# The Dark Mass Problem
Solved ?

Ll. Bel e-mail<EMAIL_ADDRESS>
###### Abstract

I discuss some of the basic properties of a potential theory derived from a
modified Newton’s law of action at a distance that includes a $1/r$ attractive
force.

##  1 Point particles

Let us start assuming that two point particles with masses $m_{1}$ and $m_{2}$
located at positions $x^{i}$ and $y^{i}$ attract each other with a force
$F^{i}$:

$F_{y|}^{i}(x)=-\frac{Gm_{1}m_{2}}{r^{3}}(x^{i}-y^{i})-\frac{G^{\prime}m_{1}m_{2}}{r^{2}}(x^{i}-y^{i}),\quad
i,j,\cdots=1,2,3$ (1)

where $r^{2}=|\overrightarrow{x}-\overrightarrow{y}|^{2}$, $G$ is Newton’s
constant and $G^{\prime}$ is a free positive physical constant with dimensions
M-1L2T-2.

This law of attraction has been considered, more or less directly, as a
candidate to solve what is known today as the Dark mass problem (more on that
below).

In considering this new law of force it is important to keep in mind that
space has three dimensions and not two, and as a consequence of this only the
Newtonian component of (1) is solenoidal.

Since:

$\frac{\partial F_{y|i}(x)}{\partial x^{j}}-\frac{\partial
F_{y|j}(x)}{\partial x^{i}}=0,$ (2)

introducing the potential function $V$ defined by:

$F_{y|i}(x)=-m_{1}\frac{\partial V_{y|}(x)}{\partial x^{i}}$ (3)

from (1) there follows that:

$\Delta V_{y|}(x)=4\pi G\rho_{y|}(x),\quad\rho_{y|}(x)\equiv
m_{2}\delta(r)+\frac{\alpha
m_{2}}{r^{2}},\quad\alpha=\frac{1}{4\pi}\frac{G^{\prime}}{G}$ (4)

Let us assume that the point particle with mass $m_{2}$ is kept fixed at the
location $y^{i}$ and that the particle $m_{1}$ is free to move under its
attraction. This simple formula above tells us that this particle will move
without friction as it would do across a cloud of dark matter with an always
positive effective density $\rho$, while obeying a potential theory formally
identical to Newton’s one.

##  2 Extended sources

If instead of a point particle with mass $m_{2}$ we consider a continuous
distribution of mass with density $\mu(y)$ then we shall have:

$F^{i}(x)=-Gm_{1}\int_{D}\mu(y)\frac{(x^{i}-y^{i})}{r^{3}}\,d^{3}y-G^{\prime}m_{1}\int_{D}\mu(y)\frac{(x^{i}-y^{i})}{r^{2}}\,d^{3}y,$
(5)

where $D$ is the domain where $\mu\neq 0$, and:

$V(x)=-G\int_{D}\mu(y)\frac{1}{r}\,d^{3}y+G^{\prime}\int_{D}\mu(y)\ln(r)\,d^{3}y,$
(6)

from where we get:

$\Delta V(x)=4\pi
G\rho(x),\quad\rho(x)\equiv\mu(x)+\alpha\int_{D}\frac{\mu(y)}{r^{2}})\,d^{3}y,$
(7)

Let us consider in particular the case where the density $\mu$ is constant
inside a sphere of radius $a$ and zero otherwise (Figure 1). In this case, $r$
being now $|\overrightarrow{x}|$, the effective density $\rho$ will be:

$\rho(r)=\mu
H(a-r)+\alpha\int_{0}^{2\pi}d\phi\int_{0}^{a}du\int_{0}^{\pi}d\theta\frac{\mu
u^{2}\sin\theta}{r^{2}+u^{2}-2ru\cos\theta}$ (8)

$H$ being the Heaviside function; or:

$\rho(r)=\mu H(a-r)+2\pi\alpha\mu\sigma(r,a)$ (9)

with:

$\sigma(r,w)=w+\frac{1}{2r}(r^{2}-w^{2})\ln\left(\frac{|r-w|}{r+w}\right)$
(10)

from where, to calculate the central potential or the force, we could proceed
as usual integrating the Poisson’s equation (7), or use their corresponding
integral definitions.

If instead we have an spherical sector with inner radius $b$ (Figure 2) then
the effective density $\rho(r)$ is:

$\rho(r)=\mu H(a-r)H(r-b)+2\pi\alpha\mu(\sigma(r,a)-\sigma(r,b))$ (11)

Notice that in the cavity defined by $r<b$, where the raw matter density is
zero, the effective density remains positive and therefore the force remains
attractive towards the center. This means that it is a legitimate speculation
to point out that if gravity includes a contribution of the type that I have
considered here, then we could sometimes be fooled to believe that a massive
object gravitates around a source that actually does not exists at all.

##  3 Comments

Notable consequences of the proposed new law are:

i) If $G^{\prime}\neq 0$ and there is ordinary matter somewhere then there is
Dark matter everywhere. Therefore this new law has a lot to say about the
rotation curves of spiral galaxies as well as micro and macro lensing. The
price that we pay for it is the non locality of the theory since the effective
density is not a point function.

ii) It predicts new surprising effects like massive bodies orbiting central
un-existing matter sources in empty cavities (Empty cavity effect), and beyond
that it promises an interesting development of the physics of voids in
cosmology..

iii) Last but not least, the transport of this proposal to General relativity
is quite natural. It suffices to use the effective density $\rho(x)$ instead
of the pure matter one $\mu$ in the source energy-momentum tensor of
Einstein’s equations. The first conceptually important implications are: 1)
that the gravitational field of a point particle becomes more singular than
what it is in Einstein’s theory and 2) that, so to speak, all exact vacuum
solutions become approximate approximate solutions of the modified theory. On
the other hand cosmology only needs to sort out what part of the total density
is matter density and what part it is dark matter.

Reference [1] considers a theory based from the beginning on the potential:

$V(r)=-\frac{Gm_{1}m_{2}}{r}+G^{\prime}\ln r$ (12)

Some aspects of this point of point view bear a similitude with mine but it is
by no means equivalent to it. It does not predict any Empty cavity effect.

References [2] and [3] are based on a Modified quantum field theory that leads
them to describe the gravity of point particles by a potential:

$V(r)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}(V(\omega)\omega^{3})\frac{\sin(\omega
r)}{\omega r}\frac{d\omega}{\omega}.$ (13)

They claim that this potential can be approximated by a $1/r^{2}$ or a $1/r$
one depending on the scale of distances. They also note ”that from a dynamical
point of view the modification of the Newton’s law of gravity can be
interpreted as if point sources lose their point-like character and acquire an
additional distribution in space”. This is also the main point in my very much
simpler proposition.

Reference [4] starts with the introduction of the potential function (6) but
the only source model that is considered is that of a thin disk of matter that
can be dealt with in the framework of potential theory in dimension 2.

## References

  * [1] W. H. Kinney and M. Brisudova, arXiv:astro-ph/0006453v1
  * [2] A. A. Kirillov, D. Turaev, Physics Letters B 532, 185-192 (2002)
  * [3] A. A. Kirillov, arXiv:astro-ph/0405623v1
  * [4] J. C. Fabris and J. Pereira Campos, arXiv:0710.3683v1 [astro-ph]


# On ${\theta}$-congruent numbers on real quadratic number fields

Ali S. Janfada  Department of Mathematics
Urmia University
Urmia
Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Sajad Salami  Instituto
da Mathemçática e Estatistica, UERJ, Brazil<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathbb{K}}={\mathbb{Q}}(\sqrt{m})$ be a real quadratic number field,
where $m>1$ is a squarefree integer. Suppose that $0<{\theta}<\pi$ has
rational cosine, say $\cos({\theta})=s/r$ with $0<|s|<r$ and $\gcd(r,s)=1$. A
positive integer $n$ is called a $({\mathbb{K}},{\theta})$-congruent number if
there is a triangle, called the $({\mathbb{K}},{\theta},n)$-triangles, with
sides in ${\mathbb{K}}$ having ${\theta}$ as an angle and $n\alpha_{\theta}$
as area, where ${\alpha_{\theta}}=\sqrt{r^{2}-s^{2}}$. Consider the
$({\mathbb{K}},{\theta})$-congruent number elliptic curve
$E_{n,{\theta}}:y^{2}=x(x+(r+s)n)(x-(r-s)n)$ defined over ${\mathbb{K}}$.
Denote the squarefree part of positive integer $t$ by ${\rm sqf}(t)$. In this
work, it is proved that if $m\neq{\rm sqf}(2r(r-s))$ and $mn\neq 2,3,6$, then
$n$ is a $({\mathbb{K}},{\theta})$-congruent number if and only if the
Mordell-Weil group $E_{n,{\theta}}({\mathbb{K}})$ has positive rank, and all
of the $({\mathbb{K}},{\theta},n)$-triangles are classified in four types.

Keywords: ${\theta}$-congruent number, elliptic curve, real quadratic number
field.

Subject class [2010]: Primary 11G05, Secondary 14H52

##  1\. Introduction

A positive integer $n$ is called a congruent number if it is the area of a
right triangle with rational sides. Finding all congruent numbers is one of
the classical problems in the modern number theory. We cite [8] for an
exposition of the congruent number problem, and [4] to see the first study of
${\theta}$-congruent numbers as a generalization of the classic one. Let
$0<{\theta}<\pi$ has rational cosine $\cos({\theta})=s/r$ with $0<|s|<r$ and
$\gcd(r,s)=1$. Let $(U,V,W)_{\theta}$ denote a triangle with an angle
${\theta}$ between sides $U$ and $V$. A positive integer $n$ is called a
${\theta}$-congruent number if there exists a triangle $(U,V,W)_{\theta}$ with
sides in ${\mathbb{Q}}$ having area $n\alpha_{\theta}$, where
${\alpha_{\theta}}=\sqrt{r^{2}-s^{2}}$. In other words, $n$ is a
${\theta}$-congruent number if it satisfies

$2rn=UV,\ \ W^{2}=U^{2}+V^{2}-\frac{2s}{r}UV.$

An ordinary congruent number is nothing but a $\pi/2$-congruent number.
Clearly, if $n$ is a ${\theta}$-congruent number, then so is $nt^{2}$, for any
positive integer $t$. We shall concentrate on squarefree numbers whenever
${\theta}$-congruent numbers concerned. Let

$E_{n,{\theta}}:\ y^{2}=x(x+(r+s)n)(x-(r-s)n)$

be the ${\theta}$-congruent number elliptic curve, where $r$ and $s$ are as
above. Theorem 2.4 gives an important connection between ${\theta}$-congruent
numbers and the Mordell-Weil group $E_{n,{\theta}}({\mathbb{Q}})$. For more
information and recent results about ${\theta}$-congruent numbers see [5, 3,
14].

The notion ${\theta}$-congruent number, which is defined over ${\mathbb{Q}}$,
can be extended in a natural way over real quadratic number fields
${\mathbb{K}}$. In this case, we refer to $n$ as a
$({\mathbb{K}},{\theta})$-congruent number and to the triangle
$(U,V,W)_{\theta}$ as a $({\mathbb{K}},{\theta},n)$-triangle. When $n$ is not
a ${\theta}$-congruent number over ${\mathbb{Q}}$, a question proposed
naturally: Is $n$ a $({\mathbb{K}},{\theta})$-congruent number for some real
quadratic number field ${\mathbb{K}}$? Tada [13] answered this question in the
case ${\theta}=\pi/2$, by studying the structure of the
${\mathbb{K}}$-rational points on the elliptic curve
$E_{n,\pi/2}:y^{2}=x(x^{2}-n^{2})$. In this paper, we answer the above
question for any$0<{\theta}<\pi$ and classify all
$({\mathbb{K}},{\theta},n)$-triangles. Through the paper we shall consider
${\mathbb{K}}={\mathbb{Q}}(\sqrt{m})$ to be a real quadratic field, where
$m>1$ is squarefree. We denote the squarefree part of any positive integer $N$
by ${\rm sqf}(N)$. The main results of this paper are the following theorems.

