpha\to\infty$. We argue that the resulting spin model has
couplings that are positive in the region where $\tau_{i}\tau_{j}=1$’s, so it
has tendency to order ferromagnetically on islands/domains, separated from
other domains by negative couplings. In effect, boundary/edge effects start to
play a role in estimates of various quantities that may characterize the order
in our system.

We present here various arguments in favor or against the spin glass order
(ferromagnetic order in overlap variables). First, we consider the original
Peierls’ argument Peierls ; Griffiths , and argue that in our situation, it
can hardly be used. We turn then to an argument, studying sensitivity of the
system to boundary conditions. This argument was originally proposed by
Thouless Thouless1 ; Thouless2 ; gang  for models of electron propagation in
the presence of disorder and subsequently adapted to study Ising models in
random magnetic fields ImryMa  (see also Chudnovsky ), and also spin glasses
Cieplak . This argument is relating the existence of the ferromagnetic phase
transition to the sensitivity to boundary conditions. It can be trivially used
for ferromagnetic spin models: it ”predicts” transition for $d\geq 2$ for
Ising model, no transitions for $d=2$ models with continuous symmetry (Mermin-
Wagner-Hohenberg theorem), and transitions for $d\geq 3$ for systems with
continuous symmetry, like $XY$ or Heisenberg models (cf. LSA17 ).

To apply this argument, we calculate $P_{+}=P(\tau_{i}=+1,\,_{\rm for\ all\
{\it i}^{\prime}s})$ on a cylinder in $d$ dimensions of cross-section
$L^{d-1}$ and length $L$, and compare it to $P_{-}=P(\\{\tau\\}={\rm
corresponding\ to\ one\ domain\ wall})$. We analyze $\delta=\log(P_{+}/P_{-})$
and argue that this quantity, within approximations used, is always positive
and proportional to $L^{d-1}$ in $d\geq 2$. We will argue that the situation
in 2D seems to be a little more complex because of the stronger interplay
between the boundary effects and the domain wall energy. This leads to
significantly higher critical temperature in 2D than in higher dimensions.

##  II Saddle point/steepest descent calculations

We focus here on the case of Gaussian disorder, since we are going to use
differential calculus. First, we rescale $K_{ij}=\alpha\Delta\kappa_{ij}$, so
that both the logarithm of the distribution of $\kappa_{ij}$, and the
logarithm of $P(\tau)$ become proportional to $\alpha^{2}$ as
$\alpha^{2}\to\infty$. The HLW formula becomes

$P(\tau)=2^{N}\left\langle\left\langle\exp{[\alpha^{2}\sum_{\langle
ij\rangle}\kappa_{ij}(1+\tau_{i}\tau_{j})]}/Z(\kappa)^{2}\right\rangle\right\rangle,$
(3)

where the average $\langle\langle\cdot\rangle\rangle$ is now with respect the
distribution $P(\kappa)=\exp(-\alpha^{2}\kappa^{2}/2)/\sqrt{2\pi/\alpha^{2}}$.

Laplace’s method. The idea is to calculate the asymptotic behavior of the
disorder average using the Laplace method, also known as the saddle
point/steepest descent (SPSD) method, which we expect to be asymptotically
accurate for $\alpha\to\infty$. The SPSD equations equating to zero the first
derivatives of the logarithm of the integrand with respect to $\kappa_{ij}$’s
read:

$0=\alpha^{2}(-\kappa_{ij}+1+\tau_{i}\tau_{j}-2\langle\sigma_{i}\sigma_{j}\rangle),$
(4)

where $\langle\sigma_{i}\sigma_{j}\rangle$ is the thermal average of the
neighboring spins correlator, calculated according to the canonical
distribution $P(\sigma)=\exp{[\alpha^{2}\sum_{\langle
ij\rangle}\kappa_{ij}\sigma_{i}\sigma_{j}]}/Z(\alpha^{2}k)$. There are two
possibilities:

  * •

$\tau_{i}\tau_{j}=1$. In this case:

$\kappa_{ij}=2(1-\langle\sigma_{i}\sigma_{j}\rangle)>0,$ (5)

so that the corresponding coupling is clearly ferromagnetic.

  * •

$\tau_{i}\tau_{j}=-1$. In this case

$\kappa_{ij}=-2\langle\sigma_{i}\sigma_{j}\rangle,$ (6)

and the situation is more delicate. For $\alpha$ large, if
$\langle\sigma_{i}\sigma_{j}\rangle)>0$, we expect the coupling $\kappa_{ij}$
to be ferromagnetic, but the above equation implies the opposite. Likewise, if
the correlation function is negative, the $\kappa_{ij}<0$ should be
ferromagnetic. The contradiction could be avoided if $\kappa_{ij}=0$, but the
true situation is more complex, as we will see below, by solving
systematically mean field equations. This contradiction is really an
expression of frustration in our system!

It follows that we can write the SPSD solutions as $\kappa_{ij}>0$ on the
domains, where neighboring $\tau_{i}\tau_{j}=1$. This solution has a very
clear meaning: the canonical ensemble that serves to calculate the correlation
functions $\langle\sigma_{i}\sigma_{j}\rangle$ corresponds to ferromagnetic
islands/domains (where $\tau_{i}\tau_{j}=1$), separated by domain walls, where
the bonds $\kappa_{ij}\leq 0$, $\tau_{i}\tau_{j}=-1$, and the correlations
between $\sigma$’s from different domain walls are still positive, but perhaps
smaller at the border.

Note that the situation we consider is not as in the standard spin glass,
where we look at $\langle\sigma_{i}\sigma_{j}\rangle$ for a fixed
configuration of random $\kappa_{ij}$’s. There, it is quite common that the
sign of $\kappa_{ij}$ is not equal to the sign of
$\langle\sigma_{i}\sigma_{j}\rangle$: this is actually how the frustration
exhibits itself basically! Here, however, we consider a different situation:
for a given configuration of $\tau$’s, we adjust the values of $\kappa_{ij}$’s
to satisfy the SPSD equations. The natural expectation is a ferromagnetic
order for $\tau$’s (i.e. SG order for $\sigma$’s) in our system, with the
energy (free energy/probability) cost of the energy wall to scale as
$L^{d-1}$, as in, say, the standard Ising ferromagnet. At the same time, we
cannot exclude the existence of other solutions of SPSD equations that would
inherit frustration more explicitly. We discuss this possibility, which goes
beyond the scope of the present paper, in the outlook.

Hessian matrix. In the zeroth order one can calculate now $P(\tau)$,
substituting for $\kappa_{ij}$’s their SPSD values. One can go one step
further calculating the Gaussian correction to the SPSD. To this aim we
calculate the Hessian matrix of the second derivatives of the logarithm of the
integrand. Let us introduce the shortened notation $(ij)=\mu$,
$(i^{\prime}j^{\prime})=\nu$, $\sigma_{i}\sigma_{j}=c_{\mu}$,
$\sigma_{i^{\prime}}\sigma_{j^{\prime}}=c_{\nu}$, etc. The Hessian matrix
reads

${\cal H}_{\mu\nu}=-\alpha^{2}[\delta_{\mu\nu}+\alpha^{2}[\langle
c_{\mu}c_{\nu}\rangle-\langle c_{\mu}\rangle\langle c_{\nu}\rangle]].$ (7)

Note that the correlations matrix

$\langle c_{\mu}c_{\nu}\rangle-\langle c_{\mu}\rangle\langle
c_{\nu}\rangle=\langle(c_{\mu}-\langle c_{\mu}\rangle)(c_{\nu}-\langle
c_{\nu}\rangle)\rangle,$

i.e. it is explicitly positively semi-definite. In effect the Hessian matrix:

$\hat{\cal H}<0,$ (8)

so that the logarithm of the integrated function, which we consider is a
strictly convex function of many variables, is expected to have one maximum,
corresponding to our SPSD solutions. Note also that eigenvalues of the Hessian
matrix are all negative and will typically be of order $\alpha^{4}$, and they
are bounded in modulus from below by $\alpha^{2}$. One should thus expect that
SPSD method should become for $\alpha\to\infty$ asymptotically very precise,
if not exact.

