On complex networks with pinning controllers

111This work is jointly supported by the National Key R$\&$D Program of China
(No. 2018AAA010030), National Natural Sciences Foundation of China under Grant
(No. 62072111).

Tianping Chen222Tianping Chen is with the School of Mathematics, Fudan
University, 200433, Shanghai, China.

Abstract In this short paper, we explore relationship between various models
of complex networks with pinning controllers.

In the paper [1], we discussed following coupling network with a single
pinning controller

$\displaystyle\left\\{\begin{array}[]{cc}\frac{dx_{1}(t)}{dt}&=f(x_{1}(t),t)+c\sum\limits_{j=1}^{m}a_{1j}x_{j}(t)\\\
&-c\varepsilon(x_{1}(t)-s(t)),\\\
\frac{dx_{i}(t)}{dt}&=f(x_{i}(t),t)+c\sum\limits_{j=1}^{m}a_{ij}x_{j}(t),\\\
&i=2,\cdots,m\end{array}\right.$ (5)

where $a_{ij}\geq 0$, if $i\neq j$ and $a_{ii}=-\sum_{j\neq i}a_{ij}$.

It is clear that (1) can be written as

$\displaystyle\frac{d(x_{i}(t)-s(t))}{dt}$
$\displaystyle=f(x_{i}(t),t)-f(s(t),t)+c\sum\limits_{j=1}^{m}\tilde{a}_{ij}(x_{j}(t)-s(t))$
(6)

where

$\tilde{A}=\left(\begin{array}[]{cccc}a_{11}-\varepsilon&a_{12}&\cdots&a_{1m}\\\
a_{21}&a_{22}&\cdots&a_{2m}\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{m1}&a_{m2}&\cdots&a_{mm}\end{array}\right)$

In [1], we gave

Proposition 1 Let $\Xi=[\xi_{1},\cdots,\xi_{n}]^{T}$ be the left eigenvector
of the matrix $A$ corresponding to eigenvalue $0$. Denote
$(\Xi\tilde{A})^{s}=\frac{\Xi\tilde{A}+\tilde{A}^{T}\Xi}{2}$. Then,
$(\Xi\tilde{A})^{s}$ is negative definite with eigenvalues
$0>\lambda_{1}\geq\cdots,\lambda_{m}$.

and proved following Theorems

Theorem A Suppose $A$ is symmetric. $0>\lambda_{1}>\cdots>\lambda_{m}$ are the
eigenvalues of $\tilde{A}$. If there are positive diagonal matrices
$P=diag\\{p_{1},\cdots,p_{n}\\}$,
$\Delta=diag\\{\Delta_{1},\cdots,\Delta_{n}\\}$ and a constant $\eta>0$, such
that QUAD condition

$(x-y)^{\top}P(f(x,t)-\Delta x-f(y,t)+\Delta y)\leq-\eta(x-y)^{\top}(x-y)$ (7)

is satisfied and $\Delta_{i}+c\lambda_{1}<0$, for $i=1,\cdots,n$. Then, the
controlled system (2) is globally exponentially synchronized to $s(t)$.

Theorem B Suppose $A$ is asymmetric. $0>\lambda_{1}>\cdots>\lambda_{m}$ are
the eigenvalues of $\\{\Xi\tilde{A}\\}^{s}$. If there are positive diagonal
matrices $P=diag\\{p_{1},\cdots,p_{n}\\}$,
$\Delta=diag\\{\Delta_{1},\cdots,\Delta_{n}\\}$ and a constant $\eta>0$, such
that QUAD condition

$(x-y)^{\top}P(f(x,t)-\Delta x-f(y,t)+\Delta y)\leq-\eta(x-y)^{\top}(x-y)$ (8)

is satisfied, and $\Delta_{i}+c\lambda_{1}<0$, for $i=1,\cdots,n$. Then, the
controlled system (2) is globally exponentially synchronized to $s(t)$.

A few years later, in [2], authors discussed

$\displaystyle\left\\{\begin{array}[]{cc}\frac{dx_{1}(t)}{dt}&=f(x_{1}(t),t)+c\sum\limits_{j=1}^{m}G_{1j}\Gamma
x_{j}(t)\\\ &-c\varepsilon\Gamma(x_{1}(t)-s(t)),\\\
\frac{dx_{i}(t)}{dt}&=f(x_{i}(t),t)+c\Gamma\sum\limits_{j=1}^{m}G_{ij}x_{j}(t),\\\
&i=2,\cdots,m\end{array}\right.$ (13)

where where $G_{ij}\geq 0$, if $i\neq j$ and $G_{ii}=-\sum_{j\neq i}G_{ij}$,
$G_{ij}=G_{ji}$. $f(x,t)$ satisfies

$(x-y)^{\top}(f(x)-f(y))-(x-y)^{\top}K\Gamma(x-y)<0$ (14)

where $\Gamma$ is positive definite and $K,\Gamma$ are commutable.

It is equivalent to

$(x-y)^{\top}(f(x)-f(y))-(x-y)^{\top}K\Gamma(x-y)<-\eta(x-y)^{\top}(x-y)$ (15)

for some constant $\eta>0$.

In [3], $G_{ij}$ is assumed as asymmetric.

Now, we will reveal that the results given in [2], [3] are direct consequences
of those given in [1].

Firstly, we give following simple proposition on the normal operators in
functional analysis.

Proposition 2 The assumption that $\Gamma$ and $K$ are commutable is
equivalent to that $\Gamma$ and $K$ have same eigenvectors.

In fact, if $\gamma_{i}$ is a eigenvalue of $\Gamma$, $x_{i}$ is the
corresponding eigenvector. Then

$\Gamma x_{i}=\gamma_{i}x_{i}$

Because $\Gamma K=K\Gamma$, then

$\Gamma Kx_{i}=K\Gamma x_{i}=\gamma_{i}Kx_{i}$

which means $Kx_{i}$ is also an eigenvector of $\Gamma$ corresponding to
$\gamma_{i}$

$Kx_{i}=k_{i}x_{i}$

$\Gamma$ and $K$ have same eigenvectors.