######  Theorem 1.1.

Let $n$ be a positive squarefree integer with $\gcd(m,n)=1$ such that $mn\neq
2,3,6$ and $m\neq{\rm sqf}(2r(r-s))$, where $m,r,s$ are as before. Then $n$ is
a $({\mathbb{K}},{\theta})$-congruent number if and only if ${\rm
rank}(E_{n,{\theta}}({\mathbb{K}}))>0$. Moreover, $n$ is a
$({\mathbb{K}},{\theta})$-congruent number if and only if either $n$ or $mn$
is a ${\theta}$-congruent number over ${\mathbb{Q}}$.

Theorem 1.1 is an extension of Part (2) of Theorem 2.4 in the following. Note
that the non-equality conditions for $mn$ and $m$ in Theorem 1.1 are
necessary. For a counterexample, when $n=1$ and ${\theta}=2\pi/3$, we have
$r=2$, $s=-1$, $\alpha_{{\theta}}=\sqrt{3}$. Now taking $m=3={\rm
sqf}(2r(r-s))$, there is a $({\mathbb{Q}}(\sqrt{3}),{\theta},1)$-triangle with
sides $(2,2,2\sqrt{3})$ and area $\sqrt{3}$ but using Theorem 2.1, ${\rm
rank}(E_{1,{\theta}}({\mathbb{Q}}(\sqrt{3})))={\rm
rank}(E_{1,{\theta}}({\mathbb{Q}}))+{\rm
rank}(E_{3,{\theta}}({\mathbb{Q}}))=0.$

The following theorem classifies all types of
$({\mathbb{K}},{\theta},n)$-triangles.

######  Theorem 1.2.

Assume that $n$ is not a ${\theta}$-congruent number over ${\mathbb{Q}}$ and
let $\sigma$ be the generator of ${\rm Gal}({\mathbb{K}}/{\mathbb{Q}})$. Then
any $({\mathbb{K}},{\theta},n)$-triangle with
$(U,V,W)\in({\mathbb{K}}^{*})^{3}$ and $(0<U\leq V<W)$ is necessarily one of
the following types:

  1. Type 1.

$U\sqrt{m}$, $V\sqrt{m}$, $W\sqrt{m}\in{\mathbb{Q}}$;

  2. Type 2.

$U$, $V$, $W\sqrt{m}\in{\mathbb{Q}}$;

  3. Type 3.

$U$, $V\in{\mathbb{K}}\backslash{\mathbb{Q}}$ such that $\sigma(U)=V$,
$W\in{\mathbb{Q}}$;

  4. Type 4.

$U$, $V\in{\mathbb{K}}\backslash{\mathbb{Q}}$ such that $\sigma(U)=-V$,
$W\in{\mathbb{Q}}$.

Let $A={\rm sqf}(r^{2}-s^{2})$, $B={\rm sqf}(2r(r-s))$ and $C={\rm
sqf}(2r(r+s))$. The following proposition shows when there is no
$({\mathbb{K}},{\theta},n)$-triangle of Types $2$, $3$ and $4$.

######  Proposition 1.3.

Let $p$ be a prime number and the pair $(m,A)$ (resp. $(m,B)$ and $(m,C)$) can
be written as $(p^{\alpha}a,p^{\beta}b)$, where $\alpha,\beta\in\\{0,1\\}$ and
$gcd(p,ab)=1$. Then there is no $({\mathbb{K}},{\theta},n)$-triangle of Type 2
(resp. Type 3 and Type 4) whenever one of the following conditions hold.

  1. (1)

$\underline{p=2:}$ $(\alpha,\beta)=(0,0)$ and
$(a,b)\mathop{\equiv}\limits^{4}(3,3)$,
$(\alpha,\beta)=(0,1)$ and
$(a,b)\mathop{\equiv}\limits^{8}(3,1),(3,5),(7,5),(7,7)$,
$(\alpha,\beta)=(1,0)$ and
$(a,b)\mathop{\equiv}\limits^{8}(1,3),(1,5),(3,5),(3,7),(5,3),(5,7),(7,3),(7,7)$,
$(\alpha,\beta)=(1,1)$ and
$(a,b)\mathop{\equiv}\limits^{8}(1,3),(1,5),(3,1),(3,3),(5,1),(5,7),(7,5),(7,7)$;

  2. (2)

$\underline{p\mathop{\equiv}\limits^{4}1:}$ $(\alpha,\beta)=(0,1)$ and
$\big{(}\frac{a}{p}\big{)}=-1$, $(\alpha,\beta)=(1,0)$ and
$\big{(}\frac{b}{p}\big{)}=-1$,
$(\alpha,\beta)=(1,1)$ and
$\big{(}\frac{a}{p}\big{)}\big{(}\frac{b}{p}\big{)}=-1$;

  3. (3)

$\underline{p\mathop{\equiv}\limits^{4}3:}$ $(\alpha,\beta)=(0,1)$ and
$\big{(}\frac{a}{p}\big{)}=-1$, $(\alpha,\beta)=(1,0)$ and
$\big{(}\frac{b}{p}\big{)}=-1$,
$(\alpha,\beta)=(1,1)$ and
$\big{(}\frac{a}{p}\big{)}\big{(}\frac{b}{p}\big{)}=1$.

The next result settles a condition on $n$ and $mn$ to be ${\theta}$-congruent
over ${\mathbb{Q}}$.

######  Theorem 1.4.

Let $n$ be a positive squarefree integer such that $\gcd(m,n)=1$ and $mn\neq
2,3,6$. Then the following statements are equivalent.

  1. (1)

There is a $({\mathbb{K}},{\theta},n)$-triangle $(U,V,W)_{\theta}$ with
$0<U\leq V<W$, $W\not\in{\mathbb{Q}}$ and $W\sqrt{m}\not\in{\mathbb{Q}}$;

  2. (2)

The integers $n$ and $mn$ are ${\theta}$-congruent numbers over
${\mathbb{Q}}$.

##  2\. Preliminaries

Consider an elliptic curve $E:y^{2}=x^{3}+ax^{2}+bx+c$ over ${\mathbb{Q}}$.
Recall that the $m$-twist $E^{m}$ of $E$ is an elliptic curve over
${\mathbb{Q}}$ defined by $y^{2}=x^{3}+amx^{2}+bm^{2}x+cm^{3}$. The next
result establishes a fact about ranks [10].

######  Theorem 2.1.

Let $E$ be an elliptic curve over ${\mathbb{Q}}$. Then

${\rm rank}(E({\mathbb{K}}))={\rm rank}(E({\mathbb{Q}}))+{\rm
rank}(E^{m}({\mathbb{Q}})).$

We denote the torsion subgroup of the groups $E({\mathbb{K}})$ and
$E^{m}({\mathbb{K}})$ by $T(E,{\mathbb{K}})$ and $T(E^{m},{\mathbb{K}})$,
respectively. Also, we write $T_{n,{\theta}}({\mathbb{K}})$ and
$T_{n,{\theta}}^{m}({\mathbb{K}})$, respectively, in the case
$E=E_{n,{\theta}}$. The following proposition and theorem have essential roles
in the proof of our results.

######  Proposition 2.2 ([9, Proposition 1]).

Let $E$ be an elliptic curve over ${\mathbb{K}}$. Then the map

$\phi:T(E,{\mathbb{K}})/T(E,{\mathbb{Q}})\rightarrow T(E^{m},{\mathbb{Q}}),\ \
\ \phi(\tilde{P}):=P-\sigma(P)$

is an injective map of abelian groups, where $\sigma$ is the generator of
${\rm Gal}({\mathbb{K}}/{\mathbb{Q}})$.

######  Theorem 2.3 ([7, Theorem 4.2]).

Let ${\mathbb{F}}$ be an algebraic number field and $E$ an elliptic curve over
${\mathbb{F}}$ defined by

$y^{2}=(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3}),\
\alpha_{1},\alpha_{2},\alpha_{3}\in{\mathbb{F}}.$

Suppose that $(x_{0},y_{0})$ be an ${\mathbb{F}}$-rational point of $E$. Then,
there exists an ${\mathbb{F}}$-rational point $(x_{1},y_{1})$ with
$2(x_{1},y_{1})=(x_{0},y_{0})$ if and only if $x_{0}-\alpha_{1},$
$x_{0}-\alpha_{2},$ $x_{0}-\alpha_{3}$ are squares in ${\mathbb{F}}$.

The next results give important information about ${\theta}$-congruent numbers
over ${\mathbb{Q}}$.

######  Theorem 2.4.

(Fujiwara, [4]) Consider $0<{\theta}<\pi$ with rational cosine.

  1. (1)

A positive integer $n$ is a ${\theta}$-congruent number if and only if
$E_{n,{\theta}}({\mathbb{Q}})$ has a point of order greater than $2$;

  2. (2)

If $n\neq 1,2,3,6$, then $n$ is a ${\theta}$-congruent number if and only if
$E_{n,{\theta}}({\mathbb{Q}})$ has positive rank.

All possibilities for the torsion subgroup of $E_{n,{\theta}}({\mathbb{Q}})$
can be found in the next result.

######  Theorem 2.5.

(Fujiwara, [5]) Let $T_{n,{\theta}}({\mathbb{Q}})$ be the torsion subgroup of
the ${\theta}$-congruent number elliptic curve $E_{n,{\theta}}$ over
${\mathbb{Q}}$.

  1. (1)

$T_{n,{\theta}}({\mathbb{Q}})\cong{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{8}$ if
and only if there exist integers $a,b>0$ such that $\gcd(a,b)=1$, $a$ and $b$
have opposite parity and satisfy either of the following conditions.

    1. (i)

$n=1$, $r=8a^{4}b^{4}$, $r-s=(a-b)^{4}$, $(1+\sqrt{2})b>a>b,$

    2. (ii)

$n=2$, $r=(a^{2}-b^{2})^{4}$, $r-s=32a^{4}b^{4}$, $a>(1+\sqrt{2})b;$

  2. (2)

$T_{n,{\theta}}({\mathbb{Q}})\cong{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{6}$ if
and only if there exist integers $u,v>0$ such that $\gcd(u,v)=1$, $u>2v$ and
satisfy one of the following conditions:

    1. (i)

$n=1$, $r=\frac{1}{2}(u-v)^{3}(u+v)$, $r+s=u^{3}(u-2v),$

    2. (ii)

$n=2$, $r=(u-v)^{3}(u+v)$, $r+s=2u^{3}(u-2v),$

    3. (iii)

$n=3$, $r=\frac{1}{6}(u-v)^{3}(u+v)$, $r+s=\frac{1}{3}u^{3}(u-2v),$

    4. (iv)

$n=6$, $r=\frac{1}{3}(u-v)^{3}(u+v)$, $r+s=\frac{2}{3}u^{3}(u-2v);$

  3. (3)

$T_{n,{\theta}}({\mathbb{Q}})\cong{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{4}$ if
and only if either of the following holds.

    1. (i)

$n=1$, $2r$ and $r-s$ are squares but not satisfy (i) of Part (1),

    2. (ii)

$n=2$, $r$ and $2(r-s)$ are squares but not satisfy (ii) of Part (1);

  4. (4)

Otherwise,
$T_{n,{\theta}}({\mathbb{Q}})\cong{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}$.