##  III Peierls and Thouless approaches

In this section we examine if the $\tau$ variables of our effective model for
two copies of the EA systems exhibite ferromagnetic order i.e. if the EA order
parameter is nonzero, signifying spin glass order. We present two approaches:
i) Peierls approach; ii) Thouless approach; in the latter case we first
discuss several analytic estimates, and then present self-consistent
calculations, using SPSD solutions for $\kappa_{ij}$’s as a point of departure
for local mean field calculations of the averages of $\tau$’s and $\tau-\tau$
correlations.

Peierls approach. Peierls considers domain walls in a square lattice in 2D,
defining them in an unambiguous way. In a ferromagnetic Ising model with the
uniform coupling $K$ (with $\beta$ absorbed into $K$), and with periodic
boundary conditions on a square of side $L$, and number of sites $N=L^{2}$,
with all spins $\tau_{i}=1$ on the boundary, all domain walls are closed. Let
$b$ denote the length of the domain’s boundary; We classify them according to
length $b$, and within a class of given length we give each a number $i$. A
wall of the length $b$ fits into a square of the side $b/4$ and area
$b^{2}/16$. Let $m(b)$ be the number of domain walls of length $b$; it is
obviously bounded by $m(b)\leq 4N3^{b-1}$. The next step is to consider the
quantity $X(b,i)=1$, if the domain wall $(b,i)$ occurs in that configuration,
and $X(b,i)=0$ otherwise. Clearly, the number of spins down fulfills:

$N_{-}/N\leq\sum_{b}(b^{2}/16)3^{b}\sum_{i}^{m(b)}X(b,i).$

Peierls estimates then the thermal average of $X(b,i)$ in the Gibbs-Boltzmann
ensemble, bounding the partition function from below by the contribution of
the configuration, in which all spins inside the considered domain were
flipped, and obtaining the bound $\langle X(b,i)\rangle\leq\exp[-2\beta Kb]$,
which leads to the desired result. Notably, it can be generalized to higher
dimensions, with a little extra effort to estimate the entropy of contours,
see Bonati  for an elementary discussion and references therein for original
work.

Unfortunately, we cannot use this reasoning, because in our case: i) couplings
are non-homogeneous; ii) their values depend on domain walls configurations,
according to SPSD equations. We can estimate that the configuration $C$, in
which the domain $(b,i)$ occurs, has contribution to the ”energy” coming from
two edges, $4\alpha^{2}\kappa_{\rm e}$, where $\kappa_{\rm e}$ is the coupling
on the edge. The configuration $\tilde{C}$, in which the spins inside the wall
are flipped, contributes to $Z(\kappa)$ with the energy larger by
$3\alpha^{2}\kappa$, with $\kappa$ being the coupling in the bulk, so that
$\langle X(b,i)\rangle\leq\exp\\{\alpha^{2}[4\kappa_{\rm e}-6\kappa]b\\}$.
Since, according to mean field, $\kappa_{\rm e}>\kappa$, the question is to be
able to estimate more precisely the interplay of the edge and bulk
contributions. To this aim we turn, however, to a simpler Thouless argument,
to decide about the existence of the magnetization, i.e. spin glass order.

Thouless argument. In order to investigate the sensitivity to boundary
conditions, we calculate $P_{+}=P(\\{\tau_{i}=1\\}_{\rm for\ all\
i^{\prime}s})$ on a cylinder in $d$ dimensions of cross-section $L^{d-1}$ and
length $L$, and compare it to $P_{-}=P(\\{\tau_{i}=1\\}_{\rm on\ the\
left},\\{\tau_{i}=-1\\}_{\rm on\ the\ right})$ with $\tau_{i}=\pm 1$ on the
left (right ) of a domain wall (DW), correspondingly. We determine the
parameter $\delta=\ln(P_{+}/P_{-})$; Ferromagnetic order for $\tau$’s (SG
order for $\sigma$’s) is indicated by $\delta>0$

We consider a lattice with coordination number $f$, with $f_{\rm out}$ bonds
sticking out at any site of any $L^{d-1}$-dimensional hyper-plane (cross-
section). As we will see, we will need to compare the effects of DW and
boundary effects, since both scale as $L^{d-1}$. To this end we will also
consider effective coordination number at the edge (boundary) hyper-planes,
$f_{\rm e}$. Geometrically, $f_{\rm e}=f-f_{\rm out}/2$ for the left and right
edge (boundary) hyper-planes – only half of $f_{\rm out}$ stick out to the
right (left) from the left (right) edge. This is evidently a good estimate for
$f_{\rm e}$ in higher dimensions, where we expect $f_{\rm out}\ll f$. On the
other hand, boundary effects do extend to more than just the edge, so it is
reasonable to approximate

$f_{\rm e}\leq f-f_{\rm out}.$ (9)

We stress that we are NOT considering here domain walls in the disordered EA
model, where they are believed to a have a very complex geometry, scaling and
maybe even effective dimension, in accordance with the seminal droplet model
Huse . We are studying here domain walls in the effective, averaged over
disorder, probability distribution of the $\tau$ variables. Just from the
construction, there are no reasons for this probability distribution to break
translation symmetry (everywhere, i.e. in $d$ dimensions, if we apply global
periodic boundary conditions, or at least in $(d-1)$ transverse dimensions, if
we apply periodic boundary conditions there). It is thus natural to look in
the first place for domain walls that are just flat hyper-planes.

##  IV Self-consistent SPSD and local MF solutions

In the following we focus on hyper-cubic lattices in $d$-dimensions, with
coordination number $f=2d$, $f_{\rm out}=2$, and $f_{\rm e}\leq 2d-1$. We
leave the preliminary discussion of other lattices to appendix C and future
publication. In this section, we estimate the bulk and the edge contributions
applying SPSD and local MF consistently from the beginning till the end. We
consider a $d$-dimensional cylinder of spins with $L$ layers with bonds
distributed according to a Gaussian distribution
$P(K)=\exp(-K^{2}/2\Delta^{2})/\sqrt{2\pi\Delta^{2}}$, at an inverse
temperature $\beta$. We denote as above $\alpha=\Delta\beta$.

