$\displaystyle\lesssim\int_{0}^{t}\Big{(}\frac{n}{N}\|v-v^{\varepsilon}\|_{L^{3/2}}+n\|w-w^{\varepsilon}\|_{C_{b}}\Big{)}\|\psi_{n}(s)\|_{L_{\bm{X}}^{2}\otimes
H^{1}(\mathbb{R}^{3n})}\|\psi_{n}^{\varepsilon}(s)\|_{L_{\bm{X}}^{2}\otimes
H^{1}(\mathbb{R}^{3n})}\mathrm{d}s\,.$

Here, we have used
$|\left\langle\varphi_{1},v\varphi_{2}\right\rangle_{L^{2}(\mathbb{R}^{3})}|\lesssim\|v\|_{L^{3/2}}\|\varphi_{1}\|_{L^{6}}\|\varphi_{2}\|_{L^{6}}\lesssim\|v\|_{L^{3/2}}\|\varphi_{1}\|_{H^{1}}\|\varphi_{2}\|_{H^{1}},$
which is a consequence of Hölder’s inequality and Sobolev’s Embedding Theorem
for $d=3$. It is also a standard exercise to check that, for each
$n\in\mathbb{N}$, the $H^{1}$ norms of $\psi_{n}(t)$ and
$\psi_{n}^{\varepsilon}(t)$ are uniformly bounded in $t\in\mathbb{R}$ and
$\varepsilon\in(0,1)$. Therefore, for each fixed $n\in\mathbb{N}$, it holds
that

$\lim_{\varepsilon\downarrow
0}\|\psi_{n}(t)-\psi_{n}^{\varepsilon}(t)\|_{L^{2}_{\bm{X}}\otimes\mathcal{F}_{n}}=0\,,\qquad\forall
t\in\mathbb{R}\,.$ (A.5)

Thus, it easily follows now from (A.3) that for all $n_{0}\in\mathbb{N}$

$\limsup_{\varepsilon\downarrow
0}\|(e^{-\mathrm{i}t\mathcal{H}_{N}^{\varepsilon}}-e^{-\mathrm{i}t\mathcal{H}_{N}})\Psi\|\,\lesssim\,\|\chi(N_{b}>n_{0})\Psi_{N,0}\|\,.$
(A.6)

The proof is complete after we take the limit $n_{0}\rightarrow\infty$. ∎

###### Proof of Lemma 3.2 .

Throughout the proof, we will use the fact that the $H^{1}$ norms of both
$\varphi_{t}$ and $\varphi_{t}^{\varepsilon}$ are uniformly bounded in $t$ and
$\varepsilon$. They will be absorbed into universal constants.

The Boson Field. First, we note that difference between the boson fields may
be written as

$\displaystyle\varphi_{t}-\varphi_{t}^{\varepsilon}$
$\displaystyle=e^{-\mathrm{i}t\Delta}(\varphi_{0}-\varphi_{0}^{\varepsilon})$
(A.7)
$\displaystyle-\mathrm{i}\int_{0}^{t}e^{-\mathrm{i}(t-s)\Delta}\Big{(}(v*|\varphi_{s}|^{2})\varphi_{s}-(v^{\varepsilon}*|\varphi_{s}^{\varepsilon}|^{2})\varphi_{s}^{\varepsilon}+\underline{w}(x,{\bm{X}}_{s})\varphi_{s}-\underline{w}^{\varepsilon}(x,{\bm{X}}_{s}^{\varepsilon})\varphi_{s}^{\varepsilon}\Big{)}\mathrm{d}s.$

Under our condition for the potential $v$, it is known that the map
$\varphi\in H^{1}(\mathbb{R}^{3})\mapsto(v*|\varphi|^{2})\varphi\in
H^{1}(\mathbb{R}^{3})$ is locally Lipschitz continuous, i.e. it satisfies
(2.23). Therefore, we apply the triangle inequality, a first order Taylor
estimate and Lipschitz continuity to find that

$\displaystyle\|\varphi_{t}-\varphi_{t}^{\varepsilon}\|_{H^{1}}$
$\displaystyle\lesssim\|\varphi_{0}-\varphi_{0}^{\varepsilon}\|_{H^{1}}+\int_{0}^{t}\|(v-v^{\varepsilon})*|\varphi_{t}^{\varepsilon}|^{2}\varphi_{t}^{\varepsilon}\|_{H^{1}}\mathrm{d}s\,.$
(A.8)
$\displaystyle+\int_{0}^{t}(\|w\|_{C_{b}^{1}}+\|\varphi_{s}\|_{H^{1}}^{2}+\|\varphi_{s}^{\varepsilon}\|_{H^{1}}^{2})(\|\varphi_{s}-\varphi_{s}^{\varepsilon}\|_{H^{1}}+|{\bm{X}}_{s}-{\bm{X}}_{s}^{\varepsilon}|+\|w-w^{\varepsilon}\|_{C_{b}^{1}})\
\mathrm{d}s$

For the term containing the difference $\delta v=v-v^{\varepsilon}$, we use
the Leibniz rule to find that

$\displaystyle\big{\|}(\delta
v*|\varphi|^{2})\varphi\big{\|}_{H^{1}}\lesssim\big{\|}(\delta
v*|\varphi|^{2})\varphi\big{\|}_{L^{2}}+\big{\|}\big{(}\delta
v*\bar{\varphi}\nabla\varphi\big{)}\varphi\big{\|}_{L^{2}}+\big{\|}(\delta
v*|\varphi|^{2})\nabla\varphi\big{\|}_{L^{2}}\,.$ (A.9)

In addition, we note than an application of Hölder’s and Young’s inequality
gives

$\big{\|}(\delta v*\varphi\phi)\,\psi\big{\|}_{L^{2}}\,\lesssim\,\|\delta
v\|_{L^{3/2}}\|\varphi\|_{L^{p_{1}}}\|\phi\|_{L^{p_{2}}}\|\psi\|_{L^{p_{3}}}\,,\qquad\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{5}{6}\,.$
(A.10)

We apply the last estimate for $(p_{1},p_{2},p_{3})=(6,2,6)$ and
$(p_{1},p_{2},p_{3})=(6,6,2)$, combined with the embedding
$H^{1}(\mathbb{R}^{3})\hookrightarrow L^{6}(\mathbb{R}^{3})$, to find that