######  Remark 2.6.

For any squarefree integer $m>1$, the $m$-twist $E_{n,{\theta}}^{m}$ of the
elliptic curve $E_{n,{\theta}}$ is defined by $y^{2}=x(x+(r+s)mn)(x-(r-s)mn)$
which is equal to $E_{mn,{\theta}}$, as seen. Therefore
$E_{n,{\theta}}^{m}({\mathbb{Q}})=E_{mn,{\theta}}({\mathbb{Q}})$, and hence
$T_{n,{\theta}}^{m}({\mathbb{Q}})=T_{mn,{\theta}}({\mathbb{Q}})$.

##  3\. Proofs

Appealing to Proposition 2.2, we first settle all possibilities for the
torsion subgroup of $E_{n,{\theta}}({\mathbb{K}})$. Let $h,k,$ and $d$ be
integers such that $2r=h^{2}{\rm sqf}(2r),r-s=k^{2}{\rm sqf}(r-s)$ and
$2r(r-s)=d^{2}m$, where $m={\rm sqf}(2r(r-s))$.

######  Proposition 3.1.

Assume that $m>1$ and $n$ are squarefree positive integers such that
$\gcd(m,n)=1$ and $mn\neq 2,3,6$. Let $T_{n,{\theta}}({\mathbb{K}})$ be the
torsion subgroup of $E_{n,{\theta}}({\mathbb{K}})$.

  1. (1)

If $m={\rm sqf}(2r(r-s))$ and $n={\rm sqf}(2r)$, then

$T_{n,{\theta}}({\mathbb{K}})=\\{\infty,(0,0),(-(r+s)n,0),((r-s)n,0),\\\
((nh)^{2}-nd\sqrt{m},\pm(\frac{d^{2}mn}{h}-n^{2}hd\sqrt{m})),((nh)^{2}+nd\sqrt{m},\pm(\frac{d^{2}mn}{h}+n^{2}hd\sqrt{m}))\\};$

  2. (2)

If $m={\rm sqf}(2r(r-s))$ and $n={\rm sqf}(r-s)$, then

$T_{n,{\theta}}({\mathbb{K}})=\\{\infty,(0,0),(-(r+s)n,0),((r-s)n,0),\\\
((nk)^{2}-nd\sqrt{m},\pm(\frac{d^{2}mn}{k}-n^{2}kd\sqrt{m})),((nk)^{2}+nd\sqrt{m},\pm(\frac{d^{2}mn}{k}+n^{2}kd\sqrt{m}))\\};$

  3. (3)

Otherwise,
$T_{n,{\theta}}({\mathbb{K}})=\\{\infty,(0,0),(-(r+s)n,0),((r-s)n,0)\\}$.

###### Proof.

The $2$-torsion subgroup of $E_{n,{\theta}}({\mathbb{K}})$ is:

$E_{n,{\theta}}[2]({\mathbb{K}})=\\{\infty,(0,0),(-(r+s)n,0),((r-s)n,0)\\}.$

Therefore, we have $T_{n,{\theta}}({\mathbb{K}})\supset
E_{n,{\theta}}[2]({\mathbb{K}})\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}.$
By Remark 2.6 and Theorem 2.5,
$T_{n,{\theta}}^{m}({\mathbb{Q}})=T_{mn,{\theta}}({\mathbb{Q}})\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}$.
Since
$T_{n,{\theta}}({\mathbb{Q}})\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}$,
by Proposition 2.2 and [9, Theorem 1] we have

$T_{n,{\theta}}({\mathbb{K}})\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}\
{\rm or}\ {\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/4{\mathbb{Z}}.$

First let
$T_{n,{\theta}}({\mathbb{K}})\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/4{\mathbb{Z}}$.
Then there exists a point $P=(x_{0},y_{0})$ of order $4$ in
$T_{n,{\theta}}({\mathbb{K}})$. Then $2P$ must be one of the points $(0,0)$,
$(-(r+s)n,0)$ and $((r-s)n,0)$. If $2P=(0,0)$ then both $(r+s)n$ and $-(r-s)n$
are squares in ${\mathbb{K}}$, which is impossible since ${\mathbb{K}}$ is a
real quadratic number field and hence $-1$ is not a square in ${\mathbb{K}}$.
Similarly, if $2P=(-(r+s)n,0)$, then $-(r+s)n$ and $-2rn$ are squares in
${\mathbb{K}}$, again a contradiction by the same reason. If $2P=((r-s)n,0)$,
then $(r-s)n$ and $2rn$ are squares in ${\mathbb{K}}$. Since $n$ is
squarefree, these integers are squares in ${\mathbb{K}}$ if $m={\rm
sqf}(2r(r-s))$. By a simple computation using the duplication formula we
obtain (1) and (2). Now,
$T_{n,{\theta}}({\mathbb{K}})\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}$
implies (3), and the proof is completed. ∎

###### Proof of Theorem 1.1.

Consider the two sets

$S=\big{\\{}(U,V,W)\in({\mathbb{K}}^{*})^{3}:\ 0<U\leq V<W,\ UV=2rn\text{ and
}U^{2}+V^{2}-2sUV/r=W^{2}\big{\\}},$ $T=\\{(u,v)\in
2E_{n,{\theta}}({\mathbb{K}})\backslash\\{\infty\\}:\ v\geq 0\\}.$

There is a one to one correspondence between the two sets $S$ and $T$ via the
two mutually inverse maps $\varphi:S\rightarrow T$ and $\psi:T\rightarrow S$
defined by

$\varphi(U,V,W):=({W^{2}}/{4},{W(V^{2}-U^{2})}/{8}),$
$\psi(u,v):=\big{(}\sqrt{u+(r+s)n}-\sqrt{u-(r-s)n},\sqrt{u+(r+s)n}+\sqrt{u-(r-s)n},2\sqrt{u}\big{)}.$

Clearly, $E_{n,{\theta}}({\mathbb{K}})\backslash
E_{n,{\theta}}[2]({\mathbb{K}})\neq\emptyset$ if and only if $S\neq\emptyset$.

Suppose that $m\neq{\rm sqf}(2r(r-s))$ and $mn\neq 2,3,6$. Then by proposition
3.1, we have $T_{n,{\theta}}({\mathbb{K}})=E_{n,{\theta}}[2]({\mathbb{K}})$.
Therefore, ${\rm rank}(E_{n,{\theta}}({\mathbb{K}}))>0$ if and only if
$E_{n,{\theta}}({\mathbb{K}})\backslash
E_{n,{\theta}}[2]({\mathbb{K}})\neq\emptyset$. So ${\rm
rank}(E_{n,{\theta}}({\mathbb{K}}))>0$ if and only if either ${\rm
rank}(E_{n,{\theta}}({\mathbb{Q}}))>0$ or ${\rm
rank}(E_{n,{\theta}}^{m}({\mathbb{Q}}))>0$, by Theorem 2.1. The second part of
the theorem follows from Remark 2.6. ∎

###### Proof of Theorem 1.2.

Assume $n$ is a $({\mathbb{K}},{\theta})$-congruent number and
$(U,V,W)_{\theta}$ is the corresponding $({\mathbb{K}},{\theta},n)$-triangle
with area $n\alpha_{\theta}$ such that $0<U\leq V<W$. As in the proof of the
Theorem 1.1, there is a point $P=(x,y)$ in
$E_{n,{\theta}}({\mathbb{K}})\backslash E_{n,{\theta}}[2]({\mathbb{K}})$ such
that $\psi(P)=(U,V,W)$. Substituting $P$ by $P+(0,0)$, $P+(-(r+s)n,0)$ or
$P+((r-s)n,0)$, if necessary, we may assume that $x>[(r+s)+\sqrt{2r(r-s)}]n$.
Putting $2P=(u,v)$ and using the map $\psi$ in the proof of Theorem 1.1, we
obtain

$U={2rnx}/{|y|},\ V={x^{2}+2snx-(r^{2}-s^{2})n^{2}}/{|y|},\
W={x^{2}+(r^{2}-s^{2})n^{2}}/{|y|},$

where $x,y\in{\mathbb{K}}$ and $|\cdot|$ is the usual absolute value induced
from the embedding $\iota:{\mathbb{K}}\hookrightarrow{\mathbb{R}}$ with
$\iota(\sqrt{m})$ positive. Suppose $\sigma$ is a generator of ${\rm
Gal}({\mathbb{K}}/{\mathbb{Q}})$ and put $\sigma(P)=(\sigma(x),\sigma(y))$.
Since $P+\sigma(P)$ is an element of $E_{n,{\theta}}({\mathbb{Q}})$ and $n$ is
not a ${\theta}$-congruent number, $P+\sigma(P)\in
T_{n,{\theta}}({\mathbb{Q}})=\\{\infty,(0,0),(-(r+s)n,0),((r-s)n,0)\\}.$
Hence, one of the following cases necessarily happens:

  1. I.

$P+\sigma(P)=\infty$. In this case, $\sigma(x)=x$ and $\sigma(y)=-y$. So,
$x,y\sqrt{m}$ and hence $U\sqrt{m},V\sqrt{m}$ and $W\sqrt{m}$ are rational and
we obtain a $({\mathbb{K}},{\theta},n)$-triangle of Type 1.

  2. II.

$P+\sigma(P)=(0,0)$. We have $\sigma(x)/x=\sigma(y)/y$, which we denote by
$\alpha$. Then,

$\sigma(y)^{2}=\alpha^{2}y^{2}=\alpha^{2}x^{3}+2sn\alpha^{2}x^{2}-(r^{2}-s^{2})n^{2}\alpha^{2}x.$

Since $\sigma(P)$ is a point on $E_{n,{\theta}}$, we get

$\sigma(y)^{2}=\sigma(x)^{3}+2sn\sigma(x)^{2}-(r^{2}-s^{2})n^{2}\sigma(x)=\alpha^{3}x^{3}+2sn\alpha^{2}x^{2}-(r^{2}-s^{2})n^{2}\alpha
x.$

Clearly, $\alpha\neq 0,1$ and $x\neq 0$, which implies $x\sigma(x)=\alpha
x^{2}=-(r^{2}-s^{2})n^{2}.$ Therefore,

$V={x(x+2sn+\sigma(x))}/{|y|},\ W\sqrt{m}={x(x-\sigma(x))\sqrt{m}}/{|y|}.$

Since $x/y=\sigma(x/y)$ and $x>[(r+s)+\sqrt{2r(r-s)}]n$, then $x/|y|$ is
rational and hence $U=2rnx/|y|$, $V$ and $W\sqrt{m}$ are rational, which gives
a $({\mathbb{K}},{\theta},n)$-triangle of Type 2.

  3. III.

$P+\sigma(P)=((r-s)n,0)$. We have $\sigma(x-(r-s)n)/(x-(r-s)n)=\sigma(y)/y,$
which we denote by $\beta$. Put $z=x-(r-s)n$. Then,

$\sigma(y)^{2}=\beta^{2}[z^{3}+(3r-s)nz^{2}+2r(r-s)n^{2}z].$

Since $\sigma(P)$ is a point on $E_{n,{\theta}}$, we get

$\sigma(y)^{2}=\beta^{3}z^{3}+(3r-s)n\beta^{2}z^{2}+2r(r-s)n^{2}\beta z.$

Now $\beta\neq 0,1$ and $z\neq 0$, which implies $\beta z^{2}=2r(r-s)n^{2}.$
Substituting this equation and $x=z+(r-s)n$ in $U$, $V$ and $W$, we obtain

$U=\frac{z(\sigma(z)+2rn)}{|y|},\ V=\frac{z(z+2rn)}{|y|},\
W=\frac{z(z+2(r-s)n+\sigma(z))}{|y|}.$

Since $z/y=\sigma(z/y)$ and $z>0$, then $z/|y|$ and hence $W$ is rational and
$\sigma(U)=V$. This time we obtain a $({\mathbb{K}},{\theta},n)$-triangle of
Type 3.