Local mean field theory. We assume translation symmetry in $d-1$ transverse
dimensions, so that magnetisation depend only on one index, $i$, enumerating
the layer, and the couplings depend on two indices, enumerating involved
single layer (two neighboring layers). Using standard mean field theory (MF),
we find the magnetization that is the thermal average of
$\langle\sigma_{i}\rangle$ at the $i$-th layer as

$\displaystyle m_{i}$ $\displaystyle=$
$\displaystyle\tanh\left[\alpha^{2}\left(2\left(d-1\right)\kappa_{i,i}m_{i}\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.\kappa_{i,i+1}m_{i+1}+\kappa_{i-1,i}m_{i-1}\right)\right],$

with

$\kappa_{i,j}=2-2m_{i}m_{j},$ (11)

and boundary conditions:

$m_{0}=m_{L+1}=0,$ (12) $\kappa_{0,1}=\kappa_{L,L+1}=0.$ (13)

Quantities to be determined. Our aim is to calculate logarithm of the
probability $P_{+}$, $P_{-}$ and $\delta=\ln(P_{+}/P_{-})$. We denote
$\ln(P_{\pm})=H_{\pm}$, and call it ”energy” in the following, so that

$H_{+}=L\ln(2)+\alpha^{2}\sum_{i,j}[2\kappa_{i,j}-\kappa_{i,j}^{2}/2]-2\ln
Z(\kappa),$ (14)

Being an extensive quantity, the energy of the system divided by the volume in
all but one dimension is

$\displaystyle\frac{H_{+}}{L^{d-1}}$ $\displaystyle=$ $\displaystyle
L\ln(2)+\alpha^{2}\left[\left(d-1\right)\sum_{i=1}^{L}[2\kappa_{i,i}-\kappa_{i,i}^{2}/2]\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{i=1}^{L-1}[2\kappa_{i,i+1}-\kappa_{i,i+1}^{2}/2]\right]-\sum_{i=1}^{L}2\ln\left[2\cosh(F_{i}(m))\right]$
$\displaystyle-$ $\displaystyle\frac{1}{2}\ln({\rm det}(\hat{H}_{+})).$

with
$F_{i}\left(m\right)=\alpha^{2}(2(d-1)\kappa_{i,i}m_{i}+\kappa_{i-1,i}m_{i-1}+\kappa_{i,i+1}m_{i+1})$.
Note that we have included in this expression the term coming from the
Gaussian fluctuations around the SPSD solution. The above quantity in the
leading order should be a linear function of the cylinder’s length,

$H_{+}/L^{d-1}=A(\alpha)L+B_{+}(\alpha).$

A similar expression holds for $P_{-}$, also including Gaussian fluctuations
terms:

$\displaystyle\frac{H_{-}}{L^{d-1}}$ $\displaystyle=$ $\displaystyle
L\ln(2)+\alpha^{2}\left[\left(d-1\right)\sum_{i=1}^{L}[2\kappa_{i,i}-\kappa_{i,i}^{2}/2]\right.$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{L/2-1}[2\kappa_{i,i+1}-\kappa_{i,i+1}^{2}/2]+\sum_{i=L/2+1}^{L-1}[2\kappa_{i,i+1}-\kappa_{i,i+1}^{2}/2]$
$\displaystyle-$
$\displaystyle\left.\kappa_{L/2,L/2+1}^{2}/2\right]-\sum_{i=1}^{L}2\ln\left[2\cosh(F_{i}(m))\right].$
$\displaystyle-$ $\displaystyle\frac{1}{2}\ln({\rm det}(\hat{H}_{-})).$

Since configuration contributing to $P_{-}$ has connection between two layers
in the middle of the cylinder given by a different expression, clearly

$H_{-}/L^{d-1}=A(\alpha)L+B_{-}(\alpha),$

with the same bulk contribution, but different boundary term; thus

$\delta=B_{+}(\alpha)-B_{-}(\alpha).$

Positive value of $\delta$ indicates ferromagnetic order for $\tau$’s and spin
glass order for $\sigma$’s.

To calculate $H_{-}/L^{d-1}$ we repeat the above calculations using the same
formulae as before, except that we use

$\kappa_{L/2,L/2+1}=-2m_{L/2}m_{L/2+1},$ (17)

.

Gaussian fluctuation terms Generally speaking, Gaussian fluctuation terms play
a sub-leading role, as expected. We approximate $\ln({\rm
det}(\hat{H}_{\pm}))=\sum_{\mu}\ln(\lambda_{\mu})\approx\sum_{\mu}\ln(\hat{H}_{\mu\mu})$,
that is the sum of logarithms of eigenvalues by the sum of logarithms of
diagonal elements of the Hessian matrix. Noting that

$\partial\langle\sigma_{i}\sigma_{j}\rangle/\partial\kappa_{ij}=\alpha^{2}\left(1-m_{i}^{2}m_{j}^{2}\right),$

we obtain

$-\frac{1}{2}\ln({\rm
det}(\hat{H}_{\pm}))\approx-\frac{1}{2}\sum_{(ij)}\ln[\alpha^{2}(1+\alpha^{2}(1-m_{i}^{2}m_{j}^{2})],$
(18)

where the SPSD solutions for $m$’s are calculated for the case $\pm$
accordingly. The above expression undergoes, obviously, further
simplifications under the translation symmetry.

High $\alpha$ regime Before going to numerical solutions, we first analyze the
asymptotic regime $\alpha\to\infty$, where $A(\alpha)$ can be estimated
analytically. We consider MF equations in the bulk of the $d$-dimensional
hyper-cubic lattice. The corresponding self-consistent equations in the bulk
are:

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle 2(1-m^{2}),$ (19)
$\displaystyle m$ $\displaystyle=$ $\displaystyle\tanh(2\alpha^{2}d\kappa m),$
(20) $\displaystyle g$ $\displaystyle=$
$\displaystyle\ln(2\cosh(2\alpha^{2}d\kappa m).$ (21)

We transform the first two into an equation of $x=2\alpha^{2}d\kappa$.

$x=4\alpha^{2}d/\cosh^{2}(x\sqrt{1-\kappa/(4\alpha^{2}d2)}).$ (22)

For large $\alpha$ we get $x=\ln(4\alpha^{2}d)/2$, and
$\kappa=\ln(4\alpha^{2}d)/(4\alpha^{2}d)$. As expected, $\kappa\to 0$ as
$\alpha\to\infty$, and $m\to 1$, but $2\alpha^{2}d\kappa$ diverges as
$\ln(4\alpha^{2}d)$. Elementary analysis leads to the result:

$A(\alpha)\simeq\ln(2)-\frac{1}{2}\ln(4\alpha^{2}d),$

i.e. as expected $\ln(P_{\pm})=H_{\pm}$ becomes negative at large $L$ (when
our analysis makes sense) and at large $\alpha$ (when SPSD should work well);
$A(\alpha)$ diverges with $\alpha$, but very slowly, only logarithmically.

In calculation of asymptotic behavior of $\alpha^{2}\kappa$. we typically set
local magnetization to 1: they indeed tend to one, but in slightly different
way in the bulk and on the ends, as the numeric illustrates below. If we set
$m_{i}=1$ in Eq. (IV), and expand for large $\alpha$, then we obtain a simple
expression for

$\displaystyle\frac{H_{+}}{L^{d-1}}$ $\displaystyle=$ $\displaystyle
L\ln(2)-2\alpha^{2}\left[\left(d-1\right)\sum_{i=1}^{L}\kappa_{i,i}\right.$
$\displaystyle+$ $\displaystyle\left.\sum_{i=1}^{L-1}\kappa_{i,i+1}\right],$

neclecting sub-leading Gaussian corrections. Since our numerical analysis in
the asymptotic regime is tough, we may and will use this expression there. The
analysis is more complex in the case of $\frac{H_{-}}{L^{d-1}}$, where we need
to take into account the dramatic change of the nature of SPSD solutions at
the domain wall.