$\displaystyle\big{\|}(\delta
v*|\varphi_{t}^{\varepsilon}|^{2})\,\varphi_{t}^{\varepsilon}\big{\|}_{H^{1}}\lesssim\|\delta
v\|_{L^{3/2}}\|\varphi_{t}^{\varepsilon}\|_{H^{1}}^{3}\lesssim\varepsilon\,.$
(A.11)

Tracer Particle Variables. Similarly as before, we may use a Taylor estimate
and the triangle inequality to find that

$\displaystyle|{\bm{X}}_{t}-{\bm{X}}_{t}^{\varepsilon}|$
$\displaystyle\lesssim\int_{0}^{t}|\dot{\bm{X}}_{s}-\dot{\bm{X}}_{s}^{\varepsilon}|\mathrm{d}s$
(A.12) $\displaystyle|\dot{\bm{X}}_{t}-\dot{\bm{X}}_{t}^{\varepsilon}|$
$\displaystyle\ \lesssim\
\int_{0}^{t}\big{(}\|w\|_{C_{b}^{2}}+\|\varphi_{s}\|_{L^{2}}^{2}+\|\varphi_{s}^{\varepsilon}\|_{L^{2}}^{2}\big{)}(\|\varphi_{s}-\varphi_{s}^{\varepsilon}\|_{L^{2}}+|{\bm{X}}_{s}-{\bm{X}}_{s}^{\varepsilon}|+\|w-w^{\varepsilon}\|_{C_{b}^{1}})\mathrm{d}s\,.$

We combine the estimates for the boson field and the tracer particle position
and apply the Grönwall inequality to the quantity
$\|\varphi_{t}-\varphi_{t}\|_{H^{1}}+|{\bm{X}}_{t}-{\bm{X}}_{t}^{\varepsilon}|+|\dot{\bm{X}}_{t}-\dot{\bm{X}}_{t}^{\varepsilon}|$
to obtain

$\|\varphi_{t}-\varphi_{t}^{\varepsilon}\|_{H^{1}}+|{\bm{X}}_{t}-{\bm{X}}_{t}^{\varepsilon}|+|\dot{\bm{X}}_{t}-\dot{\bm{X}}_{t}^{\varepsilon}|\lesssim\,e^{Ct}\,(1+t)\Big{(}\varepsilon+\|\varphi_{0}-\varphi_{0}^{\varepsilon}\|_{H^{1}}+\|w-w^{\varepsilon}\|_{C_{b}^{1}}\Big{)}\,,$
(A.13)

for some $C>0$. The proof of the lemma is finished once we apply the
definitions of the regularized objects and take the limit
$\varepsilon\downarrow 0$. ∎

##  Appendix B The Interpolation Lemma

In this appendix we give a proof of the Interpolation Lemma, stated as Lemma
5.4.

###### Proof.

Let $\Psi$ be arbitrary and assume $\lambda\in\mathbb{R}$ for simplicity.
Then, using the commutation relations we find that

$\|ABA\Psi\|^{2}=\left\langle\Psi,AB(A^{2}B)A\Psi\right\rangle=\left\langle\Psi,AB^{2}A^{3}\Psi\right\rangle-2\mathrm{i}\,\lambda\left\langle\Psi,ABA^{2}\Psi\right\rangle\,.$
(B.1)

Then, using the Cauchy-Schwarz inequality and Young’s inequality, we find that
for all $\varepsilon>0$ there is $C_{\varepsilon}$ (independent of $\Psi$)
such that

$\displaystyle|\left\langle\Psi,AB^{2}A^{3}\Psi\right\rangle|$
$\displaystyle\leqslant\varepsilon\|AB^{2}\Psi\|^{2}+C_{\varepsilon}\|A^{3}\Psi\|^{2}$
$\displaystyle|\left\langle\Psi,ABA^{2}\Psi\right\rangle|$
$\displaystyle\leqslant\varepsilon\|ABA\Psi\|^{2}+C_{\varepsilon}\lambda^{2}\|A\Psi\|^{2}.$

If we let $M(A,B)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\|A^{3}\Psi\|^{2}+\|B^{3}\Psi\|^{2}+\lambda^{2}\|A\Psi\|^{2}+\lambda^{2}\|B\Psi\|^{2}$,
we have found that

$\|ABA\Psi\|^{2}\leqslant
2\varepsilon\,(\|AB^{2}\Psi\|^{2}+\|ABA\Psi\|^{2})+2\,C_{\varepsilon}\,M(A_{,}B)\,.$
(B.2)

Using $\|A^{2}B\Psi\|^{2}\leqslant 2\|ABA\Psi\|^{2}+2\lambda^{2}\|A\Psi\|^{2}$
and $\|BA^{2}\Psi\|^{2}\leqslant 2\|ABA\Psi\|^{2}+2\lambda^{2}\|A\Psi\|^{2}$
we find (after re-updating $C_{\varepsilon}$)

$\|ABA\Psi\|^{2}+\|A^{2}B\Psi\|^{2}+\|BA^{2}\Psi\|^{2}\leqslant
10\varepsilon\,(\|AB^{2}\Psi\|^{2}+\|ABA\Psi\|^{2})+C_{\varepsilon}\,M(A_{,}B)\,.$
(B.3)

Similarly, we can interchange the roles of $A$ and $B$ to find that

$\|BAB\Psi\|^{2}+\|B^{2}A\Psi\|^{2}+\|AB^{2}\Psi\|^{2}\leqslant
10\varepsilon\,(\|BA^{2}\Psi\|^{2}+\|BAB\Psi\|^{2})+C_{\varepsilon}\,M(A_{,}B)\,.$
(B.4)

A straightforward combination of the last two inequalities, for
$\varepsilon<1/20$, finishes the proof of the lemma. ∎

Acknowledgments. I am deeply grateful to Thomas Chen for mentoring, reading
this manuscript, for holding several encouraging discussions, and for giving
me the opportunity to work on the problem of tracer particles interacting with
bosons. I would also like to thank Michael Hott for giving useful references
and insight regarding singular Hartree-type potentials. I gratefully
ackowledge support from the Provost’s Graduate Excellence Fellowship at The
University of Texas at Austin, and by the NSF grants DMS-1716198 and
DMS-2009800 through T. Chen, and by the NSF RTG Grant DMS-1840314 “Analysis of
PDE”.

