# Partitions Associated to Class Groups of Imaginary Quadratic Number Fields

Kathleen L. Petersen  and  James A. Sellers  Mathematics and Statistics
Department, University of Minnesota Duluth, Duluth, Minnesota 55812 USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We investigate properties of attainable partitions of integers, where a
partition $(n_{1},n_{2},\dots,n_{r})$ of $n$ is attainable if
$\sum(3-2i)n_{i}\geq 0$. Conjecturally, under an extension of the Cohen and
Lenstra heuristics by Holmin et. al., these partitions correspond to abelian
$p$-groups that appear as class groups of imaginary quadratic number fields
for infinitely many odd primes $p$. We demonstrate a connection to partitions
of integers into triangular numbers, construct a generating function for
attainable partitions, and determine the maximal length of attainable
partitions.

###### Key words and phrases:

partitions, triangular numbers, class groups, class numbers, Cohen-Lenstra
heuristics

###### 1991 Mathematics Subject Classification:

11P81, 11R29

##  1\. Introduction

We consider partitions $\lambda=(n_{1},n_{2},\dots,n_{r})$ of $n$ so that
$n=n_{1}+n_{2}+\dots+n_{r}$ and written so that $n_{1}\geq n_{2}\geq\dots\geq
n_{r}$. For a fixed odd prime $p$, we realize the bijection between partitions
of $n$ and abelian groups of order $p^{n}$ by associating to $\lambda$ the
abelian $p$-group

$G_{\lambda}(p)=(\mathbb{Z}/p^{n_{1}}\mathbb{Z})\times(\mathbb{Z}/p^{n_{2}}\mathbb{Z})\times\dots\times(\mathbb{Z}/p^{n_{r}}\mathbb{Z}).$

The cyclicity index of $\lambda$,

$c(\lambda)=\sum_{i=1}^{r}(3-2i)n_{i}$

measures the deviation of $G_{\lambda}(p)$ from being cyclic and governs the
size of $\mathrm{Aut}(G_{\lambda}(p))$ (see Section 2). The partition
$\lambda$ is called attainable if $c(\lambda)\geq 0.$ Holmin et. al. [6] show
that under an extension of the Cohen-Lenstra heuristics [3] for the
distribution of class numbers, an attainable partition corresponds to a family
of abelian $p$-groups that conjecturally are realized as the class groups of
imaginary quadratic number fields for infinitely many odd primes $p$. We study
attainable partitions to further understand which abelian $p$-groups should
occur in this way.

In this work, we determine a generating function for the attainable
partitions.

######  Theorem 1.1.

The generating function for the attainable partitions of $n$ is

$A(q)=\frac{1}{1-q}\prod_{i=1}^{\infty}\frac{1}{(1-q^{i(i+1)})}.$

To prove this, we show (Lemma 3.6) that the growth of the number of attainable
partitions is governed by partitions of cylicity index 0, and there are no
partitions of odd integers with cyclicity index equal to zero. Then we
demonstrate the following.

######  Theorem 1.2.

For each $m\geq 1$, partitions of $2m$ with cyclicity index 0 are in bijective
correspondence with partitions of $m$ into triangular numbers.

We also study the shape of attainable partitions. In particular, we
demonstrate that the maximal length of an attainable partition of $n$
increases like $\sqrt{n}$, where the length of a partition is the number of
terms in the partition ($r$ in the notation above).

######  Theorem 1.3.

The length of an attainable partition $(n_{1},n_{2},\dots,n_{r})$ of $n$ is at
most $r=\lfloor\tfrac{\sqrt{4n+1}+1}{2}\rfloor$. This bound is realized by
$\lambda=(n-r+1,1,\dots,1)$ with $0\leq c(\lambda)<\sqrt{4n+1}$.

The value of $r$ in Theorem 1.3 is either $\lfloor\sqrt{n}\rfloor$ or
$\lfloor\sqrt{n}\rfloor+1$. In Remark 3.4 we note that the smallest cyclicity
index of an attainable partition of $n=2m$ is 0, realized by $(m,m)$, and the
smallest cyclicity index of an attainable partition of $n=2m+1$ is 1, realized
by $(m+1,m)$.

##  2\. Background

We now give some background to motivate our study of these partitions. Based
on experimental observations, Cohen and Lenstra [6] noticed that class groups
of quadratic number fields behave like a random sequence with respect to a
probability distribution on the space of finite abelian groups. Their key
observation was that the odd part of the class group is rarely non-cyclic. The
even part of the class group is well understood due to Gauss’ genus theory
[4]. This led to a heuristic assumption that the weighting of isomorphism
classes of abelian groups should be inversely proportional to the size of
their automorphism group. When $G$ is an abelian group of odd order, this can
be stated as $\mathcal{F}(G)\approx P(G)\mathcal{F}(|G|)$ where
$\mathcal{F}(G)$ and $\mathcal{F}(h)$ are the number of fundamental
discriminants whose associated class group is $G$, and class number is $h$,
respectively. The quantity $P(G)$ is given by

$P(G)=\frac{1}{|\mathrm{Aut}(G)|}/\Big{(}\sum\frac{1}{|\mathrm{Aut}(G^{\prime})|}\Big{)}$

where the sum is over abelian groups $G^{\prime}$ of order $|G|$.

This leads to a natural question: What finite abelian groups occur as class
groups of imaginary quadratic fields? Chowla [2] showed that for large $r$,
$(\mathbb{Z}/2\mathbb{Z})^{r}$ does not occur. In fact,
$(\mathbb{Z}/n\mathbb{Z})^{r}$ does not occur for sufficiently large $r$
(depending on $n$) [1, 12, 5]. However, these results are not effective and do
not yield explicit examples of non-occurence. Watkins [11] determined all
imaginary quadratics with class number at most 100, and this work shows, for
example, none of the groups

$(\mathbb{Z}/3\mathbb{Z})^{3},(\mathbb{Z}/9\mathbb{Z})\times(\mathbb{Z}/3\mathbb{Z})^{2},(\mathbb{Z}/3\mathbb{Z})^{4}$

occurs as the class group of an imaginary quadratic field (see [6]).

Holmin et. al. [6] used the Cohen-Lenstra heuristics to predict the $p$-group
decomposition of the class group of imaginary quadratics. The automorphisms of
$G_{\lambda}(p)$ are intrinsically tied to $c(\lambda)$ since

$|\mathrm{Aut}(G_{\lambda}(p))|=p^{2n-c(\lambda)}\prod_{i=1}^{k}\prod_{j=1}^{m_{i}}\Big{(}1-\frac{1}{p^{j}}\Big{)}$

where $k$ is the number of distinct parts of $\lambda$ and $m_{i}$ is the
multiplicity of the $i$th part (see [10]). Holmin et al. establish [6,
Proposition 7.1] that

$P(G_{\lambda}(p))\sim p^{c(\lambda)-n}$

which with the Cohen-Lenstra heuristics leads to their Conjecture 1.2 that if
$c(\lambda)>0$ then as $p\rightarrow\infty$

$\mathcal{F}(G_{\lambda}(p))\sim\frac{\mathfrak{C}}{n}\cdot\frac{p^{c(\lambda)}}{\log
p}$

where $\mathfrak{C}\sim 11.317$ is a constant. Therefore, the expectation is
that when $c(\lambda)>0$ the group $G_{\lambda}(p)$ appears as a class group
for all but finitely many primes $p$. For $c(\lambda)=0$ the conjecture states
that as $x\rightarrow\infty$ then

$\sum_{p\leq
x}\mathcal{F}(G_{\lambda}(p))\sim\frac{\mathfrak{C}}{n}\frac{x}{(\log
x)^{2}}.$

So it is expected that $G_{\lambda}(p)$ occurs for infinitely many primes $p$
and also does not occur for infinitely many primes $p$. For $c(\lambda)<0$ the
conjecture states that $G_{\lambda}(p)$ appears as a class group for only
finitely many $p$. Computations determining the number of imaginary quadratics
with prescribed odd $p$-group appearing as class groups of order up to
$10^{6}$ (conditional on the GRH) in [6] support this conjecture.