”Phase transition” at moderate $\alpha$ The solution of the MF equations
change character as $\alpha$ grows from small values (when all $m_{i}=0$) to
larger values (when all $m_{i}\neq 0$). We infer the existence of this ”phase
transition” at a finite $\alpha$ by imposing that solutions get trivial at
that point, $\alpha_{T}$. This way we can approximate Eq. (IV) for
temperatures close to $\alpha_{T}$ as a series expansion for small $m_{i}$ to
get:

$m_{i}\approx\alpha^{2}\left(2\left(d-1\right)\kappa_{i}m_{i}+\kappa_{i-1,i}m_{i-1}\kappa_{i+1}m_{i+1}\right)$
(24)

To first order, $\kappa_{i,j}=2$ and $m_{i}=m$ $\forall i$, so we find the
critical temperature:

$\alpha_{T}=\frac{1}{2\sqrt{d}}$ (25)

Numerical calculations By numerically solving the system of equations
$F(m)=m_{i}$ for $1\leq i\leq L$ and taking into account that in positions
$i=\\{0,L+1\\}$ there are no spins and therefore conditions Eq. (12) and Eq.
(13) apply, we find non-trivial solutions above a certain temperature
threshold, see Fig. 1.

Figure 1: Numerical solutions of the system of Eqs. (IV) for $d=4$, $L=500$ at
various temperatures, corresponding to the case of $P_{+}$ (upper panel) and
$P_{-}$ (lower panel. In the latter case, the solutions change the sign of
$m$’s in the middle (and the keep the sign of $\kappa$’s positive). Solutions
for $d$ equal to 2 and 3 are qualitatively the same, and quantitatively very
similar.

We solve the system of equations for various lengths $L$ and fit the obtained
results in order to obtain $A$ and $B$ at different temperatures, Fig. 2. We
do so for dimensions $d=2,3,4$ and obtain similar behaviours. As expected, MF
solutions for all three systems undergo a ”phase transition” from $m=0$ to
$m\neq 0$ at their respective critical temperatures,
$\alpha_{T}^{d=(2,3,4)}=\\{\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{3}},\frac{1}{4}\\}$.

Figure 2: Numerical solutions for $A$ (upper panel) and $B_{+}$ and $B_{-}$
(lower panel) for $d$ equal 2, 3, 4, and at various temperatures. Gaussian
fluctuations contributions are included. Figure 3: The parameter $\delta$ for
$d=2,3,4$ as a function of $\alpha$ (temperature). Gaussian fluctuations
contribution is included.

The results show in accordance with analytic calculations that $A(\alpha)$
tends to $-\infty$ logarithmically. On the other hand, $B_{+}(\alpha)$ tend to
a positive constant for large $\alpha$, while $B_{-}(\alpha)$ to infinity,
indicating SG transition in $2D$ (unfortunately), $3D$ (fortunately), and $4D$
(fortunately). This is illustrated clearly in Fig. 3, Still, one observes
quite a quantitative difference in behaviour for $d=2$ and $d$ larger.

##  V Conclusions and outlook

In the short note we revised the Haake-Lewenstein-Wilkens (HLW) approach to
Edwards-Anderson (EA) model of Ising spin glass. The main results are the
following:

  * •

We have calculated the disorder averaged probability of spin configurations
for two replicas, which reduces to a probability of overlaps between spins
from the two replicas, $P(\tau)$. To this aim we used the saddle
point/steepest descent (SPSD) method which seems to be asymptotically exact in
the limit of $\alpha=\beta\Delta$ going to infinity.The integral we consider,
has an integrand, whose logarithm has a well peaked single maximum, with the
Hessian of order at least $\alpha^{2}$, if not $\alpha$. It would be
challenging to study if one can control this result rigorously.

  * •

We attempted to apply Peierls and Thouless approaches to decide whether there
exist SG order in the low temperature (large $\alpha^{2}$ limit). The results
indicate that this indeed is the case in 2D and above, but we identified the
reasons, why this does not have to be the case in 2D. Namely, the competing
boundary effects might destroy the order. Our estimates, based on mean field
theory, clearly require improvement, for instance by studying precisely the
solutions of boundary effects in SPSD equations etc. If we accept the proposed
form of the solutions of the SPSD solutions, the simulating $P_{+}$ requires
MC simulations of a finite size ferromagnetic model, while simulating $P_{-}$
– also a finite size ferromagnetic model with a domain wall and a bump/dip in
the couplings at the wall.

  * •

In a nutshell: Our results predict SG transition in EA model in $4d$, $3d$,
but unfortunately also in $2d$. There can be several reasons for that: i) SPSD
approximation is not precise enough; ii) is completely incorrect; In the first
case we can include Gaussian and maybe even beyond Gaussian corrections to
SPSD solutions. In the second case, there might be many SPSD solutions
contributing or something like that; Hessian result suggests this is not the
case, but it is not rigorous; iii) finally, local MF calculations of
edge/boundary effects might be too rough.

  * •

The paper contains 4 appendices: In Appendix A we discuss shortly the exactly
soluble 1D case, in Appendix B – the normalization of $P(\tau)$ that implies
nice properties of certain multidimensional integrals. Of course, the present
results are compatible with the expectation that there exist only one (up to
the spin flip) ground state in EA model in 2D and 3D Laundry . Another
interesting conclusion is that the existences of the SG transition in the
present picture, might depend on the connectivity of the lattice. As discussed
in Appendix C, even within our SPSD and MF domain walls have a certain width.
This might depend crucially on the dimension and even on the coordination
number (connectivity) of the lattice. Finally, alternative way of calculations
combining SPSD method with the expected behavior of $Z(\kappa)$ for large
$\alpha$ is discussed in Appendix D. This method explicitly accounts for
dependence of correlators $\langle\sigma_{i}\sigma_{j}\rangle$ on
$\kappa_{ij}$,

Clearly, this study requires further studies, but this goes beyond the present
note.

###### Acknowledgements.

ICFO group acknowledges support from ERC AdG NOQIA, State Research Agency AEI
(“Severo Ochoa” Center of Excellence CEX2019-000910-S) Plan National FIDEUA
PID2019-106901GB-I00 project funded by MCIN/ AEI /10.13039/501100011033, FPI,
QUANTERA MAQS PCI2019-111828-2 project funded by MCIN/AEI
/10.13039/501100011033, Proyectos de I+D+I “Retos Colaboración”
RTC2019-007196-7 project QUSPIN funded by MCIN/AEI /10.13039/501100011033,
Fundació Privada Cellex, Fundació Mir-Puig, Generalitat de Catalunya (AGAUR
Grant No. 2017 SGR 1341, CERCA program, QuantumCAT U16-011424, co-funded by
ERDF Operational Program of Catalonia 2014-2020), EU Horizon 2020 FET-OPEN
OPTOLogic (Grant No 899794), and the National Science Centre, Poland (Symfonia
Grant No. 2016/20/W/ST4/00314), Marie Skłodowska-Curie grant STREDCH No
101029393, “La Caixa” Junior Leaders fellowships (ID100010434), and EU Horizon
2020 under Marie Skłodowska-Curie grant agreement No 847648
(LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012).