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Also at ]INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

# From Chaos to Integrability in Double Scaled SYK

Micha Berkooz<EMAIL_ADDRESS>Nadav Brukner
<EMAIL_ADDRESS>Yiyang Jia (贾抑扬<EMAIL_ADDRESS>Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel  Ohad Mamroud
<EMAIL_ADDRESS>SISSA, via Bonomea 265, 34136 Trieste, Italy  [

###### Abstract

We study thermodynamic phase transitions between integrable and chaotic
dynamics. We do so by analyzing models that interpolate between the chaotic
double scaled Sachdev-Ye-Kitaev (SYK) and the integrable $p$-spin systems, in
a limit where they are described by chord diagrams. We develop a path integral
formalism by coarse graining over the diagrams, which we use to argue that the
system has two distinct phases: one is continuously connected to the chaotic
system, and the other to the integrable. They are separated by a line of first
order transition that ends at a critical point.

##  I Introduction

Two important universality classes in quantum mechanics are those of
_integrable_ and _chaotic_ systems, and the transition between these two
behaviors is also of major interest. In this work we will focus on a universal
dynamics which happens in such transitions in $p$-local quantum mechanical
systems, such as the SYK system.

More precisely, we will be interested in a class of models that interpolate
between chaotic and integrable $p$-local dynamics,

$H=\nu H_{\text{Chaotic}}+\kappa
H_{\text{Integrable}}\,,\quad\kappa^{2}+\nu^{2}=1\,,$ (1)

for $\kappa\in[0,1]$, and ask whether one can find any thermodynamic phase
transitions along the interpolation.

A measure of universality is obtained by focusing on systems with a _double
scaling limit_. These are systems of $N$ degrees of freedom, where we take
$N,\,p\to\infty$ while keeping $\lambda=2p^{2}/N$ fixed. The dynamics of such
systems are governed by _chord diagrams_ [1, 2, 3, 4, 5], to be defined below.
As a concrete example, we choose two well known models as our two extremes:

On the chaotic end we have the Sachdev-Ye-Kitaev (SYK) model [6, 7, 8, 9] – a
disordered quantum mechanical model of $N$ interacting Majorana fermions that
exhibits various facets of quantum chaos, from level repulsion [10, 11, 12] to
a maximal chaos exponent [6, 7, 13]. It has a rich history in nuclear physics
[14, 15] and in condensed matter physics [8, 9] where it serves as a solvable
example of a non-Fermi liquid, see [16] for a review. In the past decade it
had gained attention in the high-energy physics community as an important
controllable model of holography [6, 17, 12, 18, 19, 20, 21, 9].

On the other end we have the “commuting SYK” model [22, 23] – an integrable
system of (even) $N$ interacting fermions. It is better known as the
integrable $p$-spin model [24, 25, 26, 27] after a Jordan-Wigner
transformation is used to write it as acting on a system of qubits. In the
double scaling limit the model becomes the Random Energy model.

Below we show that the limit $\lambda\to 0$ is an emergent semi-classical
limit, in which the system exhibits a first order phase transition at low
temperatures $T$ along $\kappa\propto\sqrt{T}$. It ends at some finite
temperature in what is (presumably) a critical point. In a companion paper
[28] we elaborate on our path integral approach and show it can be used to
analyze the phase diagram of generic systems in the aforementioned double
scaling limit, amongst them the chaotic and integrable versions of the
$p$-spin system. We also find cases where both systems are chaotic, but the
interpolation exhibits a quantum phase transition at zero temperature.

##  II The microscopic Hamiltonian

We consider a quantum mechanical system of (even) $N$ Majorana fermions
$\psi_{i}$, satisfying the anti-commutation relations
$\\{\psi_{i},\psi_{j}\\}=2\delta_{ij}$. Our two Hamiltonians are

$\displaystyle H_{\text{Chaotic}}$
$\displaystyle=i^{p/2}\sum_{i_{1}<\cdots<i_{p}=1}^{N}J_{i_{1}\cdots
i_{p}}\psi_{i_{1}}\cdots\psi_{i_{p}}\,,$ (2) $\displaystyle
H_{\text{Integrable}}$
$\displaystyle=i^{p/2}\sum_{i_{1}<\cdots<i_{p/2}=1}^{N/2}B_{i_{1}\cdots
i_{p/2}}Q_{i_{1}}\cdots Q_{i_{p/2}}\,,$ (3)

with $Q_{i}=\psi_{2i-1}\psi_{2i}$ a fermion bilinear. The couplings are
independent Gaussian random variables,

$\displaystyle\left\langle J_{I}\right\rangle=0\,,\quad\left\langle
J_{I}J_{K}\right\rangle=\frac{1}{\lambda}\binom{N}{p}^{-1}{\mathbb{J}}^{2}\delta_{IK}\,,$
(4) $\displaystyle\left\langle B_{L}\right\rangle=0\,,\quad\left\langle
B_{L}B_{M}\right\rangle=\frac{1}{\lambda}\binom{N/2}{p/2}^{-1}{\mathbb{J}}^{2}\delta_{LM}\,,$
(5)

where capital letters in the subscripts denote sets of indices, and the
angular brackets denote the disorder averaging. In our normalization of the
trace $\operatorname{Tr}(\mathbbm{1})=1$ 111The natural normalization for the
trace, $\operatorname{Tr}(\mathbbm{1})=2^{N/2}$, simply multiplies the
partition function (and the moments) by that overall factor. and
$\left\langle\operatorname{Tr}(H^{2})\right\rangle={\mathbb{J}}^{2}/\lambda$.
We will be working with the units $c=\hbar=k_{B}=1$.