  4. IV.

$P+\sigma(P)=(-(r+s)n,0)$. Put $w=x+(r+s)n$. As in Case III, $w/|y|$ and

$W={w(w-2(r+s)n+\sigma(w))}/{|y|}$

are rational and $\sigma(U)=-V$, where

$U={w(2rn-\sigma(w))}/{|y|},\ V={w(w-2rn)}/{|y|}.$

Therefore, we obtain a $({\mathbb{K}},{\theta},n)$-triangle of Type 4.

∎

###### Proof of Proposition 1.3.

If we suppose that there is a $({\mathbb{K}},{\theta},n)$-triangle of Type 2,
say $(U,V,W)_{{\theta}}=(u,v,w\sqrt{m})$ with $u,v,w\in{\mathbb{Q}}^{+}$, then
$(x,y,z)=(ru-sv,v,mrw)$ is a non-zero solution of the equation

$z^{2}=mx^{2}+m(r^{2}-s^{2})y^{2}.$ (3.1)

And, if there is a $({\mathbb{K}},{\theta},n)$-triangle of Type 3, say
$(U,V,W)_{{\theta}}=(u-v\sqrt{m},u+v\sqrt{m},w)$ such that $\sigma(U)=V$, then
$(x,y,z)=(u,v,rw)$ is a non-zero solution of

$z^{2}=2r(r-s)x^{2}+2mr(r+s)y^{2}.$ (3.2)

Similarly, if $(U,V,W)_{{\theta}}=(-u+v\sqrt{m},u+v\sqrt{m},w)$ is a
$({\mathbb{K}},{\theta},n)$-triangle of Type 4 such that $\sigma(U)=-V$, then
$(x,y,z)=(u,v,rw)$ satisfies

$z^{2}=2r(r+s)x^{2}+2mr(r-s)y^{2}.$ (3.3)

By the Hasse local-global principle, the equations (3.1), (3.2) and (3.3) have
solutions in ${\mathbb{Q}}$ if and only if they have a solution in
${\mathbb{Q}}_{p}$ for every prime $p$, where ${\mathbb{Q}}_{p}$ is the field
of $p$-adic numbers. We assume that $A={\rm sqf}(r^{2}-s^{2})$, and for a
prime $p$ the pair $(m,A)$ ($(m,B)$, and $(m,C)$, resp.) can be written as
$(p^{\alpha}a,p^{\beta}b)$, where $\alpha,\beta\in\\{0,1\\}$ and
$\gcd(p,a,b)=1$. Then, using Hilbert symbols [11, Theorem 1, III], the
equations (3.1), (3.2) and (3.3) have solutions in ${\mathbb{Q}}_{2}$ if and
only if one of the following cases happens:

  1. i)

$(\alpha,\beta)=(0,0)$ and $(a,b)\mathop{\hskip
0.56905pt\not\equiv}\limits^{4}(3,3)$;

  2. ii)

$(\alpha,\beta)=(0,1)$ and $(a,b)\mathop{\hskip
0.56905pt\not\equiv}\limits^{8}(3,1),(3,5),(7,5),(7,7)$;

  3. iii)

$(\alpha,\beta)=(1,0)$ and $(a,b)\mathop{\hskip
0.56905pt\not\equiv}\limits^{8}(1,3),(1,5),(3,5),(3,7),(5,3),(5,7),(7,3),(7,7)$;

  4. iv)

$(\alpha,\beta)=(1,1)$ and $(a,b)\mathop{\hskip
0.56905pt\not\equiv}\limits^{8}(1,3),(1,5),(3,1),(3,3),(5,1),(5,7),(7,5),(7,7)$.

Also, the equations (3.1), (3.2) and (3.3) have solutions in
${\mathbb{Q}}_{p}$ with $p\mathop{\equiv}\limits^{4}1$ if and only if one of
the following happens:

  1. i)

$(\alpha,\beta)=(0,1)$ and $\big{(}\frac{a}{p}\big{)}=1$;

  2. ii)

$(\alpha,\beta)=(1,0)$ and $\big{(}\frac{b}{p}\big{)}=1$;

  3. iii)

$(\alpha,\beta)=(1,1)$ and
$\big{(}\frac{a}{p}\big{)}\big{(}\frac{b}{p}\big{)}=1$.

∎

###### Proof of Theorem 1.4.

Case 1. $n$ and $mn$ are $({\mathbb{Q}},{\theta})$-congruent numbers. Consider
the $({\mathbb{Q}},{\theta},n)$-triangle $(U_{1},V_{1},W_{1})_{\theta}$ and
the $({\mathbb{Q}},{\theta},mn)$-triangle $(U_{2},V_{2},W_{2})_{\theta}$,
where

$0<U_{1}\leq V_{1}<W_{1},\ 2rn=U_{1}V_{1},\
U_{1}^{2}+V_{1}^{2}-\frac{2sU_{1}V_{1}}{r}=W_{1}^{2},$ $0<U_{2}\leq
V_{2}<W_{2},\ 2rmn=U_{2}V_{2},\
U_{2}^{2}+V_{2}^{2}-\frac{2sU_{2}V_{2}}{r}=W_{2}^{2}.$

Hence, $(U_{2}/\sqrt{m},V_{2}/\sqrt{m},W_{2}/\sqrt{m})_{\theta}$ is a
$({\mathbb{K}},{\theta},n)$-triangle. Recall the maps $\varphi$ and $\psi$ in
the proof of Theorem 1.2 and put

$P=(u,v)=\varphi((U_{1},V_{1},W_{1}))+\varphi((U_{2}/\sqrt{m},V_{2}/\sqrt{m},W_{2}/\sqrt{m})).$

Then the additive law on $E_{n,{\theta}}({\mathbb{K}})$ implies
$u=a+b\sqrt{m}$, where

$a=\frac{m^{3}W_{1}^{2}(V_{1}^{2}-U_{1}^{2})^{2}+W_{2}^{2}(V_{2}^{2}-U_{2}^{2})^{2}}{4m(W_{2}^{2}-mW_{1}^{2})^{2}}-\big{(}\frac{W_{1}^{2}}{4}+\frac{W_{2}^{2}}{4m}+2sn\big{)}>0,$
$b=-\frac{W_{1}W_{2}(V_{1}^{2}-U_{1}^{2})(V_{2}^{2}-U_{2}^{2})\sqrt{m}}{2(W_{2}^{2}-mW_{1}^{2})^{2}}.$

We may assume $v\geq 0$. Since $(u,v)\in T$, then $\psi((u,v))\in S$ which
indicates the sides of a $({\mathbb{K}},{\theta},n)$-triangle
$(U,V,W)_{\theta}$. In fact, if we suppose
$U=u_{1}+u_{2}\sqrt{m},V=v_{1}+v_{2}\sqrt{m}$ and $W=w_{1}+w_{2}\sqrt{m},$
where $u_{1},u_{2},v_{1},v_{2},w_{1},w_{2}$ are rational, then

$w_{1}=\pm\sqrt{2(a\pm\sqrt{a^{2}-mb^{2}})},\ \ w_{2}=\frac{2b}{w_{1}},$

and

$U=(\alpha_{1}-\beta_{1})+(\alpha_{2}-\beta_{2})\sqrt{m},\ \
V=(\alpha_{1}+\beta_{1})+(\alpha_{2}+\beta_{2})\sqrt{m},$

where

$\alpha_{1}=\pm\sqrt{\frac{(a+(r+s)n)\pm\sqrt{(a+(r+s)n)^{2}-mb^{2}}}{2}},\ \
\alpha_{2}=\frac{b}{2\alpha_{1}},$
$\beta_{1}=\pm\sqrt{\frac{(a-(r-s)n)\pm\sqrt{(a-(r-s)n)^{2}-mb^{2}}}{2}},\ \
\beta_{2}=\frac{b}{2\beta_{1}}.$

Conversely, suppose to the contrary that $n$ or $mn$ is not
${\theta}$-congruent over ${\mathbb{Q}}$. First, assume $n$ is not
${\theta}$-congruent over ${\mathbb{Q}}$ but $mn$ is ${\theta}$-congruent over
${\mathbb{Q}}$. By Theorem 1.2 (1), there is no
$({\mathbb{K}},{\theta},n)$-triangle $(U,V,W)_{\theta}$ satisfying the
conditions $0<U\leq V<W$, $W\not\in{\mathbb{Q}}$ and
$W\sqrt{m}\not\in{\mathbb{Q}}$.

Case 2. $mn$ is not ${\theta}$-congruent over ${\mathbb{Q}}$ but $n$ is
$({\mathbb{K}},{\theta})$-congruent. Let $(U,V,W)_{\theta}$ denotes the sides
of the corresponding $({\mathbb{K}},{\theta},n)$-triangle. Multiplying the
three sides by $\sqrt{m}$, we get the $({\mathbb{K}},{\theta},mn)$-triangle
$(U\sqrt{m},V\sqrt{m},W\sqrt{m})_{\theta}$. For the positive integer $mn$, we
define the map $\varphi^{\prime}$ in the same way as $\varphi$. Put

$2P^{\prime}=\varphi^{\prime}((U\sqrt{m},V\sqrt{m},W\sqrt{m}))$

for some point $P^{\prime}\in E_{mn,{\theta}}({\mathbb{K}})$. For the
generator $\sigma$ of ${\rm Gal}({\mathbb{K}}/{\mathbb{Q}})$, since
$P^{\prime}+\sigma(P^{\prime})$ is an element in
$E_{mn,{\theta}}({\mathbb{Q}})$ and $mn$ is not ${\theta}$-congruent over
${\mathbb{Q}}$, we have

$P^{\prime}+\sigma(P^{\prime})\in
T_{mn,{\theta}}({\mathbb{Q}})=\\{\infty,(0,0),(-(r+s)mn,0),((r-s)mn,0)\\}.$

Therefore, by the same way as in the proof of Theorem 1.2, one of the
following cases necessarily happens:

  1. Type 1.

$U$, $V$, $W\in{\mathbb{Q}}$;

  2. Type 2.

$U\sqrt{m}$, $V\sqrt{m}$, $W\in{\mathbb{Q}}$;

  3. Type 3.

$U$, $V\in K\backslash{\mathbb{Q}}$ such that $\sigma(U)=V$,
$W\sqrt{m}\in{\mathbb{Q}}$;

  4. Type 4.

$U$, $V\in K\backslash{\mathbb{Q}}$ such that $\sigma(U)=-V$,
$W\sqrt{m}\in{\mathbb{Q}}$.

Hence, there is no $({\mathbb{K}},{\theta},n)$-triangle $(U,V,W)_{\theta}$
with $W\not\in{\mathbb{Q}}$ and $W\sqrt{m}\not\in{\mathbb{Q}}$.