It is well known that the definite matrix $\Gamma$ can be decomposed as
$\Gamma=Q^{T}\bar{\Gamma}Q$, where $Q$ is an orthogonal matrix and
$\bar{\Gamma}=diag[\gamma_{1},\cdots,\gamma_{n}]$ is a positive diagonal

Based on Proposition 2, $K$ can also be written as $K=Q^{T}\bar{K}Q$,
$\bar{K}=diag[k_{1},\cdots,k_{n}]$.

Let $\tilde{x}_{i}(t)=Qx_{i}(t)$. $\tilde{s}(t)=Qs(t)$, and

$\tilde{f}(u)=Q(f(Q^{-1}u,t)$

It is clear that

$\displaystyle(\tilde{x}-\tilde{y})^{\top}(\tilde{f}(\tilde{x})-\tilde{f}(\tilde{y}))=(x-y)^{T}(f(x)-f(y))$
(16)

Therefore, in the following, we will discuss following systems

$\displaystyle\frac{d({x}_{i}(t)-{s}(t))}{dt}$
$\displaystyle={f}({x}_{i}(t))-{f}({s}(t)))+c\sum\limits_{j=1}^{m}\tilde{G}_{ij}\bar{\Gamma}({x}_{i}(t)-{s}(t))$
(17)

where $\bar{\Gamma}=diag[\gamma_{1},\cdots,\gamma_{n}]$ is a positive diagonal
matrix.

The QUAD condition is

$\displaystyle({x}-{y})^{\top}({f}({x})-{f}(\tilde{y}))-({x}-{y})^{\top}\bar{K}\bar{\Gamma}({x}-{y})<-\eta({x}-{y})^{\top}({x}-{y})$
(18)

where $\bar{K}=diag[k_{1},\cdots,k_{n}]$ is a positive diagonal matrix.

Following the approach proposed in [1], we can prove

Theorem 1 Suppose $A$ is asymmetric. $0>\lambda_{1}>\cdots>\lambda_{m}$ are
the eigenvalues of $\\{\Xi\tilde{A}\\}^{s}$. If there is a constant $\eta>0$,
such that QUAD condition

$(x-y)^{\top}[f(x)-f(y)-\bar{\Gamma}\bar{K}(x-y)]\leq-\eta(x-y)^{\top}(x-y)$
(19)

is satisfied and $\xi_{i}\bar{k}_{j}+c\lambda_{1}<0$ for all $i=1,\cdots,m$,
$j=1,\cdots,n$. Then, the controlled system (9) is globally exponentially
synchronized to $s(t)$.

Proof Follow [1], define Lyapunov function

$\displaystyle
V(\tilde{x}(t))=\frac{1}{2}\sum\limits_{i=1}^{m}\xi_{i}({x}_{i}(t)-{s}(t))^{T}({x}_{i}(t)-{s}(t))$
(20)

Differentiating, we have

$\displaystyle\frac{d(V({x}(t)))}{dt}$
$\displaystyle=\sum\limits_{i=1}^{m}\xi_{i}({x}_{i}(t)-{s}(t))^{T}[{f}({x}_{i}(t))-{f}({s}(t))]$
$\displaystyle+c\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\xi_{i}\tilde{G}_{ij}({x}_{i}(t)-{s}(t))^{T}\bar{\Gamma}({x}_{j}(t)-{s}(t))$
$\displaystyle<\sum\limits_{i=1}^{m}\xi_{i}({x}_{i}(t)-{s}(t))^{T}\bar{K}\bar{\Gamma}({x}_{i}(t)-{s}(t))$
$\displaystyle+c\lambda_{1}\sum\limits_{i=1}^{m}({x}_{i}(t)-{s}(t))^{T}\bar{\Gamma}({x}_{i}(t)-{s}(t))$
$\displaystyle-\eta\sum\limits_{i=1}^{m}\xi_{i}({x}_{i}(t)-{s}(t))^{T}({x}_{i}(t)-{s}(t))$

In case that $\xi_{i}\bar{k}_{j}+c\lambda_{1}<0$ for all $i=1,\cdots,m$,
$j=1,\cdots,n$, we have

$\displaystyle\dot{V}({x}(t))<-\eta V({x}(t))$ (21)

and

$\displaystyle V({x}(t))=O(e^{-\eta t})$ (22)

which implies

$\displaystyle{x}_{i}(t)-{s}(t)=O(e^{-\eta t}),~{}~{}i=1,\cdots,m$ (23)

Remark 1 Proposition 1 plays the key role in the theoretical analysis. It
concludes that $\Gamma$ and $K$ have same eigenvectors.

Remark 2 The proof given in [2], [3] is questionable. A key problem is that
$K\Gamma=(\frac{K+K^{\top}}{2})\Gamma\leq||\theta||\Gamma$, where
$||\theta||=\lambda_{max}(\frac{K+K^{\top}}{2})$, is incorrect. In general,
even though both matrices $A$ and $B$ are positive definite, it is not known
whether the product $AB$ is still positive definite. That is one can not prove
$x^{T}ABx\leq||A||x^{T}Bx$. Unless $A$ and $B$ have same eigenvectors.

Remark 3 In [3], the authors proved

Lemma 2.11. Suppose that $G$ is irreducible and satisfies
$\sum_{j=1}^{m}G_{ij}=0$ with $G_{ij}\geq 0$. Then, there is a positive vector
$x$ such that $G^{T}x=0$.

It is clear that this Lemma is a special case of the following Lemma (see
Lemma 1 in [4]).

Lemma 1 If $A\in\mathbf{A}1$, then the following items are valid:

  1. 1.

If $\lambda$ is an eigenvalue of $A$ and $\lambda\neq 0$, then
$Re(\lambda)<0$;

  2. 2.

$A$ has an eigenvalue $0$ with multiplicity 1 and the right eigenvector
$[1,1,\dots,1]^{\top}$;

  3. 3.

