# Trivial extensions defined by Prüfer conditions

C. Bakkari  Department of Mathematics, Faculty of Sciences and Technology, P.
O. Box 2202, University S. M. Ben Abdellah, Fez 30000, Morocco
<EMAIL_ADDRESS>,  S. Kabbaj  Department of Mathematics, King Fahd
University of Petroleum & Minerals, P. O. Box 5046, Dhahran 31261, KSA
<EMAIL_ADDRESS>and  N. Mahdou  Department of Mathematics, Faculty of
Sciences and Technology, P. O. Box 2202, University S. M. Ben Abdellah, Fez
30000, Morocco<EMAIL_ADDRESS>
###### Abstract.

This paper deals with well-known extensions of the Prüfer domain concept to
arbitrary commutative rings. We investigate the transfer of these notions in
trivial ring extensions (also called idealizations) of commutative rings by
modules and then generate original families of rings with zerodivisors subject
to various Prüfer conditions. The new examples give further evidence for the
validity of Bazzoni-Glaz conjecture on the weak dimension of Gaussian rings.
Moreover, trivial ring extensions allow us to widen the scope of validity of
Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

###### Key words and phrases:

Prüfer domain, semihereditary ring, arithmetical ring, Gaussian ring, Prüfer
ring, weak global dimension, Gaussian polynomial, content ideal, trivial ring
extension

###### 2000 Mathematics Subject Classification:

13F05, 13B05, 13A15, 13D05, 13B25

The second author was supported by KFUPM under Research Project # MS/RING/368.
The third author would like to thank KFUPM for its hospitality

##  1\. Introduction

All rings considered in this paper are commutative with identity elements and
all modules are unital. In 1932, Prüfer introduced and studied integral
domains in which every non-zero finitely generated ideal is invertible [24].
In 1936, Krull [20] named these rings after H. Prüfer and stated equivalent
conditions that make a domain Prüfer. Since then, “Prüfer domains have assumed
a central role in the development of multiplicative ideal theory through
numeral equivalent forms. These touched on many areas of commutative algebra,
e.g., valuation theory, arithmetic relations on the set of ideals,
$*$-operations, and polynomial rings; in addition to several homological
characterizations” (Gilmer [9]).

The extension of this concept to rings with zerodivisors gave rise to five
classes of Prüfer-like rings featuring some homological apsects (Bazzoni-Glaz
[2] and Glaz [13]); namely: (1) Semihereditary ring, i.e., every finitely
generated ideal is projective (Cartan-Eilenberg [5]); (2) Ring with weak
dimension at most one (Glaz [10, 12]); (3) Arithmetical ring, i.e., every
finitely generated ideal is locally principal (Fuchs [6]); (4) Gaussian ring,
i.e., $c(fg)=c(f)c(g)$ for any polynomials $f,g$ with coefficients in the
ring, where $c(f)$ denotes the content of $f$ (Tsang [26]); (5) Prüfer ring,
i.e., every finitely generated regular ideal is invertible or, equivalently,
projective (Butts-Smith [4] and Griffin [15]).

While, in the domain context, all these forms coincide with the definition of
a Prüfer domain, Glaz [13] provided examples which show that all these notions
are distinct in the context of arbitrary rings. The following diagram of
implications summarizes the relations between them [2, 3, 12, 13, 21, 22, 26]:

Semihereditary $\Rightarrow$ weak dimension $\leq 1$ $\Rightarrow$
Arithmetical $\Rightarrow$ Gaussian $\Rightarrow$ Prüfer

It is notable that original examples -for each one of the above classes- are
rare in the literature. This paper investigates the transfer of the above-
mentioned Prüfer conditions to trivial ring extensions. Our results generate
new examples which enrich the current literature with new families of Prüfer-
like rings with zerodivisors. Particularly, we obtain further evidence for the
validity of Bazzoni-Glaz conjecture sustaining that “the weak dimension of a
Gaussian ring is 0, 1, or $\infty$” [3]. Moreover, trivial ring extensions
offer the possibility to widen the scope of validity of the content conjecture
of Kaplansky and Tsang which was extended to investigate rings where “every
Gaussian polynomial has locally principal content ideal” [1, 2, 14, 16, 21,
22, 26]. Notice that both conjectures share the common context of rings with
zerodivisors. This very fact lies behind our motivation for studying the
Gaussian condition and related concepts in trivial ring extensions.

Let $A$ be a ring and $E$ an $A$-module. The trivial ring extension of $A$ by
$E$ (also called the idealization of $E$ over $A$) is the ring
$R:=A\propto~{}E$ whose underlying group is $A\times E$ with multiplication
given by $(a,e)(a^{\prime},e^{\prime})=(aa^{\prime},ae^{\prime}+a^{\prime}e)$.
For the reader’s convenience, recall that if $I$ is an ideal of $A$ and
$E^{\prime}$ is a submodule of $E$ such that $IE\subseteq E^{\prime}$, then
$J:=I\propto E^{\prime}$ is an ideal of $R$; ideals of $R$ need not be of this
form [19, Example 2.5]. However, prime (resp., maximal) ideals of $R$ have the
form $p\propto E$, where $p$ is a prime (resp., maximal) ideal of $A$ [17,
Theorem 25.1(3)]. Suitable background on commutative trivial ring extensions
is [10, 17].

Section 2 deals with trivial ring extensions of the form $R:=A\propto~{}B$,
where $A\subseteq B$ is an extension of integral domains. Precisely, we
examine the possible transfer of Prüfer ring conditions to $R$. The main
result asserts that “$R$ is Gaussian (resp., Arithmetical) if and only if $A$
is Prüfer with $K\subseteq B$ (resp., $K=B$).” This generates new examples of
non-arithmetical Gaussian rings as well as arithmetical rings with weak
dimension strictly greater than one. Recall that classical examples of non-
semihereditary arithmetical rings stem from Jensen’s 1966 result [18] as non-
reduced principal rings, e.g., $\mathbb{Z}/n^{2}\mathbb{Z}$ for any integer
$n\geq 2$. In this line, we provide a new family of examples of non-finite
conductor arithmetical rings, hence quite far from being principal. We also
establish a result on the weak dimension of these constructions which happens
to corroborate Bazzoni-Glaz conjecture (cited above).

In their recent paper devoted to Gaussian properties, Bazzoni and Glaz have
proved that a Prüfer ring satisfies any of the other four Prüfer conditions if
and only if its total ring of quotients satisfies that same condition [3,
Theorems 3.3 & 3.6 & 3.7 & 3.12]. This fact narrows the scope of study to the
class of total rings of quotients. Section 3 investigates Prüfer conditions in
a special class of total rings of quotients; namely, those arising as trivial
ring extensions of local rings by vector spaces over the residue fields. The
main result establishes that “if $(A,M)$ is a non-trivial local ring and $E$ a
nonzero $\frac{A}{M}$-vector space, then $R:=A\propto~{}E$ is a non-
arithmetical total ring of quotients. Moreover, $R$ is a Gaussian ring if and
only if so is $A$.” This enables us to build new examples of non-arithmetical
Gaussian total rings of quotients or non-Gaussian total rings of quotients
(which are necessarily Prüfer). Further the weak dimension of these
constructions turns out to be infinite which subjects, here too, the new
examples of Gaussian rings to Bazzoni-Glaz conjecture.

A problem initially associated with Kaplansky and his student Tsang [1, 2, 14,
22, 26] and also termed as Tsang-Glaz-Vasconcelos conjecture in [16] sustained
that “every nonzero Gaussian polynomial over a domain has an invertible (or,
equivalently, locally principal) content ideal.” It is well-known that a
polynomial over any ring is Gaussian if its content ideal is locally
principal. The converse is precisely the object of Kaplansky-Tsang-Glaz-
Vasconcelos conjecture extended to those rings where “every Gaussian
polynomial has locally principal content ideal. The objective of Section 4 is
to validate this conjecture in a large family of rings distinct from the three
classes of arithmetical rings, of locally domains, and of locally
approximately Gorenstein rings, where the conjecture holds so far. This was
made possible by the main result which states that a trivial ring extension of
a domain by its quotient field satisfies the condition that “every Gaussian
polynomial has locally principal content ideal.” We end up with a conjecture
that equates the latter condition with the local irreducibility of the zero
ideal. This would offer an optimal solution to the Kaplansky-Tsang-Glaz-
Vasconcelos conjecture that recovers all previous results. The section closes
with a discussion -backed with examples- which attempts to rationalize this
statement.

##  2\. Extensions of domains

This section explores trivial ring extensions of the form $R:=A\propto~{}B$,
where $A\subseteq B$ is an extension of integral domains. Notice in this
context that $(a,b)\in R$ is regular if and only if $a\not=0$. The main result
(Theorem 2.1) examines the transfer of Prüfer conditions to $R$ and hence
generates new examples of non-arithmetical Gaussian rings and of arithmetical
rings with weak dimension $\gvertneqq 1$.

In 1969, Osofsky proved that the weak dimension of an arithmetical ring is
either $\leq 1$ or infinite [23]. In 2005, Glaz proved Osofsky’s result in the
class of coherent Gaussian rings [12, Theorem 3.3]. Recently, Bazzoni and Glaz
conjectured that “the weak dimension of a Gaussian ring is 0, 1, or $\infty$”
[3]. Theorem 2.1 validates this conjecture for the class of all Gaussian rings
emanating from these constructions. Moreover, Example 2.7 widens its scope of
validity beyond coherent Gaussian rings.

######  Theorem 2.1.

Let $A\subseteq B$ be an extension of domains and $K:=\operatorname{qf}(A)$.
Let $R:=A\propto~{}B$ be the trivial ring extension of $A$ by $B$. Then:
(1) $R$ is Gaussian if and only if $R$ is Prüfer if and only if $A$ is Prüfer
with $K\subseteq B$.
(2) $R$ is arithmetical if and only if $A$ is Prüfer with $K=B$.
(3) $\operatorname{w.dim}.dim(R)=\infty$.

The proof of the theorem involves the following lemmas of independent
interest.

######  Lemma 2.2.

Let $A$ be a ring, $E$ an nonzero $A$-module, and $R:=A\propto~{}E$. If $R$ is
Gaussian (resp., arithmetical), then so is $A$.

###### Proof.

Straightforward since the arithmetical and Gaussian properties are stable
under factor rings (here $A\cong\frac{R}{0\propto E}$). ∎

Notice that Lemma 2.2 does not hold for the Prüfer property as shown by
Example 2.8.

######  Lemma 2.3.

Let $K$ be a field, $E$ a nonzero $K$-vector space, and $R:=K\propto~{}E$.
Then $\operatorname{w.dim}.dim(R)=\infty$.

