# Mirror Symmetry for open $r$-spin invariants

Mark Gross  M. Gross:
Department of Pure Mathematics and Mathematical Statistics, University of
Cambridge, CB4 0WB, United Kingdom<EMAIL_ADDRESS>,  Tyler L. Kelly
T. L. Kelly:
School of Mathematics, University of Birmingham, Edgbaston, B15 2TT, United
Kingdom<EMAIL_ADDRESS>and  Ran J. Tessler  R. J. Tessler:
Department of Mathematics, Weizmann Institute of Science, Rehovot 76100,
Israel<EMAIL_ADDRESS>
###### Abstract.

We show that a generating function for open $r$-spin enumerative invariants
produces a universal unfolding of the polynomial $x^{r}$. Further, the
coordinates parametrizing this universal unfolding are flat coordinates on the
Frobenius manifold associated to the Landau-Ginzburg model
$(\mathbb{C},x^{r})$ via Saito-Givental theory. This result provides evidence
for the same phenomenon to occur in higher dimension, proven in the sequel
[GKT22].

##  1\. Introduction

Buryak, Clader, and Tessler recently constructed an open $r$-spin enumerative
theory [BCT21, BCT18], following the case of descendent invariants on the
moduli space of holomorphic disks developed by Pandharipande, Solomon and
Tessler in [PST14]. Roughly stated, they construct a moduli space of
$r$-stable orbidisks with $r$-spin structures that have prescribed twists at
both internal and boundary marked points. In turn, this gives a closed form
expression for the open $r$-spin invariants

$\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}$

corresponding to genus $0$ orbidisks with $k+1$ boundary marked points with
twist $r-2$ and $l$ internal marked points with twists $a_{1},\ldots,a_{l}$.

These invariants are analogous to the closed $A$-model enumerative theory for
Landau-Ginzburg models constructed in a sequence of papers [JKV01, FJR07,
FJR08, FJR13], using ideas of Witten [Wit93]. In these papers, the authors
build an enumerative theory in the case of gauged Landau-Ginzburg models
$(\mathbb{C}^{n},W,G)$ where $W$ is an invertible polynomial and $G$ is a
subgroup of the diagonal automorphism group of $W$. Berglund and Hübsch
predicted mirror pairs between such Landau-Ginzburg models. They propose that
the mirror to a gauged LG model $(\mathbb{C}^{n},W,G)$ should be a mirror
gauged LG model $(\mathbb{C}^{n},W^{T},G^{T})$, where $W^{T}$ is the so-called
transposed polynomial and $G^{T}$ is the dual group [BH92]. One such example
is the mirror pair

(1.1)
$(\mathbb{C}^{n},x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}},\mu_{r_{1}}\times\cdots\times\mu_{r_{n}})\leftrightarrow(\mathbb{C}^{n},x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}},1).$

The open $r$-spin invariants stated above correspond to the open $A$-model
invariants for the Landau-Ginzburg model $(\mathbb{C},x^{r},\mu_{r})$,
following the analogue of the closed case established in [FJR11]. Thus from
(1.1) when $n=1$, the open $r$-spin invariants should correspond to an open
$B$-model enumerative theory $(\mathbb{C}^{1},x^{r},1)$.

The (closed) $B$-model side of the story for the right-hand side of the
correspondence (1.1) was developed in [Sai83a, Sai83b, Giv96, Dub96], and was
more recently used in Landau-Ginzburg mirror symmetry in [LLSS17, HLSW15]. The
$B$-model side, put as simply as possible, is a Saito-Givental theory, which
involves calculating oscillatory integrals of the form

(1.2) $\int_{\Gamma}e^{W_{\bf t}/\hbar}f(x_{1},\ldots,x_{n},{\bf
t})dx_{1}\wedge\cdots\wedge dx_{n}.$

Here $x_{1},\ldots,x_{n}$ are coordinates on $\mathbb{C}^{n}$, $\hbar$ is a
coordinate on an auxiliary $\mathbb{C}^{*}$,
$f:\mathbb{C}^{n}\rightarrow\mathbb{C}$ is a carefully chosen regular
function, and ${\bf t}=\\{t_{ij}\,|\,1\leq i\leq n,0\leq j\leq r_{i}-2\\}$ is
a set of coordinates on the parameter space for a universal unfolding $W_{\bf
t}$ of $W$. Finally $\Gamma$ runs over some suitable non-compact cyles in
$\mathbb{C}^{n}$. The requirement on $f$ is that the form
$fdx_{1}\wedge\cdots\wedge dx_{n}$ is a so-called primitive form in the sense
of Saito-Givental theory.

While Saito-Givental theory in general gives a Frobenius manifold structure to
the universal unfolding of $W$, determining this structure can be quite
difficult. However, experience with mirror symmetry for toric Fano varieties
[Gro10, FOOO10] suggests that mirror symmetry becomes much more transparent
when a specific perturbation $W_{\bf t}$ of the original potential is used.
Previous work on the Fano/LG mirror correspondence (see e.g., [CO06],
[FOOO10], [Gro10]) found that there is a “correct” universal unfolding
$W_{\mathbf{t}}$ which is a generating function for counting Maslov index two
disks with boundary on a Lagrangian torus. The advantage of this “correct”
universal unfolding is that there is a canonical choice of primitive form
(which in our case will just be $\Omega=dx$) and that flat coordinates
coincide with a natural choice of coordinates. Using the same philosophy here,
we show that the “correct” universal unfolding is a generating function for
the open $r$-spin invariants.

######  Theorem 1.1.

The deformed potential

(1.3) $W_{\mathbf{t}}=\sum_{k\geq 0,l\geq
0}\sum_{\\{a_{i}\\}\in\mathscr{A}_{l}}(-1)^{l-1}\frac{\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}}{k!|\mathrm{Aut}(\\{a_{i}\\})|}\left(\prod_{i=1}^{l}t_{a_{i}}\right)x^{k},$

has the primitive form $\Omega=\textup{d}x$ and $t_{0},\ldots,t_{r-2}$ are
flat coordinates for the Frobenius manifold constructed via Saito-Givental
theory for the LG model $W=x^{r}$.

In this paper, we use the closed form for open $r$-spin invariants in [BCT18]
as a black box to prove the mirror Theorem 1.1. The flat structure of the
Frobenius manifold for the Landau-Ginzburg model $(\mathbb{C}^{1},x^{r},1)$
has been studied in the past in its own right in integrable systems. In the
1980s, Noumi and Yamada found algorithms for finding the flat coordinates
[Nou84, NY98]. These techniques were transcendental in nature, whereas we
prove Theorem 1.1 through purely algebraic and combinatorial means once given
the open $r$-spin invariants. We aim for this paper to serve as a link from
the integrable systems literature for flat structures to the Landau-Ginzburg
mirror symmetry analogue of the Fano/LG correspondence developed in the $n=2$
case in our sequel paper [GKT22].

In [Bur20], Buryak describes the flat structure Frobenius manifold via the
extended $r$-spin invariants introduced in [BCT19]. There, he shows that the
change of coordinates from the versal deformation to the flat coordinates can
be derived from particular differentials of a generating function built from
extended invariants (Theorem 3.1 in [Bur20]). The extended invariants are
closely related to the open $r$-spin invariants [BCT18, Theorem 1.3]. The
proof presented here is different from that in [Bur20] as the author uses
transcendental techniques from integrable systems. While it is true that, in
dimension one, one can derive the flat structure using either extended or open
$r$-spin invariants, the generating function using open invariants is more
direct and one is no longer able to use extended invariants when one considers
singularities that are not ADE. In the $n=2$ case constructed in [GKT22], we
use open FJRW invariants to construct flat coordinates for the singularity
$W=x_{1}^{r_{1}}+x_{2}^{r_{2}}$. These in turn can be used to compute the
closed extended invariants (with descendents). We aim for this paper to help
solidify the import of the perspective taken from the Fano/LG mirror
correspondence to Landau-Ginzburg mirror symmetry.

In Section 2, we outline the relevant Saito-Givental theory for the Landau-
Ginzburg models $(\mathbb{C}^{n},x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}},1)$. In
Section 3, we provide the definition of primitive forms and flat coordinates
in the context of the LG models in (1.1) and give an example. In Section 4, we
prove Theorem 1.1.

Acknowledgments. The authors would like to thank Alexander Buryak and Robert
Maher for discussions relating to this work. The first author acknowledges
support from the EPSRC under Grants EP/N03189X/1, a Royal Society Wolfson
Research Merit Award, and the ERC Advanced Grant MSAG. The second author
acknowledges that this paper is based upon work supported by the UKRI and
EPSRC under fellowships MR/T01783X/1 and EP/N004922/2. The third author,
incumbent of the Lillian and George Lyttle Career Development Chair,
acknowledges support provided by the ISF grant No. 335/19 and by a research
grant from the Center for New Scientists of Weizmann Institute.

##  2\. The $B$-Model State Space for Fermat polynomials

In this section, we describe the enumerative theory associated to a Landau-
Ginzburg $B$-model, due to Saito and Givental and described in the case of the
$B$-model of FJRW theory by He, Li, Li, Saito, Shen and Webb in [LLSS17,
HLSW15]. Let $(X,W)$ be a Landau-Ginzburg model, i.e., $X$ a variety and
$W:X\rightarrow\mathbb{C}$ a regular function. We will not allow for a group
of symmetries. We shall quickly review Saito-Givental theory in this context.
For a much more in-depth exposition in our framework and notation, see Chapter
2 of [Gro11] and references therein. A principal object of study is the
twisted de Rham complex

$(\Omega_{X}^{\bullet},d+\hbar^{-1}dW\wedge-),$

where $\Omega_{X}^{i}$ is the sheaf of algebraic $i$-forms on $X$ and
$\hbar\in\mathbb{C}^{*}$ is an auxiliary parameter. We restrict our attention
to the Landau-Ginzburg model

(2.1) $(X,W)=(\mathbb{C}^{n},W=\sum_{i}x_{i}^{r_{i}}).$

######  Proposition 2.1.

Consider the Landau-Ginzburg model $(X,W)$ in (2.1). Then the hypercohomology
group $\mathbb{H}^{n}(X,(\Omega_{X}^{\bullet},d+\hbar^{-1}dW\wedge-))$ has
dimension $\prod_{i=1}^{n}(r_{i}-1)$ and is generated by the basis

$M=\left\\{\prod_{i}x_{i}^{a_{i}}\Omega\ \middle|\ 0\leq a_{i}\leq
r_{i}-2\right\\},$

where $\Omega=dx_{1}\wedge\cdots\wedge dx_{n}$.