Suppose $\xi=[\xi_{1},\xi_{2},\cdots,\xi_{m}]^{\top}\in R^{m}$ (without loss
of generality, assume $\sum\limits_{i=1}^{m}\xi_{i}=1$) is the left
eigenvector of $A$ corresponding to eigenvalue $0$. Then, $\xi_{i}\geq 0$
holds for all $i=1,\cdots,m$; more precisely,

  4. 4.

$A\in\bf A_{2}$ if and only if $\xi_{i}>0$ holds for all $i=1,\cdots,m$;

  5. 5.

$A$ is reducible if and only if for some $i$, $\xi_{i}=0$. In such case, by
suitable rearrangement, assume that
$\xi^{\top}=[\xi_{+}^{\top},\xi_{0}^{\top}]$, where
$\xi_{+}=[\xi_{1},\xi_{2},\cdots,\xi_{p}]^{\top}\in R^{p}$, with all
$\xi_{i}>0$, $i=1,\cdots,p$, and
$\xi_{0}=[\xi_{p+1},\xi_{p+2},\cdots,\xi_{m}]^{\top}\in R^{m-p}$ with all
$\xi_{j}=0$, $p+1\leq j\leq m$. Then, $A$ can be rewritten as
$\left[\begin{array}[]{cc}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{array}\right]$
where $A_{11}\in R^{p,p}$ is irreducible and $A_{12}=0$.

For detail proof, readers can refer to [5].

Remark 4 Adaptive algorithm in Theorem 2 of [2] is identical to that reported
in Theorem 5 in the section ”Adaptive adjustment of the coupling strength” of
[1].

$\displaystyle\left\\{\begin{array}[]{cc}\frac{dx_{1}(t)}{dt}&=f(x_{1}(t),t)+c(t)\sum\limits_{j=1}^{m}a_{1j}x_{j}(t)\\\
&-c(t)\varepsilon(x_{1}(t)-s(t)),\\\
\frac{dx_{i}(t)}{dt}&=f(x_{i}(t),t)+c(t)\sum\limits_{j=1}^{m}a_{ij}x_{j}(t),\\\
&i=2,\cdots,m\\\ &\dot{c}(t)=\frac{\alpha}{2}\sum\limits_{i=1}^{m}\delta
x_{i}^{T}(t)P\delta x_{i}(t)\end{array}\right.$ (29)

where $c(0)\geq 0$ and $\alpha>0$, can synchronize the coupled system to the
given trajectory $s(t)$.

Remark 5 It is clear that Lyapunov function in Theorem 1 of [2]

$\displaystyle V(t)=\sum_{i=1}^{m}e_{i}(t)^{T}e_{i}(t)$ (30)

is a special case of the Lyapunov function

$\displaystyle V(t)=\sum_{i=1}^{m}\delta x_{i}(t)^{T}P\delta x_{i}(t)$ (31)

reported in Theorem 2 of [1].

Remark 6 It is clear that Lyapunov function in Theorem 3.1 of [3]

$\displaystyle V(t)=\sum_{i=1}^{m}\xi_{i}e_{i}(t)^{T}e_{i}(t)$ (32)

is a special case of the Lyapunov function

$\displaystyle V(t)=\sum_{i=1}^{m}\xi_{i}\delta x_{i}(t)^{T}P\delta x_{i}(t)$
(33)

reported in Theorem 3 in [1].

Remark 7 The section 4 in [3] ”Synchronization criteria via pinning control on
networks with a directed spanning tree” has been reported in [1] in detail.

See section ”C. Pin A Linearly Coupled Network With A Reducible Asymmetric
Coupling Matrix” in section II of [1].

In the following, we remove the assumption that $A$ is irreducible. In this
case, we assume

$\displaystyle A=\left[\begin{array}[]{cccc}A_{11}&0&\cdots&0\\\
A_{21}&A_{22}&\cdots&0\\\ \vdots&\ddots&\vdots&\vdots\\\
A_{p1}&A_{p2}&\cdots&A_{pp}\end{array}\right]$ (38)

where $A_{qq}\in R^{m_{q},m_{q}}$, $q=1,\cdots,p$, are irreducible. And, for
each $q$, there exists $q>k$ such that $A_{qk}\neq 0$. It is equivalent to
that the connecting graph has a spanning tree.

It is easy to see that if we add a single controller $-\epsilon(x_{1}(t)-s(t)$
to the node $x_{1}(t)$. Then, by previous arguments, we conclude that the
subsystem

$\displaystyle\left\\{\begin{array}[]{cc}\frac{dx_{1}(t)}{dt}&=f(x_{1}(t),t)+c\sum\limits_{j=1}^{m_{1}}a_{1j}x_{j}(t)\\\
&-c\varepsilon(x_{1}(t)-s(t)),\\\
\frac{dx_{i}(t)}{dt}&=f(x_{i}(t),t)+c\sum\limits_{j=1}^{m_{1}}a_{ij}x_{j}(t),\\\
&i=2,\cdots,{m_{1}}\end{array}\right.$ (43)

pins $x_{1}(t),\cdots,x_{m_{1}}$ to $s(t)$.

Now, for the subsystem $x_{m_{1}+1},\cdots,x_{m_{2}},$ we have

$\displaystyle\frac{d\delta
x_{i}(t)}{dt}=f(x_{i}(t),t)-f(s(t),t)+c\sum\limits_{j=1}^{m_{2}}a_{ij}\delta
x_{j}(t)$ $\displaystyle=$ $\displaystyle
f(x_{i}(t),t)-f(s(t),t)+c\sum\limits_{j=m_{1}+1}^{m_{2}}a_{ij}\delta
x_{j}(t)+O(e^{-\eta t})$

Because $A_{21}\neq 0$. Then, in $A_{22}$, there exists at least one row
$i_{2}$, such that

$\displaystyle a_{i_{2}i_{2}}>\sum_{j=m_{1}+1}^{m_{2}}a_{i_{2}j}$ (44)

Therefore, all eigenvalues of $A_{22}$ are negative. By the similar arguments
in the proof of theorem 4, we can pin $x_{i}(t)$, $i=m_{1}+1,\cdots,m_{2}$, to
$s(t)$.