###### Proof.

Let $\\{f_{i}\\}_{i\in I}$ be a basis of the $K$-vector space $E$ and let
$J:=0\propto E$. Consider the $R$-map $R^{(I)}\buildrel u\over{\rightarrow}J$
defined by $u((a_{i},e_{i})_{i\in I})=(0,\sum_{i\in I}a_{i}f_{i})$. Clearly,
$\operatorname{Ker}(u)=0\propto E^{(I)}$. Here we are identifying $R^{(I)}$
with $A^{(I)}\propto E^{(I)}$ as $R$-modules. We have the exact sequence of
$R$-modules:

$0\rightarrow 0\propto E^{(I)}\rightarrow R^{(I)}\buildrel
u\over{\rightarrow}J\rightarrow 0.$

We claim that $J$ is not flat. Otherwise, by [25, Theorem 3.55], we obtain

$0\propto E^{(I)}=J^{(I)}=JR^{(I)}=(0\propto E^{(I)})\cap JR^{(I)}=(0\propto
E^{(I)})J=0,$

a contradiction. Therefore the above exact sequence yields

$\operatorname{fd}(J)=\operatorname{fd}(J^{(L)})\leq\operatorname{fd}(J)-1.$

This forces the flat dimension of $J$ and hence the weak dimension of $R$ to
be infinite. ∎

###### Proof of Theorem 2.1.

(1) We need only prove the following implications:

$R$ Prüfer $\Rightarrow$ $A$ Prüfer with $K\subseteq B$ $\Rightarrow$ $R$
Gaussian

Assume $R$ is a Prüfer ring. We wish to show first that $K\subseteq B$ in the
case when $A$ is local. Let $x\not=0\in A$ and let $I:=((x,0),(x,1))R$, a
finitely generated regular ideal of $R$. Then $I$ is invertible and hence
principal (since $R$ is local too). Write $I=(a,b)R$ for some $a\in A$ and
$b\in B$. Clearly, $a=ux$ for some invertible element $u$ in $A$, hence
$I=(ux,b)R=(x,u^{-1}b)R$. Further $(x,0)\in I$ yields $u^{-1}b=b^{\prime}x$
for some $b^{\prime}\in B$. It follows that
$I=(x,b^{\prime}x)R=(x,0)(1,b^{\prime})R=(x,0)R$ since $(1,b^{\prime})$ is
invertible. But $(x,1)\in I$ yields $1=xb^{\prime\prime}$ for some
$b^{\prime\prime}\in B$. Therefore $K\subseteq B$. Next suppose $A$ is not
necessarily local and let $q\in\operatorname{Spec}(B)$ and $p:=q\cap A$.
Clearly, $S:=(A\setminus p)\times 0$ is a multiplicatively closed subset of
$R$ with the feature that $\frac{r}{1}$ is regular in $S^{-1}R$ if and only if
$r$ is regular in $R$. So finitely generated regular ideals of $S^{-1}R$
originate from finitely generated regular ideals of $R$. Hence $A_{p}\propto
B_{p}=S^{-1}R$ is a Prüfer ring. Whence $K=\operatorname{qf}(A_{p})\subseteq
B_{p}\subseteq B_{q}$. It follows that $K\subseteq B=\bigcap B_{q}$, where $q$
ranges over $\operatorname{Spec}(B)$, as desired. Now, one can easily check
that $K\subseteq B$ implies $K\propto B=Q(R)$, the total ring of quotients of
$R$. Moreover, let $f=\sum(k_{i},b_{i})x^{i}$ and
$g=\sum(k^{\prime}_{j},b^{\prime}_{j})x^{j}$ be two polynomials in $Q(R)[x]$.
If there is $i$ or $j$ such that $k_{i}\not=0$ or $k^{\prime}_{j}\not=0$, then
$(k_{i},b_{i})$ or $(k^{\prime}_{j},b^{\prime}_{j})$ is invertible, hence
$c(f)=Q(R)$ or $c(g)=Q(R)$, whence $c(fg)=c(f)c(g)$ (this is Gauss lemma which
asserts that a polynomial with unit content is Gaussian). If
$k_{i}=k^{\prime}_{j}=0$ for all $i$ and $j$, then $c(fg)=0=c(f)c(g)$.
Consequently, $Q(R)$ is a Gaussian ring and so is $R$ by [3, Theorem 3.3]. By
Lemma 2.2, $A$ is a Prüfer domain, completing the proof of the first
implication.

Assume $A$ is a Prüfer domain with $K\subseteq B$. Let $I$ be a nonzero
finitely generated ideal of $R$ minimally generated by
$(a_{1},b_{1}),\ldots,(a_{n},b_{n})$. For any $a\not=0\in A$ and any
$b,b^{\prime}\in B$, we have $(0,b^{\prime})=(a,b)(0,a^{-1}b^{\prime})$. So
minimality forces either $a_{i}=0$ for each $i$ or $a_{i}\not=0$ for each $i$.
In the first case, $I^{2}=0$ and hence $I$ is not a regular ideal. Next assume
$a_{i}\not=0$ for each $i$. It follows that $I=(\sum Aa_{i})\propto B$ since
$(a_{i},b)=(a_{i},b_{i})(1,a_{i}^{-1}(b-b_{i}))$ for each $i$ and any $b\in
B$. Since $A$ is a Prüfer domain, $J:=\sum Aa_{i}$ is invertible and $aJ^{-1}$
is an ideal of $A$ for some $a\not=0\in A$. $\begin{array}[]{lcl}\hbox{So
}(a,0)^{-1}(aJ^{-1}\propto B)I&=&(a,0)^{-1}(aJ^{-1}\propto B)(J\propto B)\\\
&=&(a,0)^{-1}(aJ^{-1}J\propto B)\\\ &=&(a,0)^{-1}(aA\propto B)\\\
&=&R.\end{array}$
Consequently, $R$ is a Prüfer ring and hence Gaussian by [3, Theorem 3.3],
completing the proof of (1).

(2) Assume $R$ is an arithmetical ring. By (1), $A$ is Prüfer with $K\subseteq
B$. So $K\propto B=Q(R)$ is arithmetical since it is a localization of $R$.
Let $b\not=0\in B$. Then $I:=((0,1),(0,b))Q(R)$ is principal and hence
$I:=(0,b^{\prime})Q(R)$ for some $b^{\prime}\not=0\in B$. Further $(0,b)\in I$
yields $b=kb^{\prime}$ for some some $k\not=0\in K$, and then
$I:=(0,k^{-1}b)Q(R)$. Moreover $(0,1)\in I$ yields $1=k^{\prime}k^{-1}b$ for
some $k^{\prime}\not=0\in K$. It follows that $b\in K$ and thus $K=B$.
Conversely, assume $A$ is Prüfer with $K=B$. By (1), $R=A\propto K$ is
Gaussian. Moreover $Q(R)=K\propto K$ is a principal ring (a fortiori
arithmetical) since it has a unique nonzero proper ideal $M:=0\propto
K=T(0,1)$. By [3, Theorem 3.5], $R$ is arithmetical, completing the proof of
(2).

(3) Let $S:=A\setminus\\{0\\}$. So $T:=S\times 0$ is a multiplicatively closed
subset of $R$. By Lemma 2.3,
$\operatorname{w.dim}.dim(T^{-1}R)=\operatorname{w.dim}.dim(K\propto
KB)=\infty$. So $\operatorname{w.dim}.dim(R)=\infty$. This completes the proof
of the theorem. ∎

The following corollary is an immediate consequence of Theorem 2.1.

######  Corollary 2.4.

Let $D$ be a domain, $K:=\operatorname{qf}(D)$, and $R:=D\propto~{}K$. Then
the following statements are equivalent:
(1) D is a Prüfer domain;
(2) $R$ is an arithmetical ring;
(3) $R$ is a Gaussian ring;
(4) $R$ is a Prüfer ring. ∎

Recall Jensen’s 1966 result: “for a ring $R$, $\operatorname{w.dim}.dim(R)\leq
1$ if and only if $R$ is arithmetical and reduced” [18]. Classical examples of
arithmetical rings with weak dimension $\gneqq 1$ stem from Jensen’s result as
non-reduced principal rings, e.g., $\mathbb{Z}/n^{2}\mathbb{Z}$ for any
integer $n\geq 2$. In this vein, Theorem 2.1 generates a new family of
examples quite far from being principal as shown below. For this purpose,
recall that a ring $R$ is finite conductor if $aR\cap bR$ and $(0:c)$ are
finitely generated for any $a,b,c\in R$ [11]. The class of finite conductor
rings properly contains the class of coherent (a fortiori, Noetherian and
hence principal) rings [11, 19].

######  Example 2.5.

Let $D$ be any Prüfer domain which is not a field and
$K:=\operatorname{qf}(D)$. Then $R:=D\propto~{}K$ is an arithmetical ring with
$\operatorname{w.dim}.dim(R)=\infty$. Moreover, $R$ is not a finite conductor
ring by [19, Thoerem 2.8] and hence not coherent.

Also, Theorem 2.1 enriches the literature with new examples of non-
arithmetical Gaussian rings, as shown below.

######  Example 2.6.

Let $K\subsetneqq L$ be a field extension. Then $R:=K\propto~{}L$ is a
Gaussian ring which is not arithmetical.

The next example shows that Theorem 2.1 widens the scope of validity of
Bazzoni-Glaz conjecture beyond the class of coherent Gaussian rings.

######  Example 2.7.

Let $\mathbb{Z}$ and $\mathbb{R}$ denote the ring of integers and field of
real numbers, respectively. Then $R:=\mathbb{Z}_{(2)}\propto~{}\mathbb{R}$
satisfies the following statements:
(1) $R$ is a Gaussian ring,
(2) $R$ is not an arithmetical ring,
(3) $R$ is not a coherent ring,
(4) $\operatorname{w.dim}.dim(R)=\infty$.

###### Proof.

Assertions (1), (2), and (4) hold by direct application of Theorem 2.1. It
remains to prove (3). Indeed, consider the following exact sequence over $R$

$0\rightarrow 0\propto~{}\mathbb{R}\rightarrow R\buildrel
u\over{\rightarrow}R(0,1)=0\propto~{}\mathbb{Z}_{(2)}\rightarrow 0$

where $u$ is defined by $u(a,b)=(a,b)(0,1)=(0,a)$. Now $0\propto~{}\mathbb{R}$
is not finitely generated as an $R$-module (otherwise $\mathbb{R}$ would be
finitely generated as a $\mathbb{Z}_{(2)}$-module). Hence
$0\propto~{}\mathbb{Z}_{(2)}$ is a finitely generated ideal of $R$ that is not
finitely presented. Whence $R$ is not coherent, as desired. ∎

The next example illustrates the failure of Theorem 2.1, in general, beyond
the context of domain extensions.

######  Example 2.8.

Let $(A,M)$ be a non-valuation local domain, $E$ a nonzero $A$-module with
$ME=0$, and $B:=A\propto~{}E$. Then $R:=A\propto~{}B$ is a Prüfer ring which
is not Gaussian.