In light of Theorem 1.3, these conjectures suggest that there are infinitely
many primes $p$ such that the group

$(\mathbb{Z}/p^{n-\lfloor\sqrt{n}\rfloor+1}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{\lfloor\sqrt{n}\rfloor-1}$

appears as the class group of an imaginary quadratic number field. The
partitions $(m,m)$ of $2m$ and $(m+1,m)$ of $2m+1$ of smallest cyclicity index
mentioned in Remark 3.4 correspond to the groups
$(\mathbb{Z}/p^{m}\mathbb{Z})^{2}$ and
$(\mathbb{Z}/p^{m+1}\mathbb{Z})\times(\mathbb{Z}/p^{m}\mathbb{Z})$, with
automorphism groups of order $p^{2n-3}(p+1)(p-1)^{2}$ and $p^{2n-3}(p-1)^{2}$,
respectively. These partitions correspond to the largest possible automorphism
group of an abelian $p$-group of order $p^{n}$ associated to an attainable
partition. By way of comparison, $(n)$ has cyclicity index $n$ and corresponds
to the cyclic group $\mathbb{Z}/p^{n}\mathbb{Z}$ which has $p^{n-1}(p-1)$
automorphisms.

Data from the aforementioned computations are available online (see [7], [8],
and [9]). For $n=5$, the odd primes with $p^{5}<10^{6}$ are $p=3,5,7,11,13$.
The only partition of negative cyclicity index that is realized for these
primes is $\lambda=(3,1,1)$ with $c(\lambda)=-1$. This is realized for $p=5$;
the group $\mathbb{Z}/5^{3}\mathbb{Z}\times(\mathbb{Z}/5\mathbb{Z})^{2}$ is
the class group for the imaginary quadratic with fundamental discriminant
$145367147$.

$\lambda$ | $G_{\lambda}(p)$ | $c(\lambda)$ |  $p=$ 3 | 5 | 7 | 11 | 13
---|---|---|---|---|---|---|---
(5) | $\mathbb{Z}/p^{5}\mathbb{Z}$ | 5 | 549 | 4610 | 19430 | 147009 | 314328
(4,1) | $\mathbb{Z}/p^{4}\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$ | 3 | 56 | 218 | 444 | 1347 | 1894
(3,2) | $\mathbb{Z}/p^{3}\mathbb{Z}\times\mathbb{Z}/p^{2}\mathbb{Z}$ | 1 | 8 | 5 | 8 | 13 | 9
Table 1. Attainable partitions of $5$ with corresponding group, cyclicity
index and, for $p\leq 13$, the values of $\mathcal{F}(G_{\lambda}(p))$

For $n=6$ no non-attainable partitions are realized for $p^{6}<10^{6}$.

$\lambda$ | $G_{\lambda}(p)$ | $c(\lambda)$ | $p=3$ | $5$ | $7$
---|---|---|---|---|---
(6) | $\mathbb{Z}/p^{6}\mathbb{Z}$ | 6 | 1512 | 19469 | 116278
(5,1) | $\mathbb{Z}/p^{5}\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$ | 4 | 177 | 1024 | 2887
(4,2) | $\mathbb{Z}/p^{2}\mathbb{Z}\times\mathbb{Z}/p^{2}\mathbb{Z}$ | 2 | 18 | 37 | 58
(3,3) | $\mathbb{Z}/p^{3}\mathbb{Z}\times\mathbb{Z}/p^{3}\mathbb{Z}$ | 0 | 2 | 2 | 3
(4,1,1) | $\mathbb{Z}/p^{4}\mathbb{Z}\times(\mathbb{Z}/p\mathbb{Z})^{2}$ | 0 | 0 | 3 | 0
Table 2. Attainable partitions of $6$ with corresponding group, cyclicity
index and, for $p\leq 7$, the values of $\mathcal{F}(G_{\lambda}(p))$

##  3\. Preliminaries

As mentioned in [6] if $\lambda$ is a partition of $n$ then $1-(n-1)^{2}\leq
c(\lambda)\leq n$, and $c(\lambda)=n$ exactly when $G_{\lambda}(p)$ is cyclic.
These extremal partitions of $n$ are readily seen to be
$\lambda_{1}=(1,1,\dots,1)$ with $c(\lambda_{1})=1-(n-1)^{2}$ and
$\lambda_{2}=(n)$ with $c(\lambda_{2})=n$, and correspond to the $p$-groups
$G_{\lambda_{1}}(p)=(\mathbb{Z}/p\mathbb{Z})^{n}$ and
$G_{\lambda_{2}}(p)=\mathbb{Z}/p^{n}\mathbb{Z}$.

We now transition to a a more detailed study of the properties of attainable
partitions.

######  Lemma 3.1.

For any partition $\lambda$ of $n$ we have $c(\lambda)\equiv n\pmod{2}$.

###### Proof.

Let $\lambda=(n_{1},n_{2},\dots,n_{r})$ be a partition of $n$. By definition

$c(\lambda)=\sum_{i=1}^{r}(3-2i)n_{i}\equiv\sum_{i=1}^{r}n_{i}\pmod{2}.$

The result follows as $n_{1}+n_{2}+\dots+n_{r}=n$. ∎

Thus, if $n$ is odd then $c(\lambda)$ cannot be zero, so there are no
partitions of odd $n$ with cyclicity index equal to zero.

######  Definition 3.2.

Given a partition $\lambda=(n_{1},n_{2},\dots,n_{r})$ of $n$ as above, we say
that the partition $\lambda^{\prime}$ of $n+1$ given by
$\lambda^{\prime}=(n_{1}+1,n_{2},\dots,n_{r})$ is a primary addition to
$\lambda$. If $n_{1}\neq n_{2}$ then we say
$\lambda^{\prime\prime}=(n_{1},n_{2}+1,n_{3},\dots,n_{r})$ is a secondary
addition to $\lambda$.

The partition $\lambda^{\prime}=(8,1,1)$ of 10 is a primary addition to the
partition $\lambda=(7,1,1)$ of 9, and the partition
$\lambda^{\prime\prime}=(7,2,1)$ of 10 is a secondary addition to $\lambda$.
We calculate $c(\lambda)=3$, $c(\lambda^{\prime})=4$ and
$c(\lambda^{\prime\prime})=2$.

######  Lemma 3.3.

  * •

If $\lambda^{\prime}$ is a primary addition to $\lambda$ then
$c(\lambda^{\prime})=c(\lambda)+1.$

  * •

If $\lambda^{\prime\prime}$ is a secondary addition to $\lambda$ then
$c(\lambda^{\prime})=c(\lambda)-1.$

###### Proof.

For the first assertion,

$c(\lambda)=\sum_{i=1}^{r}(3-2i)n_{i}=n_{1}+\sum_{i=2}^{r}(3-2i)n_{i}$

and so

$c(\lambda)+1=(n_{1}+1)+\sum_{i=2}^{r}(3-2i)n_{i}=c(\lambda^{\prime}).$

The second assertion follows similarly.

∎

This implies that if $\lambda$ is an attainable partition of $n$ then a
primary addition to $\lambda$ is an attainable partition of $n+1$. Moreover,
if $c(\lambda)=0$ then $\lambda$ cannot be obtained using a secondary
addition. The following follows directly from Lemma 3.1 and Lemma 3.3.

######  Remark 3.4.

The smallest cyclicity index of an attainable partition of $n=2m$ is 0,
realized by $(m,m)$. The smallest cyclicity index of an attainable partition
of $n=2m+1$ is 1, realized by $(m+1,m)$.

$n$ |  Attainable Partitions of $n$
---|---
1 | $(1)$
2 | $(3),(2,1)$
3 | $(3),(2,1)$
4 | $(4),(3,1),(2,2)$
5 | $(5),(4,1),(3,2)$
6 | $(6),(5,1),(4,2),(3,3),(4,1,1)$
7 | $(7),(6,1),(5,2),(4,3),(5,1,1)$
8 | $(8),(7,1),(6,2),(5,3),(4,4),(6,1,1),(5,2,1)$
9 | $(9),(8,1),(7,2),(6,3),(5,4),(7,1,1),(6,2,1)$
10 | $(10),(9,1),(8,2),(7,3),(6,4),(5,5),(8,1,1),(7,2,1),(6,3,1)$
11 | $(11),(10,1),(9,2),(8,3),(7,4),(6,5),(9,1,1),(8,2,1),(7,3,1)$
12 | $(12),(11,1),(10,2),(9,3),(8,4),(7,5),(6,6),(10,1,1),(9,2,1),(8,3,1),(7,4,1),$
| $(8,2,2),(9,1,1,1)$
13 | $(13),(12,1),(11,2),(10,3),(9,4),(8,5),(7,6),(11,1,1),(10,2,1),(9,3,1),(8,4,1),$
| $(9,2,2),(10,1,1,1)$
14 | $(14),(13,1),(12,2),(11,3),(10,4),(9,5),(8,6),(7,7),(12,1,1),(11,2,1),(10,3,1),$
| $(9,4,1),(8,5,1),(10,2,2),(9,3,2),(11,1,1,1),(10,2,1,1)$
15 | $(15),(14,1),(13,2),(12,3),(11,4),(10,5),(9,6),(8,7),(13,1,1),(12,2,1),(11,3,1),$
| $(10,4,1),(9,5,1),(11,2,2),(10,3,2),(12,1,1,1),(11,2,1,1)$
Table 3. All attainable partitions of $n$ for $n\leq 15$

######  Definition 3.5.

Let $a(n)$ denote the number of attainable partitions of $n$. Let $z_{0}(n)$
denote the number of partitions of $n$ with cyclicity index equal to zero, and
for an even number $2m$ let $z(m)=z_{0}(2m)$. We set $a(0)=z_{0}(0)=1$.