Case 3. Both $n$ and $mn$ are not ${\theta}$-congruent numbers over
${\mathbb{Q}}$, where $mn\neq 2,3,6$. If $m\neq{\rm sqf}(2r(r-s))$, by Theorem
1.1, $n$ is not $({\mathbb{K}},{\theta},n)$-congruent. If $m={\rm
sqf}(2r(r-s))$ and $n$ is $({\mathbb{K}},{\theta},n)$-congruent, we have $U=V$
for all $({\mathbb{K}},{\theta},n)$-triangles $(U,V,W)_{\theta}$. Hence, there
is no any $({\mathbb{K}},{\theta},n)$-triangle $(U,V,W)_{\theta}$ with
$W\not\in{\mathbb{Q}}$ and $W\sqrt{m}\not\in{\mathbb{Q}}$. We have completed
the proof of Theorem 1.4. ∎

##  4\. Examples

In this section, we give some examples of $({\mathbb{K}},{\theta})$-congruent
numbers and verify all four types of $({\mathbb{K}},{\theta},n)$-triangles in
Theorem 1.2 in the cases ${\theta}=\pi/3,2\pi/3.$ Given $n$, let
$(U,V,W)_{{\theta}}$ be a $({\mathbb{K}},{\theta},n)$-triangle. Then, we have

$0<U\leq V<W,\ \ UV=2rn,\ \ W^{2}=U^{2}+V^{2}-\frac{2s}{r}UV.$

For any $(U,V,W)_{{\theta}}$,
$\varphi((U,V,W))=({W^{2}}/{4},{W(V^{2}-U^{2})}/{8})$ is a point of
$2E_{n,{\theta}}({\mathbb{K}})\backslash\\{\infty\\}.$ Also, for any point
$(u,v)\in 2E_{n,{\theta}}({\mathbb{K}})\backslash\\{\infty\\},$

$\psi\big{(}(u,v)\big{)}=\big{(}(\sqrt{u+(r+s)n}-\sqrt{u-(r-s)n},\sqrt{u+(r+s)n}+\sqrt{u-(r-s)n},2\sqrt{u})\big{)}.$

In our computations we have used Cremona’s MWrank program [2] and the number
theoretic Pari software [1].

### I) Case ${\theta}=\pi/3$

In this case, we have $r=2$, $s=1$, and $\alpha_{{\theta}}=\sqrt{3}$, and
hence the area of any $({\mathbb{K}},\pi/3,n)$-triangle is $n\sqrt{3}.$

######  Example 4.1.

Take $n=3$ and $m=13$. We have the following
$({\mathbb{Q}}(\sqrt{13}),\pi/3,3)$-triangles of types 1, 2, 3 and 4 in
Theorem 1.1 and the corresponding points in the set
$2E_{3,\pi/3}({\mathbb{Q}}(\sqrt{13}))\setminus\\{\infty\\}.$

  1. Type 1.

An easy computing shows that the rank of $E_{39,\pi/3}({\mathbb{Q}})$ is 2,
and the generators of the group are $P_{1}=[-9,-216]$ and $P_{2}=[75,-720]$.
We have

$2P_{1}=[1894/16,-91805/64]\in
2E_{39,{\theta}}({\mathbb{Q}})\backslash\\{\infty\\}.$

Now, using the map $\varphi$ and $\psi$, defined in the proof of the Theorem
1.1 we get a rational $\pi/3$-triangle $\big{(}13/2,24,43/2\big{)}$ with area
$39$, which gives the following $({\mathbb{Q}}(\sqrt{13}),\pi/3,3)$-triangle
of Type 1:

$(U,V,W)_{\pi/3}=({\sqrt{13}}/{2},{24\sqrt{13}}/{13},{43\sqrt{13}}/{26})$

which corresponds to the following point
$Q=({1894}/{208},{91805\sqrt{13}}/{416})$.

  2. Type 2.

We have a $({\mathbb{Q}}(\sqrt{13}),\pi/3,3)$-triangle
$(U,V,W)_{\pi/3}=(3,4,\sqrt{13})$ of type 2 with the corresponding point
$Q=({13}/{4},{7\sqrt{13}}/{8}).$

  3. Type 3.

Let $U=u-v\sqrt{13},$ $V=u+v\sqrt{13}$ and $W=w,$ where
$u,v,w\in{\mathbb{Q}}\setminus\\{0\\}.$ Then the pair $(u,v)$ satisfies the
equation $u^{2}-13v^{2}=12.$ An easy solution of this equation is
$(u_{0},v_{0})=(5,1)$. Parametrizing $u$ and $v$ in terms of
$t\in{\mathbb{Q}}$ we obtain $u={-5t^{2}+26t-65}/{t^{2}-13}$ and
$v={t^{2}-10t+13}/{t^{2}-13}.$ By putting these into $w^{2}=u^{2}+39v^{2}$ and
taking $t=13/4$ one can see that $w^{2}=u^{2}+39v^{2}$ is a square in
${\mathbb{Q}}$. So, we obtain
$(U,V,W)_{\pi/3}=({41-11\sqrt{13}}/{3},{41+11\sqrt{13}}/{3},{80}/{3}),$ with
$({\mathbb{Q}}(\sqrt{13}),\pi/3,3)$-triangle of type 3 with the corresponding
point $Q=({1600}/{3},{18040\sqrt{13}}/{9}).$

  4. Type 4.

Let $U=-u+v\sqrt{13},$ $V=u+v\sqrt{13}$ and $W=w,$ where
$u,v,w\in{\mathbb{Q}}\setminus\\{0\\}.$ Then the pair $(u,v)$ satisfies
$13v^{2}-u^{2}=12$ with a solution $(u_{0},v_{0})=(1,1)$. A similar discussion
as in the previous step, taking $t=8$, leads us to a
$({\mathbb{Q}}(\sqrt{13}),\pi/3,3)$-triangle of Type 4, with the corresponding
point $Q=({24964}/{51},{1002352\sqrt{13}}/{51}).$

######  Example 4.2.

Let $n=11$ and $m=5$. One can see that $n$ is $\pi/3$-congruent over
${\mathbb{Q}}$ and there is a $({\mathbb{Q}},\pi/3,11)$-triangle
$(U_{1},V_{1},W_{1})=(55/12,48/5,499/60).$ Also, $nm=55$ is $\pi/3$-congruent
over ${\mathbb{Q}}$ and $(U_{2},V_{2},W_{2})=(8,55/2,49/2)$ is a rational
$\pi/3$-triangle with area $11\sqrt{3}$. Dividing its sides by $\sqrt{5}$, we
obtain a $({\mathbb{Q}}(\sqrt{5}),\pi/3,11)$-triangle

$(U_{2}/\sqrt{5},V_{2}/\sqrt{5},W_{2}/\sqrt{5})=({8\sqrt{5}}/{5},{11\sqrt{5}}/{2},{49\sqrt{5}}/{10}).$

Now, a calculations as in the proof of Theorem 1.4 leads to a
$({\mathbb{Q}}(\sqrt{5}),\pi/3,11)$-triangle

$(U,V,W)=\big{(}\frac{1}{310}(1470+499\sqrt{5}),\frac{88}{5909}(1470-499\sqrt{5}),\frac{1}{183179}(4145193-12554399\sqrt{5})\big{)}$

satisfying in Theorem 1.4.

### II) Case ${\theta}=2\pi/3$

In this case, we have $r=2$, $s=-1$, and $\alpha_{{\theta}}=\sqrt{3}.$ So, as
in the case I, the area of any $({\mathbb{K}},2\pi/3,n)$-triangle is
$n\sqrt{3}$.

######  Example 4.3.

Take $n=17$ and $m=13$. By a similar way as in Example 4.1, we find the
following $({\mathbb{Q}}(\sqrt{13}),2\pi/3,17)$-triangles with area
$17\sqrt{13}$ of types 1, 2, 3 and 4 preceding by their corresponding points
in $2E_{17,2\pi/3}({\mathbb{Q}}(\sqrt{13}))\setminus\\{\infty\\}.$

  1. Type 1.

$(U,V,W)_{2\pi/3}=({17\sqrt{13}}/{26},8\sqrt{13},{217\sqrt{13}}/{26}),$
$Q=({47089}/{16},{9325575\sqrt{13}}/{10816})$;

  2. Type 2.

$(U,V,W)_{2\pi/3}=(1,68,19\sqrt{13}),$ $Q=(13/4,7\sqrt{13}/8);$

  3. Type 3.

$(U,V,W)_{2\pi/3}=(9-\sqrt{13},9+\sqrt{13},16),$ $Q=(64,72\sqrt{13});$

  4. Type 4.

$(U,V,W)_{2\pi/3}=({-5+7\sqrt{13}}/{3},{5+7\sqrt{13}}/{3},{44}/{3}),Q=({484}/{9},{770\sqrt{13}}/{27}).$

######  Example 4.4.

Let $n=19$ and $m=6.$ Then $19$ is a $2\pi/3$-congruent number over
${\mathbb{Q}}$ and there is a $({\mathbb{Q}},2\pi/3,6)$-triangle
$(U_{1},V_{1},W_{1})=({544}/{105},{1995}/{136},{254659}/{14280})$ with area
$19\sqrt{3}$. Also, the integer $nm=114$ is a $2\pi/3$-congruent number over
${\mathbb{Q}}$ and $(U_{2},V_{2},W_{2})=(5,{912}/{10},{469}/{5})$ is a
$2\pi/3$-triangle with area $114\sqrt{3}$ from which we obtain a
$({\mathbb{Q}}(\sqrt{6}),2\pi/3,19)$-triangle

$({5\sqrt{6}}/{6},{76\sqrt{6}}/{5},{469\sqrt{6}}/{30}).$

By a similar methods as in Example 4.2, one can find a
$({\mathbb{Q}}(\sqrt{6}),2\pi/3,19)$-triangle

$(U,V,W)_{2\pi/3}=\big{(}(25449816+4838521\sqrt{6})/4683550,\\\
{20}(4145193-12554399\sqrt{6})/{28499829},\\\
{7}(3589965612532-2573211605723\sqrt{6})/{1170880474675}\big{)},$

satisfying Theorem 1.4.

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# Vectorial active matter on the lattice: polar condensates and nematic
filaments

Josué Manik Nava-Sedeño  Technische Universität Dresden, Center for
Information Services and High Performance Computing, Nöthnitzer Straße 46,
01062 Dresden, Germany  Department of Mathematics, Faculty of Sciences,
Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad
Universitaria, 04510 Mexico City, Mexico  Corresponding authors. Email:
<EMAIL_ADDRESS>Haralampos Hatzikirou  Mathematics Department,
Khalifa University, P.O. Box 127788, Abu Dhabi, UAE  Anja Voß-Böhme
Technische Universität Dresden, Center for Information Services and High
Performance Computing, Nöthnitzer Straße 46, 01062 Dresden, Germany  Fakultät
Informatik/Mathematik, Hochschule für Technik und Wirtschaft, Dresden, Germany
Lutz Brusch  Technische Universität Dresden, Center for Information Services
and High Performance Computing, Nöthnitzer Straße 46, 01062 Dresden, Germany
Andreas Deutsch  Technische Universität Dresden, Center for Information
Services and High Performance Computing, Nöthnitzer Straße 46, 01062 Dresden,
Germany  Corresponding authors. Email<EMAIL_ADDRESS>Fernando
Peruani  Laboratoire de Physique Théorique et Modélisation, UMR 8089, CY Cergy
Paris Université, 95302 Cergy-Pontoise, France  Corresponding authors. Email:
<EMAIL_ADDRESS>
###### Abstract

We introduce a novel lattice-gas cellular automaton (LGCA) for compressible
vectorial active matter with polar and nematic velocity alignment.
Interactions are, by construction, zero-range. For polar alignment, we show
the system undergoes a phase transition that promotes aggregation with strong
resemblance to the classic zero-range process (ZRP). We find that above a
critical point, the states of a macroscopic fraction of the particles in the
system coalesce into the same state, sharing the same position and momentum
(polar condensate). For nematic alignment, the system also exhibits
condensation, but there exist fundamental differences: a macroscopic fraction
of the particles in the system collapses into a filament, where particles
possess only two possible momenta. Furthermore, we derive hydrodynamic
equations for the active LGCA model to understand the phase transitions and
condensation that undergoes the system. We also show that generically the
discrete lattice symmetries – e.g. of a square or hexagonal lattice – affect
drastically the emergent large-scale properties of on-lattice active systems.
The study puts in evidence that aligning active matter on the lattice displays
new behavior, including phase transitions to states that share similarities to
condensation models.