###### Proof.

Indeed, one can easily check that $R$ is a total ring of quotients and hence a
Prüfer ring. By Lemma 2.2, $R$ is not Gaussian. ∎

##  3\. A class of total rings of quotients

In a recent paper devoted to Gaussian properties, Bazzoni and Glaz have proved
that a Prüfer ring satisfies any of the other four Prüfer conditions
(mentioned above) if and only if its total ring of quotients satisfies that
same condition [3, Theorems 3.3 & 3.6 & 3.7 & 3.12]. This fact narrows the
scope of study to the class of total rings of quotients.

This section investigates Prüfer conditions in a particular class of total
rings of quotients; namely, those arising as trivial ring extensions of local
rings by vector spaces over the residue fields. The main result (Theorem 3.1)
enriches the literature with original examples of non-arithmetical Gaussian
total rings of quotients as well as non-Gaussian total rings of quotients
(which are necessarily Prüfer). The theorem validates Bazzoni-Glaz conjecture
for the class of Gaussian rings emanating from these constructions.

######  Theorem 3.1.

Let $(A,M)$ be a local ring and $E$ a nonzero $\frac{A}{M}$-vector space. Let
$R:=A\propto~{}E$ be the trivial ring extension of $A$ by $E$. Then:
(1) $R$ is a total ring of quotients and hence a Prüfer ring.
(2) $R$ is Gaussian if and only if $A$ is Gaussian.
(3) $R$ is arithmetical if and only if $A:=K$ is a field and $\dim_{K}E=1$.
(4) $\operatorname{w.dim}.dim(R)\gvertneqq 1$. If $M$ admits a minimal
generating set, then $\operatorname{w.dim}.dim(R)=\infty$.

###### Proof.

(1) Straightforward.

(2) By Lemma 2.2, only the sufficiency has to be proved. Assume $A$ is a
Gaussian ring and let $F=\sum(a_{i},e_{i})x^{i}$ be a polynomial in $R[x]$. If
$a_{i}\notin M$ for some $i$, then $(a_{i},e_{i})$ is invertible in $R$, hence
$F$ is Gaussian. Now assume $a_{i}\in M$ for each $i$ and let
$G=\sum(a^{\prime}_{j},e^{\prime}_{j})x^{j}\in R[x]$. We may suppose, without
loss of generality, that $a^{\prime}_{j}\in M$ for each $j$. Let $f=\sum
a_{i}x^{i}$ and $g=\sum a^{\prime}_{j}x^{j}$ in $A[x]$. One can easily check
that $ME=0$ yields the following
$\begin{array}[]{lcl}c(FG)&=&c(fg)\propto c(fg)E\\\ &=&c(fg)\propto 0\\\
&=&c(f)c(g)\propto 0\\\ &=&c(F)c(G).\end{array}$
Therefore $F$ is Gaussian, as desired.

(3) Sufficiency is clear since $K\propto~{}K$ is a principal ring. Next assume
$R$ is an arithmetical ring. We claim that $A$ is a field. Deny and let
$a\not=0\in M$ and $e\not=0\in E$. Therefore the ideal $I:=R(a,0)+R(0,e)$ is
principal in $R$ (since $R$ is local). So $I=R(a^{\prime},e^{\prime})$ for
some $(a^{\prime},e^{\prime})\in R$. Clearly, $(a,0)\in I$ forces $a^{\prime}$
to be nonzero and belong to $M$. Further, $(0,e)\in I$ yields $ba^{\prime}=0$
and $e=be^{\prime}$ for some $b\in A$. Necessarily, $b\in M$ since
$a^{\prime}\not=0$. It follows that $e=be^{\prime}=0$, the desired
contradiction. Now, let $e,e^{\prime}$ be two nonzero vectors in $E$. Then
$I=R(0,e)+R(0,e^{\prime})$ is a principal ideal of $R$. Similar arguments used
in the proof of Theorem 2.1(2) yields $e=ke^{\prime}$ for some $k\in K$. So
that $\dim_{K}E=1$.

(4) Let $J:=0\propto E$ and let $\\{(f_{i})\\}_{i\in I}$ be a basis of the
$(A/M)$-vector space $E$. Consider the exact sequence of $R$-modules:

(1) $0\rightarrow Ker(u)\rightarrow R^{(I)}\buildrel
u\over{\rightarrow}J\rightarrow 0$

where $u((a_{i},e_{i})_{i\in I})=(0,\sum_{i\in I}a_{i}f_{i})$. Hence,
$Ker(u)=(M\propto E)^{(I)}$. Here, too, we identify $R^{(I)}$ with
$A^{(I)}\propto E^{(I)}$ as $R$-modules. We claim that $J$ is not flat.
Otherwise, by [25, Theorem 3.55], we obtain $J^{(I)}=(M\propto E)^{(I)}\cap
JR^{(I)}=J(M\propto E)^{(I)}=0$, absurd. By [25, Theorem 2.4],
$\operatorname{w.dim}.dim(R)\gvertneqq 1$. Next assume that $M$ admits a
minimal generating set. Then one can easily check that $M\propto E$ admits a
minimal generating set too. Let $(b_{i},g_{i})_{i\in L}$ denote a minimal
generating set of $M\propto E$ and consider the exact sequence of $R$-modules:

$0\rightarrow Ker(v)\rightarrow R^{(L)}\buildrel v\over{\rightarrow}M\propto
E\rightarrow 0$

where $v((a_{i},e_{i})_{i\in L})=\sum_{i\in L}(a_{i},e_{i})(b_{i},g_{i})$. The
minimality assumption yields (see the proof of [25, Lemma 4.43])

$Ker(v)\subseteq(M\propto E)^{(L)}.$

It follows that $\operatorname{Ker}(v)=V\propto E^{(L)}=(V\propto 0)\oplus
J^{(L)}$, where

$V:=\\{(a_{i})_{i\in L}\in M^{(L)}\ |\sum_{i\in L}a_{i}b_{i}=0\\}.$

We obtain

(2) $\operatorname{fd}\big{(}(V\propto 0)\oplus
J^{(L)}\big{)}\leq\operatorname{fd}(M\propto E).$

On the other hand, from the exact sequence in (1) we get

(3) $\operatorname{fd}(M\propto E)=\operatorname{fd}(M\propto
E)^{(I)}\leq\operatorname{fd}(J)-1.$

A combination of (2) and (3) yields
$\operatorname{fd}(J)\leq\operatorname{fd}(J)-1$. Consequently, the flat
dimension of $J$ (and a fortiori the weak dimension of $R$) has to be
infinite, completing the proof of the theorem. ∎

Theorem 3.1 generates new and original examples of rings with zerodivisors
subject to Prüfer conditions as shown below.

######  Example 3.2.

Let $(V,M)$ be a non-trivial valuation domain. Then
$R:=V\propto~{}\frac{V}{M}$ is a non-arithmetical Gaussian total ring of
quotients.

######  Example 3.3.

Let $K$ be a field and $E$ a $K$-vector space with $\dim_{K}E\geq 2$. Then
$R:=K\propto~{}E$ is a non-arithmetical Gaussian total ring of quotients.

######  Example 3.4.

Let $(A,M)$ be a non-valuation local domain. Then $R:=A\propto~{}\frac{A}{M}$
is a non-Gaussian total ring of quotients.

Recently, Bazzoni and Glaz proved that a Gaussian ring, with a maximal ideal
$M$ such that the nilradical of $R_{M}$ is non-null and nilpotent, has
infinite weak dimension [3, Theorem 6.4]. The next example widens the scope of
validity of Bazzoni-Glaz conjecture as well as illustrates the setting of this
result beyond coherent Gaussian rings.

######  Example 3.5.

Let $\mathbb{R}$ denote the field of real numbers and $x$ an indeterminate
over $\mathbb{R}$. Then $R:=\mathbb{R}\propto~{}\mathbb{R}[x]$ satisfies the
following statements:
(1) $R$ is a Gaussian ring,
(2) $R$ is not an arithmetical ring,
(3) $R$ is not a coherent ring,
(4) $R$ is local with nonzero nilpotent maximal ideal,
(5) $\operatorname{w.dim}.dim(R)=\infty$.

###### Proof.

Assertions (1) and (2) hold by direct application of Theorem 3.1. Assertion
(3) is handled by [19, Theorem 2.6(2)]. Clearly, (4) holds since the maximal
ideal of $R$ is $M:=0\propto~{}\mathbb{R}[x]$ (by [17, Theorem 25.1(3)]) with
$M^{2}=0$. Finally, (5) is satisfied by Theorem 3.1(4), [3, Proposition 6.3],
or [3, Theorem 6.4]. ∎

##  4\. Kaplansky-Tsang-Glaz-Vasconcelos conjecture

Let $R$ be a ring and $Q(R)$ its total ring of quotients. An ideal $I$ of $R$
is said to be invertible if $II^{-1}=R$, where $I^{-1}:=\\{x\in Q(R)\ |\
xI\subseteq R\\}$. A nonzero ideal is invertible if and only if it is regular,
finitely generated, and locally principal. Particularly, for finitely
generated ideals of domains, invertibility coincides with the locally
principal condition. A polynomial $f$ over $R$ is said to be Gaussian if
$c(fg)=c(f)c(g)$ holds for any polynomial $g$ over $R$.

A problem initially associated with Kaplansky and his student Tsang [1, 2, 14,
22, 26] and also termed as Tsang-Glaz-Vasconcelos conjecture in [16] sustained
that “every nonzero Gaussian polynomial over a domain has an invertible (or,
equivalently, locally principal) content ideal.” It is well-known that a
polynomial over any ring is Gaussian if its content ideal is locally
principal. The converse is precisely the object of Kaplansky-Tsang-Glaz-
Vasconcelos conjecture extended to those rings where “every Gaussian
polynomial has locally principal content ideal.”

Notice for convenience that the conjecture has a local character since the
Gaussian condition is a local property (i.e., a polynomial is Gaussian over a
ring $R$ if and only if its image is Gaussian over $R_{M}$ for each maximal
ideal $M$ of $R$). It is this very fact that has enabled a natural extension
of the conjecture from domains to rings (recall, for instance, that a Von
Neumann regular ring is locally a field).

Significant progress has been made on this conjecture. Glaz and Vasconcelos
proved it for normal Noetherian domains [14]. Then Heinzer and Huneke
established its veracity over locally approximately Gorenstein rings (see
definition below) and over locally Noetherian domains [16, Theorem 1.5 &
Corollary 3.4]. Recently, Loper and Roitman settled the conjecture for
(locally) domains [21, Theorem 4], and then Lucas extended their result to
arbitrary rings by restricting to polynomials with regular content [22,
Theorem 6]. Obviously, the conjecture is true in arithmetical rings. Moreover,
trivial ring extensions offer the possibility to widen the scope of its
validity to a large family of rings distinct from the above contexts. This
gives birth to a new class of rings containing strictly the three classes of
arithmetical rings, of locally domains, and of locally approximately
Gorenstein rings (see Figure 1). We term the new concept as follows:

######  Definition 4.1.

A ring $R$ is pseudo-arithmetical if every Gaussian polynomial over $R$ has
locally principal content ideal.