It follows from Lemma 3.1 that $z_{0}(2m+1)=0$ as there are no partitions of
an odd number with cyclicity index 0.

######  Lemma 3.6.

For any natural number $n$,

$a(n+1)=a(n)+z_{0}(n+1).$

###### Proof.

Partitions of $n+1$ obtained from partitions of $n$ by primary additions have
positive cyclicity index by Lemma 3.3. As such, it is enough to show that if
$\lambda$ is a partition of $n+1$ and $c(\lambda)>0$, then $\lambda$ can be
obtained from an attainable partition of $n$ by a primary addition. Let
$\lambda=(n_{1},n_{2},\dots,n_{r})$ be a partition of $n+1$ and since
$c(\lambda)>0$ we have $n_{1}\neq n_{2}$. Then
$\lambda^{\prime}=(n_{1}-1,n_{2},\dots,n_{r})$ is a partition of $n$ with
$c(\lambda^{\prime})=c(\lambda)-1$ by Lemma 3.3. Since $c(\lambda)>0$ we have
$c(\lambda^{\prime})\geq 0$ is an attainable partition of $n$. ∎

######  Lemma 3.7.

For $m\geq 1$ we have $a(2m+1)=a(2m)$ and all attainable partitions of $2m+1$
occur as primary additions to a partition of $2m$.

###### Proof.

Lemma 3.1 implies that $z_{0}(2m+1)=0$ and by Lemma 3.6 we conclude
$a(2m+1)=a(2m)$. By Lemma 3.3 every attainable partition of $2m$ yields an
attainable partition of $2m+1$ by a primary addition. ∎

######  Lemma 3.8.

For all $m\geq 1$ we have $a(2m+2)>a(2m+1)$.

###### Proof.

Since every primary addition to an attainable partition of $2m+1$ is an
attainable partition of $2m+2$ we have that $a(2m+2)\geq a(2m+1)$. To show
that the inequality is strict, consider the attainable partition
$\lambda^{\prime\prime}=(m+1,m+1)$ of $2m+2$. Since
$c(\lambda^{\prime\prime})=0$ it is not a primary addition to any attainable
partition of $2m+1$ by Lemma 3.3. (In fact, it is a secondary addition to
$(m+1,m)$.)

∎

Lemma 3.8 implies that there are always attainable partitions of $2m+2$ that
are not primary additions to a partition of $2m+1$. We see from the data that
we often get new attainable partitions by increasing the partition length, for
example the attainable partition $(4,1,1)$ of $6$ has length 3, but there are
no attainable length 3 partitions of $n$ for $1\leq n\leq 5$.

##  4\. Lengths of Attainable Partitions

######  Lemma 4.1.

If $\lambda=(n_{1},\dots,n_{r})$ is attainable then, for $1<k\leq r$,
$n_{k}\leq n/k(k-1)$.

###### Proof.

We have that $n=n_{1}+n_{2}+\dots+n_{r}$ so that

$\displaystyle n$ $\displaystyle\geq n_{1}+n_{2}+\dots+n_{k}$
$\displaystyle\geq(\sum_{i=2}^{k}(2i-3)n_{i})+(n_{2}+\dots+n_{k})$
$\displaystyle\geq\sum_{i=2}^{k}(2i-2)n_{k}=k(k-1)n_{k}$

where we have used the fact that $\lambda$ is attainable so $n_{1}\geq
n_{2}+3n_{3}+5n_{4}+\dots+(2r-3)n_{r}\geq
n_{2}+3n_{3}+5n_{4}+\dots+(2k-3)n_{k}$ and the fact that if $i<j$ then
$n_{i}\geq n_{j}$.

∎

######  Theorem 1.3.

The length of an attainable partition $(n_{1},n_{2},\dots,n_{r})$ of $n$ is at
most $r=\lfloor\tfrac{\sqrt{4n+1}+1}{2}\rfloor$. This bound is realized by
$\lambda=(n-r+1,1,\dots,1)$ with $0\leq c(\lambda)<\sqrt{4n+1}$.

###### Proof.

Lemma 4.1 implies that if $n/(k(k-1))<1$ then $n_{k}=0$. Therefore, the
partition length, $r$, of an attainable partition is the greatest
$r\in\mathbb{Z}$ such that $n\geq r(r-1)$. Completing the square, we see that

$r=\lfloor\sqrt{n+\tfrac{1}{4}}+\tfrac{1}{2}\rfloor=\lfloor\tfrac{\sqrt{4n+1}+1}{2}\rfloor.$

Due to the weighting of $c(\lambda)$ length is maximized by partitions of the
above shape.

The cyclicity index bound is computed directly using the inequality
$a-1<\lfloor a\rfloor\leq a$ with the definition of $c(\lambda)$. ∎

This length bound may be achievable by multiple partitions as evidenced by the
length 4 partitions $(11,1,1,1)$ and $(10,2,1,1)$ of 14. The partition of the
form stated in Theorem 1.3 does not always have the smallest cyclicity index;
$c((11,1,1,1))=2$ and $c((10,2,1,1))=0$.

It is elementary to verify that the $r$ value in the above satisfies
$\lfloor\sqrt{n}\rfloor\leq r\leq\lfloor\sqrt{n}\rfloor+1$. This lower bound
is achieved by perfect squares, and in general any $n$ where the fractional
part of $\sqrt{n}+\tfrac{1}{4}$ is less than $\tfrac{1}{2}$. The upper bound
for $n$ where $\sqrt{n}+\tfrac{1}{4}$ has fractional part greater than or
equal to $\tfrac{1}{2}$, for example integers one less than a perfect square.

##  5\. Generating Functions

In this section we prove Theorem 1.1. First, we prove Theorem 1.2 which
establishes a connection between partitions of cyclicity index 0 and
partitions of integers into triangular numbers.

From Lemma 3.6 and Lemma 3.7 we have that

$\displaystyle a(2m)$ $\displaystyle=a(2m-2)+z(m)$
$\displaystyle=a(2m-4)+z(m-1)+z(m)$ $\displaystyle\quad\vdots$
$\displaystyle=a(2)+z(2)+z(3)+\dots+z(m).$

Since $a(2)=2$ and $z(1)=1$ we have that $a(2)=a(0)+z(1)$ and we have shown
the following.

######  Lemma 5.1.

For all $m\geq 1$, $\displaystyle a(2m)=\sum_{r=0}^{m}z(r).$

$m$ | Partitions of $2m$ with Cyclicity Index 0
---|---
1 | $(1,1)$
2 | $(2,2)$
3 | $(3,3),(4,1,1)$
4 | $(4,4),(5,2,1)$
5 | $(5,5),(6,3,1)$
6 | $(6,6),(7,4,1),(8,2,2),(9,1,1,1)$
7 | $(7,7),(8,5,1),(9,3,2),(10,2,1,1)$
Table 4. All partitions of $2m$ with cyclicity index 0 for $m\leq 7$

######  Definition 5.2.

We will write $t_{i}=i(i+2)/2$ for the $i^{th}$ triangular number. We call the
numbers of the form $2t_{i}$ the oblong numbers.

Now we prove that partitions of $2m$ with cyclicity index 0 are in bijective
correspondence with partitions of $2m$ into oblong numbers, which is
equivalent to the following.

######  Theorem 1.2.

For each $m\geq 1$, partitions of $2m$ with cyclicity index 0 are in bijective
correspondence with partitions of $m$ into triangular numbers.