By induction, we prove that if we add a controller to the master subsystem
corresponding to the sub-matrix $A_{11}$, then we can ping the complex network
to $s(t)$ even if the coupling matrix is reducible.

Conclusions In this note, we point out that most of the results in papers [2]
and [3] have been reported and are direct consequences of previous papers [1],
[4]. We also correct the proof given in the papers [2] and [3].

References

[1]. Chen, T., Liu, X., and Lu,W., ”Pinning Complex Networks by a Single
Controller”, IEEE Transactions on Circuits and Systems-I: Regular Papers,
54(6), 1317-1326 (2007)

[2]. Yu.,W., Chen,G., Lu.,J., ”On pinning synchronization of complex dynamical
networks”, Automatica 45 429-435 (2009)

[3]. Yu.,W., Chen,G.,Lu.,J., and Jurgen Kurth, ”Synchronization via pinning
control on general complex networks”, SIAM J. CONTROL OPTIM. 51(2), 1395-1416
(2013)

[4]. Wenlian Lu, Tianping Chen, ”New Approach to Synchronization Analysis of
Linearly Coupled Ordinary Differential Systems”, Physica D, 213, 214-230
(2006)

[5]. Wenlian Lu and Tianping Chen, ”A New Approach to Synchronization Analysis
of Linearly Coupled Map Lattices”, Chinese Annals of Mathematics, 28B(2),
149-160 (2007)


# Positroids, Dressian and Stable polynomials

Ayush Kumar Tewari<EMAIL_ADDRESS>
###### Abstract.

Our work is motivated by the connection established between Lorentzian
polynomials and the Dressian in the seminal work of Brändén and Huh on
Lorentzian polynomials. We analyze this relation for the class of positroids,
and are able to show that in this case, we can relate a multiaffine homogenous
stable polynomial to it. Additionally, we also highlight that a conjecture for
matroids posed by Brändén and Huh is true when considered over the class of
Rayleigh matroids which strictly contain the class of positroids. We collect
these findings along with other results for further exploration.

###### Key words and phrases:

positroids, stable polynomails, Rayleigh matroids

###### 1991 Mathematics Subject Classification:

12D10, 52B40

I would like to thank Peter Brändén and Michael Joswig for going through
earlier drafts of this article and for providing valuable suggestions and
comments. I would also like to thank Kevin Purbhoo for discussing his work and
clearing any queries I had about them.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Dressian, Stable and Lorentzian Polynomials
  4. 4 Future Work

##  1\. Introduction

The combinatorics associated with electrical networks carries a lot of really
intricate details related to matroids, an initial documentation of which can
be found in [25]. In subsequent studies [11] the relation between matroids and
the geometry of polynomials is described. In recent work [8] with the
introduction of the notion of _Lorentzian polynomials_ it has been
demonstrated that highly intrinsic properties concerning matroids can be
obtained via findings that are based on the geometry of polynomials. For
example, the well-known Mason conjecture for matroids can be resolved by
methods developed in [8, 14]. These results are also extended to show negative
dependence properties for the $q$-state Potts model partition function [7].

In [11] the authors are interested in studying the _half-plane property_ for
multivariate polynomials. For matroids, this is equivalent to the case when
the basis generating polynomial is _stable_. The authors in [11] also discuss
the various examples of matroids that satisfy the half-plane matroids: uniform
matroids, sixth-root of unity matroids, and all matroids of rank and corank
two satisfy the half-plane property. Moreover, in recent work [16] the authors
provide a complete classification of which matroids on at most $8$ elements
have the half-plane property. These investigations have also now forayed into
asking slightly refined formulations, for example, the _Rayleigh_ property for
polynomials which captures negative dependence correlation between elements of
the bases in a matroid. In [10] the authors introduce the notion of _Rayleigh_
matroid, which is the class of matroids whose basis generating polynomials is
Rayleigh. They also provide examples of matroids which are Rayleigh: regular
matroids and all matroids of rank three are Rayleigh.

In this paper, our emphasis is on the class of _positroids_ , which in essence
captures positivity in the class of matroids. In one of the first results, we
build on the work done in [24] in which they show that a certain multi affine
homogenous polynomial over $\mathbb{C}$ which _represents_ a point in the
Grassmannian is stable if and only if this point represents a nonegative point
which corresponds to a positroid. We extend this result over real and real
closed fields, and use this to refine the characterization of points in the
Dressian of a positroid in the form of this result

######  Theorem 1.

Let $V$ represent a nonegative point in $Gr(k,n)$ and let $\mathcal{M}$ be a
positroid of rank $k$ on $[n]$ elements with bases $\mathcal{B}$ that
corresponds to $V$. Let $p_{B}$ represent the nonzero _Plücker coordinates_ of
$V$, which also satisfy the quadratic Plücker relations. Then for
$\mu\in\text{Trop}\>Gr(\mathcal{M})\subseteq Dr(\mathcal{M})$, the multi
affine homogenous stable polynomial $f$ that _represents_ $V$,

$f=\sum_{B\in\mathcal{B}}p_{B}\>x^{B}$

in $\mathbb{K}[x_{1},\ldots,x_{n}]$ is the one whose tropicalization is $\mu$.

This result is sort of intermediate between the weak half-plane property and
strong half-plane property; in the sense that in the case of the strong plane
property, the polynomial being considered is fixed and it is the basis
generating polynomial of the matroid, whereas, in the case of the weak half-
plane property, it guarantees only the existence of a stable polynomial whose
support is the bases of the matroid.

Additionally, we also share an observation to highlight that Conjecture 3.12
in [8] is true when considered over the class of Rayleigh matroids [10]. This
follows by the definition used in [10], and as this class of matroids strictly
contains the class of positroids [18], we refurbish this result in the
following form,

######  Corollary 2.

The following conditions are equivalent for any non-empty
$J\subseteq\\{0,1\\}^{n}$ :

  1. (i)

$J$ is the set of bases of a Rayleigh matroid on $[n]$.