We first prove a transfer result (Theorem 4.2) on trivial ring extensions.
Then Conjecture 4.5 will equate the pseudo-arithmetical notion with the local
irreducibility of the zero ideal. If true, this conjecture would offer an
optimal solution to the Kaplansky-Tsang-Glaz-Vasconcelos conjecture that
recovers all previous results.

######  Theorem 4.2.

(1) Let $R:=A\propto~{}K$ be the trivial ring extension of a domain $A$ by its
quotient field $K$. Then $R$ is a pseudo-arithmetical ring.
(2) Let $A\subseteq B$ be an extension of rings and $R:=A\propto~{}B$. If $R$
is a pseudo-arithmetical ring, then so is $A$.

###### Proof.

(1) Let $F:=\sum(a_{i},k_{i})x^{i}$ be a nonzero Gaussian polynomial in
$R[x]$. Assume $a_{i}\not=0$ for some $i$. Then $(a_{i},k_{i})$ is regular in
$R$ and so is $c(F)$ in $R$. Hence the Gaussian property forces $F$ to be
regular in $R[x]$. Then $c(F)$ is locally principal by [22, Theorem 6]. Next
assume $a_{i}=0$ for each $i$. Let $a$ be a nonzero element of $A$ such that
$ak_{i}\in A$ for each $i$ and set $F^{\prime}:=(a,0)F=\sum(0,ak_{i})x^{i}$ in
$R[x]$. We claim that $f^{\prime}:=\sum ak_{i}x^{i}$ is a (nonzero) Gaussian
polynomial of $A[x]$. Indeed, consider $g=\sum a^{\prime}_{i}x^{i}\in A[x]$
and set $G:=\sum(a^{\prime}_{i},0)x^{i}$ in $R[x]$. Then $0\propto
c(f^{\prime}g)=c(F^{\prime}G)=c(F^{\prime})c(G)$. Moreover,
$c(F^{\prime})=\sum R(0,ak_{i})=0\propto c(f^{\prime})$ and $c(G)=c(g)\propto
K$ (see proof of Lemma 2.2). It follows that $0\propto c(f^{\prime}g)=0\propto
c(f^{\prime})c(g)$ and hence $c(f^{\prime}g)=c(f^{\prime})c(g)$. Whence
$c(f^{\prime})$ is locally principal since $A$ is a domain [21]. Let
$P:=p\propto K\in\operatorname{Max}(R)$ for some maximal ideal $p$ of $A$ and
set $S:=(A\setminus p)\times 0\subseteq R\setminus P$. Since
$c(f^{\prime})A_{p}=a^{\prime}A_{p}$ for some $a^{\prime}\in A$, we get
$\begin{array}[]{lcl}(a,0)c(F)R_{P}&=&c(F^{\prime})R_{P}\\\ &=&\big{(}0\propto
c(f^{\prime})\big{)}R_{P}\\\ &=&\big{(}S^{-1}(0\propto
c(f^{\prime}))\big{)}R_{P}\\\ &=&\big{(}0\propto
c(f^{\prime})A_{p}\big{)}R_{P}\\\ &=&\big{(}0\propto
a^{\prime}A_{p}\big{)}R_{P}\\\ &=&(0,a^{\prime})R_{P}\\\
&=&(a,0)(0,\frac{a^{\prime}}{a})R_{P}.\end{array}$
Consequently, $c(F)R_{P}=(0,\frac{a^{\prime}}{a})R_{P}$ since $(a,0)$ is
regular in $R$. Thus $c(F)$ is locally principal and therefore $R$ is a
pseudo-arithmetical ring.

(2) Let $f=\sum a_{i}x^{i}$ be a Gaussian polynomial over $A$ and set
$F:=\sum(0,a_{i})x^{i}$. Let $G=\sum(a^{\prime}_{i},b_{i})x^{i}\in R[x]$ and
set $g:=\sum a^{\prime}_{i}x^{i}$ in $A[x]$. Since $f$ is Gaussian, we have
$c(F)c(G)=(0\propto c(f))c(G)=0\propto c(f)c(g)=0\propto c(fg)$. On the other
hand, one can see that $c(FG)=0\propto c(fg)$. Therefore, $c(FG)=c(F)c(G)$,
hence $F$ is a Gaussian polynomial over $R$. So $c(F)=0\propto I$ is a locally
principal ideal of $R$ where $I:=c(f)$. Now ape the proof of the arithmetical
statement in Lemma 2.2, to get that $I$ is locally principal, as desired. ∎

Obviously, a ring is arithmetical if and only if it is Gaussian and pseudo-
arithmetical. In this context, note that Examples 2.6 & 3.3 illustrate the
failure of Theorem 4.2(1) for trivial ring extensions $R:=A\propto~{}E$ with
$E\not=\operatorname{qf}(A)$.

######  Example 4.3.

Let $(A,M)$ be a local ring which is not a field and $E$ a nonzero vector
space over $\frac{A}{M}$. Then $R:=A\propto~{}E$ is a Prüfer ring which is not
pseudo-arithmetical. Indeed, Theorem 3.1 ensures that $R$ is a non-
arithmetical total ring of quotients (hence Prüfer). We claim that the
polynomial $f:=(a,0)+(0,e)x$, where $a\not=0\in M$ and $e\not=0\in E$, is
Gaussian but $c(f)$ is not principal in $R$. To see this, let $g\in R[x]$. If
$g\notin(M\times~{}E)[x]$, then Gauss lemma ensures that $c(fg)=c(f)c(g)$
since $R$ is local with ideal maximal $M\propto~{}E$. Assume
$g\in(M\times~{}E)[x]$. Then $ME=0$ yields

$c(f)c(g)=(a,0)c(g)=c((a,0)g)=c(fg).$

Now ape the proof of Theorem 3.1(3), to obtain that $c(f)$ is not principal,
and therefore $R$ is not pseudo-arithmetical.

######  Remark 4.4.

(1) Now pick any non-Prüfer domain $A$ with $K:=\operatorname{qf}(A)$ and
consider the trivial extension $R:=A\propto~{}K$. Then by Corollary 2.4, $R$
is not a Prüfer ring (a fortiori, $R$ is not arithmetical). Moreover, there
are plenty of non-regular Gaussian polynomials over $R$, e.g.,
$f:=\sum(0,k_{i})x^{i}$. However, Theorem 4.2 ensures that every Gaussian
polynomial over $R$ has locally principal content ideal (i.e., $R$ is pseudo-
arithmetical).

(2) Next we examine the Noetherian case. From [16], a local ring $(R,M)$ is
said to be approximately Gorenstein if $R$ is Noetherian and for every integer
$n>0$ there is an ideal $I\subseteq M^{n}$ such that $R/I$ is Gorenstein
(e.g., any local Noetherian ring $(R,M)$ with the $M$-adic completion
$\hat{R}$ reduced). Heinzer and Huneke proved that every locally approximately
Gorenstein ring is pseudo-arithmetical [16, Theorem 1.5]. This result combined
with [16, Remark 1.6] asserts that Noetherianity has no direct effect on the
pseudo-arithmetical notion even in low dimension, in the sense that non-
Gorenstein Artinian local rings are not pseudo-arithmetical. Finally, notice
that the above example $R:=A\propto~{}K$ is not Noetherian since it is not
coherent by [19, Theorem 2.8].

(3) From [12], a ring $R$ is called a PF ring if all principal ideals of $R$
are flat, or, equivalently, if $R$ is locally a domain [10, Theorem 4.2.2(3)].
A ring $R$ is called a PP ring or a weak Baer ring if all principal ideals of
$R$ are projective. In the class of Gaussian rings, the PP and PF properties
coincide, respectively, with the notions of semihereditary ring and ring with
weak dimension at most 1. Clearly, note that the above example
$R:=A\propto~{}K$ is not locally a domain.

In view of Example 4.3 and Remark 4.4, Figure 1 summarizes the relations
between all these classes of rings where the implications are irreversible in
general.

Figure 1.  Pseudo-arithmetical rings in perspective

From the above discussion, it turns out that the pseudo-arithmetical notion
must have a characterization that accommodates the three disparate classes of
arithmetical rings, of locally domains, and of locally approximately
Gorenstein rings (see Figure 1). This new characterization will offer a “happy
end” to the Kaplansky-Tsang-Glaz-Vasconcelos conjecture. In this vein, we
conjecture the following:

######  Conjecture 4.5.

A ring $R$ is pseudo-arithmetical if and only if the zero ideal is locally
irreducible.

######  Remark 4.6.

(1) Fuchs, Heinzer and Olberding have recently studied irreducibility in
commutative rings [7, 8] and noticed that “it is readily seen that a ring $R$
is an arithmetical ring if and only if for each proper ideal $I$ of $R$,
$I_{M}$ is an irreducible ideal of $R_{M}$ for every maximal ideal $M$ of $R$
containing $I$ [7].”

(2) Assume that Conjecture 4.5 is true. If $R$ is locally a domain or locally
approximately Gorenstein, then a polynomial over $R$ is Gaussian if and only
if its content is locally principal [21, Theorem 4] & [16, Theorem 1.5]. In
particular, a nonzero polynomial over an integral domain is Gaussian if and
only if its content is invertible. Indeed the locally domain statement follows
from the obvious fact that the zero ideal in a domain is irreducible. Next
assume $R$ is locally approximately Gorenstein. Recall that a Gaussian
polynomial $f:=\sum a_{i}x^{i}$ over a ring $R$ forces its image
$\overline{f}:=\sum\overline{a_{i}}x^{i}$ to be Gaussian over $R/I$, for every
ideal $I$ of $R$. Using this fact and the fact that the Gaussian condition is
a local property, in combination with the definition of a locally
approximately Gorenstein ring, Heinzer and Huneke showed that the proof
reduces to the case where $R$ is a zero-dimensional local Gorenstein ring (see
the beginning of the proof of [16, Theorem 1.5]). But in this setting the zero
ideal is irreducible, as desired.

## References

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# Critical-point phenomena and finite-size scaling in mean-field equal-
coupling photonic networks

Oliver Melchert<EMAIL_ADDRESS>Leibniz Universität Hannover,
Institute of Quantum Optics, Welfengarten 1, 30167 Hannover, Germany  Cluster
of Excellence PhoenixD (Photonics, Optics, and Engineering - Innovation Across
Disciplines), Welfengarten 1, 30167 Hannover, Germany

###### Abstract

The mean-field optical phase transition in multimode equal-coupling photonic
networks is studied by temporal evolution of the nonlinear equations of motion
of the coupled modes. Analogies to statistical mechanics models of interacting
classical spins, built upon the correspondence between complex-valued modes
and two-component spins, are employed to define two-component and single-
component order parameters. A comprehensive finite-size scaling analysis is
performed to estimate critical points and exponents of a second-order phase
transition, driven by the optical energy per mode. Equilibrium properties of
the system are compared to exact results whenever applicable. Considering
various parameter settings, our results confirm the mean-field nature of the
transition and establish the critical line in the nonlinearity–energy-density
phase diagram. Critical scaling leads to infer the upper critical dimension
$d_{c}=4$. Connection to thermodynamic quantities is established by means of a
kinetic temperature with appropriate zero-temperature limit. In the low
temperature phase (low energy per mode), spins align. In the high temperature
phase (large energy per mode), spins rotate independent of one another.