###### Proof.

Let $\lambda=(n_{1},n_{2},\dots,n_{r})$ be a partition of $2m$ with
$c(\lambda)=0$. Using the definition of $c(\lambda)$,

$n_{1}=n_{2}+3n_{3}+5n_{4}+\dots(2k-3)n_{r}$

and we can write

$\displaystyle 2m$ $\displaystyle=n_{1}+n_{2}+n_{3}+n_{4}+\dots+n_{r}$
$\displaystyle=(n_{2}+3n_{3}+5n_{4}+\dots(2r-3)n_{r})+n_{2}+n_{3}+n_{4}+\dots+n_{r}$
$\displaystyle=2n_{2}+4n_{3}+6n_{4}+\dots+(2r-2)n_{r}.$

We rewrite this as

$\displaystyle 2m$
$\displaystyle=2(n_{2}-n_{3})+6(n_{3}-n_{4})+12(n_{4}-n_{5})+20(n_{5}-n_{6})$
$\displaystyle\quad+\dots+2t_{r-2}(n_{r-1}-n_{r})+2t_{r-1}n_{r}.$

Since $n_{i}\geq n_{i+1}$ the $n_{i}-n_{i-1}$ terms are all non-negative, and
the final term is non-zero as $n_{r}>0$. This demonstrates that $2m$ is a sum
of non-negative multiples of oblong numbers. Dividing both sides of the
equation by 2 demonstrates that $m$ is a sum of non-negative multiples of
triangular numbers.

Conversely, assume that $n=c_{1}t_{1}+\dots+c_{r-1}t_{r-1}$ with $c_{i}\geq 0$
for $i=1,\dots,r-2$ and $c_{r-1}>0$. We let $n_{r}=c_{r-1}$, and for
$i=2,\dots r-1$ let

$n_{i}=c_{i-1}+c_{i}+\dots+c_{r-1}$

and define $n_{1}=n_{2}+3n_{3}+5n_{4}+\dots(2r-3)n_{r}$. As such, $n_{i}>0$
and $n_{i+1}\geq n_{i}$ for $i=1,\dots,r-1$ so that
$\lambda=(n_{1},n_{2},\dots,n_{r})$ is a partition of $2n$ with
$c(\lambda)=0$.

∎

$n$ | Partitions of $n$ into Triangular Numbers
---|---
1 | $(1)$
2 | $(1,1)$
3 | $(3),(1,1,1)$
4 | $(3,1),(1,1,1,1)$
5 | $(3,1,1),(1,1,1,1,1)$
6 | $(6),(3,3),(3,1,1,1),(1,1,1,1,1,1)$
7 | $(6,1),(3,3,1),(3,1,1,1,1),(1,1,1,1,1,1,1)$
Table 5. All partitions of $n$ into triangular numbers for $n\leq 7$

######  Example 5.3.

Consider the partition $(8,5,1)$ of 14, which has cyclicity index 0. We have

$14=8+5+1=(5+3\cdot 1)+5+1=2\cdot 5+4\cdot 1=2(5-1)+6(1-0)$

which corresponds to writing $7=1\cdot 4+3\cdot 1$ giving the summation
$7=1+1+1+1+3$ in terms of triangular numbers.

Now consider $7=1+3+3$ as a different sum of triangular numbers. Writing
$t_{1}=1$ as the first triangular number and $t_{2}=3$ as the second, in the
notation above $7=c_{1}t_{1}+c_{2}t_{2}$ with $c_{1}=1$ and $c_{2}=2$. We have
$n_{3}=c_{2}=2$, $n_{2}=c_{1}+c_{2}=3$ and $n_{1}=n_{2}+3n_{3}=9$. This gives
us the partition $(9,3,2)$ of 14 with cyclicity index 0.

In this correspondence the partition $(1,1,1,\dots,1)$ of $n$ into triangular
numbers corresponds to the partition $(n,n)$ of $2n$ with cyclicity index 0.

######  Corollary 5.4.

The generating function for $z(n)$ is

$\sum_{n=0}^{\infty}z(n)q^{n}=\prod_{i=1}^{\infty}\frac{1}{1-q^{i(i+1)/2}}.$

###### Proof.

By Theorem 1.2 it suffices to determine a generating function for partitions
into triangular numbers, which is the generating function above.

∎

Since $z_{0}(2m)=z(m)$, by Corollary 5.4, the generating function for
$z_{0}(2m)$ is

$\sum_{m=0}^{\infty}z_{0}(2m)q^{2m}=\prod_{i=1}^{\infty}\frac{1}{(1-q^{i(i+1)})}.$

By Lemma 5.1 we have

$a(2m)=\sum_{j=0}^{m}z(j)=\sum_{j=0}^{m}z_{0}(2j)$

and the generating function of $a(2m)$ is

$\sum_{m=0}^{\infty}a(2m)q^{2m}=\frac{1}{(1-q^{2})}\prod_{i=1}^{\infty}\frac{1}{(1-q^{i(i+1)})}.$

Because $a(2m+1)=a(2m)$ the generating function for $a(2m+1)$ is
$\sum_{m=0}^{\infty}a(2m)q^{2m+1}$. Putting this together, the generating
function for $a(n)$ has the form

$\displaystyle\sum_{n=0}^{\infty}a(n)q^{n}$
$\displaystyle=\sum_{m=0}^{\infty}a(2m)q^{2m}+\sum_{m=0}^{\infty}a(2m+1)q^{2m+1}$
$\displaystyle=(1+q)\sum_{m=0}^{\infty}a(2m)q^{2m}.$

From above we have

$\sum_{n=0}^{\infty}a(n)q^{n}=\frac{(1+q)}{(1-q^{2})}\prod_{i=1}^{\infty}\frac{1}{(1-q^{i(i+1)})}=\frac{1}{1-q}\prod_{i=1}^{\infty}\frac{1}{(1-q^{i(i+1)})}$

proving Theorem 1.1.

## References

  * [1] David W. Boyd and H. Kisilevsky, _On the exponent of the ideal class groups of complex quadratic fields_ , Proc. Amer. Math. Soc. 31 (1972), 433–436. MR 289454
  * [2] S. Chowla, _An extension of heilbronn’s class number theorem_ , Quarterly J. Math. 5 (1934), 304–307.
  * [3] H. Cohen and H. W. Lenstra, Jr., _Heuristics on class groups of number fields_ , Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082
  * [4] D.F. Gauss, _Disquisitiones arithmeticae_ , (1801).
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  * [7] S. Holmin and P. Kurlberg, _List of $\mathcal{F}(g)$ for all noncyclic abelian groups of odd prime power order $<10^{6}$_, Available online at: (https://people.kth.se/ $\sim$kurlberg/ class$\underline{\ }$group$\underline{\ }$data/ noncyclic$\underline{\ }$class$\underline{\ }$groups.txt).
  * [8] by same author, _List of $\mathcal{F}(g)$ for all noncyclic abelian groups of odd prime power order $<10^{6}$_, Available online at: (https://people.kth.se/ $\sim$kurlberg/ class$\underline{\ }$group$\underline{\ }$data/ class$\underline{\ }$group$\underline{\ }$orders.txt).
  * [9] by same author, _List of $\mathcal{F}(g)$ for all noncyclic abelian groups of odd prime power order $<10^{6}$_, Available online at: (https://people.kth.se/ $\sim$kurlberg/ class$\underline{\ }$group$\underline{\ }$data/ discriminants$\underline{\ }$of$\underline{\ }$noncyclic$\underline{\ }$groups.txt).
  * [10] Arthur Ranum, _The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group_ , Trans. Amer. Math. Soc. 8 (1907), no. 1, 71–91. MR 1500775
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  * [12] P. J. Weinberger, _Exponents of the class groups of complex quadratic fields_ , Acta Arith. 22 (1973), 117–124. MR 313221


# Intertwined quantum phase transitions in the zirconium and niobium isotopes

N. Gavrielov  GANIL, CEA/DSM–CNRS/IN2P3, Bd Henri Becquerel, BP 55027, F-14076
Caen Cedex 5, France<EMAIL_ADDRESS>
###### Abstract

Nuclei in the $A\approx 100$ region exhibit intricate shape-evolution and
configuration crossing signatures. Exploring both even-even and their adjacent
odd-mass nuclei gives further insight on the emergence of deformation and
shape-phase transitions. We employ the algebraic frameworks of the interacting
boson model with configuration mixing and the new interacting boson-fermion
model with configuration mixing in order to investigate the even-even
zirconium with neutron number 52–70 (40Zr) and odd-mass niobium (41Nb)
isotopes with 52–62. We compare between the evolution in energy levels,
configuration and symmetry content of the wave functions, two neutron
separation energies and $E2$ transition rates. The comparisons between the two
chains of isotopes denote the occurrence of intertwined quantum phase
transitions (IQPTs) in both chains. Such a situation occurs when two
configurations, normal and intruder, cross through the critical point of a
Type II quantum phase transition (QPT), and the intruder configuration
undergoes on its own a Type I shape-evolution QPT from a spherical shape (weak
coupling scenario) to axially deformed rotor (strong coupling scenario) in the
Zr (Nb) isotopes.