###  II.1 Chord diagrams

In the double scaling limit the annealed 222For SYK, the system self averages
to leading order in $N$ [40]. For the integrable system there is a glassy
phase transition [27], but in the double scaling limit it happens at
temperature of order ${\mathbb{J}}/(\lambda N)\to 0$, and so the annealed
averaging is justified. We assume that there is no glassy phase at any point
along the interpolation. averaged partition function has a convenient
diagrammatic expansion. First, the partition function with inverse temperature
$\beta$, $Z=\left\langle\operatorname{Tr}\left(e^{-\beta
H}\right)\right\rangle$, is expanded into a series in the moments
$m_{k}=\left\langle\operatorname{Tr}\left(H^{k}\right)\right\rangle$. The
disorder average of each moment can be computed by Wick’s theorem and
represented diagrammatically [1, 3, 2, 4, 5] – draw a circle with $k$ nodes on
its boundary, each representing an insertion of the Hamiltonian. Divide the
nodes into $k/2$ pairs, and draw a chord connecting each pair. This is a
_chord diagram_. The sum over all Wick contractions becomes a sum over all
possible chord diagrams. In our case, since we have two types of random
couplings, $J$ and $B$, we have two types of chords, which we call $n$\- and
$z$-chords for the chaotic and integrable Hamiltonians respectively. An
illustration appears in Figure 1.

Figure 1: A chord diagram with three $n$-chords and one $z$-chord. It
contributes
$\left(\frac{{\mathbb{J}}^{2}}{\lambda}\right)^{4}\nu^{6}\kappa^{2}q^{3}$ to
the moment $m_{8}$.

Following [3], and as explained in Appendix B of the companion paper [28], in
the double scaling limit the weight of the diagram is found as follows. Every
$n$-chord contributes a factor of $(\nu^{2}{\mathbb{J}}^{2}/\lambda)$ from the
disorder averaging and the two Hamiltonian insertions associated with it. Any
$z$-chord contributes $(\kappa^{2}{\mathbb{J}}^{2}/\lambda)$. In addition, Any
intersection between $n$-chords contributes a factor of $q=e^{-\lambda}$, as
do those between $n$-chords and $z$-chords. Intersections between $z$-chords
are weighted by $1$. Denoting the number of intersections between $i$\- and
$j$-chords in a diagram by $X_{ij}$, then the exact expression in the double
scaled limit is

$m_{2k}=\sum_{n+z=k}\sum_{C^{k}_{n,z}}\left(\frac{{\mathbb{J}}^{2}}{\lambda}\right)^{k}\nu^{2n}\kappa^{2z}q^{X_{nn}+X_{nz}}\,,$
(6)

where $C^{k}_{n,z}$ represents the set of all chord diagrams with $k$ chords,
out of which $n$ are $n$-chords and $z$ are $z$-chords. Odd moments vanish due
to the disorder average.

##  III Path integral over chords

In order to compute the partition function of the model we need to evaluate
the sum over all chord diagrams with two types of chords, which is quite a
formidable task. Luckily, the sum simplifies in the limit $\lambda\to 0$ – an
emergent semi-classical limit controlled by diagrams with many chords. Our
approach would be to coarse grain the system by dividing the boundary of the
diagram into segments, and write the partition function as a sum over the
number of chords of each kind that stretch between the different segments.
Each term in the sum is associated with many diagrams, and we will have to
compute their collective weight. At the end, we will take a continuum limit
where the number of segments, $s$, is very large. In this limit the
$i^{\text{th}}$ segment is identified with the Euclidean time $\tau_{i}$,
where $i/s=\tau_{i}/\beta$. This will result in a path integral expression for
the partition function.

Let’s begin. Arbitrarily divide the $2k$ Hamiltonian insertions in $m_{2k}$
into $s$ segments, each of length $k_{i}$. The moment computed using this
division is denoted by $m_{\vec{k}}$, but since the division is arbitrary,
$m_{\vec{k}}=m_{2k}$ for any $\vec{k}$. We average over the different
divisions using
$\frac{1}{s^{2k}}\sum_{\vec{k}}\binom{2k}{k_{i},\cdots,k_{s}}=1$,

$Z=\sum_{k=0}^{\infty}\frac{\beta^{2k}}{\left(2k\right)!}m_{2k}=\sum_{\vec{k}=0}^{\infty}\left[m_{\vec{k}}\prod_{i=1}^{s}\frac{\left(\beta/s\right)^{k_{i}}}{k_{i}!}\right]\,.$
(7)

The insertions in each segments fall into different types: some belong to
$n$\- and $z$-chords that connect the $i^{\text{th}}$ segment with the
$j^{\text{th}}$ segment, and hence we define

$n_{ij},z_{ij}\text{ -- No. of chords between {\hbox{i^{\text{th}}}},
{\hbox{j^{\text{th}}}} segments.}$ (8)

Eventually we will write the partition function as a sum over these variables.
In total, each segment has $n_{i}=\sum_{j}n_{ij}$ $n$-chords and
$z_{i}=\sum_{j}z_{ij}$ $z$-chords connecting it to other segments. There are
also chords connecting the segment to itself, $\hat{n}_{i}$ and $\hat{z}_{i}$
chords of each kind, which come with twice as many insertions.

The coarse grained classes of chord diagrams are characterized by different
possible divisions of each $k_{i}$ into these kinds of chords, constrained by
$k_{i}=n_{i}+z_{i}+2\hat{n}_{i}+2\hat{z}_{i}$. To compute $m_{\vec{k}}$ we
therefore need to sum over these divisions, and we do so with the following
weights, illustrated in Figure 2. We will evaluate the leading non-trivial
order for each of these factors in the limit $\lambda\to 0$.

(a) The chords of the $i^{\text{th}}$ segment, with
$\hat{n}_{i}=\hat{z}_{i}=z_{i}=1$, and $n_{i}=3$.

(b) A whole diagram, including the crossing between chords that connect
different segments.

Figure 2: An illustration of the terms that are considered in each part of the
coarse graining procedure.

The first factor counts all the possible ways to generate the chords that
leave the segment, i.e. their possible placement within the segment and their
possible intersections with the chords that connect the segment to itself.
There are $\binom{k_{i}}{n_{i},z_{i},2\hat{n}_{i},2\hat{z}_{i}}$ ways of
choosing which nodes belong to each group of chords in each segment. Then,
there are
$\left(2\hat{n}_{i}-1\right)!!=\frac{\left(2\hat{n}_{i}\right)!}{2^{\hat{n}_{i}}\hat{n}_{i}!}$
ways of connecting the $2\hat{n}_{i}$ nodes, as the first could be connected
to any of the other $2\hat{n}_{i}-1$, the third to any of the remaining
$2\hat{n}_{i}-3$ and so on. An additional factor of $(2\hat{z}_{i}-1)!!$ comes
from the $z$-nodes. We also remind the reader that the factors associated with
the each insertion give
$\left(\frac{\nu{\mathbb{J}}}{\sqrt{\lambda}}\right)^{n_{i}+2\hat{n}_{i}}\left(\frac{\kappa{\mathbb{J}}}{\sqrt{\lambda}}\right)^{z_{i}+2\hat{z}_{i}}$
for each segment.