###### Proof.

First, since $\mathbb{C}^{n}$ is affine, the cohomology of the sheaves
$\Omega_{\mathbb{C}^{n}}^{i}$ vanishes in degree at least one. Thus by the
hypercohomology spectral sequence, it is enough to compute the cohomology of
the complex

$0\rightarrow\Omega_{\mathbb{C}^{n}}^{0}\overset{\delta}{\longrightarrow}\cdots\overset{\delta}{\longrightarrow}\Omega_{\mathbb{C}^{n}}^{n-1}\overset{\delta}{\longrightarrow}\Omega_{\mathbb{C}^{n}}^{n}\rightarrow
0,$

where $\delta=d+\hbar^{-1}dW\wedge-$. Thus the hypercohomology we are
interested in can be written as

(2.2)
$\mathbb{H}^{n}(X,(\Omega_{X}^{\bullet},d+\hbar^{-1}dW\wedge-))=\Omega_{\mathbb{C}^{n}}^{n}/\delta(\Omega_{\mathbb{C}^{n}}^{n-1}).$

First note that

(2.3) $\delta(x_{1}^{a_{1}}\cdots x_{i-1}^{a_{i-1}}x_{i+1}^{a_{i+1}}\cdots
x_{n}^{a_{n}}dx_{1}\wedge\cdots\wedge\widehat{dx_{i}}\wedge\cdots\wedge
dx_{n})=(-1)^{i-1}\hbar^{-1}r_{i}x_{1}^{a_{1}}\cdots
x_{i-1}^{a_{i-1}}x_{i}^{r_{i}-1}x_{i+1}^{a_{i+1}}\cdots x_{n}^{a_{n}}\Omega.$

Next, when $a_{1}\ldots a_{n}\in\mathbb{Z}_{\geq 0}$ and $a_{i}>0$ we have
that

(2.4) $\displaystyle\delta(x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}dx_{1}\wedge\cdots\wedge\widehat{dx_{i}}\wedge\cdots\wedge
dx_{n})=$ $\displaystyle{}(-1)^{i-1}a_{i}x_{1}^{a_{i}}\cdots
x_{i}^{a_{i}-1}\cdots x_{n}^{a_{n}}\Omega$
$\displaystyle{}+(-1)^{i-1}\hbar^{-1}r_{i}x_{1}^{a_{1}}\cdots
x_{i}^{r_{i}+a_{i}-1}\cdots x_{n}^{a_{n}}\Omega$

Thus we have in the quotient
$\Omega_{\mathbb{C}^{n}}^{n}/\delta(\Omega_{\mathbb{C}^{n}}^{n-1})$ the
relations

(2.5) $\displaystyle x_{1}^{a_{1}}\cdots
x_{i-1}^{a_{i-1}}x_{i}^{r_{i}-1}x_{i+1}^{a_{i+1}}\cdots x_{n}^{a_{n}}\Omega$
$\displaystyle=0$ $\displaystyle x_{i}^{r}x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\Omega$
$\displaystyle=-\hbar\tfrac{a_{i}+1}{r_{i}}x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\Omega.$

Thus, any section of $\Omega_{\mathbb{C}^{n}}^{n}$ of the form

$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\Omega$

with $a_{i}\geq r_{i}$ can have its exponent reduced by $r_{i}$ and any
section with $a_{i}=r_{i}-1$ vanishes, yielding the result. ∎

######  Remark 2.2.

Note that the right-hand side of (2.2) is the exact expression for the
formally completed version of the Brieskorn lattice $\mathcal{H}_{W}^{(0)}$
associated to $W$ stated in §3.1 of [LLSS17].

Using Proposition 2.1, there is a vector bundle ${\mathcal{R}}^{\vee}$ on the
$\hbar$-plane $\mathbb{C}$ whose fibre over $\hbar\in\mathbb{C}^{*}$ is the
cohomology group
$\mathbb{H}^{n}(X,(\Omega_{X}^{\bullet},d+\hbar^{-1}dW\wedge-))$, and $M$
yields a frame for $\mathcal{R}^{\vee}$ which extends across the origin; see
[Gro11], §2.2.2 and in particular Definition 2.37.

There is a homology group dual to the hypercohomology group
$\mathbb{H}^{n}(X,(\Omega_{X}^{\bullet},d+\hbar^{-1}dW\wedge-))$ given by the
relative homology

$H_{n}(X,\operatorname{Re}W/\hbar\ll 0;\mathbb{C}).$

Indeed, there is a natural perfect pairing

(2.6) $\displaystyle H_{n}(X,\operatorname{Re}W/\hbar\ll
0;\mathbb{C})\times\mathbb{H}^{n}(X,(\Omega_{X}^{\bullet},d+\hbar^{-1}dW\wedge-))$
$\displaystyle\longrightarrow\mathbb{C},$ $\displaystyle(\Xi,\omega)$
$\displaystyle\longmapsto\int_{\Xi}e^{W/\hbar}\omega.$

Thus there must be a dual basis for $H_{n}(X,\operatorname{Re}W/\hbar\ll
0;\mathbb{C})$ for any basis of the hypercohomology group.

######  Example 2.3.

Consider the Landau-Ginzburg model $(\mathbb{C},x^{r})$. Given a fixed value
of $\hbar\in\mathbb{C}^{*}$, set
$\Psi_{j}:=\\{te^{((\pi+\arg\hbar)\sqrt{-1}+2\pi j\sqrt{-1})/r}\ |\
t\in\mathbb{R}_{\geq 0}\\}$. Note this requires a choice of $\arg\hbar$: if
$\arg\hbar$ is replaced with $\arg\hbar+2\pi$, $\Psi_{j}$ becomes
$\Psi_{j+1}$. This will yield a multi-valuedness for the cycles $\Xi_{j}^{r}$
constructed below. In the formulas below, we make a specific choice of branch
of $\arg\hbar$, which will then give a well-defined choice of $\hbar^{1/r}$.

We now show there exists cycles $\Xi^{r}_{j}\in
H_{1}(X,\operatorname{Re}(x^{r}/\hbar)\ll 0;\mathbb{C})$ for $0\leq i\leq r-2$
so that

(2.7) $\int_{\Xi^{r}_{j}}x^{k}e^{x^{r}/\hbar}dx=\delta_{jk}$

for all $0\leq j\leq r-2$, where $\delta_{jk}$ is the Kronecker delta
function.

First note that $W/\hbar$ is real and negative on $\Psi_{j}$, going to
$-\infty$ in the unbounded direction of $\Psi_{j}$. Thus
$\Psi_{j+1}-\Psi_{j}\in H_{1}(X,\operatorname{Re}W/\hbar\ll 0;\mathbb{C})$. By
a direct computation of exponential integrals, we find that

$A_{jk}:=\int_{\Psi_{j+1}-\Psi_{j}}x^{k}e^{x^{r}/\hbar}dx=\frac{1}{r}C_{k}\zeta^{j(k+1)},$

where $\zeta:=e^{2\pi\sqrt{-1}/r}$ and

$C_{k}=e^{\pi
i\frac{k+1}{r}}\hbar^{(k+1)/r}\Gamma\left(\frac{k+1}{r}\right)\left(e^{2\pi
i(k+1)/r}-1\right).$

We find the inverse of the matrix
$A=(A_{jk})\in\operatorname{Mat}_{(r-1)\times(r-1)}(\mathbb{C}(\hbar))$ to
explicitly compute the cycles $\Xi_{j}$ as a linear combination of the
$(\Psi_{j+1}-\Psi_{j})$. Let $C=\operatorname{diag}(C_{k})_{k=0}^{r-2}$. If we
postmultiply by $C^{-1}$, we obtain the matrix

$B:=AC^{-1}=\left(\frac{1}{r}\zeta^{j(k+1)}\right)_{0\leq j,k\leq r-2}.$

The inverse matrix here can be computed to be
$(B^{-1})_{jk}=(\zeta^{-k(j+1)}-\zeta^{j+1}).$ So then

$(A^{-1})_{jk}=(C^{-1}B^{-1})_{jk}=\frac{1}{C_{j}}(\zeta^{-k(j+1)}-\zeta^{j+1}).$

We then have an explicit description of $\Xi_{j}^{r}$:

(2.8) $\Xi_{j}^{r}:=\frac{1}{e^{\pi
i\tfrac{j+1}{r}}\hbar^{(j+1)/r}\Gamma(\tfrac{j+1}{r})(\zeta^{j+1}-1)}\sum_{k=0}^{r-2}(\zeta^{-k(j+1)}-\zeta^{j+1})(\Psi_{k+1}-\Psi_{k})$

The following more general integrals will also be important for us in proving
Theorem 1.1:

######  Lemma 2.4.

For all $r\in\mathbb{N}$ and $n\in\mathbb{Z}$:

(2.9)
$\displaystyle\begin{split}\int_{\Xi_{d}}x^{nr+k}e^{x^{r}/\hbar}dx={}&(-1)^{n}\hbar^{n}\left(\prod_{i=1}^{n}(i-1+\frac{k+1}{r})\right)\int_{\Xi_{d}}x^{k}e^{x^{r}/\hbar}dx\\\
={}&(-1)^{n}\hbar^{n}\frac{\Gamma\left(n+\frac{k+1}{r}\right)}{\Gamma\left(\frac{k+1}{r}\right)}\int_{\Xi_{d}}x^{k}e^{x^{r}/\hbar}dx.\end{split}$

###### Proof.

This is proven by iterating integration by parts or the relation given in
(2.5). ∎

Using this example, we can describe a similar set of cycles for the Fermat
Landau-Ginzburg model

$(\mathbb{C}^{n},W_{0}),\qquad W_{0}:=x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}}.$

Define the set $D:=\\{\mu=(\mu_{1},\ldots,\mu_{n})\in\mathbb{Z}_{\geq 0}^{n}\
|\ 0\leq\mu_{i}\leq r_{i}-2\\}$. For any $\mu\in D$, we then can define the
cycles

(2.10)
$\Xi_{\mu}:=\Xi_{\mu_{1}}^{r_{1}}\times\cdots\times\Xi_{\mu_{n}}^{r_{n}}$

as products of those defined in the previous example. Then we have that

(2.11)
$\int_{\Xi_{\mu}}x^{\mu^{\prime}}e^{(x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}})/\hbar}\Omega=\delta_{\mu\mu^{\prime}}$

with $\Omega=dx_{1}\wedge\cdots\wedge dx_{n}$.

When $n=1$ and $W=x^{r}$, we will suppress notation and take
$\Xi_{j}:=\Xi^{r}_{j}$.

##  3\. Flat coordinates for Fermat polynomials

We restrict to the case where
$(X,W_{0})=(\mathbb{C}^{n},x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}})$. Note that the
elements $x^{\mu}$ for $\mu\in D$ are a basis for the Jacobian ring
$\mathbb{C}[x_{1},\ldots,x_{n}]/(\partial W_{0}/\partial x_{1},\ldots,\partial
W_{0}/\partial x_{n})$. Thus we may write a universal unfolding of $W_{0}$
parameterized by a germ $\mathcal{M}$ of the origin in the Jacobian ring,
viewed as a vector space. We use coordinates $y_{\mu}$ on $\mathcal{M}$,
parameterized by $\mu\in D$. Here $\\{y_{\mu}\\}$ is the dual basis to
$\\{x^{\mu}\\}$. The versal deformation $W$ for $W_{0}$ on $\mathcal{M}\times
X$ is then given by

$W=x_{1}^{r_{1}}+\cdots+x_{n}^{r_{n}}+\sum_{\mu\in D}y_{\mu}x^{\mu}.$

For a polynomial function $W:\mathbb{C}^{n}\rightarrow\mathbb{C}$ , we now
consider the following collection of oscillatory integrals, each of which we
can view as a formal power series in the variables $y_{\mu}$:

$\int_{\Xi_{\mu}}e^{W/\hbar}f\Omega=\sum_{j=-\infty}^{\infty}\varphi_{\mu,j}(\mathbf{y})\hbar^{-j}$

where $\varphi_{\mu,j}(\mathbf{y})\in\mathbb{C}[[y_{\mu}]]$.

The following is an oversimplification of Saito’s theory of primitive forms,
but is sufficient for our purposes:

######  Definition 3.1.

If $\varphi_{\mu,j}\equiv 0$ for all $j<0$ and
$\varphi_{\mu,0}=\delta_{\mu\mathbf{0}}$, then we say that $f\Omega$ is a
_primitive form_. Further, in this case, $\varphi_{\mu,1}$ form a set of
coordinates on the universal unfolding $\mathcal{M}$ called _flat
coordinates_.

######  Notation 3.2.

We typically will use the variables $t_{\mu}:=\varphi_{\mu,1}$ for the flat
coordinates.

######  Example 3.3.

Consider the Landau-Ginzburg model $(\mathbb{C},x^{4})$. In this case, we have
$M=\\{1,x,x^{2}\\}$ and we consider the versal deformation

$W=x^{4}+y_{2}x^{2}+y_{1}x+y_{0}.$

Using Lemma 2.4, we obtain the following expansions showing the lowest degree
terms in $\hbar^{-1}$:

(3.1) $\displaystyle\int_{\Xi_{0}}e^{W/\hbar}\Omega$
$\displaystyle=1+(y_{0}-\tfrac{1}{8}y_{2}^{2})\hbar^{-1}+O(\hbar^{-2})$
$\displaystyle\int_{\Xi_{1}}e^{W/\hbar}\Omega$
$\displaystyle=y_{1}\hbar^{-1}+O(\hbar^{-2})$
$\displaystyle\int_{\Xi_{2}}e^{W/\hbar}\Omega$
$\displaystyle=y_{2}\hbar^{-1}+O(\hbar^{-2})$

Thus in this case $\Omega$ is already a primitive form, and flat coordinates
are given by $t_{0}=y_{0}-y_{2}^{2}/8$, $t_{1}=y_{1}$, $t_{2}=y_{2}$. However,
we may then rewrite the universal unfolding using this change of variables,
obtaining

$W_{\mathbf{t}}=x^{4}+t_{2}x^{2}+t_{1}x+t_{0}+\tfrac{1}{8}t_{2}^{2}.$

Then

$\int_{\Xi_{d}}e^{W_{\mathbf{t}}/\hbar}\Omega=\delta_{0d}+t_{d}\hbar^{-1}+\dots$

for all $d\in\\{0,1,2\\}$.

######  Remark 3.4.

He, Li, Shen and Webb in [HLSW15] use Saito’s general framework for
constructing Frobenius manifold structures on the universal unfoldings of
potentials (see [Sai83a, Sai83b] or Section III.8 of [Man99]) to construct the
$B$-model Frobenius manifold for LG models $(\mathbb{C}^{n},W)$, where $W$ is
an invertible polynomial. Our definition of flat coordinates coincides with
the definition given in Equation (13) in §2.2.2 of [HLSW15], with different
notation.

##  4\. Open Landau-Ginzburg Mirror Symmetry in dimension 1

In the case of the mirror pair of Landau-Ginzburg models

$(\mathbb{C},x^{r},\mu_{r})\longleftrightarrow(\mathbb{C},x^{r}),$

the relevant open invariants for the Landau-Ginzburg model
$(\mathbb{C},x^{r},\mu_{r})$, i.e., open $r$-spin invariants, have been
already constructed in [BCT21, BCT18]. Recall the following theorem:

######  Theorem 4.1 (Theorem 1.2 of [BCT18]).

Take $k,l\geq 0$ and $0\leq a_{1},\ldots,a_{l}\leq r-1$. Suppose we consider
the open $r$-spin invariant associated to $k+1$ boundary marked points with
twist $r-2$ and $l$ internal marked points with twists $a_{1},\ldots a_{l}$.
Then we have

$\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}=\begin{cases}\frac{(k+l-1)!}{(-r)^{l-1}},&\text{if
$k\geq 0$ and $\frac{(r-2)k+2\sum_{i}a_{i}}{r}=2l+k-2$};\\\
0,&\text{otherwise}.\end{cases}$

######  Proposition 4.2.

If the open $r$-spin invariant
$\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}$
is nonzero, then $k\leq r$ and $l\leq r$. In particular, the number of nonzero
primary open $r$-spin invariants is finite.

###### Proof.

The constraint in Theorem 4.1

(4.1) $\frac{(r-2)k+2\sum_{i}a_{i}}{r}=2l+k-2$

requires that $2r=2(lr-\sum_{i}a_{i})+2k$. Since $lr-\sum_{i}a_{i}\geq l$,
there are at most $r+1$ boundary marked points and at most $r$ internal marked
points. ∎

We also have the following lemma:

######  Lemma 4.3.

Suppose $I=\\{a_{1},\ldots,a_{l}\\}$ is a nonempty multiset of internal
marking twists with $0\leq a_{i}\leq r-2$. Let $r(I)\in\\{0,\ldots,r-1\\}$ be
such that $\sum_{i}a_{i}\equiv r(I)\pmod{r}$. Then the open $r$-spin invariant
$\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}$
is nonzero if and only if $k=r(I)$ and $\sum_{i}a_{i}=r(I)+(l-1)r$.

###### Proof.

By Observation 2.1 of [BCT21], in order for the moduli space corresponding to
the open $r$-spin invariant
$\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}$
to be nonempty, the following integrality condition must hold:

$e:=\frac{2\sum_{i}a_{i}+k(r-2)}{r}=\frac{2(\sum_{i}a_{i}-k)}{r}+k\in\mathbb{Z}.$

Moreover, by Observation 2.10 of [BCT21], we must also have that $e\equiv
k\pmod{2}$. This implies that $\frac{2(\sum_{i}a_{i}-k)}{r}$ is an even
integer, hence $\sum_{i}a_{i}-k\in r\mathbb{Z}$, so $k\equiv r(I)\pmod{r}$.

By Proposition 4.2, we have $k\leq r$, so $k=r(I)$ unless $k=r$. If $k=r$,
then, by (4.1), we must have $\sum_{i}a_{i}=rl$ hence $l=0$. However, we are
assuming the set of internal markings is non-empty, so this does not occur.
Thus $k=r(I)$.

Now, suppose that $\sum_{i}a_{i}=r(I)+pr$ for some $p$. Then we have that

(4.2) $\frac{(r-2)k+2(r(I)+pr)}{r}=\frac{(r-2)r(I)+2r(I)+2pr}{r}=r(I)+2p.$

Note by Theorem 4.1 we must have $r(I)+2p=2l+k-2$ in order to have a nonzero
open $r$-spin invariant, but since $k=r(I)$ this implies that $2p=2l-2$, or
$p=l-1$ as desired.