##  I  Introduction

In statistical mechanics it is usually the goal to describe system behavior in
the thermodynamic limit, i.e. under the condition that the size of an
elementary building block of a system is miniscule in comparison to the
overall size of the system [1, 2, 3]. Much effort is thus spend on designing
algorithms and numerical strategies that permit to efficiently simulate
systems with a large but finite number $N$ of building blocks [4, 2, 3], and
to reliably extrapolate towards $N\to\infty$, e.g., employing the heuristical
statistical mechanics theory of finite-size scaling (FSS) [5, 6, 1, 7]. In
case of photonic networks, however, systems of interest naturally exhibit a
finite number of building blocks, and effects caused by their finite size may
even become an important feature. This is especially important for the study
of collective phenomena that require a system-wide cooperation of building
blocks, such as phase transitions.

Based on these considerations we study fully connected, equal-coupling
photonic networks (ECPNs), consisting of $N$ complex-valued modes that are
subject to nonlinear effects and interact pairwise via uniform couplings. For
such long-range interacting ECPNs, recent studies provided evidence for a
second-order phase transition [8]. This optical phase transition can be
expressed in terms of statistical mechanics spin-model terminology, if a modes
phase is interpreted as a two-component spin vector, and its amplitude as the
length of that vector. This geometric picture exploits analogies to models of
interacting classical spins [1, 3, 7], as, e.g., the XY model [9, 10, 11, 12,
13]. The analogy of light propagating through nonlinear two-dimensional (2D)
arrays of weakly coupled waveguides and the 2D short-range interacting XY
model has previously been used to study an optical analog of the Berezinskii-
Kosterlitz-Thouless transition [14]. On long-range interacting systems, the XY
model exhibits a temperature driven, continuous phase transition with a
ferromagnetic phase at low temperature and a paramagnetic phase at high
temperature [12, 13]. In case of the fully connected ECPN model, working in
the microcanonical ensemble, the transition is observed as a function of the
energy density per mode. If the energy per mode is large, the system is in a
disordered state in which the angular degrees of freedom of the spins are
active and where there is no orientational order amongst the spins. If the
average energy in each mode decreases, the system undergoes a second order
phase transition to an ordered phase [8], in which alignment forces induce
global order. Previously, Ref. [8] demonstrated the role of the nonlinearity
in achieving this optical phase transition and reported the critical point and
verified the mean-field scaling of a magnetization-like order parameter for a
particular parameter setting.

Here, we perform a more detailed analysis, based on numerical simulations of
ECPNs of finite size, employing FSS to thoroughly characterize the optical
phase transition in terms of critical points and critical exponents. Since
fully connected ECPNs represent a long-range interacting system, the
transition can be expected to be in the mean-field universality class,
consistent with the mean-field theory for photonic networks developed in Ref.
[8]. Due to the inevitably finite number of building blocks of real photonic
systems, a thorough study of finite-size effects in such systems seems
rewarding.

In Sec. II we introduce equal-coupling photonic networks in more detail, and
we introduce classical statistical mechanics quantities that allow to
characterize the state of the system. Section III contains a detailed analysis
of equilibrium features of the model and of the optical phase transition,
focusing on a parameter range considered previously in Ref. [8]. We first
discuss emergence and loss of order in the system (Sec. III.1), demonstrating
that a symmetry of the Hamiltonian enables coherent phase-rotations in the
ordered phase, and we derive exact results for time-averaged quantities in the
disordered phase, even accounting for finite-size effects. Subsequently we
employ FSS to precisely locate the critical point and determine the critical
exponents from information on systems of finite size (Sec. III.3). Correlation
times (Sec. III.2), and scaling at criticality (Sec. III.4) are discussed
separately. In Sec. III.5 we utilize a kinetic Temperature based on the
angular momenta of the spins to analyze the thermodynamic signature of the
transition within the considered dynamical system. Finally, we summarize
results obtained for various other parameter settings in terms of a phase
diagram and make conclusions in Sec. IV.

##  II  Model and Methods

Fully connected, equal coupling photonic networks, consisting of $N$ complex-
valued modes $\Psi\equiv(\psi_{1},\ldots,\psi_{N})$ that interact pairwise via
uniform couplings $J_{0}>0$, are described by the Hamiltonian [8]

$\mathcal{H}[\Psi]=-\frac{J_{0}}{N}\sum_{\ell,j\neq\ell}\psi_{\ell}^{*}\psi_{j}+\frac{\chi}{2}\sum_{\ell}|\psi_{\ell}|^{4},$
(1)

where the second term models a Kerr-type nonlinearity with nonlinearity
parameter $\chi>0$. The total optical power of a mode configuration $\Psi$
reads

$\mathcal{A}[\Psi]=\sum_{\ell}|\psi_{\ell}|^{2},$ (2)

and the time evolution of the system (1) is governed by the nonlinear
equations of motion

$\displaystyle
i\dot{\psi}_{\ell}=-\frac{J_{0}}{N}\sum_{j\neq\ell}\psi_{j}+\chi|\psi_{\ell}|^{2}\psi_{\ell},$
(3)

i.e. $N$ coupled first-order differential equations where the dot denotes
derivative with respect to time. They conserve the energy density
$h\equiv\mathcal{H}[{\Psi}]/N$ and the optical power per mode
$a\equiv\mathcal{A}[\Psi]/N$. The norms $|\psi_{\ell}|$ of the individual
modes are not conserved. The equations of motion (3) are obtained in the
framework of time-dependent coupled mode theory [8, 15]. Equation (1) is
invariant under the transformation
$\psi_{\ell}\to\psi_{\ell}\exp\\{i\theta\\}$ for $\ell=1,\ldots,N$, indicating
a continuous symmetry that causes degeneracy of distinct states: a global
phase-rotation $\theta$ will conserve the energy density $h$. This symmetry is
also manifested in the dynamics of the system, see Sec. III.1. Instances
$\Psi$, evolving according to the equations of motion Eq. (3), approach
equilibrium dynamics for $h$ in the range from $h_{\rm min}=-J_{0}a+\chi
a^{2}/2$ to $h_{\rm{max}}=\chi a^{2}$ [8]. In the limit of weak nonlinearity
it is directly possible to test whether a system exhibits equilibrium
dynamics: the framework of optical thermodynamics facilitates comparison of
the average optical power within the different eigenmodes of the system to
theoretical predictions [8, 16, 17, 18]. Moreover, in weakly nonlinear
systems, the thermalization process can be controlled by dispersion
engineering [15]. Values of $\chi$ considered in this work do not qualify as
“weak”, hence we will need to use a different strategy to assess whether a
simulation run has reached dynamical equilibrium, see Sec. III.2.

In analogy to statistical mechanics models of interacting $n$-dimensional
classical spins [9, 1], and, more specifically, in analogy to the XY model
(obtained for $n=2$) [9, 10, 12, 13], the modes $\psi_{\ell}$ can be viewed as
_photonic soft-spins_

$\mathbf{s}_{\ell}=(s_{x,\ell},s_{y,\ell})\equiv(\mathsf{Re}[\psi_{\ell}],\mathsf{Im}[\psi_{\ell}]),$
(4)

vectors of variable length in the
$\mathsf{Re}[\psi]$-$\mathsf{Im}[\psi]$–plane. The _softness_ of a spin, i.e.
the variable length of the spin vector, is a feature that distinguishes
photonic networks from classical spin models [11]. Inspired by such
statistical mechanics models we introduce the two-component magnetization-like
_order parameter_ [1, 19, 12, 20, 2]

$\mathbf{m}=(m_{x},m_{y})=\frac{1}{N}\sum_{\ell}\mathbf{s}_{\ell},$ (5)

to analyze the soft-spins collective dynamics. Let us note that for the fully
connected photonic network, Eq. (3) can be expressed in terms of
${\mathbf{s}}_{\ell}$ and ${\mathbf{m}}$ as

$i\dot{\mathbf{s}}_{\ell}=-J_{0}\,\left(\mathbf{m}-\frac{\mathbf{s}_{\ell}}{N}\right)+\chi|\mathbf{s}_{\ell}|^{2}\mathbf{s}_{\ell},$
(6)

wherein $J_{0}\mathbf{m}$ serves as a mean field guiding the time evolution of
individual spins, and where the second term couples the components of the spin
vector. To investigate emergence of order in the ECPN, we further consider the
single-component order parameter $m\equiv|\mathbf{m}|$. Its long-time average

$\displaystyle\langle
m\rangle=\frac{1}{t_{\mathrm{obs}}-t_{\mathrm{eq}}}\int_{t_{\mathrm{eq}}}^{t_{\mathrm{obs}}}~{}m(t)~{}{\mathrm{d}}t,$
(7)

with equilibration time $t_{\mathrm{eq}}$ and total observation time
$t_{\mathrm{obs}}$, allows to analyze equilibrium properties as function of
the _control parameter_ $h$, and to probe the nature of the optical phase
transitions supported by Eq. (1). While the quantity $m$ has also been used in
a previous study [8], we here show that the two-component order parameter
${\mathbf{m}}$ plays a leading role. It permits much deeper insight into the
dynamics, allowing to identify collective rotational modes, and to derive
exact results for several quantities in the limit $h\to h_{\mathrm{max}}$.

Subsequently, we fix the optical power per mode to $a=1$ and consider
parameter values $J_{0}=1.2$ (fixing the value of $J_{0}$ is equivalent to
fixing time units), and $\chi=1$. The equilibration range is bounded by
$h_{\mathrm{min}}=-0.7$ and $h_{\mathrm{max}}=1$. This allows to reproduce and
built on previous results presented in Ref. [8]. Numerical simulations for
other parameter choices are pointed out explicitly in Sec. IV. In the
presented study, the equations of motion Eq. (3) are solved for configurations
$\Psi$ of up to $N=384$ modes, for long observation times up to
$t_{\mathrm{obs}}=10^{7}$ time units. Numerical integration of Eq. (3) is
performed using a step size controlled Runge-Kutta method of high order [21,
22, 23]. Working in the microcanonical ensemble, the (constant) energy density
and optical power of the system is set by the initial condition. In order to
prepare intitial mode-configurations for specified optical power per mode $a$
and energy density $h$, we utilize an effective optimization heuristic that
incrementally improves a random trial configuration, see Supplemental Material
[24]. The initial $t_{\rm{eq}}=10^{6}$ time units are reserved for the system
to approach equilibrium. This equilibration phase is much longer than the
correlation times for even the largest system, see Sec. III.2. No measurements
are taken during that time.