The second factor comes from the different ways we can split in each segment
the $n_{i}$ outgoing chords into the subsets $\left\\{n_{ij}\right\\}$, i.e.
to decide which $n$-chords connect to every other segment, and similarly for
the $z$-chords. There are
$\binom{n_{i}}{n_{i1},\cdots,n_{is}}\binom{z_{i}}{z_{i1},\cdots,z_{is}}$ ways
of doing so.

The third counts the ways we can reorder the $n_{ij}$, $z_{ij}$ chords
connecting every two segments – the ways in which they could intersect
themselves. There are $n_{ij}!z_{ij}!$ such ways, and this factor needs to be
counted once for every pair of segments $i$ and $j$.

Finally, the last factor weights the bulk crossings of the chords that connect
different segments, see Figure 2(b),

$\prod_{i<k<j<\ell}q^{n_{ij}n_{k\ell}+n_{ij}z_{k\ell}+n_{k\ell}z_{ij}}\,.$ (9)

Putting all of these factors together and performing the sum over the chords
that are restricted to a single segment, $\hat{n}_{i}$ and $\hat{z}_{i}$, we
find

$Z=e^{\frac{(\beta{\mathbb{J}})^{2}}{2\lambda
s}}\sum_{\left\\{n_{ij}\right\\},\left\\{z_{ij}\right\\}}\Bigg{\\{}q^{\sum_{i<k<j<\ell}\left[n_{ij}n_{k\ell}+n_{ij}z_{k\ell}+z_{ij}n_{k\ell}\right]}\\\
\times\prod_{j>i=1}^{s}\frac{1}{n_{ij}!z_{ij}!}\left(\frac{\nu^{2}\beta^{2}{\mathbb{J}}^{2}}{\lambda
s^{2}}\right)^{n_{ij}}\left(\frac{\kappa^{2}\beta^{2}{\mathbb{J}}^{2}}{\lambda
s^{2}}\right)^{z_{ij}}\Bigg{\\}}\,.$ (10)

Note that we have not taken the strict $\lambda\to 0$ limit in every factor.
The reason for that comes from the continuum limit we will soon take, where
the segments are very small. The bulk crossings term (9) encodes correlations
between regions with large (Euclidean) time separation, and we keep their
$\lambda$ dependence. On the other hand, the other factors are related to
chords that are very close to each other, and therefore captures in some sense
the short time behavior of the chords. The number of intersections here can be
made small in the subsequent continuum limit, and we take $\lambda\rightarrow
0$. Our approach balances between these behaviors.

The partition function simplifies considerably by taking two limits. First,
assuming a large number of chords, $n_{ij},z_{ij}\gg 1$. This is self
consistent at the saddle point when $\lambda\rightarrow 0$. The limit allows
us to approximate the factorials by Stirling’s formula,
$n!\xrightarrow{n\to\infty}e^{n\left(\log n-1\right)}$. Second, we take the
continuum limit $s\to\infty$, as alluded to before. By introducing the
continuum chord density fields, which are now functions of the Euclidean
times,

$n\left(\tau_{i},\tau_{j}\right)=\frac{\lambda s^{2}}{\beta^{2}}n_{ij}\,,\quad
z\left(\tau_{i},\tau_{j}\right)=\frac{\lambda s^{2}}{\beta^{2}}z_{ij}\,,$ (11)

we end up with a path integral expression for the partition function,

$Z=\int{\mathcal{D}}n{\mathcal{D}}z\,e^{-\frac{1}{\lambda}S\left[n,z\right]}\,,$
(12)

with a flat measure over all non-negative, symmetric, and periodic functions
of two variables, and with the action

$S=\frac{1}{4}\int_{0}^{\beta}d\tau_{1}\int_{0}^{\beta}d\tau_{2}\int_{\tau_{1}}^{\tau_{2}}d\tau_{3}\int_{\tau_{2}}^{\tau_{1}}d\tau_{4}\,\Big{\\{}n(\tau_{1},\tau_{2})n(\tau_{3},\tau_{4})+2n(\tau_{1},\tau_{2})z(\tau_{3},\tau_{4})\Big{\\}}\\\
+\frac{1}{2}\int_{0}^{\beta}d\tau_{1}\int_{0}^{\beta}d\tau_{2}\,\left\\{n(\tau_{1},\tau_{2})\left[\log\left(\frac{n(\tau_{1},\tau_{2})}{\nu^{2}{\mathbb{J}}^{2}}\right)-1\right]+z(\tau_{1},\tau_{2})\left[\log\left(\frac{z(\tau_{1},\tau_{2})}{\kappa^{2}{\mathbb{J}}^{2}}\right)-1\right]\right\\}\,.$
(13)

The $\tau$’s live on a circle of length $\beta$, and flipping the limits of
the integration is understood as integrating over the other side of the
circle. The pre-factors correct the overcounting when switching $\tau_{1}$ and
$\tau_{2}$, and when switching the pairs $\tau_{1,2}$ and $\tau_{3,4}$.