The converse is a straightforward computation from Theorem 4.1. ∎

######  Corollary 4.4.

Consider a multiset $I$ as in Lemma 4.3 so that $\sum_{i}a_{i}=r(I)+(|I|-1)r$.

  1. (a)

For any $I^{\prime}\subset I$, we have $\sum_{i\in
I^{\prime}}a_{i}=r(I^{\prime})+(|I^{\prime}|-1)r$. Moreover, there exists a
unique $k_{I^{\prime}}\in\\{0,\ldots,r-1\\}$ so that $\left\langle\prod_{i\in
I^{\prime}}\tau_{0}^{a_{i}}\sigma^{k_{I^{\prime}}+1}\right\rangle^{\tfrac{1}{r},o}$
is nonzero.

  2. (b)

For a multiset partition $I=I_{1}\cup\cdots\cup I_{h}$, we have
$r(I_{1})+\cdots+r(I_{h})=r(I)+(h-1)r$.

###### Proof.

We first prove item (a). Note the condition that $\sum_{i\in
I^{\prime}}a_{i}=r(I^{\prime})+(|I^{\prime}|-1)r$ for any $I^{\prime}\subseteq
I$ is equivalent to $\sum_{i\in I^{\prime}}(r-a_{i})=r-r(I^{\prime})$. As
$r(I^{\prime})\in\\{0,\ldots,r-1\\}$, this is equivalent to $0<\sum_{i\in
I^{\prime}}(r-a_{i})\leq r$. The left-hand inequality is automatic as
$a_{i}<r$ for all $i$. We have $\sum_{i\in I}(r-a_{i})\leq r$ by assumption,
and hence $\sum_{i\in I^{\prime}}(r-a_{i})\leq r$ for any subset
$I^{\prime}\subseteq I$. Thus $\sum_{i\in
I^{\prime}}a_{i}=r(I^{\prime})+(|I^{\prime}|-1)r$.

For (b), note that $r(I)+(|I|-1)r=\sum_{i\in I}a_{i}=\sum_{j}\sum_{i\in
I_{j}}a_{i}=\sum_{j}(r(I_{j})+(|I_{j}|-1)r)=(|I|-h)r+\sum_{j}r(I_{j})$, from
which the result follows. ∎

For any $l\geq 0$, set $[l]:=\\{1,2,\ldots,l\\}$ and let $\mathscr{A}_{l}$ be
the set of multisets $\\{a_{1},\ldots,a_{l}\\}$ of integers where $0\leq
a_{i}\leq r-2$. If $A\in\mathscr{A}_{l}$, we define $\text{Aut}(A)$ to be the
group of permutations $\sigma:[l]\rightarrow[l]$ such that
$a_{i}=a_{\sigma(i)}$ for all $i\in[l]$. For example, the multiset
$A=\\{1,2,2,3,3,3\\}$ has $|\text{Aut}(A)|=1!\cdot 2!\cdot 3!=12$.

We are now ready to prove Theorem 1.1 which we repeat here:

######  Theorem 4.5.

With the potential

(4.3) $W_{\mathbf{t}}=\sum_{k\geq 0,l\geq
0}\sum_{\\{a_{i}\\}\in\mathscr{A}_{l}}(-1)^{l-1}\frac{\left\langle\prod_{i=1}^{l}\tau_{0}^{a_{i}}\sigma^{k+1}\right\rangle^{\tfrac{1}{r},o}}{k!|\mathrm{Aut}(\\{a_{i}\\})|}\left(\prod_{i=1}^{l}t_{a_{i}}\right)x^{k}\in\mathbb{C}[t_{0},\ldots,t_{r-2},x],$

$\Omega=\textup{d}x$ is a primitive form and $t_{0},\ldots,t_{r-2}$ are flat
coordinates. Further, if $\mathcal{I}_{d}$ denotes the ideal in
$\mathbb{C}[t_{0},\ldots,t_{r-2}]$ generated by all degree $d$ monomials, then

(4.4) $W_{\mathbf{t}}=x^{r}+t_{r-2}x^{r-2}+\ldots+t_{0}\mod\mathcal{I}_{2}.$

###### Proof.

Note that by Lemma 4.3, all terms of $W_{\mathbf{t}}$ arising from invariants
with internal markings are of degree at most $r-1$ in the variable $x$. On the
other hand, by Theorem 4.1, the only non-zero invariant
$\langle\sigma^{k+1}\rangle^{{1\over r},o}$ with no internal markings has
$k=r$ and the invariant is $-r!$. Thus $x^{r}$ appears in $W_{\mathbf{t}}$ and
is the only monomial in $W_{\mathbf{t}}$ with degree greater than or equal to
$r$. That is,

$W_{\mathbf{t}}\equiv x^{r}\mod\mathcal{I}_{1}.$

A similar argument from Theorem 4.1 for invariants with one internal marked
point then gives (4.4).

We now expand

(4.5)
$\int_{\Xi_{d}}e^{W_{\mathbf{t}}/\hbar}\Omega=\int_{\Xi_{d}}\left(\sum_{l\geq
0}\frac{(W_{\mathbf{t}}-x^{r})^{l}}{l!\hbar^{l}}\right)e^{x^{r}/\hbar}\Omega.$

Any monomial summand in the formal power series $\sum_{l\geq
1}\frac{(W-x^{r})^{l}}{l!\hbar^{l}}$ will be of the form $cx^{nr+k}\hbar^{-l}$
for some $c\in\mathbb{C}[t_{0},\ldots,t_{r-2}]$, $n\in\mathbb{Z}_{\geq 0}$ and
$k\in\\{0,\ldots,r-1\\}$. By Lemma 2.4, it thus follows that

$\int_{\Xi_{d}}cx^{nr+k}\hbar^{-l}e^{x^{r}/\hbar}\Omega=\tilde{c}\hbar^{n-l}\int_{{\Xi_{d}}}x^{k}e^{x^{r}/\hbar}\Omega=\tilde{c}\hbar^{n-l}\delta_{dk}$

for some $\tilde{c}\in\mathbb{C}[t_{0},\ldots,t_{r-2}]$. Since
$nr+k\leq(r-1)l$, we have that $n<l$. Thus, the highest power of $\hbar$
appearing in $\int_{\Xi_{d}}\left(\sum_{l\geq
1}\frac{(W_{\mathbf{t}}-x^{r})^{l}}{l!\hbar^{l}}\right)e^{x^{r}/\hbar}\Omega$
is $-1$. Hence

(4.6)
$\int_{\Xi_{d}}e^{W_{\mathbf{t}}/\hbar}\Omega=\delta_{0d}+\varphi_{d}\hbar^{-1}+\cdots$

where $\varphi_{d}\in\mathbb{C}[[t_{0},\ldots,t_{r-2}]]$, for all $d$. Note
that the $\delta_{0d}$ term comes from (2.7) and the $l=0$ term of the Taylor
expansion in (4.5).

We will prove that

(4.7) $\varphi_{d}\equiv t_{d}\bmod\mathcal{I}_{l}$

for all $l$, and hence $\varphi_{d}=t_{d}$. First, by (4.4), it is clear that
$\varphi_{d}=t_{d}\bmod\mathcal{I}_{2}$.

Consider a multiset $I=\\{a_{1},\ldots,a_{l}\\}$. Suppose that there is a
multiset partition $I=\bigcup_{j=1}^{h}I_{j}$ of $I$ such that there exists a
non-negative integer $k_{I_{j}}$ with $\langle\prod_{i\in
I_{j}}\tau^{a_{i}}\sigma^{k_{I_{j}}+1}\rangle^{\tfrac{1}{r},o}\neq 0$ for all
$j$. Then we obtain in the expansion of $(W-x^{r})^{h}$ a term of the form

$cx^{\sum_{j=1}^{h}k_{I_{j}}}\hbar^{-h}\prod_{j=1}^{h}\langle\prod_{i\in
I_{j}}\tau^{a_{i}}\sigma^{k_{I_{j}}+1}\rangle^{\tfrac{1}{r},o}$

where $c\in\mathbb{C}[t_{0},\ldots,t_{r-2}]$. Note that $k_{I_{j}}=r(I_{j})$
by Lemma 4.3, so the above term, after taking the integral, will contribute a
term of the form $\hbar^{-1}$ if and only if
$\sum_{j=1}^{h}r(I_{j})=r(I)+(h-1)r$. So let us assume this is the case, as we
are not interested in contributions of the form $\hbar^{-d}$, $d\geq 2$. Then,
again by Lemma 4.3 and this assumption,

$\sum_{i\in
I}a_{i}=\sum_{j=1}^{h}\left[r(I_{j})+(|I_{j}|-1)r\right]=r(I)+(|I|-1)r$

and the hypotheses of Corollary 4.4 hold.

On the way to proving that Equation (4.7) holds for all $l$, we will introduce
the following notation. Let $\mathrm{Part}_{h}(I)$ denote the unordered
partitions of $I$ into $h$ multisets. For example, with $h=2$,

$\mathrm{Part}_{h}(\\{1,1,2,2\\})=\bigg{\\{}\\{\\{1\\},\\{1,2,2\\}\\},\\{\\{2\\},\\{1,1,2\\}\\},\\{\\{1,1\\},\\{2,2\\}\\},\\{\\{1,2\\},\\{1,2\\}\\}\bigg{\\}}.$

On the other hand, we write $\mathrm{Part}_{h}([l])$ for partitions of $[l]$
into $h$ disjoint sets. There is a map

${\bf a}:\mathrm{Part}_{h}([l])\rightarrow\mathrm{Part}_{h}(I)$

given by ${\bf a}(\\{Q_{1},\ldots,Q_{h}\\})=\\{I_{1},\ldots,I_{h}\\}$, where
$I_{j}=\\{a_{i}\,|\,i\in Q_{j}\\}$. This map is surjective but not injective:
e.g., in the above example,

${\bf a}(\\{\\{1,3\\},\\{2,4\\}\\})={\bf
a}(\\{\\{1,4\\},\\{2,3\\}\\})=\\{\\{1,2\\},\\{1,2\\}\\}.$

For $\\{I_{1},\ldots,I_{h}\\}\in\mathrm{Part}_{h}(I)$, we write
$\text{Aut}(\\{I_{1},\ldots,I_{h}\\})$ for the set of permutations
$\sigma:[h]\rightarrow[h]$ with $I_{\sigma(i)}=I_{i}$.