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##  Appendix A Exact solution 1D

Calculation of $P(\tau)$ in 1D are elementary. We observe first that

$Z(\kappa)=\prod_{i=1}^{i=L-1}2\cosh(\alpha^{2}\kappa_{i,i+1}),$

so that $P(\tau)$ can be written as

$\displaystyle P(\tau)$ $\displaystyle=$
$\displaystyle\langle\langle\frac{2^{L}}{Z^{2}(\kappa)}\prod_{i=1}^{i=L-1}\left[\cosh(\alpha^{2}\kappa_{i,i+1})+\sinh(\alpha^{2}\kappa_{i,i+1})\right]$
(26)
$\displaystyle\left[\cosh(\alpha^{2}\kappa_{i,i+1})+\tau_{i}\tau_{i+1}\sinh(\alpha^{2}\kappa_{i,i+1})\right]\rangle\rangle.$

Since we average over the even distributions the terms $\cosh(.)\sinh(.)$
average zero, and we get

$P(\tau)=2\prod_{i=1}^{i=L-1}\left[1+\tau_{i}\tau_{i+1}\langle\langle\tanh^{2}(\alpha^{2}\kappa)\rangle\rangle\right],$
(27)

where we skipped the subscript of $\kappa$. We can again estimate
$\langle\langle\tanh^{2}(\alpha^{2}\kappa)\rangle\rangle$ using SPSD. Saddle
point value for $2\alpha^{2}\kappa$ diverges again as $\ln(4\alpha^{2})$, so
the 1D system exhibits a ”phase transition” at zero temperature
($\alpha\to\infty$) with diverging correlation length $\xi\propto\alpha^{2}$.

##  Appendix B Normalization issues - amazing formulae

Note that if we observe that $P(\tau)$, by definition is normalized

$P(\tau)=2^{N}\left\langle\left\langle\exp{[\beta\sum_{\langle
ij\rangle}K_{ij}(1+\tau_{i}\tau_{j})]}/Z(K)^{2}\right\rangle\right\rangle,$
(28)

then by tracing over $\tau$’s we obtain

$1=2^{N}\left\langle\left\langle\exp{[\beta\sum_{\langle
ij\rangle}K_{ij}]}/Z(K)\right\rangle\right\rangle.$ (29)

The above expression is true for any even distribution of $K$’s, Gaussian or
not, discrete or continuous. It can be generalized to certain matrix models
with couplings invariant with respect to local unitary transformations. The
independent proof of this formula employs the fact that

$2^{N}=\sum_{\sigma}1.$

Using the above formula and then incorporating each of the configurations of
$\sigma$’s into the averaging over disorder, gives the desired identity.

##  Appendix C Domain wall width

It is worth noticing the domain walls in the case of $P_{-}$ have a finite
width. This means that local magnetization $m_{i}$ does not jump from nearly
one to nearly minus one (see Fig. 4). In effect, $\kappa$’s in the domain wall
regions are not so close to zero, and the terms $\alpha^{2}\kappa$ simply
behave in this region as $\alpha^{2}$. This explain the rapid growth of
$B_{-}$ in Fig. 2.

Figure 4: Numerical solutions for magnetization in the domain wall region for
$d=2$ (upper panel), $d=3$ (middle panel), and for $d=4$ (lower panel) at
indicated temperatures.

Sharpening of the domain wall to the configuration that $m=\pm 1$ in the bulk,
and $m=0$ at the domain wall edges would, would lead presumably to instability
of the ferromagnetic phases. In fact we have originally postulated (incorrect)
solutions of SPSD equations with $\kappa=0$ at the walls. Such solution leads
to $B_{-}=2B_{+}$ – it still predicts the ferromagnetic order, but with very
different, much more milder behavior of $\delta$. Conversely, widening the
wall, more in the spirit of the ”droplet model” might also lead to unexpected
behavior, since the assumption that $\ln(P_{\pm 1})=AL+B_{\pm 1}$ would then
cease to hold.

Our numerical findings with the SPSD and MF approximations indicate that: i)
for fixed $\alpha^{2}$, the domain wall reaches an $L$-independent limit for
$L$ large; ii) for fixed $L$, the domain wall shinks from $L$ (below phase
transtion, where all $m$’s are zero), to a very small values dictated by the
very fast growth of $|m|$’s toward one, in accordance with the MF laws.

##  Appendix D Alternative approach

Here we propose alternative way of calculating $\delta$ based on expected
behaviour of $\ln(Z(K))$ for low temperatures. Namely, we expect that

$-2\ln(Z(K))=2\beta F\simeq 2\beta\langle U\rangle,$

where $U$ is the internal energy. That means that in the SPSD method we need
to analyse the logarithm of the integral kernel:

$\alpha^{2}\sum_{\langle
ij\rangle}[\kappa_{ij}(1+\tau_{i}\tau_{j}-2\langle\sigma_{i}\sigma_{j}\rangle)-\kappa_{ij}^{2}/2]$
(30)

The equations for $\kappa$’s are modified due to the explicit dependence of
$\langle\sigma_{k}\sigma_{l}\rangle$ on $\kappa_{ij}$; in fact one easily gets

$\partial\langle\sigma_{k}\sigma_{l}\rangle/\partial\kappa_{ij}=\alpha^{2}\left(\langle\sigma_{k}\sigma_{l}\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{k}\sigma_{l}\rangle\langle\sigma_{i}\sigma_{j}\rangle\right).$

Fortunately, most of these correlators are negligible: in fact they vanish in
the MF approximation for distinct, non-overlaping pairs $(k,l)$ and $(i,j)$.
The non-vanishing and non-trivial are

$\partial\langle\sigma_{i}\sigma_{j}\rangle/\partial\kappa_{ij}=\alpha^{2}\left(1-m_{i}^{2}m_{j}^{2}\right),$

and

$\partial\langle\sigma_{i}\sigma_{l}\rangle/\partial\kappa_{ij}=\alpha^{2}\left(m_{l}m_{j}(1-m_{i}^{2})\right),$

and its variations. We obtain then modified equations for $\kappa_{ij}$ that
have now to be solved in an iterative manner,

$\kappa_{ij}=\frac{1+\tau_{i}\tau_{j}-2m_{i}m_{j}-2\Delta\kappa_{ij}}{1+2\alpha^{2}(1-m_{i}^{2}m_{j}^{2})},$

where

$\Delta\kappa_{ij}=\alpha^{2}\left(\sum_{l={\rm
n.n.}}\kappa_{il}m_{l}m_{j}(1-m_{i}^{2})+\sum_{k={\rm
n.n.}}\kappa_{kj}m_{i}m_{k}(1-m_{j}^{2})\right),$

where $l$’s ($k$’s) and neighbors of $i$ ($j$), different from $j$ ($i$).