In our $\lambda\to 0$ limit the integral is controlled by its saddle point.
After introducing the fields

$\displaystyle
g_{n}(\tau_{1},\tau_{2})=-\int_{\tau_{1}}^{\tau_{2}}d\tau_{3}\int_{\tau_{2}}^{\tau_{1}}d\tau_{4}\,n(\tau_{3},\tau_{4})\,,$
(14) $\displaystyle
g_{z}(\tau_{1},\tau_{2})=-\int_{\tau_{1}}^{\tau_{2}}d\tau_{3}\int_{\tau_{2}}^{\tau_{1}}d\tau_{4}\,z(\tau_{3},\tau_{4})\,,$
(15)

that have period $\beta$ in either variable and satisfy

$g_{n}(\tau,\tau)=g_{z}(\tau,\tau)=0\,,$ (16)

the saddle point equations read

$\displaystyle\partial_{\tau_{1}}\partial_{\tau_{2}}g_{n}(\tau_{1},\tau_{2})$
$\displaystyle=-2{\mathbb{J}}^{2}\nu^{2}e^{g_{n}(\tau_{1},\tau_{2})+g_{z}(\tau_{1},\tau_{2})}\,,$
(17)
$\displaystyle\partial_{\tau_{1}}\partial_{\tau_{2}}g_{z}(\tau_{1},\tau_{2})$
$\displaystyle=-2{\mathbb{J}}^{2}\kappa^{2}e^{g_{n}(\tau_{1},\tau_{2})}\,.$
(18)

When $\kappa=0$ the equations of motion and the resulting partition function
reproduce those of the $G\Sigma$ approach in the large interaction length [7]
and double scaling [12, 31] limits, but it’s important to note that our off-
shell action is different than theirs.

##  IV From chaos to integrability

The saddle point equations can be solved numerically, and are found to have
two types of solutions: one, the chaotic phase, is continuously connected to
the saddle of the chaotic system at $\kappa=0$. The other, which we call the
quasi-integrable phase, is connected to the saddle point of the integrable
system at $\kappa=1$. We stress that the names only reflect the relation to
the pure integrable and chaotic solutions, and nothing more. At low enough
temperatures both solutions exist for some range of $\kappa$ and change their
dominance as we increase it, which results in a first order phase transition,
see Figure 3.

Figure 3: Comparison of the numerics vs leading order approximation for the
action of the two phases.

At low temperatures, $\beta{\mathbb{J}}\ll 1$, the existence of the phase
transition can be explained by the perturbative solutions around the two
extremes. To leading order in both $\kappa$ and $\beta{\mathbb{J}}$, the
solution around $\kappa=0$ is

$\displaystyle g_{n}^{\kappa=0}(\tau_{1},\tau_{2})$
$\displaystyle=2\log\left[\frac{\pi}{\beta{\mathbb{J}}\cos\left[\pi\left(\frac{1}{2}-\frac{|\tau_{1}-\tau_{2}|}{\beta}\right)\right]}\right]\,,$
(19) $\displaystyle g^{\kappa=0}_{z}(\tau_{1},\tau_{2})$ $\displaystyle=0\,,$
(20)

On the other hand, around $\kappa=1$ the solution is

$\displaystyle g_{n}^{\kappa=1}(\tau_{1},\tau_{2})$ $\displaystyle=0\,,$ (21)
$\displaystyle g^{\kappa=1}_{z}(\tau_{1},\tau_{2})$
$\displaystyle=-({\mathbb{J}}\kappa)^{2}\tau(\tau-\beta)\,.$ (22)

The two solutions have the actions

$S_{\kappa=0}=-2\beta{\mathbb{J}}\,,\quad
S_{\kappa=1}=-\frac{1}{2}(\beta{\mathbb{J}}\kappa)^{2}\,.$ (23)

We give more details about the perturbative solutions in the companion paper
[28]. Importantly, we show that the subleading corrections around $\kappa=1$
are suppressed at low temperatures. This enhances the range of validity for
the approximation, and allows us to connect it to the one around $\kappa=0$.

The phase transition happens when the two actions are equal. The above
approximations set this point at

$\kappa_{*}=\frac{2}{\sqrt{\beta{\mathbb{J}}}}\,.$ (24)

This low-temperature estimate fits the numerical results summarized in Figure
4 quite well – the power law is precisely matched, and the numerical
coefficient agrees to within $10\%$. At fixed $\kappa$, the low temperature
phase is the integrable phase. At higher temperatures our analytic
approximations are no longer accurate, and the numerics show that the phase
transition line ends at finite temperature and $\kappa$, presumably in a
second order fixed point.

Figure 4: Phase diagram in the $\kappa-1/\beta\mathbb{J}$ plane. The red and
yellow dots are first-order phase transition points. The former are in the low
temperature region, and only they are used to obtain the power-law fit (blue).
The first-order transition line terminates at the black dot.

##  V Discussion

The double scaling limit is a robust universal limit for many $p$-local
systems. The results of this paper show that in this class of models,
described by chord diagrams, there is a semi-classical description whose
saddle point is controlled by coupled Liouville equations. This presents a
generalization of the collective field ($G\Sigma$) approach to the case of
multiple Hamiltonians in the double scaling limit.

It is interesting that coupled Liouville are the generic description of such
systems. Systems of coupled Liouvilles often have additional symmetries, and
it would be interesting to see if there are additional structures like a
generalized Schwarzian. The strong exponential dependence of the field almost
guarantees that the system will be governed by first order phase transitions
since it is difficult (but not impossible) to balance the exponentials.

Indeed, we find that there is a first order phase transition that separates
the chaotic phase from a quasi-integrable phase. Perhaps the abrupt transition
between the two phases should not surprise us, as previous studies have found
a discontinuous behavior of the chaos exponent when deforming integrable
systems in quantum mechanical and field theoretical models [32, 33], which is
somewhat reminiscent of the KAM theorem in classical systems.

One might wonder what happens to these systems away from the double scaling
limit, at finite $p$. Other works have studied integrable deformations of the
SYK model, [34, 35, 36, 37], in part as controllable models for transitions
between thermalizing and many body localized systems. In the finite $p$
version of our Hamiltonian (1) there is also sometimes a glassy phase, and so
it could provide an arena to study the interplay between chaos, integrability
and the glass transition.

Finally, we note that SYK-like systems could also be studied experimentally
[38] or by quantum simulations [39]. It is therefore important to verify that
such realizations are in the correct phase. It is exciting to hope that the
phase transition itself could be studied in the lab.