With this notation, note that

(4.8) $|\text{Aut}(I)|=|{\bf
a}^{-1}(\\{I_{1},\ldots,I_{h}\\})|\cdot|\text{Aut}(\\{I_{1},\ldots,I_{h}\\})|\cdot\prod_{j=1}^{h}|\text{Aut}(I_{j})|.$

Using Equations (4.3) and (4.5), we can see that the summand in the integral
that corresponds to the coefficient of the monomial $t_{I}:=\prod_{a_{i}\in
I}t_{a_{i}}$ is:

$t_{I}\sum_{h=1}^{|I|}\frac{1}{h!}\sum_{\\{I_{1},\ldots,I_{h}\\}\in\mathrm{Part}_{h}(I)}\frac{h!}{\text{Aut}(\\{I_{1},\ldots,I_{h}\\})}x^{\sum_{j}r(I_{j})}\hbar^{-h}\prod_{j=1}^{h}(-1)^{|I_{j}|-1}\frac{\langle\prod_{i\in
I_{j}}\tau^{a_{i}}\sigma^{k_{I_{j}}+1}\rangle^{\tfrac{1}{r},o}}{k_{I_{j}}!|\text{Aut}(I_{j})|},$

where $k_{I_{j}}=r(I_{j})$ as above.

Using Corollary 4.4(b) combined with Lemma 2.4, we can see that the only
integral that will have a nonzero contribution is that corresponding to the
cycle $\Xi_{r(I)}$ and that the $\hbar^{-1}$ term will have a summand of the
form $\Lambda_{I}t_{I}$ where

(4.9)
$\Lambda_{I}:=\sum_{h=1}^{|I|}\frac{1}{h!}\sum_{\\{I_{1},\ldots,I_{h}\\}\in{\mathrm{Part}}_{h}(I)}\frac{h!}{\text{Aut}(\\{I_{1},\ldots,I_{h}\\})}(-1)^{h-1}\frac{\Gamma(\tfrac{1+\sum_{j}k_{I_{j}}}{r})}{\Gamma(\frac{1+k_{I}}{r})}\prod_{j=1}^{h}(-1)^{|I_{j}|-1}\frac{\langle\prod_{i\in
I_{j}}\tau^{a_{i}}\sigma^{k_{I_{j}}+1}\rangle^{\tfrac{1}{r},o}}{k_{I_{j}}!|\text{Aut}(I_{j})|}.$

We remind the reader that $k_{I}=r(I)$ by Lemma 4.3. By using the closed form
for the open $r$-spin invariants given Theorem 4.1 and cancelling signs, we
then have that

(4.10)
$\Lambda_{I}=\sum_{h=1}^{|I|}\frac{1}{h!}\sum_{\\{I_{1},\ldots,I_{h}\\}\in{\mathrm{Part}}_{h}(I)}\frac{h!}{\text{Aut}(\\{I_{1},\ldots,I_{h}\\})}(-1)^{h-1}\frac{\Gamma(\tfrac{1+\sum_{j}k_{I_{j}}}{r})}{\Gamma(\frac{1+k_{I}}{r})}\prod_{j=1}^{h}\frac{(k_{I_{j}}+|I_{j}|-1)!}{k_{I_{j}}!|\text{Aut}(I_{j})|r^{|I_{j}|-1}}$

By (4.8),

(4.11)
${1\over|\text{Aut}(\\{I_{1},\ldots,I_{h}\\})|\cdot\prod_{j=1}^{|I|}|\text{Aut}(I_{j})|}={|{\bf
a}^{-1}(\\{I_{1},\ldots,I_{h}\\})|\over|\text{Aut}(I)|}$

and hence by replacing the sum over elements of $\mathrm{Part}_{h}(I)$ with
the larger sum over elements of $\mathrm{Part}_{h}([l])$ and multiplying by
$|\text{Aut}(I)|$, we obtain the following simplification:

(4.12)
$|\text{Aut}(I)|\Lambda_{I}=\sum_{h=1}^{|I|}\sum_{\\{Q_{1},\ldots,Q_{h}\\}\in\mathrm{Part}_{h}([l])}(-1)^{h-1}\frac{\Gamma(\tfrac{1+\sum_{j}k_{I_{j}}}{r})}{\Gamma(\frac{1+k_{I}}{r})}\prod_{j=1}^{h}\frac{(k_{I_{j}}+|I_{j}|-1)!}{k_{I_{j}}!r^{|I_{j}|-1}},$

We claim that $|\text{Aut}(I)|\Lambda_{I}$ vanishes. To prove this, we will
instead first view the twists $a_{i}$ as formal variables and then change
coordinates to new formal variables $b_{i}:=r-a_{i}$. We set the notation
$b_{A}:=\sum_{i\in A}b_{i}$. Note that, since $\sum_{i\in
A}a_{i}=r(A)+(|A|-1)r$ for all subsets $A\subset I$, $b_{A}=r-r(A)$ and the
number $b_{A}$ is in the set $\\{0,\ldots,r\\}$.

Note now that we can rewrite $|\text{Aut}(I)|\Lambda_{I}$ in the following
way:

(4.13)
$|\text{Aut}(I)|\Lambda_{I}=\sum_{h=1}^{|I|}\sum_{\\{Q_{1},\ldots,Q_{h}\\}\in\mathrm{Part}_{h}([l])}(-1)^{h-1}\frac{\Gamma(h+\tfrac{1-b_{I}}{r})}{\Gamma(\frac{1+r-b_{I}}{r})}\prod_{j=1}^{h}\frac{(r-1+|I_{j}|-b_{I_{j}})!}{(r-b_{I_{j}})!r^{|I_{j}|-1}}.$

This is further rearranged as:

(4.14) $\displaystyle|\text{Aut}(I)|\Lambda_{I}$
$\displaystyle=\sum_{h=1}^{|I|}\frac{(-1)^{h-1}}{r^{l-1}}\sum_{\\{Q_{1},\ldots,Q_{h}\\}\in\mathrm{Part}_{h}([l])}\left(\prod_{j=1}^{h-1}(jr+1-b_{I})\right)\left(\prod_{j=1}^{h}\frac{(r-1+|I_{j}|-b_{I_{j}})!}{(r-b_{I_{j}})!}\right)$
$\displaystyle=\frac{1}{r^{l-1}}\sum_{h=1}^{|I|}(-1)^{h-1}\sum_{\\{Q_{1},\ldots,Q_{h}\\}\in\mathrm{Part}_{h}([l])}\left(\prod_{j=1}^{h-1}(jr+1-b_{I})\right)\left(\prod_{j=1}^{h}\prod_{i=1}^{|I_{j}|-1}(r-b_{I_{j}}+i)\right).$

From (4.14), it is transparent that $|\text{Aut}(I)|\Lambda_{I}$ is a
polynomial in the variables $b_{i}$. If we show that
$|\text{Aut}(I)|\Lambda_{I}$ is the zero polynomial in the variables $b_{i}$,
then this implies that $\Lambda_{I}$ vanishes. Therefore, the theorem reduces
to showing that $|\text{Aut}(I)|\Lambda_{I}$ is the zero polynomial for any
$I$ with cardinality at least two. We proceed by induction.

Let $l=2$ and consider the multiset $I=\\{a_{1},a_{2}\\}$. Then
$\mathrm{Part}_{1}([l])=\\{I\\}$ and $\mathrm{Part}_{2}([l])$ only contains
the one (unordered) partition $\\{\\{a_{1}\\},\\{a_{2}\\}\\}$. In this case,
(4.14) reduces to:

$|\text{Aut}(I)|\Lambda_{I}=\frac{1}{r}(r-b_{I}+1)-\frac{1}{r}(r+1-b_{I})=0.$

Moving to the case where $I$ has cardinality greater than two, by the
induction hypothesis, we know for any subset $I^{\prime}\subset I$ of
cardinality above $1$ that $\Lambda_{I^{\prime}}=0$ viewed as a polynomial in
the variables $b_{i}$.

Note that $|\text{Aut}(I)|\Lambda_{I}$ is a symmetric polynomial of degree at
most $l-1$ in $l$ variables. Recall that for an arbitrary polynomial $P_{l}$
of degree at most $l-1$ in $l$ variables $b_{i}$ that if
$P_{l}|_{\\{b_{i}=0\\}}=0$ for all $i$ then the polynomial $P_{l}$ is the zero
polynomial. Since $\Lambda_{I}$ is symmetric, it suffices to show that
$\Lambda_{I}|_{b_{l}=0}=0$. Note that $\Lambda_{I}|_{b_{l}=0}$ is a polynomial
in $l-1$ variables, and we will now work to relate it to the polynomial
$\Lambda_{I\setminus\\{a_{l}\\}}$.