[tb!]  Equilibrium dynamics of fully connected equal-coupling photonic
networks. Configurations of $N=16$ soft-spins, demonstrating order and
disorder at $h=-0.42$ (left), and $h=1$ (right), respectively. Clip shows
temporal evolution of both configurations for an interval of $1000$ time units
with increment $\Delta t=10$ between frames.

##  III  Results

Observing the time evolution of mode preparations $\Psi$ at different values
of the control parameter $h$, allows to identify phases with distinct
dynamics. Exemplary equilibrium dynamics, giving an account of order and
disorder within an ensemble of $N=16$ soft spins, are shown in Vid. II. Both
phases can be distinguished via the long-time average $\langle m\rangle$ of
the order parameter. For $J_{0}=1.2$ and $\chi=1$, a previous qualitative
analysis suggested a threshold value $h_{c}\approx 0.75$, separating an
ordered phase with nonzero spontaneous magnetization ($h<h_{c}$), from a
disordered phase with an asymptotically ($N\to\infty$) zero value of $\langle
m\rangle$ ($h>h_{c}$) [8]. Below, we derive exact results for several
quantities in the limit $h\to h_{\mathrm{max}}$, and perform a thorough FSS
analysis, locating the asymptotic critical point, and giving the critical
exponents of the transition.

Figure 1:  Distribution of the two-component order parameter ${\mathbf{m}}$
for $N=64$ at (a) $h=0.6$, (b) $h=0.75$, and, (c) $h=1.0$. (d) Distribution of
$m=|{\mathbf{m}}|$ corresponding to (a-c). $p_{G}$ is a Gaussian with mean
$\mu=0.367$ and variance $\sigma^{2}=0.002$. $p_{R}$ is a Rayleigh
distribution with scale parameter $\sigma_{R}=0.083$. (e) Normalized
distribution of only the $x$-component of ${\mathbf{m}}$ at $h=1$ for $N=384$.
$p_{G}$ is a Gaussian with $\mu=0$ and $\sigma^{2}=1$.

###  III.1  Angular excitations and exact results

In the ordered phase, as $h\to h_{\mathrm{min}}$, the spins align and loose
their softness. This becomes evident right at $h_{\mathrm{min}}$, where in
absence of any length-fluctuations $|{\mathbf{s}}_{\ell}|=1$ [24]. In this
limit, the spins equation of motion [Eq. (6)] take the approximate form
$i\dot{{\mathbf{s}}}_{\ell}\approx-J_{0}{\mathbf{m}}+\chi{\mathbf{s}}_{\ell}$,
where the right-hand side guides individual spins towards their average
direction. Time-evolution proceeds mostly by directional changes [see left
panel of Vid. II], and it is safe to assume
$\dot{\mathbf{s}}_{\ell}\approx|\mathbf{s}_{\ell}|\dot{\theta}_{\ell}\,{\mathbf{e}}_{\theta,\ell}$,
with polar angle $\theta_{\ell}$ and unit vector
${\mathbf{e}}_{\theta,\ell}\equiv-\sin(\theta_{\ell})\,{\mathbf{e}}_{x}+\cos(\theta_{\ell})\,{\mathbf{e}}_{y}$.
For the entire system it follows that
$\dot{\mathbf{m}}=(1/N)\sum_{\ell}\dot{\mathbf{s}}_{\ell}\approx
m\dot{\theta}\,{\mathbf{e}}_{\theta}$, where $\theta$ is the polar angle of
${\mathbf{m}}$. Since Eq. (1) is invariant with respect to global phase
rotation, the momentum corresponding to $\theta$ is conserved. Hence, we
expect the ensemble of spins to be able to sustain collective rotational modes
with constant angular frequency $\dot{\theta}$, allowing the system to move
between distinct equilibrium configurations at fixed $h$. This can be seen in
Vid. II for $h=-0.42$, where $\dot{\theta}\approx-0.05$ ($\theta$ is measured
in counterclockwise direction). Consequently, ${\mathbf{m}}$ completes a full
rotation every $2\pi/|\dot{\theta}|\approx 125$ time units. These angular
excitations represent the Goldstone modes of this classical long-range
interacting system [1, 25, 26]. Such coherent rotations can be suppressed by
preparing the system in initial states comprised of real-valued modes [8]. The
angular momenta of the soft-spins relative to the collective rotational mode
can be used as a measure of “thermal” excitation of the spins angular degrees
of freedom, see Sec. III.5. Let us note that the emergence of a nonzero
magnetization within the ordered phase is signaled by both, ${\mathbf{m}}$
[Fig. 1(a-c)], and $m$ [Fig. 1(d)]. The distribution of ${\mathbf{m}}$
exhibits a circular shape with radius $m$. In particular, at $h=0.6$, the
distribution of $m$ compares well to a Gaussian with nonzero mean $\langle
m\rangle\approx 0.37$ [Fig. 1(d)]. This spontaneous magnetization indicates
the breakdown of the inherent isotropy, exhibited by equilibrium
configurations within the disordered phase, see Vid. II for $h=1$.

Deep in the disordered phase, as $h\to h_{\rm{max}}$, the components $s_{x}$
and $s_{y}$ of the individual soft-spins have the properties of i.i.d.
(independent and identically distributed) normal random variables with zero
mean and variance $1/2$. (Practically any random mode configuration, obtained
using the simple sampling approach discussed in the Supplemental Material
[24], will have these properties. Note that the distribution of energy
densities obtained via simple sampling is also strongly peaked at
$h=h_{\mathrm{max}}$.) Denoting a normal distribution with mean $\mu$ and
variance $\sigma^{2}$ by $\mathcal{N}(\mu,\sigma^{2})$, it is
$s_{x,\ell}\sim\mathcal{N}(0,1/2)$ [and $s_{y,\ell}\sim\mathcal{N}(0,1/2)$]
for $\ell=1,\ldots,N$. The above assumption was justified _a posteriori_ by
performing numerical tests, see Supplemental Material [24]. As a result,
$m_{x}\sim\frac{1}{N}\mathcal{N}(0,N/2)$ [and
$m_{y}\sim\frac{1}{N}\mathcal{N}(0,N/2)$]. Thus, for given system size $N$,
the time-average of the components of ${\mathbf{m}}$ yield $\langle
m_{x}\rangle\approx 0$ and $\langle m_{y}\rangle\approx 0$. At a given instant
of time, however, $m_{x}$ and $m_{y}$ are normal i.i.d. random variables with
variance $\sigma^{2}=1/(2N)$ [since
$c\,\mathcal{N}(\mu,\sigma^{2})=\mathcal{N}(\mu,c^{2}\sigma^{2})$]. Both
features are demonstrated in Fig. 1(e), where the probability density of
$m_{x}$ for a system of size $N=384$ is shown to fall onto a Gaussian function
with zero mean and unit variance, if scaled by the standard deviation
$\sigma_{N}=1/\sqrt{2N}$. As a result, ${\mathbf{m}}$ is a two-component
normal random vector and Eq. (6) can be cast into the form
$i\dot{\mathbf{s}}_{\ell}={\mathbf{m}}_{{\mathrm{rand}},\ell}+\chi|{\mathbf{s}}_{\ell}|^{2}{\mathbf{s}}_{\ell}$,
describing an ensemble of effectively uncoupled nonlinear oscillators, driven
by fast-varying local random fields
${\mathbf{m}}_{{\mathrm{rand}},\ell}=-J_{0}({\mathbf{m}}-{\mathbf{s}}_{\ell}/N)$.
The normal random nature of ${\mathbf{m}}$ has direct implications for $m$: it
specifies the length of a two-dimensional random vector, thus following a
Rayleigh-distribution
$p_{R}(m)=m\sigma_{R}^{-2}\exp\\{-m^{2}/(2\sigma_{R}^{2})\\}$ with scale
parameter $\sigma_{R}=1/\sqrt{2N}$ [27], see Fig. 1(d). From this, the $n$-th
moment about the origin $\mu_{n}^{\prime}\equiv\int
m^{n}p_{R}(m)~{}{\mathrm{d}}m$ of the order parameter distribution can be
found explicitly [27]: it is $\mu_{1}^{\prime}=\sqrt{\pi/4/N}$,
$\mu_{2}^{\prime}=1/N$, and $\mu_{4}^{\prime}=2/N^{2}$. With these results,
the time-average of $m$ in the limit $h\to h_{\mathrm{max}}$ is found as
$\langle m\rangle_{N}^{\star}=\mu_{1}^{\prime}\approx 0.89/\sqrt{N}$. These
exact limiting behaviors are tested in Sec. III.3, and Sec. III.4 below.
Having discussed the role of the magnetization-like parameter ${\mathbf{m}}$
in the limits $h\to h_{\mathrm{min}}$ and $h\to h_{\mathrm{max}}$, it is now
in order to study how systems composed of a large but finite number of
photonic soft-spins gradually loose orientational order upon increasing the
energy density $h$.

###  III.2  Correlation times

To assess under which conditions a given simulation run allows to obtain
equilibrium properties of $m$, we study the time-displaced autocorrelation
function [2]

$C(t)=\int_{t>t_{\mathrm{eq}}}\left[m(t^{\prime})m(t^{\prime}+t)-\langle
m\rangle^{2}\right]~{}{\mathrm{d}}t^{\prime}.$ (8)

It takes a non-zero value if, on average, the fluctuations of $m(t^{\prime})$
and its time-displaced value $m(t^{\prime}+t)$ are correlated. It will be zero
if they are uncorrelated. The analysis below is restricted to a control
parameter in the vicinity of the critical point, where correlations are
notoriously persistent [2]. Specifically, we consider the point $h=0.75$,
motivated by a previous study [8]. Figure 2 shows autocorrelation functions
for different system sizes $N$. The time-scale $\tau$ on which the
autocorrelation decays is a measure of the correlation time, indicating how
long it takes for small disturbances in $m$ to relax [3]. We find a good fit
to the two-exponential decay function
$C_{\mathrm{fit}}(t)\propto\exp(-t/\tau)\left[1+b\exp(-t/\tau_{\mathrm{corr}})\right]$
[28, 3]. Results for $\tau$, obtained by least-squares fits to
$C_{\mathrm{fit}}$, are shown in the inset of Fig. 2. We assume that if a
system has evolved for several multiples of the correlation time, it is
practically in equilibrium. Irrespective of $N$, both, the initial
equilibration time $t_{\mathrm{eq}}$ as well as the total observation time
$t_{\mathrm{obs}}$ exceed the equilibration time by far.

Figure 2: Dependence of the autocorrelation function $C(t)$ on time-
displacement $t$ for different number of modes $N$ at $h=0.75\approx h_{c}$.
Inset shows autocorrelation times $\tau$ obtained by least-squares fits of $C$
to a parameterized decay function (see text for details).