  2. (ii)

The generating function $f_{J}$ is a homogeneous $1-$Rayleigh polynomial.

Essentially, Conjecture 3.12 [8] is true for the class of Rayleigh matroids.

We conclude with an extension of results relating Lorentzian polynomials and
Dressians to the case of Flag Dressians and we also list some problems which
we aim to work on in the future.

##  2\. Preliminaries

We try to introduce the basic notions in this section which we use in our
subsequent sections. We refer the reader to [22] for details on matroid
theory.

A _matroid_ $\mathcal{M}$ of rank $k$ on the set $E$ is a nonempty collection
$\mathcal{B}\subseteq\binom{E}{k}$ of $k$-element subsets of $E$, called
_bases_ of $\mathcal{M}$, that satisfies the basis exchange axiom: For any
$I,J\in\mathcal{B}$ and $a\in I$, there exists $b\in J$ such that
$I\setminus\\{a\\}\cup\\{b\\}\in\mathcal{B}$.

A matroid is called _representable_ if it can be represented by columns of a
matrix over some field $\mathbb{K}$. We index the columns of a $k\times n$
matrix by the set $[n]$.

The generating function of a matroid $\mathcal{M}$ with bases $\mathcal{B}$
(also referred to as the basis enumeration polynomials in [18]) is defined as
the polynomial

(1) $M(x)=\sum_{B\in\mathcal{B}}x^{B}$

A slightly general notion of _generating function_ of a subset
$J\subseteq\mathbb{N}^{n}$, is defined as follows

$f_{J}=\sum_{\alpha\in
J}\frac{x^{\alpha}}{\alpha!}\quad\text{where}\quad\alpha!=\prod_{i=1}^{n}\alpha_{i}!$

and we see that these two notions agree when $J\subseteq\\{0,1\\}^{n}$ .

The _Grassmannian_ $Gr(k,n)$ is the parameterization of the family of all
$k$-dimensional subspaces of $n$-dimensional vector space in $\mathbb{K}^{n}$.
It enjoys a smooth projective variety structure, corresponding to the
vanishing set of the _Plücker ideal_.

An element $V$ in the Grassmannian $Gr(k,n)$ is understood as a collection of
$n$ vectors $v_{1},\ldots,v_{n}\in\mathbb{K}^{k}$ spanning the space
$\mathbb{K}^{k}$ modulo the simultaneous action of $GL(k,n)$. Let $A$ be a
$k\times n$-matrix consisting of column vectors $v_{1},v_{2},\ldots,v_{n}$. We
call $A$, a _representative_ of $V$. This defines a matroid $\mathcal{M}_{V}$
whose bases are the $k$-subsets $I\subseteq[n]$ such that
$\text{det}(A[I])\neq 0$. It is important to note that this is independent of
the choice of $A$, and only depends on $V$. Here, $\text{det}(A[I])$ denotes
the determinant of the $k\times k$ submatrix of $A$ with the column set
$I\in\binom{[n]}{k}$.

A _positroid_ $P$ of rank $k$ is a matroid that can be represented by a
$k\times n$-matrix $A$ such that the maximal minor $p_{I}$ is non-negative for
each $I\in\binom{[n]}{k}$.

We now introduce the definition of _Lorentzian_ polynomials [8]. We mostly
rely on [8] and [19] for our definitions. Let $n$ and $d$ be nonnegative
integers, and set $[n]=\\{1,\ldots,n\\}$ and $H^{d}_{n}$ to be the set of
degree $d$ homogeneous polynomials in $\mathbb{R}[w_{1},\ldots,w_{n}]$. Let
$f$ be a polynomial in $\mathbb{R}[w_{1},\cdots,w_{n}]$,

$f(x)=\sum_{\alpha\in\mathbb{N}^{n}}\frac{c_{\alpha}}{\alpha!}x^{\alpha}$

The _support_ of the polynomial $f$ is the subset of $\mathbb{N}^{n}$ n
defined by

$\text{supp}(f)=\\{\alpha\in\mathbb{N}^{n}\>\>|\>\>c_{\alpha}\neq 0\\}$

Let $P^{d}_{n}\subset H^{d}_{n}$ be the open subset of polynomials all of
whose coefficients are positive. The _Hessian_ of
$f\in\mathbb{R}[w_{1},\cdots,w_{n}]$ is the symmetric matrix

$\mathcal{H}_{f}(w)=(\partial_{i}\partial_{j}f)^{n}_{i,j=1}$

where $\partial$ denotes the partial derivative $\frac{\partial}{\partial
w_{i}}$.

######  Definition 3.

Set $L^{0}_{n}=P^{0}_{n},L^{1}_{n}=P^{1}_{n}$ , and

$L^{2}_{n}=\\{f\in P^{2}_{n}\>\>|\>\>\mathcal{H}_{f}\>\>\text{is nonsingular
and has exactly one positive eigenvalue}\\}$

For $d\geq 2$, $L^{d}_{n}$ is defined recursively as follows

$L^{d}_{n}=\\{f\in P^{d}_{n}\>\>|\>\>\partial_{i}f\in L^{d-1}_{n}\>\>\text{for
all}\>\>i\in[n]\\}$

The polynomials in $L^{d}_{n}$ are called _strictly Lorentzian_ , and the
limits of strictly Lorentzian polynomials are called _Lorentzian_.

$L^{d}_{n}$ can also be identified with the set of $n\times n$ symmetric
matrices with positive entries that have the _Lorentzian signature_
$(+,-,...,-)$.