Take $\mathrm{Part}([l])=\bigcup_{h=1}^{l}\mathrm{Part}_{h}([l])$ and let
$\mathrm{Part}([l-1])=\bigcup_{h=1}^{l-1}\mathrm{Part}_{h}([l-1])$. We then
can define a surjective function
$f:\mathrm{Part}([l])\rightarrow\mathrm{Part}([l-1])$ by:

(4.15)
$f(\\{Q_{1},\ldots,Q_{h}\\})=\begin{cases}\\{Q_{1},\ldots,Q_{j}\setminus\\{l\\},\ldots,Q_{h}\\}&\text{if
$l\in Q_{j}$ and $|Q_{j}|>1$};\\\
\\{Q_{1},\ldots,\widehat{Q_{j}},\ldots,Q_{h}\\}&\text{if
$Q_{j}=\\{l\\}$}.\end{cases}$

We will denote by $\mathrm{Cont}(\\{Q_{1},\ldots,Q_{h}\\})$ the polynomial

$\mathrm{Cont}(\\{Q_{1},\ldots,Q_{h}\\}):=(-1)^{h-1}\left(\prod_{j=1}^{h-1}(jr+1-b_{I})\right)\left(\prod_{j=1}^{h}\prod_{i=1}^{|I_{j}|-1}r-b_{I_{j}}+i\right),$

which is the contribution to $\Lambda_{I}$ given by the partition
$\\{Q_{1},\ldots,Q_{h}\\}$, where

$\\{I_{1},\ldots,I_{h}\\}={\bf a}(\\{Q_{1},\ldots,Q_{h}\\}).$

We compare the contribution $\mathrm{Cont}(Q)$ for $Q\in\mathrm{Part}([l-1])$
with the sum $\sum_{Q^{\prime}\in f^{-1}(Q)}\mathrm{Cont}(Q^{\prime})$. We
claim:

$\sum_{Q^{\prime}\in
f^{-1}(Q)}\mathrm{Cont}(Q^{\prime})|_{b_{l}=0}=(l-2)\mathrm{Cont}(Q).$

Indeed, take $Q=\\{Q_{1},\ldots,Q_{h}\\}\in\mathrm{Part}_{h}([l-1])$, with
${\bf a}(Q)=\\{I_{1},\ldots,I_{h}\\}$. Its inverse image under $f$ consists of
exactly $h+1$ partitions, namely $Q^{0}=\\{Q_{1},\ldots,Q_{h},\\{a_{l}\\}\\}$
and $Q^{j}=\\{Q_{1},\ldots,Q_{j}\cup\\{l\\},\ldots,Q_{h}\\}$ for $1\leq j\leq
h$. Note that

(4.16) $\displaystyle\mathrm{Cont}(Q^{0})|_{b_{l}=0}$
$\displaystyle=-(hr+1-b_{I\setminus\\{a_{l}\\}})\mathrm{Cont}(Q)$
$\displaystyle\mathrm{Cont}(Q^{j})|_{b_{l}=0}$
$\displaystyle=(r-b_{I_{j}}+|Q_{j}|)\mathrm{Cont}(Q)\text{ for $1\leq j\leq
h$}.$

Hence, by using that $\sum_{j=1}^{h}|Q_{j}|=l-1$, we see that

$\sum_{Q^{\prime}\in
f^{-1}(Q)}\mathrm{Cont}(Q^{\prime})|_{b_{l}=0}=\sum_{j=0}^{h}\mathrm{Cont}(Q^{j})|_{b_{l}=0}=(l-2)\mathrm{Cont}(Q).$

We now can use this recursion with Equation (4.14), the surjectivity of $f$,
and the induction hypothesis to see that

(4.17) $\displaystyle(|\text{Aut}(I)|\Lambda_{I})|_{b_{l}=0}$
$\displaystyle=\frac{1}{r^{l-1}}\sum_{Q\in\mathrm{Part}([l])}\mathrm{Cont}(Q)|_{b_{l}=0}$
$\displaystyle=\frac{1}{r^{l-1}}\sum_{Q\in\mathrm{Part}([l-1])}(l-2)\mathrm{Cont}(Q)$
$\displaystyle=\frac{1}{r^{l-1}}\left(r^{l-2}(l-2)|\text{Aut}(I\setminus\\{a_{l}\\})|\Lambda_{I\setminus\\{a_{l}\\}}\right)$
$\displaystyle=0$

This consequently implies that $\Lambda_{I}=0$ for all $|I|=l\geq 2$.
Therefore, $\Omega$ is a primitive form and the coordinates
$t_{0},\ldots,t_{r-2}$ are flat coordinates. ∎

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# Spin squeezing from bilinear spin-spin interactions: two simple theorems

Tommaso Roscilde, Fabio Mezzacapo and Tommaso Comparin  Univ Lyon, Ens de
Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

We demonstrate two simple theorems about squeezing induced by bilinear spin-
spin interactions that conserve spin parity – including a vast majority of
quantum spin models implemented by state-of-the-art quantum simulators. In
particular we show that squeezing captures the first form of quantum
correlations which are produced: 1) at equilibrium, by adiabatically turning
on the spin-spin interactions starting from a factorized state aligned with an
external, arbitrary field; 2) away from equilibrium, by evolving unitarily the
same state with the interacting Hamiltonian.

##  I Introduction

Entanglement Horodecki et al. (2009) is the fundamental trait of quantum many-
body states, making them at once very hard to efficiently store in a classical
computer; as well as very promising for possible applications in _e.g._
quantum information processing Nielsen and Chuang (2010) and quantum metrology
Pezzè et al. (2018). The controlled preparation and certification of entangled
quantum many-body states is a central challenge of fundamental research, and
it is the pre-requisite for many future quantum technologies Acín et al.
(2018).

Among the many forms of entangled states for ensembles of $S=1/2$ spins or
qubits, spin-squeezed states Ma et al. (2011) occupy a special position, as
they are characterized uniquely by the properties of the collective spin
variable. Introducing the collective spin operator
${\bm{J}}=\sum_{i=1}^{N}{\bm{S}}_{i}$ of an ensemble of $N$ qubits, spin
squeezed states have the property that the spin squeezing parameter
$\xi_{R}^{2}<1$ Wineland et al. (1994), where

$\xi_{R}^{2}=\frac{N\min_{\perp}{\rm
Var}(J^{\perp})}{|\langle\bm{J}\rangle|^{2}}~{}.$ (1)

Here the minimization is taken over the spin components transverse to the
average spin direction $\langle\bm{J}\rangle$. The presence of spin squeezing
$\xi_{R}^{2}<1$ acts as an entanglement witness, as all separable states have
$\xi_{R}^{2}\geq 1$ Sørensen et al. (2001). The metrological use of spin-
squeezed states is also straightforward, since spin squeezing allows one to
surpass the standard quantum limit for phase estimation Pezzè et al. (2018)
within Ramsey inteferometry.

The paradigmatic scheme for the generation of spin-squeezed states via
Hamiltonian dynamics is offered by the one-axis twisting (OAT) dynamics
Kitagawa and Ueda (1993), namely by the evolution of a coherent spin state
$|{\rm CSS}\rangle=\otimes_{i=1}^{N}|\uparrow_{z}\rangle_{i}$ (where
$|\uparrow_{z}\rangle$ is the $+1/2$ eigenstate of the $S^{z}$ operator) with
the collective-spin Hamiltonian ${\cal H}_{\rm
OAT}=\frac{\chi}{N}(J^{x})^{2}$, which is equivalent to an Ising Hamiltonian
with infinite-range couplings. Such an Hamiltonian has been realized in
seminal experiments on Bose-Einstein condensates Estève et al. (2008); Riedel
et al. (2010); an approximation to the OAT dynamics is offered by long-range
Ising interactions realized _e.g._ in ensembles of trapped ions Bohnet et al.
(2016). A second paradigmatic scheme is offered by the two-axis-
countertwisting (TACT) dynamics Kitagawa and Ueda (1993), generated by the
Hamiltonian ${\cal H}_{\rm
TACT}=\frac{\chi}{iN}\left[(J^{+})^{2}-(J^{-})^{2}\right]$, which still awaits
an experimental realization.

In this paper we show that spin squeezing is in fact the early form of quantum
correlations generated by a large class of bilinear quantum spin Hamiltonians
relevant for quantum simulation platforms – of which the OAT and TACT
Hamiltonians and variants thereof are just special cases. The same
Hamiltonians can produce spin squeezing either via non-equilibrium dynamics,
or via adiabatic dynamics starting from a Hamiltonian which stabilizes the
coherent spin state (CSS) as ground state. Our findings apply to bilinear spin
Hamiltonians that conserve parity in the computational basis of simultaneous
eigenstates of the $S_{i}^{z}$ operators, namely that commute with the
operator ${\cal P}=\prod_{i=1}^{N}(2S_{i}^{z})$. Such Hamiltonians take the
general form

${\cal H}={\cal H}_{\perp}+{\cal H}_{z}$ (2)

where

$\displaystyle{\cal
H}_{\perp}=\sum_{ij}\left(K_{ij}S_{i}^{+}S_{j}^{+}+J_{ij}S_{i}^{+}S_{j}^{-}+{\rm
h.c.}\right)$ (3)

in which $J_{ij}$ and $K_{ij}$ are generic, complex-valued Hermitian matrices
with finite matrix elements. Moreover ${\cal H}_{z}$ is an arbitrary
Hamiltonian term which is diagonal in the computational basis.

A central role in the discussion will be played by the function

${\cal
K}(\\{\theta_{i}\\})=\sum_{ij}e^{i(\theta_{i}+\theta_{j})}K_{ij}=R(\\{\theta_{i}\\})+iI(\\{\theta_{i}\\})$
(4)

whose real and imaginary parts are given by the real functions $R$ and $I$
respectively.