  * February 2024

##  1 Introduction

Nuclei in the region of $A\approx 100,Z\approx 40$ have said to undergo an
abrupt change in the structure of their ground and non-yrast states as one
varies the number of neutrons [1, 2]. The situation has been ascribed to
multiple shell model configurations that mix through the proton-neutron
interaction and cross [3], while also changing their intrinsic shape [4]. Such
scenarios, shape evolution and configuration crossing, have been termed as,
respectively, Type I and type II quantum phase transitions (QPTs).

QPTs are zero temperature phase transitions that occur from quantum
fluctuations as a function of coupling constants in the Hamiltonian, which
serve as the control parameters [5]. In nuclear physics, where this concept
has been presented for the first time [6, 7], an intrinsic shape parameter
serves as the order parameter and thus one terms it often shape-phase
transitions. The concept of QPTs in atomic nuclei has been separated into two
Types, I and II, which are said to occur in different situations.

Figure 1: Schematic illustration for the evolution with nucleon number of
energies (in arbitrary units) of the lowest $J^{+}$ states of one or two
configurations, A and B. (a) Type I QPT: shape changes within a single
configuration (small and large circles denote weak and strong deformation,
respectively). (b) Type II QPT: coexisting and possibly crossing of two
configurations, usually, with strong mixing. The dashed lines depict the
mixing, as in a two states mixing scenario. (c) IQPTs: abrupt crossing of two
configurations, with weak mixing, accompanied by a pronounced gradual shape
evolution within each configuration.

The first QPT, Type I, is considered when nucleons are added in a single shell
model configuration and a shape evolution occurs as one adds more nucleons, as
shown schematically in Figure 1(a). The circles denote the amount of
deformation and as more nucleons are added it increases, which lowers the
absolute energy of the ground state. A Type I QPT can be described by the
following Hamiltonian

$\hat{H}=(1-\xi)\hat{H}_{1}+\xi\hat{H}_{2}~{}.$ (1)

As the control parameter $\xi$ varies from 0 to 1, the equilibrium shape and
symmetry of the Hamiltonian vary from those of $\hat{H}_{1}$ to those of
$\hat{H}_{2}$.

The second QPT, Type II, is considered when nucleons are added in multiple
shell model configurations that interact and cross. This is in the sense that
higher lying states that are associated with nucleons residing in a higher
configuration are lowered to become the ground state. This is shown
schematically in Figure 1(b) where one observes what appears to resemble a two
state mixing scenario. In this case, Type II QPTs can be described by a matrix
Hamiltonian [8]

$\hat{H}=\left[\begin{array}[]{cc}\hat{H}^{\rm
A}(\xi^{\rm(A)})&\hat{W}(\omega)\\\ \hat{W}(\omega)&\hat{H}^{\rm
B}(\xi^{\rm(B)})\end{array}\right]$ (2)

given here for two configurations, denoted by the indices $A$ and $B$, and
$\hat{W}$ denotes their coupling.

Recently, studies have shown the occurrence of such crossing and shape
evolution in the even-even zirconium (40Zr) isotopes [9, 10, 11] and the
adjacent odd-$A$ niobium (41Nb) isotopes [12, 13]. This unique scenario has
been termed as intertwined quantum phase transitions (IQPTs), which denotes
the identification of both Types of QPTs, I and II, in the same chain of
isotopes.

##  2 Theoretical framework

For the study of QPTs involving configuration crossing we shall use the
algebraic framework of the interacting boson model with configuration mixing
[14, 15, 16] for the even-even Zr isotopes and the interacting boson-fermion
model with configuration mixing for the odd-mass Nb isotopes [12, 13].

###  2.1 IBM-CM

The IBM for a single shell model configuration has been widely used [14] to
describe low-lying collective states in nuclei in terms of $N$ monopole ($s$)
and quadrupole ($d$) bosons, representing valence nucleon pairs. The
Hamiltonian interactions are Hermitian, rotationally invariant and conserve
the total number of $s$ and $d$ bosons,

$\hat{N}=\hat{n}_{s}+\hat{n}_{d}=s^{\dagger}s+\sum_{\mu}d^{\dagger}_{\mu}d_{\mu}~{}.$
(3)

The number of bosons (3) is fixed by the microscopic interpretation of the IBM
[17] to be the total number of proton and neutron particle or hole pairs
counted from the nearest closed shell.

For excitations involving $n$-particles and $m$-holes ($n$p-$m$h) from
multiple shell model configurations, one can extend the IBM to IBM with
configuration mixing (IBM-CM) by associating each $n$p-$m$h excitations to a
different $N$-boson space, such as 0p-0h ($N$), 2p-2h ($N+2$), 4p-4h ($N+4$),
etc. In such a case, the Hamiltonian named $\hat{H}_{b}$ is written in matrix
form, which for two configurations is written as in Eq. 2, where
$\hat{H}_{b}^{\rm A}(\xi^{(\rm A)})$ represents the normal configuration ($N$
boson space, denoted henceforth by A) and $\hat{H}_{b}^{\rm B}(\xi^{(\rm B)})$
represents the intruder configuration ($N\\!+\\!2$ boson space, denoted
henceforth by B), corresponding to 2p-2h excitations across the (sub-) shell
closure (the subscript $b$ denotes that these are boson Hamiltonians). In this
work, their form is

$\displaystyle\hat{H}_{b}^{\rm A}$
$\displaystyle=\epsilon^{\rm(A)}_{d}\hat{n}_{d}+\kappa^{\rm(A)}\hat{Q}_{\chi}\cdot\hat{Q}_{\chi}~{},$
(4a) $\displaystyle\hat{H}_{b}^{\rm B}$
$\displaystyle=\epsilon^{\rm(B)}_{d}\hat{n}_{d}+\kappa^{\rm(B)}\hat{Q}_{\chi}\cdot\hat{Q}_{\chi}+\kappa^{\rm\prime(B)}\hat{L}\cdot\hat{L}+\Delta~{},$
(4b)

where $\Delta$ is the off-set energy between configurations A and B, the
quadrupole operator is
$\hat{Q}_{\chi}=d^{\dagger}s+s^{\dagger}\tilde{d}\\!+\\!\chi(d^{\dagger}\tilde{d})^{(2)}$,
where $\tilde{d}_{\mu}\\!=\\!(-)^{\mu}d_{-\mu}$, and the mixing term is
$\hat{W}_{\rm
b}\\!=\\!\omega[(d^{\dagger}d^{\dagger})^{(0)}+(s^{\dagger})^{2}]+\rm{H.c.}$,
where H.c. stands for Hermitian conjugate.

The Hamiltonian $\hat{H}_{b}^{i}$, with $i=\rm A,B$, can interpolate between
three forms of dynamical symmetry (DS) limits using the control parameters
$(\epsilon^{(i)}_{d},\kappa^{(i)},\chi)$ ($\kappa^{\prime(\rm B)}$ and
$\Delta$ of $\hat{H}_{b}^{\rm B}$ do not affect the DS structure), where U(6)
serves as its spectrum generating algebra and SO(3) its symmetry algebra

$\rm{U(6)}\supset\left\\{\begin{array}[]{cc}&\mkern-18.0mu\rm{U(5)}\supset\rm{SO(5)}\supset\rm{SO(3)}\\\
&\mkern-72.0mu\rm{SU(3)}\supset\rm{SO(3)}\\\
&\rm{SO(6)}\supset\rm{SO(5)}\supset\rm{SO(3)}\end{array}\right.$ (4e)

In such a case, the spectrum is completely solvable and resembles known
paradigms of collective motion: spherical vibrator [U(5)], axial deformed
rotor [SU(3)] and $\gamma$-soft deformed rotor [SO(6)].

####  2.1.1 Boson $E2$ transitions operator.

For $E2$ transitions, the boson operator reads

$\hat{T}_{b}(E2)=e^{(\rm A)}\hat{Q}^{(N)}_{\chi}+e^{(\rm
B)}\hat{Q}^{(N+2)}_{\chi}~{},$ (4f)

where $e^{(\rm A)},e^{(\rm B)}$ are the boson effective charges for
configuration A and B, respectively, and superscript $(N)$ denotes a
projection onto the $[N]$ boson space.