###  III.3  Finite-size scaling in the critical region

To investigate the emergence of “magnetic ordering” through soft-spin
alignment, equilibrium properties of ECPNs as a function of the control
parameter $h$ for systems of different size $N$ are considered. As evident
from Fig. 3(a), and in agreement with earlier results [8], an asymptotically
nonzero magnetization appears below $h_{c}\approx 0.75$, and the transition
from the disordered phase ($h>h_{c}$) to the ordered phase ($h<h_{c}$) becomes
more clear as the size $N$ of the system increases. This is supported by the
scaling behavior of other standard statistics mechanics quantities, such as
the finite-size susceptibility $\chi_{m}$, and the Binder fourth-order
cumulant $b_{m}$, given by

$\displaystyle\chi_{m}$ $\displaystyle\equiv N\left(\langle
m^{2}\rangle-\langle m\rangle^{2}\right),$ (9a) $\displaystyle b_{m}$
$\displaystyle\equiv 1-\frac{\langle m^{4}\rangle}{3\langle
m^{2}\rangle^{2}},$ (9b)

shown in Figs. 3(b,c), respectively. FSS theory asserts [5, 29, 30, 31, 1],
that the scaling behavior of the above quantities satisfies the scaling forms

$\displaystyle\langle m\rangle$
$\displaystyle=N^{-\beta/\bar{\nu}}f_{\beta}(x),\quad$ $\displaystyle
f_{\beta}(x)$ $\displaystyle\stackrel{{\scriptstyle x\to
0}}{{\sim}}(-x)^{\beta},$ (10a) $\displaystyle\chi_{m}$
$\displaystyle=N^{\gamma/\bar{\nu}}f_{\gamma}(x),\quad$ $\displaystyle
f_{\gamma}(x)$ $\displaystyle\stackrel{{\scriptstyle x\to
0}}{{\sim}}x^{-\gamma},$ (10b) $\displaystyle b_{m}$
$\displaystyle=f_{0}(x),\quad$ $\displaystyle f_{0}(x)$
$\displaystyle\stackrel{{\scriptstyle x\to 0}}{{\sim}}{\mathrm{const.}},$
(10c)

with scaling variable $x\equiv(h/h_{c}-1)N^{1/\bar{\nu}}$. $\bar{\nu}$ is a
critical exponent governing the divergence $N_{c}\sim|h-h_{c}|^{-\bar{\nu}}$
of a coherence number $N_{c}$ for the fully connected system in the limit
$N\to\infty$ [30]. The exponent $\bar{\nu}$ is related to the mean-field
critical exponent $\nu$ through the relation $\bar{\nu}=d_{c}\nu$ [32, 33],
where $d_{c}$ is the upper critical dimension [1], i.e. the dimension above
which short-range interacting systems have the same critical exponents as the
mean-field model. $\beta$ is a critical exponent describing the vanishing of
the order parameter $\langle m\rangle$, and $\gamma$ describes the divergence
of $\chi_{m}$ at the transition point. The scaling functions $f_{\beta}$,
$f_{\gamma}$, and $f_{0}$ are expected to approach constant values for
$x\to\infty$. The scaling assumptions Eqs. (10) are valid in the limit
$N\to\infty$ with systematic deviations for systems of finite size [3].
Consequently, the FSS analysis below is performed for systems of size $N\geq
96$. Different methods for performing FSS are used throughout the literature,
based, e.g., on the minimization of a local linearity function [34], a refined
quality function [35, 36, 37], considered in the presented work, and Bayesian
inference methods [38, 39]. Since the Hamiltonian Eq. (1) describes a long-
range interacting system, we expect our results to be in accord with the usual
mean field exponents $\nu=1/2$, $\beta=1/2$, and $\gamma=1$ [1, 7]. In
particular, the relation $\bar{\nu}=d_{c}/2$ will allow to estimate the upper
critical dimension of ECPNs.

####  III.3.1 Binder fourth order cumulant

To locate $h_{c}$, and simultaneously determine $\bar{\nu}$, the dimensionless
Binder parameter $b_{m}$ [Eq. (9b)] is considered first. At the critical
point, according to Eq. (10c), the Binder parameter for the equilibrium system
should be independent of $N$ [5]. Figure 3(c) shows that indeed, as function
of $h$, curves at different $N$ intersect at a common point. In the limit
$x\to\infty$, the numerical results are in reasonable agreement with
$b_{m}^{\star}=1-\mu_{4}^{\prime}/[3(\mu_{2}^{\prime})^{2}]=\frac{1}{3}$
[dash-dotted line in Fig. 3(c)], expected on basis of the exact results
derived in Sec. III.1. Let us note that this is different from the
corresponding high-temperature limit for the fully connected Ising model,
where $b_{m}\approx 0$ [40, 32]. Imposing the scaling form Eq. (10c), and
minimizing an objective function measuring the quality of the _data collapse_
[35, 41, 37], yields the optimal parameters $h_{c}=0.756(1)$ and
$\bar{\nu}=2.04(8)$. (An independent analysis, performed by minimizing a local
linearity function for Eq. (10c) [34], results in the supporting estimates
$h_{c}=0.756$ and $\bar{\nu}=1.97$.) The excellent data collapse achieved by
these parameter values is demonstrated in Fig. 3(f).

Figure 3:  Results of the finite-size scaling analysis at $J_{0}=1.2$ and
$\chi=1$. (a-c) Scaling behavior as function of energy density $h$. (a) Order
parameter $\langle m\rangle$. Solid line indicates $\langle
m\rangle\sim(h_{c}-h)^{1/2}$ for $h_{c}=0.756$. (b) Finite-size susceptibility
$\chi_{m}$. (c) Binder cumulant $b_{m}$. (d-f) Data collapse of scaled
quantities as function of $x\equiv(h/h_{c}-1)N^{1/\bar{\nu}}$ (for the
definition of the parameters see text). (d) Scaled order parameter. (e) Scaled
susceptibility. (f) Binder cumulant. In (a-c), dashed lines indicate
$h=0.756$, and dash-dotted lines indicate values expected for $h\to
h_{\mathrm{max}}$. Errorbars are smaller than symbol sizes.

####  III.3.2 Order parameter

To locate $h_{c}$ and simultaneously determine the critical exponents $\beta$
and $\bar{\nu}$, the finite-size scaling of the order parameter is considered
next. In the ordered phase close to the critical point, it is well described
by $\langle m\rangle\propto|h-h_{c}|^{\beta}$. This is shown by the dashed
line in Fig. 3(a), obtained for _a priori_ choice $h_{c}=0.756$, and
$\beta=1/2$, in qualitative agreement with previous results [8]. In the limit
$x\to\infty$, our numerical results are in excellent agreement with the exact
result $\langle m\rangle^{\star}_{N}=\sqrt{\pi/4/N}$ obtained in Sec. III.1
[dash-dotted line in Fig. 3(a)]. By imposing Eq. (10a), the optimal data
collapse quality is achieved for $h_{c}=0.755(1)$, $\bar{\nu}=2.00(4)$, and
$\beta=0.52(4)$ [35, 41, 37]. (Minimizing the local linearity function for Eq.
(10a) [34], gives $h_{c}=0.752$, $\bar{\nu}=2.03$, and $\beta=0.51$.) The
corresponding data collapse is shown in Fig. 3(d).

####  III.3.3 Finite-size susceptibility

The finite-size susceptibility $\chi_{m}$ [Eq. (9a)] allows to define a
sequence of pseudocritical points $h_{c}^{\prime}$. For a given system size
$N$, $h_{c}^{\prime}$ is determined by the peak-location of $\chi_{m}$, see
Fig. 3(b). At $N=64$ the peak is located at $h_{c}^{\prime}=0.736(2)$. For
increasing system size these pseudocritical points shift to larger values of
$h$, e.g., at $N=384$ it is $h_{c}^{\prime}=0.744(4)$, approaching the
previous estimates of $h_{c}$ from below. In the limit $x\to\infty$, the exact
results of Sec. III.1 lead to expect
$\chi_{m}^{\star}=N[\mu_{2}^{\prime}-(\mu_{1}^{\prime})^{2}]=1-\pi/4\approx
0.21$ [dash-dotted line in Fig. 3(b)], in good agreement with our numerical
results. Imposing Eq. (10b), the optimal data collapse is achieved for
$h_{c}=0.750(2)$, $\bar{\nu}=2.00(4)$, and $\gamma=0.98(6)$ [35, 41, 37], see
Fig. 3(e). (Minimizing the local linearity function for Eq. (10b) [34], yields
$h_{c}=0.753$, $\bar{\nu}=2.06$, and $\gamma=0.99$.) Thus, approaching the
critical point from above, the susceptibility obeys the Curie-Weiss law
$\chi_{m}\propto|h-h_{c}|^{-1}$.

Figure 4: Scaling behavior at selected values of $h$. (a) Order parameter
$\langle m\rangle$ in the ordered phase ($h=0.5$), at criticality
($h=0.75\approx h_{c}$), and in the disordered phase ($h=0.94$). (b) Same for
the finite-size susceptibility $\chi_{m}$. Short-dashed lines at $h=0.5$
indicate scaling laws accounting for corrections to scaling (see text). Dashed
lines at $h=0.75$ indicate asymptotic power-law scaling. Dashed dotted lines
at $h=0.94$ indicate exact results. Errorbars are smaller than symbol size.
Figure 5: Angular displacement and soft-spin temperature. (a) Probability
density function $p(\vartheta)$ of the angular displacement $\vartheta$ of the
soft-spins relative to the average spin direction. $p_{G}$ is a Gaussian with
zero mean and variance $\approx 0.085$. (b) Mean-square angular displacement
$\langle\vartheta^{2}\rangle$ as function of energy density $h$. Vertical
dashed line indicates transition point $h_{c}=0.754$. $\pi^{2}/3$ is the
variance of a uniform distribution in range $(-\pi,\pi)$. (c) Soft-spin
temperature $T_{S}$ as function of energy density. Inset shows the order
parameter as function of soft-spin temperature. The transition occurs at
$T_{c}\approx 1.48$.