######  Definition 4.

A subset $J\subset\mathbb{N}^{n}$ to be $M$-convex if it satisfies any one of
the following equivalent conditions:

  * •

For any $\alpha,\beta\in J$ and any index $i$ satisfying
$\alpha_{i}>\beta_{i}$, there is an index $j$ satisfying

$\alpha_{j}<\beta_{j}\quad\text{and}\quad\alpha-e_{i}+e_{j}\in J$

  * •

For any $\alpha,\beta\in J$ and any index $i$ satisfying
$\alpha_{i}>\beta_{i}$, there is an index $j$ satisfying

$\alpha_{j}<\beta_{j}\quad\text{and}\quad\alpha-e_{i}+e_{j}\in
J\quad\text{and}\quad\beta-e_{i}+e_{j}\in J$

Let $\mu$ be a function from $\mathbb{N}^{n}$ to $\mathbb{R}\cup\\{\infty\\}$.
The effective domain of $\mu$ is, by definition,

$\text{dom}(\mu)=\\{\alpha\in\mathbb{N}^{n}\>\>|\>\>\mu(\alpha)<\infty\\}$

######  Definition 5.

A function $\mu$ from $\mathbb{N}^{n}$ to $\mathbb{R}\cup\\{\infty\\}$ is said
to be $M$-convex if it satisfies the _symmetric exchange property_ :

  * •

For any $\alpha,\beta\in\text{dom}(\mu)$ and any $i$ satisfying
$\alpha_{i}>\beta_{i}$ , there is $j$ satisfying

$\alpha_{j}<\beta_{j}\quad\text{and}\quad\mu(\alpha)+\mu(\beta)\geq\mu(\alpha-
e_{i}+e_{j})+\mu(\beta-e_{j}+e_{i}).$

The effective domain of an $M$-convex function on $\mathbb{N}^{n}$ is an
$M$-convex subset of $\mathbb{N}^{n}$. When the effective domain of a function
$\mu$ is is $M$-convex, the symmetric exchange property for $\mu$ is
equivalent to the following _local exchange property_ ,

  * •

For any $\alpha,\beta\in\text{dom}(\mu)$ with $|\alpha-\beta|_{1}=4$, there
exist $i$ and $j$ satisfying

$\alpha_{i}>\beta_{i},\alpha_{j}<\beta_{j}\quad\text{and}\quad\mu(\alpha)+\mu(\beta)\geq\mu(\alpha-
e_{i}+e_{j})+\mu(\beta-e_{j}+e_{i}).$

A function $\mu:\mathbb{N}^{n}\rightarrow\mathbb{R}\cup\\{\infty\\}$ is said
to be $M$-concave if $-\mu$ is $M$-convex. The effective domain of an
M-concave function $\mu$ is,

$\text{dom}(\mu)=\\{\alpha\in\mathbb{N}^{n}\>\>|\>\>\mu(\alpha)>-\infty\\}$

A _valuated matroid_ on $[n]$ is an $M$-concave function on $\mathbb{N}^{n}$
whose effective domain is a nonempty subset of $\\{0,1\\}^{n}$ . The effective
domain of a valuated matroid $\mu$ on $[n]$ is the set of bases of a matroid
on $[n]$, the _underlying matroid_ of $\mu$.

Let $M^{d}_{n}(\mathbb{K})$ denote the set of all all degree $d$ homogeneous
polynomials in $K_{\geq 0}[w_{1},\cdots,w_{n}]$ whose support is $M$-convex.

Although, the standard definition of Lorentzian polynomials is over
$\mathbb{R}$ in [8] itself the authors provide a definition over any field
$\mathbb{K}$ as follows,

######  Definition 6.

Set
$L^{0}_{n}(\mathbb{K})=M^{0}_{n}(\mathbb{K}),L^{1}_{n}(\mathbb{K})=M^{1}_{n}(\mathbb{K})$,
and

$L^{2}_{n}(\mathbb{K})=\\{f_{t}\in M^{2}_{n}(\mathbb{K})\>\>|\>\>\text{The
Hessian of}\>\>f_{t}\>\>\text{has at most one eigenvalue
in}\>\>\mathbb{K}_{\geq 0}\\}$

For $d\geq 3$, we define $L^{d}_{n}(\mathbb{K})$ as follows

$L^{d}_{n}(\mathbb{K})=\\{f_{t}\in
M^{d}_{n}(\mathbb{K})\>\>|\>\>\partial^{\alpha}f_{t}\in
L^{2}_{n}(\mathbb{K})\>\>\text{for all}\>\>\alpha\in\Delta^{d-2}_{n}\\}$

where $\Delta^{d-2}_{n}$ represents the discrete $(d-2)-$th simplex.

We now discuss the background of the definition of stable polynomials and how
they are related to the study of Lorentzian polynomials. In earlier works, the
definition of stable polynomials is referred to as the _half plane property_
[11]. Let $\mathcal{H}=\\{x\in\mathbb{C}\>\>|\>\>\text{Re}x>0\\}$ denote the
open upper half plane.

######  Definition 7.

A polynomial $f$ in $\mathbb{R}[w_{1},\cdots,w_{n}]$ is called _stable_ if $f$
is nonvanishing on $\mathcal{H}^{n}$ or identically zero.

In this case, $f$ is also said to satisfy the _half plane property_.

######  Remark 8.

In [11], the half-plane property is also defined for a matroid. $\mathcal{M}$
is said to satisfy the _half plane property_ if the basis generating
polynomial of $\mathcal{M}$ satisfies the half-plane property. Additionally,
$\mathcal{M}$ is said to satisfy the _weak half plane_ property if there
exists a polynomial that satisfies the half-plane property and whose support
is the basis of $\mathcal{M}$ [11].

Let $S^{d}_{n}$ be the set of degree $d$ homogeneous stable polynomials in $n$
variables with nonnegative coefficients and by Hurwitz Theorem
$S^{d}_{n}\subset H^{d}_{n}$. We refer the reader to [31] to explore the rich
theory of stable polynomials. Any polynomial $f\in S^{d}_{n}$ is the limit of
_strictly stable polynomials_ [20].

Lorentzian polynomials are a generalization of stable polynomials, i.e., all
stable polynomials are Lorentzian, moreover, any homogeneous stable polynomial
is a constant multiple of a Lorentzian polynomial [8].