### Keywords

Velocity alignment; swarming; nematic order; cellular automaton

##  1 Introduction

Self-organized patterns of self-propelled entities are found at every scale in
biology: from in vitro cytoskeletal extracts [1, 2, 3], bacterial systems [4,
5, 6, 7, 8], and insect swarms [9] to schools of fish [10], herds of mammals
[11], and human crowds [12]. Collective phenomena are also observed in
artificial active systems such as synthetic swimmers [13, 14], phoretic
colloids [15, 16], and active rollers [17, 18, 19]. Active systems in the
absence of torques – often referred as scalar active matter – can phase
separate [20, 21, 22]. Other complex, self-organized, collective motion
patterns exhibit either polar or nematic (orientational) order. Given the
vectorial nature of orientational order, active systems sharing such a feature
are often referred to as “vectorial active matter” [23]. It is widely believed
that self-organized pattern formation in vectorial active matter requires
particles to interact via some type of velocity alignment mechanism [24, 25],
though alternative mechanisms exist [26], such as combined attraction-
repulsion and active forces [27]. The best known example of an active system
with polar velocity alignment is the Vicsek model [28], where complex, active
polar flows spontaneously emerge. A different class of active systems is
obtained by changing the velocity alignment in the Vicsek model from polar to
nematic [29, 30]. Self-propelled (point-mass) particles with nematic alignment
do not generate polar flows, but self-organize into high-density, nematic
bands [30]. The theoretical understanding of vectorial active matter has been
strongly based on the study of off-lattice Vicsek-like models by means of
numerically costly, particle-based simulations [28, 30, 31, 32, 33, 34] and by
the derivation of hydrodynamic equations for these systems [35, 36, 37, 38,
39, 40, 41]; for comprehensive reviews, see [24, 25]. There is, however, a
renewed interest in understanding the behavior of active matter on lattices
given the suitability of efficient numerical schemes, and in particular due to
the large repertoire of available analytical tools to study lattice models
that are expected to help to improve our theoretical understanding of active
matter physics [42, 43, 44, 45, 46, 47].

Figure 1: LGCA dynamics. Particles reside in velocity channels within lattice
sites. During the reorientation step (R), particles within each lattice site
are assigned new velocity channels according to the transition probability
$T_{\vartheta}\left(\mathbf{s}\right)$. Subsequently, during the migration
step (M), particles are deterministically moved to neighboring lattice sites.

Here, we introduce an active lattice-gas cellular automaton (LGCA) for
compressible (i.e. without volume exclusion) vectorial active matter,
including both polar and nematic velocity alignment, where interactions are,
by construction, zero-range. We show that the introduced LGCA with polar
alignment undergoes a non-equilibrium phase transition that is fundamentally
different from the non-local, incompressible active LGCA, where the onset of
order is characterized by the presence of travelling polar bands [48] as
occurs in the off-lattice Vicsek model [49]. In contrast, in the present model
(with polar alignment), the emergence of order occurs via a process by which a
macroscopic fraction of the particles coalesce into the same state, and thus
sharing the same position and momentum, i.e. a polar condensate. For nematic
aligment, we find that the system displays a condensation, but with a
fundamental difference: a macroscopic fraction of the particles in the system
collapses into a nematic filament. Note that recently, condensation behavior
was found in scalar active matter in the presence of an external potential by
a mechanism known edge-diffusivity [50, 51]. In scalar active matter – i.e. in
self-propelled disks – it has been observed that phase separation into dense
and dilute phases can result from volume exclusion interactions among the
particles, a phenomenon known as motility-induced phase separation (MIPS); see
[52] for a review). Our study shows that in vectorial active matter,
condensation behavior is based on velocity alignment and does not require an
external potential or external field, even in the absence of volume exclusion
interactions. Thus, the emergence of polar and nematic order in the active
LGCA is fundamentally different from the onset of order in other models [28,
35, 31, 43, 47, 44, 45, 46, 42]. Similarly, phase separation in the active
LGCA, that takes the form of a condensation, is fundamentally different from
MIPS [52]. At high interaction strengths and low effective temperatures, these
models show a great variety of collective behaviors such as homogeneous,
polar-ordered states, formation of spatially-extended clusters, emergence of
long-range order, filament-linked clusters, phase separation into extended
jammed clusters, the formation of moving bands, and rippling; whereas our
active LGCA models consistently show only two phases: a disordered,
homogeneous state, and an aligned, highly condensed state in either single
lattice sites, or in filaments of a single lattice site in width, depending on
the nature of the alignment interaction. Furthermore, the hydrodynamic
equations for the introduced LGCA allows us to show that generically the
discrete lattice symmetries – at least for square and hexagonal lattices –
have a strong impact on the emergent large-scale properties of on-lattice
active systems. In summary, our study unveils the existence of condensed
phases in vectorial active matter on the lattice, without attractive or
excluded volume interactions.

##  2 Model definition

We study an hexagonal LGCA, where each node possesses six velocity channels,

$\mathbf{c}_{\vartheta}=\left(\cos[(\vartheta-1)\frac{\pi}{3}]\,,\sin[(\vartheta-1)\frac{\pi}{3}]\right)$

with $\vartheta\in[1,2,\cdots,6]$, which can be occupied by a number
$s_{\vartheta}$ of cells moving in the direction specified by
$\mathbf{c}_{\vartheta}$. In contrast to standard passive [53] and active [48,
54] LGCA that consider that each velocity channel of a node can either be
unoccupied or occupied by at most one cell, i.e. $s_{\vartheta}\in\\{0,1\\}$ –
similar to the Pauli exclusion principle – here we assume particles without
volume exclusion by letting any velocity channel $\mathbf{c}_{\vartheta}$ of a
node contain an unrestricted number of particles $s_{\vartheta}\in\mathbb{N}$:
i.e. many particle can share the same state (position and momentum). At each
discrete time step, as illustrated in Fig. 1, two processes take place:

(i) Particles modify their velocity by transitioning to a channel $\vartheta$,
according to the transition probability $T_{\vartheta}$:

$T_{\vartheta}=\frac{1}{Z}\exp\left[\beta\,H_{\vartheta}(\mathbf{r},k)\right]$
(1)

with $H_{\vartheta}(\mathbf{r},k)$ defined as:

$H_{\vartheta}(\mathbf{r},k)=\sum^{6}_{\alpha=1}s_{\alpha}(\mathbf{r},k)\left(\mathbf{c}_{\vartheta}\cdot\mathbf{c}_{\alpha}\right)^{m}$
(2)

where
$\mathbf{s}(\mathbf{r},\\!k)=[s_{1}(\mathbf{r},\\!k),s_{2}(\mathbf{r},\\!k),\cdots,s_{6}(\mathbf{r},\\!k)]$
is the configuration of node $\mathbf{r}$ at time $k$, which is given by the
number of particles $s_{\alpha}(\mathbf{r},k)$ occupying channel $\alpha$
(with $\alpha\in[1,2,\cdots,6]$), the strength of the velocity alignment is
denoted by $\beta\in\mathbb{R}_{+}$, and $Z$ is the node normalization
constant. Finally, $m=1$ corresponds to polar and $m=2$ to nematic velocity
alignment. These transition probabilities are the volume exclusion-free
analogues of those derived previously [55], which were obtained as
discretized, steady state distributions of flocking off-lattice Langevin
equation models with pairwise velocity alignment. Note that Eq. 1 is a
Maxwell-Boltzmann distribution, where the parameter $\beta$ is inversely
proportional to the effective temperature. Indeed, as seen in [55], $\beta$ is
proportional to the interaction strength, and inversely proportional to the
noise amplitude of off-lattice, stochastic differential equation models. Thus,
in the limit $\beta\rightarrow 0$ (corresponding to infinitely high
temperatures or noise amplitudes), Eq. 1 reduces to a discrete, uniform
distribution, with particles exhibiting no preference in their reorientation.
On the other hand, in the limit $\beta\rightarrow\infty$ (corresponding to an
absolute zero temperature or no noise), $T_{\vartheta}=1$ for a single
orientation $\vartheta$ and $T_{\vartheta}=0$ for all other orientations,
resulting in completely deterministic behavior, for $m=1$, and two opposite
orientations with $T_{\vartheta}=\frac{1}{2}$, and $T_{\vartheta}=0$ for all
other orientations, resulting in a one-dimensional, symmetric random walk
along a single axis, for $m=2$. We stress that Eqs. (1) and (2) define a
purely local velocity alignment process: the reorientation dynamics does not
consider the configuration of neighboring nodes. Note that the number of
particles at node $\mathbf{r}$ at time $k$ is given by
$n(\mathbf{r},k)=\sum_{\alpha}s_{\alpha}(\mathbf{r},\\!k)$. This implies that
Eq. (2) for $m=1$ can be expressed as
$H(\mathbf{r},k)_{\vartheta}=n(\mathbf{r},k)\mathbf{P}(\mathbf{r},k)\cdot\mathbf{c}_{\vartheta}$
where
$\mathbf{P}(\mathbf{r},k)=\left[\sum_{\alpha}s_{\alpha}(\mathbf{r},k)\mathbf{c}_{\alpha}\right]/n(\mathbf{r},k)$.

(ii) Particles move spatially according to the following rule: particles in
node $\mathbf{r}$ at channel $\vartheta$ are transported into the same channel
into the neighboring node $\mathbf{r}+\mathbf{c}_{\vartheta}$; see Fig. 1. It
should be noted that lattice-gas cellular automata are mesoscopic, rather than
microscopic, models. Therefore, time and space are coarse-grained. At this
level of resolution, the reorientation processes that occur continuously in
microscopic models are assumed to have reached a steady state at every
discrete time step [55], and thus can only move at a fixed speed of one
lattice site per time step.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 2: Dynamics of the polar, BA-LGCA. 2(a). Simulation snapshot; node
occupancy is color coded, with dark color indicating large particle number.
The inset shows the state of a node. See [61] for movies. 2(b). Global polar
order $\left\|\mathbf{P}\right\|$ vs. average occupation per node $n$. 2(c).
Probability of a particle escaping a polarized condensate given its particle
number. Inset: Probability of a particle staying in a polarized cluster. 2(d)
Histogram of the number of particles in all lattice sites at two different
time steps. 2(e). Mean cluster size (blue) and polar order (red) as function
of time; the dotted line corresponds to $t^{1/2}$, the characteristic growth
in classic ZRPs [56], while the dashed line corresponds to $t^{3/4}$. Average
over 200 simulations with $\beta=4.3$ and $\bar{\rho}=12$. 2(f). Polar order
$P_{g}$ and condensate size $N_{c}$ as a function of $\beta\bar{\rho}$ for
average $\bar{\rho}=12$. 2(g). Global polar order $P_{g}$ and local polar
order $P_{\ell}$ as a function of $\beta\bar{\rho}$ for average
$\bar{\rho}=12$. 150 simulations were averaged. A lattice size of $60\times
60$ was used for all simulations. Parameter values were recorded after 1000
time steps.

##  3 Emergence of polar condensates

For polar alignment ($m=1$), we observe, in strong contrast to other polar
active systems whose polar ordered phase consists of either a traveling polar
band or an spatially homogeneous polar phase [28, 35, 31], the emergence of
polar, moving condensates; see Fig. 2(a). In the following, we explain the
mechanism that leads to the emergence of such condensates.