Our findings can be cast in the form of two simple _theorems_ , valid for
Hamiltonians of the type of Eq. (3):

  * •

_Theorem 1_. Consider the initial state $|\psi(0)\rangle=|{\rm CSS}\rangle$,
and the infinitesimally evolved state

$|\psi(dt)\rangle=(\mathbb{1}-i{\cal H}dt)|\psi(0)\rangle+{\cal
O}(dt)^{2}~{}.$ (5)

The state $|\psi(dt)\rangle$ exhibits spin squeezing iff

$I_{\rm max}=\max_{\\{\theta_{i}\\}}I(\\{\theta_{i}\\})>0~{},$ (6)

with squeezing parameter

$\xi_{R}^{2}[dt]=1-\frac{4I_{\max}}{N}~{}dt+{\cal O}(dt)^{2}~{}.$ (7)

  * •

_Theorem 2_. Consider the perturbed Hamiltonian

${\cal H}^{\prime}(\lambda)=\lambda{\cal H}-H\sum_{i}S_{i}^{z}$ (8)

and be $|\psi_{0}(\lambda)\rangle$ its ground state. The state
$|\psi_{0}(d\lambda)\rangle$ exhibits spin squeezing iff

$R_{\rm max}=\max_{\\{\theta_{i}\\}}R(\\{\theta_{i}\\})>0~{},$ (9)

with squeezing parameter

$\xi_{R}^{2}[d\lambda]=1-\frac{2R_{\max}}{HN}~{}d\lambda+{\cal
O}(d\lambda)^{2}~{}.$ (10)

Recent works Wang et al. (2001); Hazzard et al. (2013, 2014); Foss-Feig et al.
(2016); Perlin et al. (2020); Comparin et al. (2021) have pointed out the
robustness of spin-squeezing dynamics in models that deviate from one-axis-
twisting Hamiltonians, and that strong squeezing can be found at Ising quantum
critical points Frérot and Roscilde (2018). Our theorems include and
generalize the Hamiltonians considered in these works. They apply in
particular to spin models relevant for quantum simulations, such as the XYZ
models with interactions of arbitrary range – realized using trapped ions
Monroe et al. (2021), Rydberg atoms Browaeys and Lahaye (2020), spinful
ultracold atoms Fukuhara et al. (2013, 2015); Mazurenko et al. (2017); Jepsen
et al. (2020), superconducting circuits Kjaergaard et al. (2020), etc.
Therefore all these platforms can act as sources of spin-squeezed states,
either adiabatically or dynamically.

##  II Proof of Theorem 1

It is easy to see that the average collective-spin vector
$\langle\psi(t)|{\bm{J}}|\psi(t)\rangle$ has (at most) only one finite
component, namely the $z$-component $\langle J^{z}\rangle[t]$. Indeed the
bilinear interactions cannot generate any net rotation of the collective spin
orientation with respect to the initial state.

The most general collective spin component transverse to the $z$ axis can be
built by defining locally the $x$ and $y$ axes, namely

$\displaystyle S_{i}^{x^{\prime}}$ $\displaystyle=$
$\displaystyle\cos\theta_{i}S_{i}^{x}+\sin\theta_{i}S_{i}^{y}=\frac{1}{2}\left(e^{-i\theta_{i}}S_{i}^{+}+e^{i\theta_{i}}S_{i}^{-}\right)$
$\displaystyle S_{i}^{y^{\prime}}$ $\displaystyle=$
$\displaystyle-\sin\theta_{i}S_{i}^{x}+\cos\theta_{i}S_{i}^{y}$ (11)

From there, we can define (without loss of generality)
$J_{\perp}(\\{\theta_{i}\\})=\sum_{i}S_{i}^{x^{\prime}}$. The minimization
involved in the definition of the squeezing parameter is then performed with
respect to the local angles $\\{\theta_{i}\\}$ . We observe that

$\displaystyle\langle\psi(dt)|J^{z}|\psi(dt)\rangle=$ (12) $\displaystyle=$
$\displaystyle\frac{N}{2}-i\langle{\rm CSS}|[J^{z},{\cal H}]|{\rm CSS}\rangle
dt+{\cal O}(dt)^{2}$ $\displaystyle=$ $\displaystyle\frac{N}{2}+{\cal
O}(dt)^{2}$

where the linear terms in $dt$ vanish because $|{\rm CSS}\rangle$ is an
eigenstate of $J^{z}$, so that the expectation value of the commutator of
$J^{z}$ with any operator vanishes on that state.

On the other hand, since $\langle J_{\perp}\rangle=0$ at all times, we have
that

$\displaystyle{\rm Var}(J_{\perp})[dt]$ $\displaystyle=$
$\displaystyle\frac{N}{4}-i\langle{\rm CSS}|[J_{\perp}^{2},{\cal H}]|{\rm
CSS}\rangle dt+{\cal O}(dt)^{2}~{}.$ (13)

Moreover we can expand the expression of $J_{\perp}^{2}$ as

$\displaystyle J_{\perp}^{2}=\frac{N}{4}$ $\displaystyle+$
$\displaystyle\frac{1}{4}\sum_{l\neq
m}\Big{(}e^{-i(\theta_{l}+\theta_{m})}S_{l}^{+}S_{m}^{+}$ (14)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+e^{-i(\theta_{l}-\theta_{m})}S_{l}^{+}S_{m}^{-}+{\rm
h.c.}\Big{)}$

and take the commutator $[J_{\perp}^{2},{\cal H}]$ piecewise in terms of pairs
of spin operators. When doing so, it is easy to observe that, in the
expectation value on the $|{\rm CSS}\rangle$ state, only the latter
commutators between products of two spin operators are non zero:

$\displaystyle\langle{\rm CSS}|[S^{+}_{l}S^{+}_{m},S_{i}^{-}S_{j}^{-}]|{\rm
CSS}\rangle=\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}$ (15)
$\displaystyle=$ $\displaystyle-\langle{\rm
CSS}|[S^{-}_{l}S^{-}_{m},S_{i}^{+}S_{j}^{+}]|{\rm CSS}\rangle$

All the other commutators vanish on average over the initial state because of
the action of a $S^{+}$ operator before that of a $S^{-}$ operator on the same
site. Therefore we can conclude that $\langle{\rm CSS}|[J_{\perp}^{2},{\cal
H}]|{\rm CSS}\rangle=-iI(\\{\theta_{i}\\})$ where $I$ is defined as in Eq.
(4). The squeezing parameter (before minimization) takes therefore the form

$\displaystyle\tilde{\xi}_{R}^{2}[dt]$ $\displaystyle=$
$\displaystyle\frac{N^{2}/4-NI(\\{\theta_{i}\\})dt+{\cal
O}(dt)^{2}}{N^{2}/4+{\cal O}(dt)^{2}}$ (16) $\displaystyle=$ $\displaystyle
1-\frac{4I(\\{\theta_{i}\\})}{N}dt+{\cal O}(dt)^{2}~{}.$

Minimizing the expression over the angles $\theta_{i}$ amounts to maximizing
the $I(\\{\theta_{i}\\})$ function. Squeezing is therefore present in the
state iff $I_{\rm max}=\max_{\\{\theta_{i}\\}}I(\\{\theta_{i}\\})>0$.

##  III Discussion of theorem 1

It would be useful to understand for which Hamiltonians of the class of Eq.
(3) the condition of Eq. (6) is satisfied. This class is actually very broad,
and it encompasses most of the relevant models for quantum simulators. A most
general class of Hamiltonians implemented in the experiments is provided by
the XYZ model with arbitrary interactions

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\sum_{ij}\left({\cal
J}^{x}_{ij}S_{i}^{x}S_{j}^{x}+{\cal J}^{y}_{ij}S_{i}^{y}S_{j}^{y}+{\cal
J}^{z}_{ij}S_{i}^{z}S_{j}^{z}\right)$ (17) $\displaystyle=$
$\displaystyle\sum_{ij}\left(\frac{{\cal J}^{x}_{ij}-{\cal
J}^{y}_{ij}}{4}~{}S_{i}^{+}S_{j}^{+}+\frac{{\cal J}^{x}_{ij}+{\cal
J}^{y}_{ij}}{4}~{}S_{i}^{+}S_{j}^{-}+{\rm h.c.}\right)$ $\displaystyle+$
$\displaystyle\sum_{ij}{\cal J}^{z}_{ij}~{}S_{i}^{z}S_{j}^{z}$

so that $K_{ij}=({\cal J}^{x}_{ij}-{\cal J}^{y}_{ij})/4\in\mathbb{R}$. We do
not require any spatial symmetry nor condition on the range of the interaction
parameters ${\cal J}_{ij}^{\alpha}$ ($\alpha=x,y,z$).

For this class of models one can easily prove that the condition of Eq. (6)
can be fulfilled provided that

$R_{0}=R(\\{\theta_{i}=0\\})=\sum_{ij}K_{ij}\neq 0~{}.$ (18)

This is a very generic condition unless the Hamiltonian is especially fine
tuned to have couplings summing up to zero. The condition of Eq. (18) makes it
also clear that the Hamiltonian cannot be $U(1)$ symmetric with symmetry axis
along $z$ (in which case one would have ${\cal J}^{x}_{ij}={\cal
J}^{y}_{ij}$), but this is obvious from the start, because otherwise the
initial state $|{\rm CSS}\rangle$ would be an Hamiltonian eigenstate.

Indeed, once the condition Eq. (18) is satisfied, one can choose angles
$\theta_{i}$ such that

$\theta_{i}=\frac{\pi}{4}{\rm sign}(R_{0})~{}~{}~{}~{}\forall i.$ (19)

Since $\exp[i\frac{\pi}{2}{\rm sign}(R_{0})]=i~{}{\rm sign}(R_{0})$, we have
that

$I\left(\left\\{\frac{\pi}{4}{\rm sign}(R_{0})\right\\}\right)=|R_{0}|$ (20)

which proves that $I$ can be positive, so that _a fortiori_ $I_{\rm max}$ will
be positive as well. Even if the condition of Eq. (18) were not satisfied, it
is conceivable that it exists a choice of the angles $\theta_{i}$ less trivial
than the one indicated above, still leading to the conclusion that $I_{\rm
max}>0$.