####  2.1.2 Boson wave functions.

The eigenstates $\ket{\Psi;L}$ of the boson Hamiltonian with angular momentum
$L$, are linear combinations of the wave functions, $\Psi_{\rm A}$ and
$\Psi_{\rm B}$, in the two spaces $[N]$ and $[N+2]$,
$\ket{\Psi;L}=a\ket{\Psi_{\rm A};[N],L}+b\ket{\Psi_{\rm B};[N\\!+\\!2],L}$,
with $a^{2}+b^{2}=1$. Each part of the wave functions with a given boson
number $N$ and angular momentum $L$ can be expanded in terms of the DS bases
of the IBM in the following manner

$\displaystyle\ket{\Psi;[N],L}$
$\displaystyle=\sum_{n_{d},\tau,n_{\Delta}}C^{(N,L)}_{n_{d},\tau,n_{\Delta}}\ket{N,n_{d},\tau,n_{\Delta},L}$
$\displaystyle\qquad\left[\rm{U(5)}\right]$ (4g)
$\displaystyle=\sum_{(\lambda,\mu),K}C^{(N,L)}_{(\lambda,\mu),K}\ket{N,(\lambda,\mu),K,L}$
$\displaystyle\qquad\left[\rm{SU(3)}\right]$
$\displaystyle=\sum_{\sigma,\tau,n_{\Delta}}C^{(N,L)}_{\sigma,\tau,n_{\Delta}}\ket{N,\sigma,\tau,n_{\Delta},L}$
$\displaystyle\qquad\left[\rm{SO(6)}\right]$

where $N,n_{d},(\lambda,\mu),\sigma,\tau,L$ label the irreducible
representations of U(6), U(5), SU(3), SO(6), SO(5) and SO(3), respectively,
and $n_{\Delta},K$ are multiplicity labels. The coefficients
$C^{(N,L)}_{\alpha}$, with quantum numbers $\alpha$, give the weight of each
component in the wave function. They can therefore give us the probability of
having definite quantum numbers of a given symmetry. For example, for the U(6)
DS the boson number $N$, which reveals the amount of normal-intruder mixing,
and for the U(5) DS the $d$-boson number, $n_{d}$, which reveals the amount of
deformation

$\displaystyle P^{(N_{i},J)}$
$\displaystyle=~{}~{}\sum_{n_{d}}P^{(N_{i},L)}_{n_{d}}~{};\quad i={\rm
A,B}\quad\left(P^{(N_{\rm A},L)}+P^{(N_{\rm B},L)}=1\right)$ (4ha)
$\displaystyle P^{(N_{i},L)}_{n_{d}}$
$\displaystyle=\sum_{\tau,n_{\Delta},L}|C^{(N_{i},L)}_{n_{d},\tau,n_{\Delta},L}|^{2}~{},$
(4hb)

where $N_{\rm A}=N$ and $N_{\rm B}=N+2$.

###  2.2 IBFM-CM

The IBFM for a single shell model configuration has been widely used [18] to
describe low-lying collective states of odd-mass nuclei, where a fermion is
coupled to the boson Hamiltonian, which serves as a core. For two shell model
configurations, 0p-0h and 2p-2h, the IBFM-CM framework [12, 13] Hamiltonian
can be written in the following matrix form

$\displaystyle\hat{H}=\hat{H}_{\rm b}+\hat{H}_{\rm f}+\hat{V}_{\rm bf}$
$\displaystyle\qquad=\left[\begin{array}[]{cc}\hat{H}_{b}^{\rm
A}(\xi^{\rm(A)})&\hat{W}_{b}(\omega)\\\ \hat{W}_{b}(\omega)&\hat{H}_{b}^{\rm
B}(\xi^{\rm(B)})\end{array}\right]+\left[\begin{array}[]{cc}\sum_{j}\epsilon_{j}\hat{n}_{j}&0\\\
0&\sum_{j}\epsilon_{j}\hat{n}_{j}\end{array}\right]+\left[\begin{array}[]{cc}\hat{V}^{\rm
A}_{bf}(\zeta^{(\rm A)})&\hat{W}_{bf}(\omega_{j})\\\
\hat{W}_{bf}(\omega_{j})&\hat{V}^{\rm B}_{bf}(\zeta^{(\rm
B)})j\end{array}\right]$ (4ho)

The first part of $\hat{H}$ is the boson core with entries from Equations
(4a)-(4b), the second part is the fermion Hamiltonian with single particle
energies $\epsilon_{j}$ and the fermion number operator $\hat{n}_{j}$, the
third part is the Bose-Fermi interaction. The latter has off diagonal parts
that combine boson and fermion mixing terms and a diagonal part that typically
takes the following form of monopole, quadrupole and exchange interactions
$\hat{V}^{(i)}_{\rm bf}(A_{0},\Gamma_{0},\Lambda_{0})=V^{{\rm MON}(i)}_{\rm
bf}(A_{0})+V^{{\rm QUAD}(i)}_{\rm bf}(\Gamma_{0})+V^{{\rm EXC}(i)}_{\rm
bf}(\Lambda_{0})$, with $i=\rm A,B$ [18]. These Bose-Fermi interactions depend
on the strengths $A_{0}$, $\Gamma_{0}$ and $\Lambda_{0}$, respectively, given
from the microscopic theory of the IBFM (see [13] for more details).

####  2.2.1 Bose-Fermi $E2$ transitions

For $E2$ transitions, the boson-fermion operator reads

$\hat{T}(E2)=\hat{T}_{b}(E2)+\hat{T}_{f}(E2)~{},$ (4hp)

where the boson part, $\hat{T}_{b}(E2)$, is given in Equation (4f). The
fermion part, $\hat{T}_{f}(E2)$, is given by

$\hat{T}_{\rm
f}(E2)=\sum_{jj^{\prime}}f^{(2)}_{jj^{\prime}}[a^{\dagger}_{j}\times\tilde{a}_{j^{\prime}}]^{(2)},$
(4hq)

with
$f^{(2)}_{jj^{\prime}}=-\frac{e_{f}}{\sqrt{5}}\braket{j}{Y^{(2)}_{lm}}{j^{\prime}}$,
where $e_{f}$ is the effective charge for $E2$ transitions.

####  2.2.2 Bose-Fermi wave functions

The eigenstates of the Hamiltonian (2.2), $\ket{\Psi;J}$, are linear
combinations of wave functions (4g), involving bosonic basis states in the two
spaces $\ket{[N],\alpha,L}$ and $\ket{[N+2],\alpha,L}$, where $\alpha$ denotes
additional quantum numbers characterizing the boson basis used. The boson
($L$) and fermion ($j$) angular momenta are coupled to a total $J$ and the
combined wave function takes the form

$\displaystyle\ket{\Psi;J}\\!=\\!\sum_{\alpha,L,j}C^{(N,J)}_{\alpha,L,j}\ket{\Psi_{\rm
A};[N],\alpha,L,j;J}+\sum_{\alpha,L_{,}j}C^{(N+2,J)}_{\alpha,L,j}\ket{\Psi_{\rm
B};[N+2],\alpha,L,j;J}~{},$ (4hr)

For such a wave function, similarly as in Equations (4ha) and (4hb), it is
possible to examine the probability of the quantum labels of the U(6) boson
DS, $N$, and the U(5) boson DS, $n_{d}$,

$\displaystyle P^{(N_{i},J)}$
$\displaystyle=~{}~{}\sum_{n_{d}}P^{(N_{i},J)}_{n_{d}}~{},\quad i={\rm
A,B}\quad\left(P^{(N_{\rm A},J)}+P^{(N_{\rm B},J)}=1\right)$ (4hsa)
$\displaystyle P^{(N_{i},J)}_{n_{d}}$
$\displaystyle=\sum_{\tau,n_{\Delta},j,L}|C^{(N_{i},J)}_{n_{d},\tau,n_{\Delta},L,j}|^{2}~{}$
(4hsb)

##  3 IQPTs in the zirconium and niobium chains

To describe the 40Zr isotopes in the IBM-CM framework, we consider
${}_{40}^{90}$Zr as a core and valence neutrons in the 50–82 major shell. The
normal A configuration corresponds to having no active protons above
$Z\\!=\\!40$ sub-shell gap, and the intruder B configuration corresponds to
two-proton excitation from below to above this gap, creating 2p-2h states (see
[11] for more details). To describe the 41Nb isotopes in the IBFM-CM framework
we couple a proton to the respective 40Zr cores with neutron number 52–62. In
the present contribution, we focus on the positive-parity states in the Nb
isotopes, which reduces to a single-$j$ calculation with the $1g_{9/2}$ orbit
(a multi-$j$ calculation for negative-parity states can be found in [13]). The
model space for 99Nb is shown schematically in Figure 2, where 98Zr serves as
the boson core.