###  III.4 Scaling behavior at selected energy densities

In the ordered phase ($h<h_{c}$) the spontaneous magnetization exhibits the
usual scaling behavior of the form $\langle m\rangle=m_{\infty}(h)+cN^{-x}$
[31]. A least-squares fit at $h=0.5$ for $N>20$ yields
$m_{\infty}(0.5)=0.446(1)$, $c=0.77(6)$, and $x=0.97(3)$, see Fig. 4(a). Close
to the asymptotic critical point ($h\approx h_{c}$), power-law scaling
$\langle m\rangle=cN^{-x}$, with $c=0.50(1)$, and $x=0.248(5)$ is observed to
hold for $N>30$. This is consistent with the FSS assumption [Eq. (10a) for
$x\to 0$], supporting an asymptotic exponent ratio $\beta/\bar{\nu}=x=1/4$
[Fig. 4(a)]. Deep in the disordered phase ($h>h_{c}$), the time-averaged order
parameter is expected to scale as $\langle m\rangle^{\star}=m_{0}/\sqrt{N}$,
where $m_{0}=\sqrt{\pi/4}\approx 0.89$. Figure 4(a) verifies this scaling law
for the parameter choice $h=0.94$. A least squares fit yields
$m_{0,{\mathrm{fit}}}=0.88(1)$, in excellent agreement with the expected
scaling behavior. Reference [8] reports the fit results
$\bar{m}_{\rm{min}}\approx 0.75/\sqrt{N}$, where, at a given system size $N$,
${\bar{m}}_{\mathrm{min}}$ is the asymptotic value to which $\langle m\rangle$
converges in the disordered phase. A reason for the smaller scaling-factor
might be found by observing that the analysis in Ref. [8] is based mostly on
systems of size $N<100$, where corrections to scaling might still be large.
For comparison, in the presented study, restricting the fit at $h=0.94$ to
systems of size $N<100$ yields $m_{0,{\mathrm{fit}}}=0.81(2)$, indeed trending
towards a smaller value.

Similar scaling laws govern the behavior of the finite-size susceptibility
$\chi_{m}$, see Fig. 4(b). In the ordered phase at $h=0.5$, the scaling law
$\chi_{m}=\chi_{m,\infty}(h)+cN^{-x}$ yields $\chi_{m,\infty}(0.5)=0.080(1)$,
$c=-0.16(1)$, and $x=0.66(2)$. At the critical point, $\chi_{m}=c(N+\Delta
N)^{-x}$, where $c=0.042(3)$, $\Delta N=5(2)$, and, $x=0.49(1)$, in support of
an asymptotic exponent ratio $\gamma/\bar{\nu}=x=0.5$ [cf. Eq. 10b for $x\to
0$]. In the disordered phase, saturation at $\chi_{m}^{\star}=1-\pi/4\approx
0.21$ is clearly evident for $N>100$.

###  III.5 Soft-spin temperature

While in the limit of weak nonlinearity, the framework of optical
thermodynamics can be employed to relate the equilibrium properties to a
thermodynamic temperature [8, 16, 17, 18], no obvious corresponding relation
is available in the present case. There exist, however, various temperature
estimators for use under equilibrium conditions in computer simulations. For
example, a kinetic temperature is defined in molecular dynamics simulations
[42], kinetic and spin temperatures are considered for ensembles of spins in
semiclassical Langevin dynamics [43, 44], and a configurational temperature is
used as diagnostic tool in Monte Carlo simulations [45]. In the microcanonical
ensemble, it is further possible to measure temperature in terms of a
dynamical, geometrical approach [46, 47, 48].

Table 1: Critical points and exponents of the optical phase transition on fully connected, equal-coupling photonic networks. From left to right: mode-mode coupling strength ($J_{0}$), nonlinearity parameter ($\chi$), critical point ($h_{c}$), coherence number exponent ($\bar{\nu}$), order parameter exponent ($\beta$), and, susceptibility exponent ($\gamma$). Results in the first row are reproduced after Ref. [8]. $J_{0}$ | $\chi$ | $h_{c}$ | $\bar{\nu}$ | $\beta$ | $\gamma$
---|---|---|---|---|---
1.2 | 1.0 | 0.75 | – | $\frac{1}{2}$ | –
1.2 | 0.6 | 0.47(1) | 2.0(1) | 0.49(1) | 0.92(8)
1.2 | 0.8 | 0.619(5) | 2.00(4) | 0.52(1) | 0.95(5)
1.2 | 1.0 | 0.754(4) | 2.00(4) | 0.52(4) | 0.98(6)
1.2 | 1.2 | 0.880(1) | 2.00(2) | 0.50(1) | 0.99(2)

Here we introduce an expression for the temperature, based entirely on the
angular momenta $p_{\ell}\equiv|{\mathbf{s}}_{\ell}|\,\dot{\vartheta}_{\ell}$
of the photonic soft-spins, where $\vartheta_{\ell}\equiv\theta_{\ell}-\theta$
measures the angular displacement of soft-spin ${\mathbf{s}_{\ell}}$ relative
to ${\mathbf{m}}$. The corresponding soft-spin temperature $T_{S}$ is computed
as the time-averaged, squared angular momentum per spin

$\displaystyle T_{S}=\frac{1}{N}\sum_{\ell}\langle p_{\ell}^{2}\rangle,$ (11)

with the system in equilibrium. Deep in the ordered phase, the distribution of
the angles $\vartheta_{\ell}$ is a narrow Gaussian with zero mean, e.g., at
$h=-0.5$ its variance is $\langle\vartheta^{2}\rangle\approx 0.085$, see Fig.
5(a). For increasing energy density, the variance increases. In the disordered
phase, close to $h_{\rm{max}}$, it compares well to a uniform distribution
$p_{\rm{u}}(\vartheta)=1/(2\pi)$ in range $-\pi\leq\vartheta\leq\pi$. The
increase of $\langle\vartheta^{2}\rangle$ with $h$ is shown in Fig. 5(b). In
the disordered phase, the variance
$\int_{-\pi}^{\pi}\vartheta^{2}\,p_{\rm{u}}(\vartheta)~{}{\mathrm{d}}\vartheta=\pi^{2}/3$
is approached as $N\to\infty$. The inflection points of the curves are located
close to the transition point $h_{c}$. It is not surprising that the angular
displacements $\vartheta_{\ell}$ are a sensitive measure of orientational
order in the system: in case of the XY model, at low temperatures, a direct
relation between a quantity similar to $1-\vartheta^{2}$ and the magnetization
can be established [10, 49]. Similar to the temperature considered in earlier
studies of the XY model [20], $T_{S}$ is associated with kinetic information
contained in the rotational degrees of freedom of the system. Figure 5(c)
indicates a monotonous increase of the soft-spin temperature with energy
density. In the limit $h\to h_{\rm{min}}$, $T_{S}\to 0$. Further, for
decreasing soft-spin temperature, the length fluctuations of the spins are
increasingly suppressed, strengthening the analogy to the XY model.
Considering $T_{S}$ as control parameter, we find that in the vicinity of the
critical point $\langle m\rangle\propto|T_{S}-T_{c}|^{1/2}$, see inset of Fig.
5(c), according to which the order parameter vanishes with exponent
$\beta=1/2$, in agreement with the mean-field theory for photonic networks
developed in Ref. [8]. This is explained by $T_{S}$ being a linear function of
$h$ for $h\lessapprox h_{c}$. Hence, a consistent description of the phase
transition is also obtained in terms of the soft-spin temperature.

Figure 6: Finite-size scaling results at different values of the nonlinearity
parameter $\chi$. The main figure shows the rescaled magnetization as function
of the scaling variable $x\equiv\left[h/h_{c}(\chi)-1\right]N^{1/\bar{\nu}}$.
The inset shows a phase diagram in the $\chi$-$h$ plane, indicating the
$\chi$-dependence $h_{c}(\chi)=0.754\,\chi^{0.88}$ of the transition point.

##  IV  Discussion and conclusions

In the present study, we build upon a recent work [8], wherein a
magnetization-like order parameter $m$ was considered for studying optical
phase transitions in photonic networks. Going beyond previous results, we
showed that the two-component quantity $\mathbf{m}$, a two-component order
parameter inspired by classical statistical mechanics models, provides a
deeper insight into the equilibrium dynamics. Within the ordered phase it has
the additional benefit of distinguishing between non-identical equilibrium
configurations at fixed energy density, revealing collective angular
excitations sustained by the photonic soft-spins. These are a manifestation of
a symmetry of the Hamiltonian in the system dynamics. Within the disordered
phase, it even allows to obtain exact results for several quantities of
interest. Specifically, it predicts the order parameter to follow a Rayleigh-
distribution, and even yields an expression for the finite-size scaling
behavior of $\langle m\rangle$ in the limit $h\to h_{\mathrm{max}}$, both in
excellent agreement with our numerical simulations. Qualitatively we find that
the large-$N$ behavior of the order parameter in equal-coupling photonic
networks is consistent with that of statistics mechanics mean-field spin
systems, such as the fully connected Ising model [31], and the XY model on
small-world networks [12], and long-range interacting one-dimensional systems
[13].

Let us note that, in the vicinity of the asymptotic critical point, the order
parameter distribution is not Gaussian and has a maximum at a nonzero value of
$m$, see Figs. 1(b,d) and the Supplementary Material [24]. In case of the
fully connected Ising model this has led to the observation of anomalous mean-
field scaling [32], where, at criticality, the mean value of the magnetization
and the most probable magnetization were found to scale differently. In the
present case we verified that the scaling of the mean value and the most
probable value of $m$ are identical (not shown).

In addition to the FSS analysis reported in Sec. III.3, we performed further,
similar analysis for other choices of the nonlinearity parameter $\chi$. To
account for these results, Fig. 6 shows the data collapse achieved for the
order parameter, when taking into account a $\chi$-dependent critical point
$h_{c}(\chi)$. In either case, the critical exponents where found to be in
good agreement with the above analysis, see Tab. 1. We find that the location
of the critical point is well represented by the critical line
$h_{c}(\chi)=0.754\cdot\chi^{x}$, with sub-linear exponent $x\approx 0.88$,
see the $\chi$-$h$ phase diagram shown in the inset of Fig. 6.

In conclusion, we have studied fully connected ECPNs using dynamic simulations
of the nonlinear equations of motion of the coupled modes. For several choices
of the nonlinearity parameter $\chi$, critical points and exponents have been
determined by imposing appropriate scaling assumptions on a magnetization-like
order parameter, its associated Binder cumulant and finite-size
susceptibility. For all parameter settings considered in this work, the FSS
analysis of the optical phase transition yields mean-field critical exponents
and allows to infer the upper critical dimension $d_{c}=4$. Overall,
corrections to scaling where found to affect the scaling behavior for systems
of size up to $N\approx 100$. Generally, for systems of finite size, finite-
size effects result in effective transition points that differ slightly from
the asymptotic values quoted in Tab. 1, also see the discussion in Sec.
III.3.3. These findings may prove to be useful, when considering instances of
photonic networks with finite (small) number of building blocks in optical
technologies. We further established a connection between the energy density
and a kinetic temperature, measuring the activation of the angular degree of
freedom of the photonic soft spins, in terms of which a consistent signature
of the transition could be obtained. Finally, let us note that in Ref. [8], a
mean-field theory for photonic networks with finite coordination number was
formulated. From the point of view of statistical mechanics it would be
rewarding to test this mean-field theory by studying the equal-coupling
photonic network on random regular graphs [50, 51], an ensemble of random
graphs with fixed coordination number.

###### Acknowledgements.

I would like to thank Ayhan Demircan for helpful discussions. Funding by the
Deutsche Forschungsgemeinschaft within the Cluster of Excellence PhoenixD
(Photonics, Optics, and Engineering–Innovation Across Disciplines) (EXC 2122,
projectID 390833453) is gratefully acknowledged.

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