These expressions get simplified upon translation symmetry,

$\displaystyle\Delta\kappa_{i,i}$ $\displaystyle=$ $\displaystyle
2\alpha^{2}\left(\kappa_{i,i}(2d-3)m_{i}^{2}(1-m_{i}^{2})\right.$
$\displaystyle+$
$\displaystyle\left.m_{i}(\kappa_{i,i+1}m_{i+1}+\kappa_{i-1,i}m_{i-1})(1-m_{i}^{2})\right),$
$\displaystyle\Delta\kappa_{i,i+1}$ $\displaystyle=$
$\displaystyle\alpha^{2}\left(2(d-1)\kappa_{i,i+1}m^{2}_{i+1}(1-m_{i}^{2})\right.$
$\displaystyle+$ $\displaystyle 2(d-1)\kappa_{i,i+1}m_{i}^{2}(1-m_{i+1}^{2})$
$\displaystyle+$
$\displaystyle\left.\kappa_{i-1,i}m_{i-1}m_{i+1}(1-m_{i}^{2})+\kappa_{i,i+1}m_{i}m_{i+2}(1-m_{i+1}^{2})\right).$

For $P_{+}$ and $P_{-}$ (away from the wall) we get

$\kappa_{ij}=\frac{2-2m_{i}m_{j}-2\Delta\kappa_{ij}}{1+2\alpha^{2}(1-m_{i}^{2}m_{j}^{2})},$

and at the wall

$\kappa_{L/2,L/2+1}=\frac{-2m_{L/2}m_{L/2+1}-2\Delta\kappa_{L/2,L/2+1}}{1+2\alpha^{2}(1-m_{L/2}^{2}m_{L/2+1}^{2})}.$

Otherwise, all other expressions are valid. The calculations of $P_{+}$ and
$P_{-}$ reduces now to evaluation of

$\displaystyle\frac{H_{+}}{L^{d-1}}$ $\displaystyle=$ $\displaystyle
L\ln(2)+\alpha^{2}\left[\left(d-1\right)\sum_{i=1}^{L}[2\kappa_{i,i}(1-m_{i}^{2})-\kappa_{i,i}^{2}/2]\right.$
$\displaystyle+$
$\displaystyle\left.\sum_{i=1}^{L-1}[2\kappa_{i,i+1}(1-m_{i}m_{i+1})-\kappa_{i,i+1}^{2}/2]\right].$

and

$\displaystyle H_{-}/L^{d-1}$ $\displaystyle=$
$\displaystyle\alpha^{2}\sum^{\prime}_{\langle
ij\rangle}[2\kappa_{ij}-\kappa_{ij}^{2}/2-2\kappa_{ij}m_{i}m_{j}]$
$\displaystyle+$
$\displaystyle\alpha^{2}[-\kappa_{L/2,L/2+1}^{2}/2-2\kappa_{L/2,L/2+1}m_{L/2}m_{L/2+1}].$

Similarly,

$\displaystyle\frac{H_{-}}{L^{d-1}}$ $\displaystyle=$ $\displaystyle
L\ln(2)+\alpha^{2}\left[\left(d-1\right)\sum_{i=1}^{L}[2\kappa_{i,i}(1-m_{i}^{2})-\kappa_{i,i}^{2}/2]\right.$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{L/2-1}[2\kappa_{i,i+1}(1-m_{i}m_{i+1})-\kappa_{i,i+1}^{2}/2]$
$\displaystyle+$
$\displaystyle\sum_{i=L/2+1}^{L-1}[2\kappa_{i,i+1}(1-m_{i}m_{i+1})-\kappa_{i,i+1}^{2}/2]$
$\displaystyle-$
$\displaystyle\left.\kappa_{L/2,L/2+1}^{2}/2-\kappa_{L/2,L/2+1}m_{L/2}m_{l/2+1}\right].$

We have calculated $\delta$, using the present approach, in which we neglected
contributions from $\Delta\kappa$’s terms, leaving only the effect due to
$\partial\langle\sigma_{i}\sigma_{j}\rangle/\partial\kappa_{ij}=\alpha^{2}\left(1-m_{i}^{2}m_{j}^{2}\right)$.
The end results are quantitatively and qualitatively the very similar to those
obtained with the ”pure” SPSD method.


# Preservation of complete Baireness

Sergey Medvedev  South Ural State University, Chelyabinsk, Russia
<EMAIL_ADDRESS>
###### Abstract.

The main result is the following. Let $f\colon X\rightarrow Y$ be a continuous
mapping of a completely Baire space $X$ onto a hereditary weakly Preiss-Simon
regular space $Y$ such that the image of every open subset of $X$ is a
resolvable set in $Y$. Then $Y$ is completely Baire.

The classical Hurewicz theorem about closed embedding of the space of rational
numbers into metrizable spaces is generalized to weakly Preiss-Simon regular
spaces.

###### Key words and phrases:

resolvable set, completely Baire space, weakly Preiss-Simon space, rational
numbers

###### 2010 Mathematics Subject Classification:

54H05, 03E15

One of the main tasks considered in topology is to find the properties of
topological spaces that are preserved when moving to the image of the space
with this property. For example, Sierpiński [11] and Vainštain [12] have
established that if $Y$ is an image of a completely metrizable space $X$ under
a closed mapping or an open mapping, then $Y$ is completely metrizable. A
natural question to ask is which maps preserve complete metrizability.

Ostrovsky [10] proved that if a continuous mapping from a zero-dimensional
Polish (separable, completely metrizable) space $X$ onto a metrizable space
$Y$ takes any clopen set $W\subset X$ to the union of an open set and a closed
set, then $Y$ is completely metrizable. He raised the question whether the
same is true when the images are the intersection of an open set and a closed
set. Gao and Kieftenbeld [4] showed that if the image under $f\colon
X\rightarrow Y$ of every open set or every closed set in a Polish space $X$ is
a resolvable set in $Y$, then $Y$ is Polish. Holický and Pol [6] established
similar results for the nonseparable case. The latter result solves
Ostrovsky’s problem for metrizable spaces.

It is notorious [3, Theorem 4.3.26] that a topological space is completely
metrizable if and only if it is a Čech-complete metrizable space. Holický [5,
Corollary 3.3] proved among other things that if $f\colon X\rightarrow Y$ is a
continuous mapping from a Čech-complete space $X$ onto a metrizable space $Y$
such that $f$ takes open sets in $X$ to resolvable sets in $Y$, then $Y$ is
completely metrizable.

Let us pass now from a Čech-complete metrizable space to a completely Baire
space. Suppose we are given a continuous mapping $f\colon X\rightarrow Y$ from
a zero-dimensional metrizable space $X$ onto a metrizable space $Y$ such that
$f$ takes clopen sets in $X$ to resolvable sets in $Y$. Holický and Pol [6,
Corollary 4] showed that if $X$ is completely Baire, then so is $Y$. Our main
Theorem 3 generalizes this statement to regular spaces. In order to obtain it,
we strengthen the Hurewicz theorem [7] about closed embedding of the space of
rational numbers into metrizable spaces, see Theorem 1.

Notation. For all undefined terms see [3].

A set $E$ in a space $X$ is said to be resolvable [8, § 12] if it can be
represented as

$E=(F_{1}\setminus F_{2})\cup(F_{3}\setminus
F_{4})\cup\ldots\cup(F_{\xi}\setminus F_{\xi+1})\cup\ldots,$

where $\langle F_{\xi}\rangle$ forms a decreasing transfinite sequence of
closed sets in $X$. Equivalently, $E$ is resolvable if, for every closed set
$F\subset X$, the boundary of $F\bigcap E$ relative to $F$ is nowhere dense in
$F$. Resolvable sets form a field of subsets of $X$ which contains closed
sets.

A topological space $X$ is called a Baire space if the intersection of
countably many dense open sets in $X$ is dense. We call a space completely
Baire (or hereditarily Baire cf. [6]) if every closed subspace of it is Baire.
By Hurewicz’s theorem [7], a metrizable space is completely Baire if and only
if it contains no closed homeomorphic copy of rational numbers.