Figure 2: Schematic representation of the two coexisting shell-model
configurations (A and B) for ${}^{99}_{41}$Nb58, where ${}^{98}_{40}$Zr58
serves as a core. The corresponding numbers of proton bosons ($N_{\pi}$) and
neutron bosons ($N_{\nu}$), are listed for each configuration and
$N=N_{\pi}+N_{\nu}$.

In order to understand the change in structure of the Zr and Nb isotopes, it
is insightful to examine the evolution of observables along the chain. In this
contribution, the observables include energy levels, two-neutron separation
energies and $E2$ transition rates.

###  3.1 Evolution of energy levels

Figure 3: Panels (a) and (c): comparison between experimental and calculated
energy levels of Zr isotopes. Panels (b) and (d): comparison between energy
levels of Nb isotopes. Empty (filled) symbols indicate a states dominated by
the normal A configuration (intruder B configuration), with assignments based
on Equation (4ha) (Zr) and Equation (4hsa) (Nb). In panels (a) and (c), the
shape of the symbols [$\circ,\triangle,\diamondsuit$], indicates the closest
dynamical symmetry [U(5), SU(3), SO(6)] to the level considered, based on
Equation (4hb). Note that the calculated values start at neutron number 52,
while the experimental values include the closed shell at 50\. References for
the data are given at [11] (for Zr) and [13] (for Nb).

A comparison between experimental and calculated levels of Zr and Nb is shown
in Figure 3, along with assignments to configurations based on Equation (4ha),
for Zr, and Equation (4hsa), for Nb. For Zr, we also add symbols denoting the
most dominant DS component in the wave function based on the decompositions of
Equation (4hb), for each state (the definitions to the other SU(3) and SO(6)
DS decompositions can be found in [11]).

For Zr isotopes, panels (a) and (c), in the region between neutron number 50
and 56, there appear to be two configurations, one spherical (seniority-like),
A, and one weakly deformed, B, as evidenced by the ratio $R_{4/2}$, which is
at 52–56, $R^{\rm(A)}_{4/2}\cong 1.6$ and $R^{\rm(B)}_{4/2}\cong 2.3$. Their
trend decreases at neutron number 52–54, away from the closed shell, and rise
again at 56 due to the $\nu(2d_{5/2})$ subshell closure. From neutron number
58, there is a pronounced drop in energy for the states of configuration B and
at 60, the two configurations exchange their role indicating a Type II QPT. At
this stage, the intruder B configuration appears to be at the critical point
of a U(5)-SU(3) Type I QPT, as evidenced by the low value of the excitation
energy of the first excited $0^{+}$ state of this configuration (the
$0^{+}_{3}$ state in 100Zr shown in Figure 3 of [9]). Beyond neutron number
60, the intruder configuration (B) is strongly deformed, as evidenced by the
small value of the excitation energy of the state $2_{1}^{+}$,
$E_{2_{1}^{+}}\\!=\\!139.3$ keV and by the ratio $R^{\rm(B)}_{4/2}\\!=\\!3.24$
in 104Zr. At still larger neutron number 66, the ground state band becomes
$\gamma$-unstable (or triaxial) as evidenced by the close experimental energy
of the states $2_{2}^{+}$ and $4_{1}^{+}$, $E_{2_{2}^{+}}\\!=\\!607.0$ keV,
$E_{4_{1}^{+}}\\!=\\!476.5$ keV, in 106Zr, and especially by the experimental
$E_{4^{+}_{1}}\\!=\\!565$ keV and $E_{2^{+}_{2}}\\!=\\!485$ keV in 110Zr, a
signature of the SO(6) symmetry. In this region, the ground state
configuration undergoes a crossover from SU(3) to SO(6).

For Nb isotopes, panels (b) and (d), we see a quintuplet of states
($5/2^{+}_{1},7/2^{+}_{1},11/2^{+}_{2},13/2^{+}_{1},9/2^{+}_{2}$). For neutron
number 52–56, their energy varies due to the filling of the $\nu(2d_{5/2})$
shell. At neutron number 58, there is a pronounced drop in energy for the
states of the B configuration, due to the onset of deformation. At 60, the two
configurations cross, indicating a Type II QPT, and the ground state changes
from $9/2^{+}_{1}$ to $5/2^{+}_{1}$, becoming the bandhead of a $K=5/2^{+}$
rotational band composed of
$5/2^{+}_{1},7/2^{+}_{1},9/2^{+}_{1},11/2^{+}_{1},13/2^{+}_{1},\ldots$ states.
At this point the intruder B configuration also undergoes a Type I QPT from a
weak to a strong coupling scenario. Beyond neutron number 60, the intruder B
configuration remains strongly deformed and the band structure persists. The
above trend is similar to that encountered in the even-even 40Zr cores.

The comparability to the even-even Zr cores comes from the fact that four of
the states in Figure 3, ($5/2^{+}_{1},7/2^{+}_{1},11/2^{+}_{2},13/2^{+}_{1}$),
have the same trend as the $2^{+}_{1}$ state of the adjacent Zr isotopes and
one state, ($9/2^{+}_{2}$), like the $0^{+}_{2}$. For 52–58, the
correspondence occurs since the Zr $L=2^{+}_{1}$ state is coupled to single
$j=9/2$ orbit through a weak coupling scenario giving a total angular momentum
of $|L-j|\leq J\leq|L+j|$ (note that the $J=9/2^{+}$ which is part of this
quintuplet is the $9/2^{+}_{3}$ and is not shown in the figure). In such a
scenario, the “center of gravity” [19] of the Nb multiplet is centered around
the energy of the $2^{+}_{1}$, as described in [13]. The same occurs for the
$9/2^{+}_{2}$ of Nb and the $0^{+}_{2}$ of Zr.

###  3.2 Evolution of configuration and symmetry content

\begin{overpic}[width=303.53267pt]{fig_Nb_Zr_SDANCA23_nd_decomp_3.pdf}
\put(10.0,25.0){\scriptsize\fcolorbox{black}{white}{(a) Intruder component}}
\put(48.7,46.0){\scriptsize\fcolorbox{black}{white}{(b) $n_d=0$ component}}
\end{overpic} Figure 4: Percentage of (a) the intruder B component, Equations
(4ha) and (4hsa), (b) the boson U(5) $n_{d}=0$ component, Equations (4hb) and
(4hsb), of 52-110Zr (red lines) and 93-103Nb (blue lines) isotopes,
respectively. On top of panel (b) shown are the boson DS indicating the
transitions in symmetry within the intruder B configuration. For Nb isotopes,
the ground state $J^{+}_{gs}$ ($J^{+}_{B}$) is $9/2^{+}_{1}$ ($9/2^{+}_{2}$)
for 93-99Nb and $5/2^{+}_{1}$ ($5/2^{+}_{2}$) for 101,103 Nb and

To understand the occurrence of both Type II and I QPTs in the chain of Zr and
Nb isotopes, we examine explicitly the evolution of the intruder and of the
U(5) symmetry components of the wave function.

For the Type II QPT, we examine in Figure 4(a) the intruder component of the
wave function of the ground state of Zr ($P^{(N_{\rm B},0^{+}_{gs})}$) and Nb
($P^{(N_{\rm B},J^{+}_{gs})}$) isotopes, Equations (4ha) and (4hsa),
respectively. We observe the same trend, where from neutron number 52 to 58
this component is small and at 60 it jumps to about 87% in Zr and 99% in Nb
and remains at about 99% for the heavier isotopes in both chains. This is the
occurrence of the Type II QPT, where the ground state changes its
configuration content from normal (at neutron number 58) to intruder (at 60).

For the Type I QPT within configuration B, we examine in Figure 4(b) the
$n_{d}=0$ component of the configuration B part of $0^{+}_{B}=0^{+}_{2}$ for
92-98Zr and $0^{+}_{B}=0^{+}_{1}$ for 100-110Zr ($P^{(N_{\rm
B},0^{+}_{B})}_{n_{d}=0}$) and of $J^{+}_{B}=9/2^{+}_{2}$ for 93-99Nb and
$J^{+}_{B}=5/2^{+}_{1}$ for 101-103Nb ($P^{(N_{\rm B},J^{+}_{B})}_{n_{d}=0}$).
The circles represent the percentage of the U(5) $n_{d}\\!=\\!0$ component in
the wave function, Equations (4hb) and (4hsb). For Zr, it is large ($\approx
90\%$) for neutron number 52–58 and drops drastically ($\approx 30\%$) at 60.
The drop means that other $n_{d}\\!\not=\\!0$ components are present in the
wave function and therefore this state becomes deformed. Above neutron number
60, the $n_{d}\\!=\\!0$ component drops almost to zero (and rises again a
little at 70), indicating the state is strongly deformed. For Nb we see the
exact same trend, however, the total value of $P^{(N_{\rm
B},J^{+}_{B})}_{n_{d}=0}$ is smaller, indicating that for odd-mass nuclei
coupling the fermion to the boson core increases deformation.