###### Acknowledgements.

We would like to thank Ahmed Almheiri, Dionysios Anninos, Andreas Blommaert,
Ronny Frumkin, Damian Galante, Antonio García-García, Vladimir Narovlansky,
and Josef Seitz for useful discussions. YJ also thanks Chi-Ming Chang and
Zhenbin Yang for their invitation to visit Yau Mathematical Sciences Center
(YMSC) and Institute for Advance Studies at Tsinghua University (IASTU), where
part of this work is done. This work was supported in part by an Israel
Science Foundation (ISF) center for excellence grant (grant number 2289/18),
by ISF grant no. 2159/22, by the Minerva foundation with funding from the
Federal German Ministry for Education and Research, by the German Research
Foundation through a German-Israeli Project Cooperation (DIP) grant
“Holography and the Swampland”. YJ is also supported by the Koshland
postdoctoral fellowship and by a research grant from Martin Eisenstein. OM is
supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative dynamics of
quantum fields: from new deconfined phases of matter to quantum black holes”,
by the MUR-FARE2020 grant No. R20E8NR3HX “The Emergence of Quantum Gravity
from Strong Coupling Dynamics”. OM is also partially supported by the INFN
“Iniziativa Specifica GAST”.

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A.I. Ennis PNe as tracers of stellar population properties

Planetary Nebulae: a Universal Toolbox in the Era of Precision Astrophysics
2023 10.1017/xxxxx 384

Proceedings IAU Symposium

# Planetary nebulae as tracers of stellar population properties: unlocking
their potential with integral-field spectroscopy

Ana Inés Ennis1,2 Johanna Hartke3,4 Magda Arnaboldi5 Claudia Pulsoni6 Fuyan
Bian7 Chiara Spiniello8,9 1 Waterloo Centre for Astrophysics, University of
Waterloo, Canada  2Perimeter Institute, Canada  3Finnish Centre for Astronomy
with ESO (FINCA), FI-20014 University of Turku, Finland  4Tuorla Observatory,
Department of Physics and Astronomy, FI-20014 University of Turku, Finland
5European Southern Observatory, Garching bei München, Germany  6Max-Planck-
Institut für Extraterrestrische Physik (MPE), Garching, Germany  7European
Southern Observatory, Alonso de Córdova, Vitacura, Santiago, Chile  8INAF –
Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131 - Napoli,
Italy  9Department of Physics, University of Oxford, Denys Wilkinson Building,
Keble Road, Oxford OX1 3RH, UK

###### Abstract

Planetary nebulae (PNe) are essential tracers of the kinematics of the diffuse
halo and intracluster light where stellar spectroscopy is unfeasible, due to
their strong emission lines. However, that is not all they can reveal about
the underlying stellar population. In recent years, it has also been found
that PNe in the metal-poor halos of galaxies have different properties
(specific frequency, luminosity function), than PNe in the more metal-rich
galaxy centers. A more quantitative understanding of the role of age and
metallicity in these relations would turn PNe into valuable stellar-population
tracers. In order to do that, a full characterization of PNe in regions where
the stellar light can also be analysed in detail is necessary. In this work,
we make use of integral-field spectroscopic data covering the central regions
of galaxies, which allow us to measure both stellar ages and metallicities as
well as to detect PNe. This analysis is fundamental to calibrate PNe as
stellar population tracers and to push our understanding of galaxy properties
at unprecedented galactocentric distances.

###### keywords:

Catalogs, Surveys, galaxies: elliptical and lenticular, cD, planetary nebulae

††editors: O. De Marco, A. Zijlstra, R. Szczerba, eds.

##  1 Introduction

For almost thirty years planetary nebulae (PNe) have been used as kinematic
tracers of the extended haloes of galaxies (Arnaboldi et al. 1994; Napolitano
et al. 2000; Coccato et al. 2009; Spiniello et al. 2018; Pulsoni et al. 2023).
The distinct optical spectral features of PNe, characterized by a flat
continuum and prominent emission lines (see Figure 1), render them detectable
at large distances from the centre of galaxies, where the emitted light is too
faint for absorption line spectroscopy and stellar population analysis.
Furthermore, the luminosity-specific PN number ($\alpha=N_{PN}/L_{gal}$ ratio)
has been shown to vary with the age and metallicity of the stellar population
of the host galaxy (Buzzoni et al. 2006). A comprehensive understanding of
this relationship would unlock the full potential of PNe as tracers of stellar
populations, enabling us to characterize age and metallicity in the extended
haloes at greater distances than ever before.

Given that stellar population properties vary throughout the full radial span
of a galaxy, it is reasonable to expect these differences may also appear in
the PN population if they are connected. Currently, there are very few
galaxies for which a joint analysis has been performed of the radial
variations of $\alpha$ and the age and metallicity of its stellar populations.
In Hartke et al. (2017), the $\alpha$-parameter of M49 was shown to vary with
colour, presenting lower values in the inner, red, metal-rich halo. Similarly,
for M105 in Hartke et al. (2020) two distinct values of $\alpha$ were
estimated for different areas of the galaxy, identifying a diffuse outer
component with old halo metal-poor stars. Since these studies rely on
photometric data, the relation is estimated in terms of colours, which do not
translate directly into age and metallicity.

To resolve this degeneracy, it is necessary to analyse PNe in regions of the
galaxy where high SNR spectroscopic data of the stellar population can be
obtained in order to estimate their age and metallicity independently. A
challenge arises, however, as wherever the stellar population is bright enough
to allow this, PNe become much harder to detect using traditional methods. The
central regions of galaxies, which are perfect candidates for detailed
analysis of stellar populations, are usually blind spots for PNe detection
techniques.

In this work, we present a way to overcome these limitations by using IFU data
from MUSE and the DELF technique (Differential Emission-Line Filter, Roth et
al. 2021). This enables us to detect PNe while simultaneously analysing the
underlying stellar population in the same area. By linking $\alpha$-parameters
directly to age and metallicity for a sample of galaxies, we aim to gain a
thorough understanding of these relations. This takes us one step closer to
PNe becoming reliable tracers of stellar population properties. We aim to
perform this analysis on a sample of early-type galaxies which was built using
two criteria: that they have MUSE observations publicly available in the ESO
Archive, and that they have an available PNe catalogue for their outer halo,
for future comparison. We show in these proceedings the results process
applied to one of the galaxies in this sample, NGC 1387.

Figure 1: Top: full spectrum for a PNe detected in NGC 1387. Bottom: zoom-in
of the same spectrum in the [OIII] doublet region.