A topological space $X$ is called Preiss-Simon at a point $x\in X$ if for any
subset $A\subset X$ with $x\in\overline{A}$ there is a sequence $\langle
U_{n}\colon n\in\omega\rangle$ of nonempty open subsets of $A$ that converges
to $x$ in the sense that each neighborhood of $x$ contains all but finitely
many sets $U_{n}$. A space $X$ is called a Preiss-Simon space if $X$ is
Preiss-Simon at each point $x\in X$. It is clear that every first countable
space is Preiss-Simon and each Preiss–Simon space is Frèchet–Urysohn.

We shall say that a space $X$ is weakly Preiss-Simon at a point $x\in X$ if
there is a sequence $\langle U_{n}\colon n\in\omega\rangle$ of nonempty open
subsets of $X$ that converges to $x$. Denote by $wPS(X)$ the set of points
$x\in X$ at which $X$ is weakly Preiss-Simon. A space $X$ is called weakly
Preiss-Simon space if the set $wPS(X)$ is dense in $X$. A space $X$ is called
hereditary weakly Preiss-Simon space if every nonempty closed subset of $X$ is
a weakly Preiss-Simon space, i. e., the set $wPS(F)$ is dense in $F$ for every
nonempty closed set $F\subset X$. Clearly, every Preiss-Simon space is
hereditary weakly Preiss-Simon.

######  Lemma 1.

Let $F$ be a nowhere dense closed subset of a regular space $X$. Suppose $X$
is weakly Preiss-Simon at a non-isolated point $x$. Then for every
neighborhood $O$ of $x$ there is a sequence $\langle U_{n}\colon
n\in\omega\rangle$ of nonempty open sets such that:

  1. (a)

$\langle U_{n}\colon n\in\omega\rangle$ converges to $x$,

  2. (b)

$\overline{U_{n}}\subset O\setminus\\{x\\}$ and $\overline{U_{n}}\bigcap
F=\emptyset$ for every $n\in\omega$,

  3. (c)

$\overline{U_{n}}\bigcap\overline{U_{k}}=\emptyset$ provided $n\neq k$,

  4. (d)

$\overline{\bigcup\\{U_{n}\colon
n\in\omega\\}}=\\{x\\}\cup\bigcup\\{\overline{U_{n}}\colon n\in\omega\\}$.

###### Proof.

Take a sequence $\langle V_{k}\colon k\in\omega\rangle$ of nonempty open sets
in $X$ that converges to $x$. The neighborhood $O$ of $x$ contains all but
finitely many sets $V_{k}$. Choose the least number $k_{0}$ with
$V_{k_{0}}\subset O$. The point $x$ is non-isolated in $X$. Then there is a
point $x_{0}\in V_{k_{0}}\setminus(\\{x\\}\bigcup F)$ and its neighborhood
$U_{0}$ satisfying $\overline{U_{0}}\subset V_{k_{0}}\setminus(\\{x\\}\bigcup
F)$. Similarly, choose the least $k_{1}$ with $V_{k_{1}}\subset
O\setminus\overline{U_{0}}$. Find a point $x_{1}\in V_{k_{1}}\setminus\\{x\\}$
and its neighborhood $U_{1}$ satisfying $\overline{U_{1}}\subset
V_{k_{1}}\setminus(\\{x\\}\bigcup F)$. Clearly,
$\overline{U_{0}}\bigcap\overline{U_{1}}=\emptyset$. Next, choose the least
$k_{2}$ with $V_{k_{2}}\subset
O\setminus(\overline{U_{0}}\bigcup\overline{U_{1}})$. Find a point $x_{2}\in
V_{k_{2}}\setminus\\{x\\}$ and its neighborhood $U_{2}$ satisfying
$\overline{U_{2}}\subset V_{k_{2}}\setminus(\\{x\\}\bigcup F)$, and so on. One
can check that all conditions (a)–(d) hold. ∎

The following statement generalizes the Hurewicz theorem [7] about closed
embedding of the space of rational numbers $\mathbb{Q}$ into metrizable
spaces. Note that Theorem 1 was proved in [9] for a regular space $X$ of the
first category with a dense first-countable subset.

######  Theorem 1.

Let $X$ be a regular space of the first category such that the set $wPS(X)$ is
dense in $X$. Then $X$ contains a closed copy of the space of rational
numbers.

###### Proof.

Let $X=\bigcup\\{X_{n}\colon n\in\omega\\}$, where each $X_{n}$ is a closed
nowhere dense subset of $X$. Without loss of generality, $X_{0}\subset
X_{1}\subset\ldots$.

By induction on the tree $\omega^{<\omega}$, we shall construct points
$x_{t}\in wPS(X)$, open sets $U_{t}$, and sequences $\langle
U_{t\hat{\,}k}\colon k\in\omega\rangle$ of open sets such that the following
conditions hold for each $t\in\omega^{<\omega}$:

  1. (a)

$U_{t}$ is a neighborhood of the point $x_{t}$,

  2. (b)

$\overline{U_{t}}\bigcap X_{n}=\emptyset$ whenever $t\in\omega^{n+1}$,

  3. (c)

$\overline{U_{t\hat{\,}k}}\subset U_{t}\setminus\\{x_{t}\\}$ for each
$k\in\omega$,

  4. (d)

the sequence $\langle U_{t\hat{\,}k}\colon k\in\omega\rangle$ converges to
$x_{t}$,

  5. (e)

$\overline{U_{t\hat{\,}i}}\bigcap\overline{U_{t\hat{\,}k}}=\emptyset$ provided
$i\neq k$,

  6. (f)

$\overline{\bigcup\\{U_{t\hat{\,}k}\colon
k\in\omega\\}}=\\{x_{t}\\}\cup\bigcup\\{\overline{U_{t\hat{\,}k}}\colon
k\in\omega\\}$.

The space $X$ has no isolated points as a set of the first category.

For the base step of the induction, take a point $x_{\emptyset}\in wPS(X)$ and
$U_{\emptyset}=X$. By Lemma 1 with $F=X_{0}$ and $O=X$, there is a sequence
$\langle U_{k}\colon k\in\omega\rangle$ of nonempty open sets satisfying
conditions (a)–(f) for $n=0$.

Suppose the points $x_{s}$ and the sequences $\langle U_{s\hat{\,}k}\colon
k\in\omega\rangle$ of open sets have been constructed for every
$s\in\omega^{j}$ with $j\leq n$. Fix $s\in\omega^{n}$. The set $X_{n+1}$ is
nowhere dense in the open set $U_{s\hat{\,}i}$ for every $i\in\omega$. Then we
can find a point $x_{s\hat{\,}i}\in wPS(X)\bigcap(U_{s\hat{\,}i}\setminus
X_{n+1})$. By Lemma 1 with $F=X_{n+1}$ and $O=U_{s\hat{\,}i}$, there is a
sequence $\langle U_{s\hat{\,}i\hat{\,}k}\colon k\in\omega\rangle$ of nonempty
open sets satisfying conditions (a)–(f) for $t=s\hat{\,}i$. This completes the
induction step.