We make use of a mean-field argument and define the probability of finding a
particle in channel $\vartheta$ of a certain node at time $k$ as
$q_{\vartheta}(k)=s_{\vartheta}(k)/n(k)$, with
$n(k)=\sum_{\alpha\in\Theta}s_{\alpha}(k)$ being the number of particles in
the node. The node configuration, after reorientation, takes the form:

$\displaystyle q_{\vartheta}(k+1)=\frac{1}{Z}\exp\left[\beta
n\sum_{\alpha\in\Theta}\cos(\alpha-\vartheta)q_{\alpha}(k)\right].$ (3)

Let us assume a steady state condition: $q_{\vartheta}(k+1)=q_{\vartheta}(k)$
for all $\vartheta$. Under this assumption, the transcendental Eq. (3) can be
easily solved; see Figs. 2(b) and 2(c). Note that it is possible to recast
Eq.(3) as $q_{\vartheta}=\exp\left[\beta
n\mathbf{P}\cdot\mathbf{c}_{\vartheta}\right]/Z$ in order to obtain an
equation for the polar order parameter of the node:
$\mathbf{P}=\sum_{\alpha\in\Theta}q_{\alpha}\mathbf{c}_{\alpha}$. And thus,
$\mathbf{P}=\sum_{\alpha\in\Theta}\frac{1}{Z}\exp\left[\beta
n\mathbf{P}\cdot\mathbf{c}_{\vartheta}\right]\mathbf{c}_{\vartheta}$. This
transcendental equation describes a phase transition. The disorder state,
given by $||\mathbf{P}||=0$, and corresponding to $q_{\vartheta}=1/6$ for all
$\vartheta$, is a solution of the above transcendental equation. This solution
is stable for a number of particle $n$ below the critical $n^{*}$ value. This
is evidenced by the behavior of $||\mathbf{P}||$ in Fig.2(b). Above $n^{*}$,
$q_{\vartheta}=1/6$ is not stable and particles accumulate preferentially in
one velocity channel and polar order in the node emerges. It is very important
to realize that the probability of finding all particles in a node occupying
the same velocity channel, for $n>n^{*}$ grows exponentially with the number
of particles $n$ in the node. This implies that once all particles in a node
occupy the same velocity channel, the probability that single particles
transition to other velocity channels decays exponentially with $n$, as does
the probability of not finding the $n$ particles in the same node. This
mechanism leads to a condensation of particles. To understand this, assume,
without loss of generality, that all particles are initially located in
channel $0$; then $q_{0}=e^{\beta n}/Z=(1+e^{\beta
n(\gamma_{1}-1)}+\dots+e^{\beta n(\gamma_{5}-1)})^{-1}$, with
$\gamma_{j}=\cos(-{\pi\,j/3})$ so $\gamma_{j}-1<0$. The probability that
particles change channel is $1-q_{0}=\sum_{j=1}^{5}e^{\beta
n(\gamma_{j}-1)}/[1+\sum_{j=1}^{5}e^{\beta n(\gamma_{j}-1)}]$, which decays
exponentially fast with $n$, while the probability of finding all particles
again in channel $0$ after one time step behaves as $(1+e^{\beta
n(\gamma_{1}-1)}+\dots+e^{\beta n(\gamma_{5}-1)})^{-n}$. Consequently, the
probability for a particle in a condensate of mass $n\gg n^{*}$ to leave the
condensate decays exponentially with $n$; Fig. 2(c). This feature is shared by
the standard zero-range process (ZRP) [56]. It should be mentioned that, in
this case, randomness decreases exponentially with particle number $n$ due to
the pairwise, additive definition of $H_{\vartheta}(\mathbf{r},k)$ (Eq. 2),
which increases linearly as more aligned particles are present. Conversely,
other population models may exhibit similar fluctuations originating from
multiplicative noise terms [57], which we do not consider in this model.
Particle condensation can be evidenced by looking at the distribution of
particles per node $P(n,t)$; Fig. 2(d) (cf. [58]). The second peak that shifts
over time is indicative of an aggregation dynamics. Note that here, it is the
alignment mechanism that holds particles together. The emergent condensate
moves ballistically in contrast to the processes studied in [56, 58]. This
means that the condensation dynamics is not solely mediated by particle
exchange with the bulk, but results from the motion of the condensate: as the
condensate moves cross the system collects clusters in its way and grows in
size, Figs. 2(d) and 2(e). This is further supported by the fact that the mean
condensation time increases with system size (see Fig. S8 of the Supplementary
Material), since clusters need to travel longer distances to accumulate
particles in distant lattice sites. Note that, as argued in [59], in the
absence of velocity correlations among clusters (and only in this scenario),
polar order becomes intimately connected with the number and distribution of
moving clusters. As expected, in Fig. 2(f) we observe that as particles
coalesce and the mean cluster size grows with time, the global order parameter
$P_{g}$ – defined as
$P_{g}=||\sum_{\mathbf{r}}\sum_{\alpha\in\Theta}q_{\alpha}(\mathbf{r})\mathbf{c}_{\alpha}||$
also does it. Furthermore, note that the mean cluster size scales with time
with an exponent larger than $1/2$; cf. [56]. Moreover, Fig. 2(f) shows that
the condensate size $N_{c}$, defined as
$N_{c}=\left\langle\mathrm{max}\left\\{s_{\vartheta}\left(\mathbf{r}\right):\vartheta\in\Theta,\mathbf{r}\in\mathcal{L}\right\\}\right\rangle$,
and $P_{g}$ exhibit almost identical dependency with $\beta\bar{\rho}$, with
$\bar{\rho}=\left\langle n\left(\mathbf{r}\right)\right\rangle=N/L^{2}$, where
$N$ is the number of particles in a system of linear size $L$. An additional
observable obtained from simulations was the local polar order, defined as
$P_{\ell}=\sum_{\mathbf{r}}||\sum_{\alpha\in\Theta}q_{\alpha}(\mathbf{r})\mathbf{c}_{\alpha}||/N_{o}$,
where $N_{o}$ is the number of nodes $r\in\mathcal{L}$ with at least one
particle in any channel. Thus, $P_{\ell}$ is the average of the polar order
within individual nodes, across all occupied nodes in the lattice. It is
evident that, if particles are homogeneously distributed among all channels in
every non-empty node, then $P_{\ell}=0$. Conversely, if for every node, all
particles are contained within a single channel, then $P_{\ell}=1$. Fig. 2(g)
shows a plot of the global order $P_{g}$ and the local polar order $P_{\ell}$,
as a function of the model parameters after several time steps. After the
transition, the local polar order $P_{\ell}$ saturates up to its maximum
value, while the global polar order $P_{g}$ exhibits non-monotonic behavior,
reaching a maximum exactly at the transition point. This means that, after the
transition, most occupied nodes accumulate particles within a single node. At
the transition point, few condensates form and therefore the condensate with
the most particles increases the overall global polar order. After the
transition, several condensates with their own orientation form at once. The
difference among the condensates’ orientation drives the goal polar order
down. Thus, disconnected condensates are present for all parameter values
after the transition.

##  4 Emergence of nematic filaments

The strict local nature imposed by zero-range interactions seems to prevent
nodes to exchange orientational information in order to establish extended
ordered structures. This is true for polar alignment: the polar order of two
neighboring nodes is not coupled. Counterintuitively, for nematic alignment,
and in contrast to polar alignment, nodes can exchange orientational
information and particles self-organize into stable, elongated, nematic
filaments that lead to global nematic order; Fig. 3(a). We stress that these
nematic filaments should not be confused with, and are different from the
nematic bands observed in off-lattice models [30]: i) the width of these
filaments is always of one node and does not vary with the distance to the
critical point as occurs in nematic bands [30], and ii) as consequence of
this, the transversal instability observed for nematic bands, precluding
global nematic order, is not present [60, 39].

How can an extended nematic filament emerge from zero-range interactions?
Picture an initial configuration where a large number $n$ of particles are
placed on a single channel of a lattice site. As in the polar case,
$q_{\vartheta}=1/6$ is not stable for $n>n^{*}$ and particles accumulate in
two opposite velocity channels. In the next time step, particles move to
neighboring sites associated to these two velocity channels. If the density in
each new lattice site is high, the same process occurs in the new sites, and
thus information on the initial orientational order, propagates forward and
backward, as shown in the sketch of Fig. 3(c). If we consider that this
process starts taking place in an area surrounded by a disordered population
of particles, then as the nematic filament expands enlarging its scattering
cross-section, these particles feed the growing filament until it eventually
percolates. The described mechanism leads to global nematic order as evidenced
by the global nematic order parameter
$|Q|=\left|\sum_{\mathbf{r}}\sum_{\alpha\in\Theta}q_{\alpha}\left(\mathbf{r}\right)e^{2i\alpha}\right|$.
Note that the emergent filament contains a macroscopic fraction of the
particles in the system, as evidenced by the behavior of the filament density
$N_{f}$ with $\beta$, which is similar to the one of $N_{c}$ for polar
alignment; Fig. 3(d). Furthermore, we observe that for fixed $\bar{\rho}$, the
$N$ grows as $L^{2}$, and thus $N_{f}$ as $L$, which implies that for
$L\to\infty$, this quantity diverges: the nematic filament acts as a
condensate with a macroscopic particle fraction located at it moving in two
opposite directions. The local nematic order,
$Q_{\ell}=\sum_{\mathbf{r}}\left|\sum_{\alpha\in\Theta}q_{\alpha}\left(\mathbf{r}\right)e^{2i\alpha}\right|/N_{o}$
was also observed in simulations. Here as well, we observed a monotonic
increase of the local nematic order and a non-monotonic behavior of the global
nematic order; Fig. 3(b). As before, this indicates the formation of few
filaments with most particles at the transition point, driving polar order up,
and several filaments with different orientations far above the transition
point, mantaining a high local nematic order, but inhibiting nematic order at
a global level.

##  5 Hydrodynamic equations

The onset of the (macroscopic) symmetry breaking, while the system remains
near the disordered phase, can be described in an approximate manner by
calculating the expected occupation numbers of lattice velocity channels.
However, note that well-developed condensates and filaments cannot be
described by smooth PDE solutions. As a first step, let us define
$f_{\vartheta}\left(\mathbf{r},k\right)\coloneqq\left\langle
s_{\vartheta}\left(\mathbf{r},k\right)\right\rangle$. By performing an in-node
mean-field approximation and employing the binomial statistics of velocity
channel occupation numbers, one obtains the lattice-Boltzmann equation (LBE):
$f_{\vartheta}\left(\mathbf{r}+\mathbf{c}_{\vartheta},k+1\right)=\rho\left(\mathbf{r},k\right)T_{\vartheta}\left(\mathbf{r},k\right),$
where
$\rho\left(\mathbf{r},k\right)\coloneqq\sum_{\vartheta\in\Theta}f_{\vartheta}\left(\mathbf{r},k\right)=\left\langle
n(\mathbf{r},k)\right\rangle$ is the density, equal to the expected value of
the total number of particles in each lattice site, and
$T_{\vartheta}\left(\mathbf{r},k\right)\coloneqq\frac{1}{Z}\exp\left[\beta\sum_{\alpha\in\Theta}f_{\alpha}\left(\mathbf{r},k\right)\left(\mathbf{c}_{\vartheta}\cdot\mathbf{c}_{\alpha}\right)^{m}\right]$
is the mean-field alignment probability. We perform a discrete Fourier
transform on $\vartheta$ to arrive to hydrodynamic fields that depend on
position and time. For polar alignment ($m=1$), the relevant hydrodynamic
fields reduce to the local density
$\rho\coloneqq\sum_{\vartheta\in\Theta}f_{\vartheta}$ and local polar order
(or momentum)
$\mathbf{M}\coloneqq\sum_{\vartheta\in\Theta}f_{\vartheta}\left(\cos\vartheta,\sin\vartheta\right)$.
The temporal evolution of these fields, in dimensionless form, leading to unit
active speed, is given by:

$\displaystyle\partial_{t}\rho+\nabla\cdot\mathbf{M}=0$ (4a)
$\displaystyle\begin{split}\partial_{t}\mathbf{M}+\frac{1}{2}\left[\nabla\varrho-\mathcal{D}\left(\nabla\varrho\right)\right]+\\\
\frac{\beta}{\rho}\left[\left(\mathbf{M}\cdot\nabla\right)\mathbf{M}-\frac{1}{2}\nabla
M^{2}+\left(\nabla\cdot\mathbf{M}\right)\mathbf{M}\right]\\\
=\rho\left\langle\mathbf{u}\right\rangle-\mathbf{M},\end{split}$ (4b)

where
$\langle\mathbf{u}\rangle\coloneqq\sum_{\vartheta\in\Theta}T_{\vartheta}\left(\cos\vartheta,\sin\vartheta\right)$,
$\mathcal{D}\left(\nabla\rho\right)\coloneqq\frac{1}{\rho^{2}}\left[\mathbf{M}\left(\mathbf{M}\cdot\nabla\rho\right)+\mathbf{M}\times\left(\mathbf{M}\times\nabla\rho\right)\right]$.
For detailed derivation, see [61]. The term $\langle\mathbf{u}\rangle$ can be
approximated, assuming $\beta\ll 1$, by a Taylor series. If we approximate it
up to third order in $\beta$, we find
$\left\langle\mathbf{u}\right\rangle=\frac{\beta}{2}\mathbf{M}\left(1-\frac{\beta^{2}}{8}\mathbf{M}^{2}\right)$,
and by replacing this expression into Eq. (4b), we obtain an equation that is
invariant under arbitrary rotations of $\mathbf{M}$ as occurs in Toner-Tu
theory [35]. However, if the expansions is performed up to 5th order, we
uncover that steady solution of the form
$\mathbf{M}=M(\cos\theta,\sin\theta)$, with $M$ a scalar constant and $\theta$
an angle, cannot exist for arbitrary values of $\theta$. The only possible
solutions correspond to $\theta=\frac{a\pi}{3}$, with $a\in[0,1,\cdots,5]$,
i.e. to one of the possible 6 directions of the lattice; for details see [61].

Differences with the standard off-lattice Vicsek model can be shown even using
the third order expansion in $\beta$ mentioned above. There are, as expected,
two spatially homogeneous solutions, corresponding to $M=0$ (disorder) and
$M>0$ (order). Performing a linear stability analysis around the disordered
homogeneous steady state, and assuming perturbations along a single direction,
we find that the system is stable against uniform perturbations for an initial
average density $\beta\bar{\rho}<2$, showing that the product
$\beta\bar{\rho}$ is the effective bifurcation parameter. The second steady
state corresponds to spatially homogeneous polar order state with
$M=\left[8\left(1-\frac{2}{\bar{\beta\rho}}\right)\right]^{1/2}$. The ordered
homogeneous state only exists when $\beta\bar{\rho}>2$, where the disordered
state becomes unstable. For non-uniform perturbations, the spatially
homogeneous ordered state, that is never observed in the LGCA, is stable only
for the parameter range $2.564\lessapprox\beta\bar{\rho}<3$. Otherwise, the
real parts of the eigenvalues grow with increasing wavenumber, indicating that
small wavelength perturbations dominate, as confirmed by direct integration of
the hydrodynamic equations. This instability indicates a drastic density
increase in localized areas of the space, which can be associated to the
emergence of condensates at this level of description. Numerical evidence
strongly suggests that spatially homogeneous ordered states, and ordered polar
bands are not possible in the LGCA model, even when when forcing the system by
using these configurations as initial conditions (see SI).

(a)

(b)

(c)

(d)

Figure 3: Dynamics of the nematic, BA-LGCA. 3(a). Snapshot of a simulation,
with high node occupation numbers displayed in dark color; inset shows the
state of a node. See [61] for movies. 3(b). Global nematic order $|Q|$ and
local nematic order $Q_{\ell}$ as a function of $\beta\bar{\rho}$ for average
$\bar{\rho}=12$. 3(c). Schematic depicts the filament formation process.
Arrows indicate the direction of particle groups, with colors, from red to
magenta, decreasing particle number. 3(d). Filament density vs.
$\beta\bar{\rho}$ averaged over 150 simulations for fixed $\bar{\rho}=0.6$. a
Lattice of size $60\times 60$ was used for all simulations. Parameter values
were recorded after 1000 time steps.

For nematic alignment ($m=2$), the relevant hydrodynamic fields include
nematic order, defined for simplicity as
$Q\coloneqq\sum_{\vartheta\in\Theta}f_{\vartheta}\left(\cos 2\vartheta+i\sin
2\vartheta\right)$, and the hydrodynamic equations read:

$\displaystyle\partial_{t}\rho=\mathfrak{d}\mathfrak{d}^{\ast}\rho+\mathfrak{R}\left(\mathfrak{d}^{2}Q^{\ast}\right)$
(5a)
$\displaystyle\begin{split}\partial_{t}Q=\frac{1}{2}\left\\{\mathfrak{d}^{2}\rho+\mathfrak{d}\mathfrak{d}^{\ast}Q+2\mathfrak{d}^{\ast}\left[\mathfrak{R}\left(\mathfrak{d}Q\right)\right]\right\\}\\\
+\frac{\beta\rho}{4}\left[Q+\frac{\beta}{8}Q^{\ast}\left(Q^{\ast}-\frac{\beta}{4}Q^{2}\right)\right]-Q,\end{split}$
(5b)

where $\mathfrak{d}\coloneqq\partial_{x}+i\partial_{y}$, ∗ indicates complex
conjugate, $\mathfrak{R}\left(\cdot\right)$ denotes real part, and fast
relaxation of $\mathbf{M}$ was assumed. It is worth stressing, as we explain
below, that according to Eq. (5) nematic order is constrained to exist, as
occurs for $\mathbf{M}$, on the orientations allowed by the lattice structure.
Eq. (5) possesses spatially homogeneous disordered and nematically ordered
solutions. The disordered state is stable when $\beta\bar{\rho}<4$, where the
bifurcation depends only on the product $\beta\bar{\rho}$, as before. The
order in the spatially homogeneous steady state is given by $Q=\left(2\pm
2\sqrt{9-\frac{32}{\beta\bar{\rho}}}\right)e^{i\frac{2k\pi}{3}}$,
$k\in\left\\{0,1,2\right\\}$. Two features are worth discussing. First, the
ordered state exists when $\beta\bar{\rho}>\frac{32}{9}$, i.e. it exists even
when the disordered state is stable, indicating a subcritical bifurcation.
Second, while for $\frac{32}{9}<\beta\bar{\rho}<4$ three pairs of solutions
with different moduli, but same argument
$\mathrm{arg}\left(Q\right)\in\left\\{\frac{2k\pi}{3},k\in\mathbb{N}\right\\}$
exist, for $\beta\bar{\rho}>4$, six different solutions exist, each one with a
different argument corresponding to one of the lattice directions
$\vartheta\in\Theta$. For $\beta\bar{\rho}>4$, solutions with
$\mathrm{arg}\left(Q\right)=\frac{2k\pi}{3}$ are stable, while those with
$\mathrm{arg}\left(Q\right)=\frac{\left(2k+1\right)\pi}{3}$ are unstable. This
is a consequence of the constraints imposed on $Q$ by the lattice, such that
nematic structures can only emerge along the three hexagonal lattice axes, as
observed in the LGCA simulations. Again, simulations suggest that spatially
homogeneous ordered states, as well as spatially extended bands are not
stable, and consistently break into nematic filaments, or revert back to the
homogeneous disordered state. In this case, however, ordered states are much
less unstable than in the polar case, and need long times in order to break
apart (see SI).

##  6 Discussion

The original LGCA was introduced with the intention of reproducing basic fluid
mechanics properties. However, the initial study performed on a square lattice
[62] displayed a series of anomalies due to the use a finite number of
velocity channels. Later on, it was found that such anomalies vanish in
hexagonal lattices [53], where the phenomenology of Navier–Stokes equations is
recovered, despite the use of only 6 velocity channels. Similarly, it is
tempting to speculate that an active LGCA with polar velocity alignment in an
hexagonal lattice (and short-range interactions) has to reproduce the large-
scale properties predicted by the Toner-Tu equations [35] for a (compressible)
active polar fluid. After all, it is expected that most microscopic details
are, at macroscopic scales, irrelevant except for the range and symmetry of
the interactions. However, our study shows that spatially homogeneous ordered
phases (cf. Toner-Tu phase [35]) or polar bands (cf. [31, 46]) cannot exist in
the LGCA model defined in this work. Moreover, in this LGCA, order emerges via
a fundamentally different phase transition from the ones previously reported
[28, 35, 31, 47, 46, 48, 30] in active matter. In LGCA, the emergence of
global order is intrinsically related to a condensation transition – that
occurs in the absence of an external potential – which shares similarities
with the ZRP. For the polar LGCA, it is associated to the formation of a polar
macroscopic condensate, while for the nematic LGCA to the collapse of a
macrocopic number of particles in the systems into a nematic filament (that
leads to global nematic order; cf. [30, 39]).

After their introduction as fluid-dynamical models, LGCA have been used to
model biological systems, specifically for modeling collective cell migration
(see [63] for a review). The original LGCA formulation considers an exclusion
principle, which limits the maximum number of cells which may be present at a
single spatial point at a given time. On the one hand, our model could be
considered as a 2D projection of a 3D system. For example, the accumulation of
particles at certain lattice sites could correspond to the formation of an
elongated structure by the piling up of agents, such as during the fruiting
body and slug formation in _D. discoideum_ [64]. On the other hand, cells are
quite compressible, and are able to accumulate in great numbers.
Experimentally, it has been found that cells can greatly reduce their volume
during tissue growth [65, 66]. Cells also regularly change their size to
maintain epithelial homeostasis [67]. For example, the photoreceptor
epithelium shows a wide variety of receptor packing densities, corresponding
to photoreceptor cells of varying sizes [68]. Furthermore, invasive cancer
cells can be ten times as compressible as their healthy counterparts [69], and
cell compressibility has been shown to correlate with invasive potential [70].
Moreover, it has been shown that densely packed cells confined to narrow
environments can still move collectively, as long as they mantain sufficiently
high adhesive contacts and velocity correlation levels [71]. The present LGCA
model extension, which does not consider an exclusion principle, represents a
first step towards a better mathematical representation of these cellular
aggregation and compression processes.

Finally, we strongly suspect that the condensation reported here is related to
the recent findings that indicate that here exist fundamental differences
between additive and non-additive interactions in vectorial active matter [72,
38]. In particular, we speculate that the emergent macroscopic behavior can be
affected, not only by modifying the symmetry or range of the interactions (see
[48] for an analogous model with extended range and an exclusion principle),
as it is usually expected, but by eliminating the dependency on
$n(\mathbf{r},k)$ in Eq. (2), for instance by defining for $m=1$,
$H(\mathbf{r},k)_{\vartheta}=\mathbf{P}(\mathbf{r},k)\cdot\mathbf{c}_{\vartheta}$.
This will be the focus of future investigations.

## Author Contributions

JMNS: Formal analysis, investigation, methodology, software, visualization,
writing (original draft), and writing (review and editing). HH: Funding
acquisition, supervision, and writing (review and editing). AVB and LB:
supervision, methodology, and writing (review and editing). AD: Funding
acquisition, supervision, project administration, and writing (review and
editing). FP: Conceptualization, formal analysis, investigation, methodology,
project administration, supervision, and writing (review and editing).

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

F.P. acknowledges financial support from CY Initiative through the Ambition
project “Collective Intelligence” and Labex MME-DII. JMNS acknowledges support
from the PAPIIT-UNAM grant, project IA104821.

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