##  2 Detection and classification of sources

Traditionally, the main detection method for PNe is based on obtaining on and
off images, i.e. a narrowband filter image centered in the [OIII] line, and a
broadband filter image which should capture continuum emission (Ford et al.
1973; Ford & Jenner 1975; Jacoby et al. 1992). Because PNe have a negligible
continuum, they will not show up in the broadband filter, but they will be
very bright in the narrow-band one, making them easy to identify among other
sources. The main idea behind DELF is that this is easily emulated using IFU
data since we can create images based on any filter, and we can enhance our
signal by having access to the entire spectrum with high resolution, thus
being able to build customised filters in the most convenient way for our
purposes.

As a first step, we subtract the continuum emission from the cubes by fitting
a polynomial function to each pixel and creating a ”continuum cube”, which we
then subtract to create an ”emission-line cube” (Spriggs et al. 2020). From
the emission-line cube, we extract our on images. We create our filter
considering that PNe have velocity dispersions of about $\pm 500$ km/s, so the
line can be detected in $5007.8\pm 7.5$Å. Rather than creating a narrowband
filter with this width, we split this range into 12 slices with a width of
$1.25\AA$, which improves our detection capabilities. We run Source Extractor
(Bertin & Arnouts 1996) in each of the slices and require sources to be
detected in three consecutive slices in order to be considered positive
detections. Once we build our catalogue of detected sources, we filter out
spurious detections by fitting a Gaussian profile to the flux across all
slices. This ensures that we are not detecting extended emission across the
entire wavelength range, or potential artifacts in the spectra. Finally, we
extract the full spectrum for each source, and perform a visual check on the
lines, as seen in Figure 1.

Since we have access to the full MUSE spectrum for each object, we are able to
estimate the flux of multiple emission lines. This makes it possible to
construct diagnostic diagrams that separate PNe from other emission-line
sources such as supernova remnants and HII regions. We calculate the [OIII]
magnitude using the expression from Jacoby (1989):

$m_{\textrm{OIII}}=-2.5\,log_{10}(F_{\textrm{OIII}})-13.74$ (1)

In Figure 2 we show the diagnostic diagram for NGC 1387, where PNe are
identified in the region determined by Ciardullo et al. (2002).

Figure 2: Diagnostic diagram for sources in NGC 1387. Pink triangles indicate
PNe, and blue dots, contaminants. The limits shown in light blue solid lines
are defined in Ciardullo et al. (2002)

##  3 Calculation of the luminosity-specific PNe number

As a first step, we perform a completeness analysis by injecting artificial
stars on the slices using a PSF modeled with a Moffat profile and then
repeating the same detection process as before. We compare the fraction of
recovered sources to the injected sources in terms of magnitude range and
distance to the galactic center.

We also look into the luminosity function of PNe, which is widely known to be
a distance indicator due to the universal absolute magnitude value of its
bright cut-off (Ciardullo et al. 1989). Due to the small number of PNe given
the limited size of our field of view, we do not aim at measuring distances.
However, the PNLF also allows us to estimate what proportion of PNe we are
detecting within our completeness limit, and to extrapolate towards fainter
magnitudes since its shape has been widely studied. In Figure 3 we show the
PNLF for NGC 1387.

Figure 3: PNLF for NGC 1387. Dashed blue lines show the original histogram,
while grey solid lines show the completeness corrected distribution. The red
line shows the function as determined by the distance obtained through SBF
method for this galaxy. Figure 4: Surface brightness profile of NGC 1387 in a
solid black line, with three pink stars indicating the numeric density of PNe
at the corresponding radius, offset by the respective amount.

The luminosity-specific PN number, $\alpha$, is connected to the total number
of PNe, $N_{PN}$, and the total bolometric luminosity $L_{bol}$ through the
following expression:

$N_{PN}=\alpha L_{bol}$ (2)

We bin PNe into elliptical bins, with their geometry defined by the isophotal
properties of the host galaxy, and apply the completeness correction. We then
measure the PN logarithmic number density profile as

$\mu_{PN}(r)=-2.5log_{10}\left(\frac{N_{PN,corr}(r)}{A(r)}\right)+\mu_{off}$
(3)

where $A(r)$ is the area of the respective ellipse. The offset $\mu_{off}$ is
fit to match the surface brightness profile, and we refer to Hartke et al.
(2020) for a thorough explanation of how to estimate $\alpha_{2.5}$ from it.
For NGC 1387, we obtain a value of $\alpha_{2.5}=4.39\times
10^{-8}\,N_{PNe}\times L_{bol}^{-1}$.

We use the GIST pipeline to analyse the stellar population in the cubes. GIST
utilizes Penalized PiXel-Fitting (Cappellari 2023) in order to obtain
kinematics and stellar population properties (i.e. age and metallicity). In
Figs. 5 and 6 we show the stellar population properties of NGC 1387 obtained
with this method.

Figure 5: Age estimates for the stellar population of NGC 1387. Figure 6:
Metallicity estimates for the stellar population of NGC 1387. Figure 7:
Luminosity-specific PN number of a sample of galaxies from Buzzoni et al.
(2006) in relation to their metallicity. The symbols correspond to different
morphologies (circles - ellipticals, squares - spirals, diamonds -
irregulars). The unfilled circle is our pilot galaxy shown in this work, NGC
1387.

In Figure 7 we show the relation between $\alpha$ and the mean metallicity of
the host galaxy for the sample studied by Buzzoni et al. (2006), covering a
wide range of morphologies, and indicate the position that the inner region of
NGC 1387 has in this relation. Adding the remaining nine galaxies in our
sample will allow us to populate the relation, examining how it behaves when
the measurements of $\alpha$ and stellar properties are performed in the same
regions.

##  4 Conclusions

In this work, we have shown preliminary results for the analysis of PNe in the
inner regions of galaxies. We describe the method and show the results for one
of the galaxies in our sample. In an upcoming paper, this analysis will be
applied to the full sample of ten galaxies, which will enable us to interpret
the relation between $\alpha$ and stellar population properties in larger
detail.

##  5 Acknowledgements

A.I.E. acknowledges the financial support from the visitor and mobility
program of the Finnish Centre for Astronomy with ESO (FINCA), funded by the
Academy of Finland grant nr 306531. This research was supported in part by
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported by the Government of Canada through the Department of Innovation,
Science and Economic Development and by the Province of Ontario through the
Ministry of Research and Innovation.

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