Proving the validity of the condition Eq. (6) for generic Hamiltonians,
including ones with complex couplings, might be harder; but the theorem
ensures that the condition Eq. (6) is necessary and sufficient for squeezing
to be generated. For instance, if ${\rm Im}(K_{ij})\neq 0$ for some pairs
$(ij)$, then we can require that

$I_{0}=I(\\{\theta_{i}=0\\})=\sum_{ij}{\rm Im}(K_{ij})\neq 0$ (21)

– once again, a generic property away from fine tuning. Under this condition,
the uniform choice of angles $\theta_{i}=0$ if ${\rm sign}(I_{0})=1$ and
$\theta_{i}=\pi/2$ if ${\rm sign}(I_{0})=-1$ guarantees that
$I(\\{\theta_{i}\\})>0$, so that _a fortiori_ $I_{\rm max}>0$.

##  IV Proof of Theorem 2

The state $|\psi_{0}(d\lambda)\rangle$ can be obtained via first-order
perturbation theory Landau and Lifshitz (1981)

$|\psi_{0}(d\lambda)\rangle=|{\rm CSS}\rangle-d\lambda\sum_{n\neq{\rm
CSS}}\frac{\langle n|{\cal H}|{\rm
CSS}\rangle}{E_{n}+H\frac{N}{2}}|n\rangle+{\cal O}(d\lambda)^{2}$ (22)

where $|n\rangle$ are the eigenstates of the unperturbed Hamiltonian
$-H\sum_{i}S_{i}^{z}$ with energies $E_{n}$. Given that
$S_{i}^{+}S_{j}^{-}|{\rm CSS}\rangle=S_{i}^{-}S_{j}^{+}|{\rm
CSS}\rangle=S_{i}^{+}S_{j}^{+}|{\rm CSS}\rangle=0$, we have that only the
$S_{i}^{-}S_{j}^{-}$ in the Hamiltonian Eq. (3) has a non-vanishing action on
the CSS, so that

$|\psi_{0}(d\lambda)\rangle=|{\rm
CSS}\rangle-\frac{d\lambda}{2H}\sum_{ij}K^{*}_{ij}|{\rm CSS}_{ij}\rangle+{\cal
O}(d\lambda)^{2}$ (23)

where $|{\rm CSS}_{ij}\rangle=S_{i}^{-}S_{j}^{-}|{\rm CSS}\rangle$.

For this state we have that

$\langle
J^{z}\rangle[d\lambda]=\langle\psi_{0}(d\lambda)|J^{z}|\psi_{0}(d\lambda)\rangle=\frac{N}{2}+{\cal
O}(d\lambda)^{2}$ (24)

since the $J^{z}$ operator is diagonal in the unperturbed Hamiltonian
eigenbasis and the state perturbation in Eq. (22) is orthogonal to the CSS.

On the other hand, using the same definition of $J_{\rm perp}$ as in Theorem
1, we have that $\langle J_{\perp}\rangle[d\lambda]=0$ given that
$|\psi_{0}(d\lambda)\rangle$ contains only states $|n\rangle$ with the same
parity as that of the CSS, while $J_{\perp}$ is by definition a parity-
changing operator. Therefore

$\displaystyle{\rm Var}(J_{\perp})[d\lambda]$ $\displaystyle=$
$\displaystyle\frac{N}{4}-\frac{d\lambda}{H}~{}{\rm
Re}\left(\sum_{ij}K_{ij}\langle{\rm CSS}_{ij}|J_{\perp}^{2}|{\rm
CSS}\rangle\right)$ (25) $\displaystyle+$ $\displaystyle{\cal
O}(d\lambda)^{2}$

Given that

$\langle{\rm CSS}_{ij}|J_{\perp}^{2}|{\rm
CSS}\rangle=\frac{e^{i(\theta_{i}+\theta_{j})}}{2}$ (26)

we conclude that

$\displaystyle{\rm Var}(J_{\perp})[d\lambda]$ $\displaystyle=$
$\displaystyle\frac{N}{4}-\frac{d\lambda}{2H}R(\\{\theta_{i}\\})+{\cal
O}(d\lambda)^{2}$ (27)

where $R$ is defined as in Eq. (4). Given that the squeezing parameter (before
minimization) takes the form

$\tilde{\xi}_{R}^{2}=1-\frac{2R(\\{\theta_{i}\\})}{HN}+{\cal
O}(d\lambda)^{2}~{},$ (28)

we can conclude that the perturbed ground state $|\psi_{0}(d\lambda)\rangle$
features squeezing iff $R_{\rm
max}=\max_{\\{\theta_{i}\\}}R(\\{\theta_{i}\\})>0$.

##  V Discussion of theorem 2

Similarly to the discussion of Sec. III, we can immediately prove that the
theorem applies to the XYZ Hamiltonians of Eq. (17) under the condition of Eq.
(18) on the Hamiltonian couplings, which turns out to imply the condition of
Eq. (9). Indeed, choosing $\theta_{i}=0$ if ${\rm sign}(R_{0})=1$ and
$\theta_{i}=\pi/2$ if ${\rm sign}(R_{0})=-1$, we obtain that $R=|R_{0}|>0$, so
that $R_{\rm max}$ will be positive _a fortiori_. Similarly to what discussed
in Sec. III, the satisfaction of the condition of Eq. (18) is sufficient to
conclude that the condition of validity of the theorem (Eq. (9)) is satisfied,
but it is not at all necessary – and it is conceivable to find a combination
of angles $\theta_{i}$ less trivial than the one indicated above, and which
still leads to the conclusion that $R_{\rm max}>0$.

Similarly to the case of Theorem 1, Theorem 2 can also apply to more general
parity-conserving Hamiltonians than those of the XYZ class by exhibiting a set
of angles that leads to the satisfaction of the condition Eq. (9). If ${\rm
Im}(K_{ij})\neq 0$ for some pairs $(ij)$, then we can require the condition
Eq. (21) to be satisfied, which immediately implies that the choice of angles
$\theta_{i}=-\frac{\pi}{4}{\rm sign}(I_{0})$ leads to $R=|I_{0}|$, proving
that _a fortiori_ $R_{\rm max}>0$.

##  VI Generalization to different initial states

The above theorems can be straightforwardly generalized to arbitrary initial
states $|\sigma_{1}\sigma_{2},....\sigma_{N}\rangle$ in the computational
basis, where $\sigma=+1,-1$ (corresponding to $\uparrow_{z},\downarrow_{z}$
respectively). For that purpose, we can introduce the modified collective spin
${\bm{J}}^{\prime}=(J^{x^{\prime}},J^{y^{\prime}},J^{z^{\prime}})$ where

$J^{x^{\prime}}=\sum_{i}S_{i}^{x^{\prime}}~{}~{}~{}J^{y^{\prime}}=\sum_{i}\sigma_{i}S_{i}^{y^{\prime}}~{}~{}~{}J^{z^{\prime}}=\sum_{i}\sigma_{i}S_{i}^{z}$
(29)

Introducing the unitary transformation $U=\prod_{i}\exp(i(1-\sigma_{i})\pi
S_{i}^{x}/2)$ we obtain that
$U|\sigma_{1}\sigma_{2},....\sigma_{N}\rangle=|{\rm CSS}\rangle$, and
$UJ^{z^{\prime}}U^{\dagger}=J^{z}$. Moreover the transformed Hamiltonian
${\cal H}_{U}=U{\cal H}U^{\dagger}$ maintains the same property of parity
conservation – the only effect of the transformation is that of flipping the
signs of some of the couplings. Therefore the theorems apply to squeezing of
the modified collective spin ${\bm{J}}^{\prime}$ provided that the conditions
Eq. (6) and Eq. (9) apply to the couplings $K_{ij}$ after the sign changes
imposed by the unitary transformation.

##  VII Conclusions

We have demonstrated via two theorems that, for parity-conserving bilinear
Hamiltonians, the first form of entanglement induced by the unitary evolution
of a (collinear) coherent spin state of $S=1/2$ spins, or by an adiabatic
evolution of the same state induced by the Hamiltonian as a perturbation, is
captured by spin squeezing. This result suggests that spin squeezing is an
ubiquitous form of quantum correlation, which can be induced by using nearly
all current platforms of quantum simulation of quantum spin Hamiltonians.
Squeezing appears linearly in time or in the Hamiltonian perturbation, and the
rate of buildup of squeezing is uniquely determined by an appropriate sum of
the Hamiltonian couplings weighted by phase factors. What happens to the
dynamical or adiabatic buildup of squeezing after the initial linear growth
depends on further details of the Hamiltonian interactions, such as the number
of spatial dimensions in which the Hamiltonian is defined, and the spatial
decay of the interactions. These details will dictate whether squeezing – as
quantified by the inverse squeezing parameter $\xi_{R}^{-2}$ – grows up to a
value which scales with system size (as in the case of the dynamics induced by
one-axis-twisting or two-axis-countertwisting Hamiltonians Kitagawa and Ueda
(1993), or in the case of the adiabatic ground state of Ising Hamiltonians
exhibiting quantum-critical points Frérot and Roscilde (2018)); or whether it
saturates instead to a size-independent value.

###### Acknowledgements.

_Acknowledgements._ We acknowledge useful discussions with B. Laburthe, M.
Robert de Saint Vincent and L. Vernac. This work is supported by ANR (“EELS”
project) and by QuantERA (“MAQS” project).

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