###  3.3 Evolution of two neutron separation energies

Figure 5: Comparison of the deformed part of the two neutron separation
energies, ($S^{\rm def}_{2n}$), between Zr and Nb experiment (a) and (b) [20]
and Zr and Nb calculation (c) and (d), respectively.

An observable that portrays both types of QPTs is two neutron separation
energy, defined as

$S_{2n}=2M_{n}+M(N-2,Z)-M(N,Z),$ (4hst)

where $M(N,Z)$ is the mass of a nuclei with $N,Z$ neutrons and protons,
respectively, and $M_{n}$ is the neutron mass. In the IBM and IBFM, it is
convenient to transcribe the $S_{2n}$ as

$S_{2n}=-\tilde{A}-\tilde{B}N_{v}\pm S^{\rm def}_{2n}-\Delta_{n},$ (4hsu)

where $N_{v}$ is half the number of valence particles in the boson core and
$S^{\rm def}_{2n}$ is the contribution of the deformation, obtained by the
expectation value of the Hamiltonian in the ground state. The $+$ sign applies
to particles and the $-$ sign to holes. The $\Delta_{n}$ parameter takes into
account the neutron subshell closure at 56, $\Delta_{n}=0$ for 50–56, and
$\Delta_{n}=2$ MeV for 58–70 for Zr (58–62 for Nb). Its value is adapted from
[21]. For the Zr (Nb) isotopes, the chosen values in Eq. 4hsu are
$\tilde{A}=-17.25$ ($-16.5$) and $\tilde{B}=0.758$ MeV. The value of
$\tilde{A}$ is taken to fit 90Zr (91Nb), and the value of $\tilde{B}$ is taken
from a fit to the binding energies of 92,94,96Zr. In Figure 5, the Zr and Nb
experimental and calculated deformed part, $S^{\rm def}_{2n}$ [22, 23], are
shown in red open circles and lines, respectively. $S^{\rm def}_{2n}$ is
obtained by subtracting the linear part and $\Delta_{n}$ from the experimental
and calculated $S_{2n}$. One can clearly see the onset of deformation going
from neutron number 52–56, where $S^{\rm def}_{2n}$ is close to zero, to
58–62, where it jumps and rises in both Zr and Nb isotopes. For Zr, we also
see after neutron number 62 a gradual decrease in $S^{\rm def}_{2n}$ due to
the crossover towards SO(6) symmetry.

In order to denote the occurrence of both Type I and II QPTs, in addition to
Equation (4hsu), it is also possible using Equation (4hst) to estimate two
neutron separation energies for excited states, by using the mass of an
excited state $M(N,Z)\\!\equiv\\!M_{exc}(N,Z)=M_{gs}(N,Z)+E_{exc}(N,Z)$, where
$M_{gs}(N,Z)$ is the mass for the ground state and $E_{exc}(N,Z)$ is the
energy of the excited state [13]. We choose the lowest configuration B state
for this case in both Zr and Nb isotopes. The experimental and calculated
results, $S^{\rm def}_{2n;B}$, are given in blue triangles and lines, in
Figure 5. For both Zr and Nb isotopes, it is seen that for neutron number
54–56 $S^{\rm def}_{2n;B}$ is close to zero, then at 58 it jumps due to the
onset of deformation, then it flattens at 60. This behavior denotes the Type I
QPT of shape evolution within configuration B of Zr (Nb) from spherical (weak
coupling) to axially-deformed (strong coupling). For neutron number 54–56,
$S^{\rm def}_{2n;B}$ (triangles) is close to the value of $S^{\rm def}_{2n}$
(circles), as configuration B is more spherical. At 58, there is a larger jump
than $S^{\rm def}_{2n}$ since configuration B is more deformed than A, which
continues at 60. For 62 (and 64–70 for Zr), both $S^{\rm def}_{2n;B}$ and
$S^{\rm def}_{2n}$ coincide since the ground state is configuration B, which
denotes the Type II QPT.

###  3.4 Evolution of $E2$ transition rates

Figure 6: $B(E2)$ values in W.u. (a) for $2^{+}\to 0^{+}$ transitions in the
Zr chain and (b) for $7/2^{+}_{1}\to J^{+}_{gs}$ in the Nb chain, where
$J^{+}_{gs}$ is given in the panel above (b). On top of panel (a) shown are
the boson dynamical symmetries indicating the transitions in symmetry within
the intruder B configuration. The solid line (symbols
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\CIRCLE},~{}{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\blacksquare},~{}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\blacktriangle},~{}{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\blacklozenge}$)
denote calculated results (experimental results). In (a), dotted lines denote
calculated $E2$ transitions within a configuration. References for the data
can be found in [11].

The occurrence of both Type I and II QPTs are also stressed by an analysis of
$B(E2)$ values. As shown in Fig. 6(a) for Zr, the calculated $2^{+}_{A}\to
0^{+}_{A}$ transition rates coincide with the empirical $2^{+}_{1}\to
0^{+}_{1}$ rates for neutron numbers 52–56. The calculated $2^{+}_{B}\to
0^{+}_{B}$ transition rates coincide with the empirical $2^{+}_{2}\to
0^{+}_{2}$ rates for neutron numbers 52–56, with the empirical $2^{+}_{1}\to
0^{+}_{2}$ rates at 58 and with the empirical $2^{+}_{1}\to 0^{+}_{1}$ rates
at 60–64. The large jump in $B(E2;2^{+}_{1}\\!\to\\!0^{+}_{1})$ between 58 and
60 reflects the passing through a critical point, common to a Type II QPT
involving a crossing of two configurations. The further increase in
$B(E2;2^{+}_{1}\\!\to\\!0^{+}_{1})$ for neutron numbers 60–64 is as expected
from a U(5)-SU(3) Type I QPT and reflects an increase in the deformation in a
spherical to deformed shape-phase transition within configuration B. The
subsequent decrease from the peak at neutron number 64 towards 70 is expected
from an SU(3) to SO(6) crossover. The same trend at neutron number 52–62 in
the $B(E2)$ value Zr can be observed in Fig. 6(b) for the Nb isotopes.
Although there is less data, the $B(E2)$ values from the first excited to the
ground state are small for neutron number 52–58 and at 60 there is large jump,
which continues to increase towards 62, possibly suggesting even stronger
deformation than in the adjacent Zr, although the experimental error bar is
large.

##  4 Conclusions and outlook

The general framework of the interacting boson model and interacting boson-
fermion model with configuration mixing, IBM-CM and IBFM-CM, respectively, has
been presented, allowing a quantitative description of shape-coexistence
(configuration-mixing) and related QPTs in even-even and odd-mass nuclei. By
employing such a comparison, odd-mass nuclei can serve as a further test case
for shape coexistence in even-even nuclei by verifying whether the even-even
core is the right one to describe the adjacent odd-mass nucleus.

A quantal analysis for the chain of the even-even 40Zr and odd-mass 41Nb
isotopes involving positive-parity states was performed for neutron number
52–70 (Zr) and 52–62 (Nb). It examined the evolution of energy levels, two-
neutron separation energies and $E2$ transition rates along both chains.
Special attention has been devoted to changes in the configuration-content and
U(5) content of wave functions.

The results of the analysis suggest a complex phase structure yet similar in
both chains of these isotopes, involving two configurations. The two
configurations cross near neutron number 60, and the ground state changes from
configuration A to configuration B, demonstrating a Type II QPT. They are
weakly mixed and retain their purity before and after the crossing. Alongside
the Type II QPT, the intruder B configuration undergoes a spherical-U(5) (weak
coupling) to axially-deformed-SU(3) (strong coupling) QPT within the boson
core (Bose-Fermi Hamiltonian), with a critical point near $A\approx 100$, thus
demonstrating, IQPTs in both even-even and odd-mass nuclei.

The author acknowledges support from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No. 101107805. The results reported are based on work done in
collaboration with F. Iachello (Yale University) and A. Leviatan (The Hebrew
University).

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