In [11] the authors envisaged that for large classes of matroids, the _basis
generating polynomials_ are stable. Subsequent studies also concentrate on
certain relaxations of this class, the most prominent of them being _Rayleigh_
polynomials,

######  Definition 9.

Let $c$ be a fixed positive real number, and let $f$ be a polynomial in
$\mathbb{R}[w_{1},\cdots,w_{n}]$. $f$ is called $c$-_Rayleigh_ if $f$ has
nonnegative coefficients and

$\partial^{\alpha}f(x)\>\partial^{\alpha+e_{i}+e_{j}}f(x)\leq
c\>\partial^{\alpha+e_{i}}f(x)\>\partial^{\alpha+e_{j}}f(x)\quad\text{for
all}\>\>i,j\in[n],\alpha\in\mathbb{N}^{n},x\in\mathbb{R}^{n}_{\geq 0}$

When $f$ is multi-affine, that is, when $f$ has degree at most one in each
variable, the $c$-Rayleigh condition for $f$ is equivalent to

$f(x)\partial^{i}\partial^{j}f(x)\leq
c\>\>\partial^{i}f(x)\partial^{j}f(x)\>\>\text{for all
distinct}\>\>i,j\in\mathbb{N}^{n},\>\>\text{and}\>\>x\in\mathbb{R}^{n}_{\geq
0}$

A multi-affine polynomial $f$ is said to be _strongly Rayleigh_ if

$f(x)\partial^{i}\partial^{j}f(x)\leq
c\>\>\partial^{i}f(x)\partial^{j}f(x)\>\>\text{for all
distinct}\>\>i,j\in\mathbb{N}^{n},\>\>\text{and}\>\>x\in\mathbb{R}^{n}$

A multi-affine polynomial is stable if and only if it is strongly Rayleigh [6,
Theorem 5.6]. Based on the definition of Rayleigh polynomials, the notion of
_Rayleigh_ matroids [10] was introduced, where a matroid $\mathcal{M}$ is a
Rayleigh matroid if its basis generating polynomial is Rayleigh. Similarly,
there also exists the class of _strongly Rayleigh matroid_ , where a matroid
is a _strongly Rayleigh matroid_ if its basis generating polynomial is
strongly Rayleigh. It is straightforward to see from [6, Theorem 5.6] that all
strongly Rayleigh matroids satisfy the half-plane property.

We now define notions related to tropical geometry that appear in our
discussion and we refer the reader to [17] for further details. Tropical
geometry is the study of vanishing sets of polynomials defined over the
_tropical_ semiring
$\mathbb{T}=\\{\mathbb{R}\cup\\{-\infty\\},\text{max}=\oplus,+=\odot\\}$. A
_tropical_ polynomial is a polynomial defined over $\mathbb{T}$ with the
binary operations replaced by $\oplus$ and $\odot$. $\text{Trop}(f)$ denotes
the _tropical hypersurface_ associated with $f$ which is the collection of all
points where the maxima is achieved at least twice.

The _tropical Grassmannian_ $\text{TropGr}(k,n)$ is the intersection of the
tropical hypersurfaces $\text{Trop}(f)$, where $f$ ranges over all elements of
the _Plücker ideal_ $\mathcal{I}_{k,n}$ which is generated by the _quadratic
Plücker relations_ , and therefore it is also a tropical variety [17]. The
_Dressian_ $Dr(k,n)$ is the intersection of the tropical hypersurfaces
$\text{Trop}(f)$, where $f$ ranges over all three-term Plücker relations, that
generate the _Plücker ideal_ and hence it possesses the structure of a
tropical prevariety [17]. The underlying matroid for the definitions of the
tropical Grassmannian and Dressian is the _uniform matroid_
$\mathcal{U}_{k,n}$. However, the notion of Dressian has been extended to
arbitrary matroids with the idea of the _local Dressian_. The _local Dressian_
$Dr(M)$ is defined as the tropical pre-variety given by the set of quadrics
obtained from the three-term Plücker relations by setting the variables
$p_{B}$ to zero, where $B$ is not a basis of $M$ [21].

######  Remark 10.

We acknowledge that in some instances, all Plücker relations (not necessarily
only the 3-term Plücker relations) are considered for the definition of the
Dressian. However, by the work in [3] we know that these two notions coincide
over the tropical hyperfield and hence our definition would remain consistent
for our discussion.

##  3\. Dressian, Stable and Lorentzian Polynomials

Since the introduction of the notion of a Dressian, it has been noted that the
points residing in the Dressian enjoy multiple avatars in which they can be
treated and satisfy multiple properties. The subject for initial findings was
the Dressian $Dr(k,n)$ with the underlying matroid being the uniform matroid
$\mathcal{U}_{k,n}$. We collect the various previously known [17, 29, 8, 30]
equivalent notions for a point residing in the Dressian $Dr(k,n)$ in the form
of Theorem 11.

######  Theorem 11.

Let $\mu$ be a point in $Dr(k,n)$. Then the following are equivalent

  1. (i)

$\mu$ is a valuated matroid with the underlying matroid $\mathcal{U}_{k,n}$.

  2. (ii)

$\mu$ defines a $M-convex$ function on the matroid $\mathcal{U}_{k,n}$.

  3. (iii)

$\mu$ satisfies the tropical three-term Plücker relations.

  4. (iv)

$\mu$ as a weight vector induces a regular matroidal subdivision of
$\Delta(k,n)$.

  5. (v)

$\mu=\text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on
the bases set $\mathcal{B}$ of $\mathcal{U}_{k,n}$.

  6. (vi)

$\mu$ defines a metric for a generalised metric tree arrangement.

These equivalences can also be extended to the local Dressian
$Dr(\mathcal{M})$ [21], which can be defined for an arbitrary matroid
$\mathcal{M}$, and it generalizes the notion of Dressian $Dr(k,n)$, and we
list the corresponding equivalences for a point residing in the local Dressian
in Theorem 12,

######  Theorem 12.

Let $\mu$ be a point in $Dr(\mathcal{M})$. Then the following are equivalent

  1. (i)

$\mu$ is a valuated matroid with the underlying matroid $\mathcal{M}$.

  2. (ii)

$\mu$ defines a $M-convex$ function on the matroid $\mathcal{M}$.