Clearly, the set $Z=\bigcup\\{x_{t}\colon t\in\omega^{<\omega}\\}$ is
countable and has no isolated points. For every $t\in\omega^{<\omega}$ from
conditions (d) and (e) it follows that the sequence $\langle
V_{t\hat{\,}n}\colon n\in\omega\rangle$ forms a countable base at the point
$x_{t}$ with respect to $Z$, where

$V_{t\hat{\,}n}=Z\bigcap\left(U_{t}\setminus\cup\\{\overline{U_{t\hat{\,}k}}\colon
k<n\\}\right).$

Hence, $Z$ is a second-countable space. The Urysohn theorem [3, Theorem 4.2.9]
implies that $Z$ is metrizable. According to the Sierpiński theorem [11] (see
also [3, Exercise 6.2.A]), $Z$ is homeomorphic to the space of rational
numbers.

It remains to verify that $Z$ is closed in $X$. For each $n\in\omega$ define
the sets

$\displaystyle Z_{n}$ $\displaystyle=\bigcup\left\\{x_{s}\colon
s\in\omega^{j},j\leq n\right\\},$ $\displaystyle P_{n}$
$\displaystyle=\bigcup\left\\{\overline{U_{t\hat{\,}k}}\colon
t\in\omega^{n},k\in\omega\right\\}.$

Then $Z=\bigcup\\{Z_{n}\colon n\in\omega\\}$, $Z_{n}\cap P_{n}=\emptyset$, and
$Z\subset Z_{n}\bigcup P_{n}$ for each $n$.

By induction on $n\in\omega$, let us check that the union $Z_{n}\cup P_{n}$ is
closed in $X$ for each $n$.

For $n=0$, the set $Z_{0}\cup P_{0}$ is closed in $X$ by Lemma 1. We now show
that $Z_{n+1}\cup P_{n+1}$ is closed under the induction assumption on
closeness of $Z_{n}\cup P_{n}$. Suppose that, on the contrary, there is a
point

$x\in\overline{(Z_{n+1}\cup P_{n+1})}\setminus(Z_{n+1}\cup P_{n+1}).$

Since $(Z_{n+1}\setminus Z_{n})\cup P_{n+1}\subset P_{n}$, we have
$x\in\overline{Z_{n}\cup P_{n}}=Z_{n}\cup P_{n}$. Then $x\in P_{n}\setminus
Z_{n+1}$. From (c) and (e) it follows that there is a unique index
$t\hat{\,}k\in\omega^{n+1}$ with $x\in\overline{U_{t\hat{\,}k}}\subset P_{n}$.
According to (f), we have

$\overline{P_{n+1}}\cap\overline{U_{t\hat{\,}k}}=\\{x_{t\hat{\,}k}\\}\cup\bigcup\left\\{\overline{U_{t\hat{\,}k\hat{\,}i}}\colon
i\in\omega\right\\}.$

By (e), $x\in\overline{U_{t\hat{\,}k\hat{\,}j}}$ for some $j$. Then $x\in
P_{n+1}$, a contradiction.

Thus, the set $Z_{n}\cup P_{n}$ is closed in $X$ for each $n$.

By construction, $Z\subset Z_{n}\cup P_{n}$ for each $n$. Then
$\overline{Z}\subset\bigcap\\{Z_{n}\cup P_{n}\colon n\in\omega\\}$.

Striving for a contradiction, suppose there is a point
$x\in\overline{Z}\setminus Z$. Fix the least $i$ with $x\in X_{i}$. Obviously,
$x\notin Z_{i}$. Condition (b) implies that $P_{i}\cap X_{i}=\emptyset$. Then
$x\notin Z_{i}\cup P_{i}$, a contradiction. ∎

Let us recall the well-known theorem of van Douwen [2]. For a first countable
regular space $X$, he proved that every closed subspace of $X$ is a Baire
space if and only if $X$ has no countable closed crowded subspace. Here a
space is said to be crowded if it has no isolated points. Note that up to
homeomorphism the space of rational numbers is the only countable first-
countable crowded regular space [11]. In other words, for a first countable
regular space $X$, van Douwen proved that $X$ is completely Baire if and only
if the space of rational numbers does not embed as a closed subspace into $X$.

Now we obtain a generalization of this result.

######  Theorem 2.

A hereditary weakly Preiss-Simon regular space $X$ is completely Baire if and
only if the space of rational numbers does not embed as a closed subspace into
$X$.

###### Proof.

Clearly, if $X$ contains a closed copy of the space of rational numbers, then
it cannot be completely Baire. For the other direction, aiming for a
contradiction, suppose that $X$ is not completely Baire. Then $X$ contains a
closed subset $F$ of the first category. It remains to apply Theorem 1 to $F$.
∎

######  Lemma 2.

Let $f\colon X\rightarrow\mathbb{Q}$ be a continuous mapping of a Baire space
$X$ onto the space of rational numbers $\mathbb{Q}$.

Then there is an open set $U\subset X$ such that $f(U)$ is not resolvable.

###### Proof.

Denote by $X^{\circ}_{q}$ the interior of $f^{-1}(q)$ for every point
$q\in\mathbb{Q}$. Put $\mathbb{Q}^{\circ}=\\{q\in\mathbb{Q}\colon
X^{\circ}_{q}\neq\emptyset\\}$.

Let us verify that the open set $A=\bigcup\\{X^{\circ}_{q}\colon
q\in\mathbb{Q}^{\circ}\\}$ is dense in $X$. Suppose the contrary and choose a
nonempty open set $W\subset X$ missing $A$. For each point
$q\in\mathbb{Q}\setminus\mathbb{Q}^{\circ}$ the intersection $f^{-1}(q)\cap W$
is a closed nowhere dense subset of $W$. For each point
$q\in\mathbb{Q}^{\circ}$ the intersection $f^{-1}(q)\cap W=(f^{-1}(q)\setminus
A)\cap W$ is the same. This implies that $W$ is a set of the first category on
itself. On the other hand, $W$ is a Baire space as an open subset of the Baire
space $X$, a contradiction.

Then $\mathbb{Q}^{\circ}=f(A)$ is dense in $\mathbb{Q}$ because $f$ is
continuous.

Take $B\subset\mathbb{Q}^{\circ}$ such that $B$ is dense, co-dense in
$\mathbb{Q}$. Clearly, $B$ is not a resolvable subset of $\mathbb{Q}$. The
open set $U=\bigcup\\{X^{\circ}_{q}\colon q\in B\\}$ is as required. ∎

Now we are ready to give the main result of the paper.

######  Theorem 3.

Let $f\colon X\rightarrow Y$ be a continuous mapping of a completely Baire
space $X$ onto a hereditary weakly Preiss-Simon regular space $Y$ such that
the image of every open subset of $X$ is a resolvable set in $Y$. Then $Y$ is
completely Baire.

###### Proof.

Striving for a contradiction, suppose that $Y$ is not completely Baire. By
Theorem 2, $Y$ contains a closed copy $Q$ of the space of rational numbers.
The inverse image $f^{-1}(Q)$ is a Baire space as a closed subset of $X$. By
Lemma 2, there is an open in $f^{-1}(Q)$ set $U\subset f^{-1}(Q)$ such that
$f(U)$ is not resolvable. Take an open set $U^{*}\subset X$ with $U^{*}\bigcap
f^{-1}(Q)=U$. Then $f(U^{*})$ is not resolvable because the closed subset
$Q\bigcap f(U^{*})=f(U)$ of a resolvable set must be resolvable. A
contradiction. ∎

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