  3. (iii)

$\mu$ satisfies the tropical three-term Plücker relations.

  4. (iv)

$\mu$ as a weight vector induces a regular matroidal subdivision of
$\mathcal{P}_{M}$.

  5. (v)

$\mu=\text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on
the bases set $\mathcal{B}$ of $\mathcal{M}$.

The categorization given in (v) is one of the most recent ones, proven in [8,
Theorem 3.20].

Our aim now is to possibly refine the characterization of the Dressian,
concerning Lorentzian polynomials. We know that the class of Loentzian
polynomials contains the class of stable polynomials [8]. Of interest in this
regard is the work in [24], in which the author relates the notions of
stability and total nonnegativity. We first briefly describe the setup used in
[24] to make ideas clearer. Consider the matrix $M\in\text{Mat(k,n)}$ of rank
$k$ over the field of complex number. The column space of the matrix provides
us a k-dimensional subspace $V$ which lies in the $Gr(k,n)$. Let $M[I]$ denote
the $k\times k$ submatrix of $M$ with row set $I\in\binom{n}{k}$. The _Plücker
coordinates_ of $V$ are the maximal minors
$\Bigl{\\{}\text{det}(M[I]):I\in\binom{n}{k}\Bigr{\\}}$. The homogenous multi-
affine polynomial,

(2) $f=\sum_{I\in\binom{n}{k}}\text{det}(M[I])\>x^{I}$

where $x^{I}=\prod_{i\in I}x_{i}$, is said to _represent_ $V$ in $Gr(k,n)$
[24]. Additionally, a necessary and sufficient condition for a polynomial to
represent $V$ is that the coefficients satisfy the quadratic _Plücker_
relations, which are the defining equations for the $Gr(k,n)$, and provide it
the structure of a projective variety. We recall the following result from
[24] concerning the multi affine polynomial in Equation 2

######  Theorem 13 (Theorem 1.1 [24]).

Let $f\in\mathbb{C}[x]$ be a multi-affine homogenous polynomial of degree $k$
that represents a point $V$ in $Gr(k,n)$. Then, $f$ is stable if and only if
$V$ is totally nonnegative.

We state the above result over the field of real numbers,

######  Corollary 14.

Let $f\in\mathbb{R}[x]$ be a multi-affine homogenous polynomial of degree $k$
that represents a point $V$ in $Gr(k,n)$. Then, $f$ is stable if and only if
$V$ is totally nonnegative. Moreover, if $\mathbb{K}$ is a real closed field
and $f\in\mathbb{K}[x]$ be a multi-affine homogenous polynomial of degree $k$
that represents a point $V$ in $Gr(k,n)$ then, $f$ is stable if and only if
$V$ is totally nonnegative.

###### Proof.

With the well-known “phase theorem” [11, Theorem 6.1] we know that every
homogeneous stable polynomial has non-negative real coefficients up to scalar
multiples. Hence, the stability of the polynomial in Theorem 13 over
$\mathbb{C}$ is equivalent to stability of over $\mathbb{R}$ [23].

Since, the field of real numbers also allows quantifier elimination, therefore
by invoking the Tarski principle as done in [8] , we can extend Corollary 14
over real closed fields as well. ∎

We also know that $V\in Gr(k,n)$ determines a representable matroid of rank
$k$ on the set $[n]$, by taking the bases to be the indices of the nonzero
Plücker coordinates. If $V$ is totally nonnegative, this matroid is a
positroid [24]. With this observation, we want to consider a restriction of a
result on matroids [8, Theorem 3.20] to the class of positroids. We first
recall this result,

######  Theorem 15 (Theorem 3.20 [8]).

The following conditions are equivalent for any function
$\mu:\Delta(k,n)\rightarrow\mathbb{Q}\cup\\{\infty\\}$

  1. (i)

the function $\mu$ is M-convex;

  2. (ii)

there is a Lorentzian polynomial in $\mathbb{K}[w_{1},\ldots,w_{n}]$ whose
tropicalization is $\mu$.

We also elaborate on the background setup for this statement used in [8]. In
this statement, the field $\mathbb{K}$ is the real convergent Puiseux series

$\mathbb{K}=\bigcup_{k\geq 1}\mathbb{R}((t^{1/k}))_{\text{conv}}$

which is a _real closed field_ and its corresponding algebraic closure is the
field

$\overline{\mathbb{K}}=\bigcup_{k\geq 1}\mathbb{C}((t^{1/k}))_{\text{conv}}$

The definition of Lorentzian polynomials over $\mathbb{K}$ are also provided
in [8, Definition 3.17]. We propose the following refinement of Theorem 15,
when we restrict ourselves to the class of positroids, in which case we see
that we obtain a relation to homogenous stable polynomials which are a
subclass of Lorentzian polynomials.

######  Theorem 16.

Let $V$ represent a nonnegative point in $Gr(k,n)$ and let $\mathcal{M}$ be a
positroid of rank $k$ on $[n]$ elements with bases $\mathcal{B}$ that
corresponds to $V$. Let $p_{B}$ represent the nonzero _Plücker coordinates_ of
$V$, which also satisfy the quadratic Plücker relations. Then for
$\mu\in\text{Trop}\>Gr(\mathcal{M})\subseteq Dr(\mathcal{M})$, the multi
affine homogenous stable polynomial $f$ that _represents_ $V$,

$f=\sum_{B\in\mathcal{B}}p_{B}\>x^{B}$

in $\mathbb{K}[x_{1},\ldots,x_{n}]$ is the one whose tropicalization is $\mu$.

###### Proof.

We consider the homogenous multi-affine polynomial,

(3) $f=\sum_{B\in\mathcal{B}}p_{B}\>x^{B}$

where $x^{B}=\prod_{b\in B}x_{b}$, which _represents_ $V$. By Corollary 14,
this polynomial is stable over $\mathbb{K}$ with coefficients satisfying the
quadratic Plücker relations. We also realize that the tropicalization
$\text{trop}(f)$ is defined by the tropicalization of the coefficients which
are just Plücker coordinates. However, these tropical Plücker relations are
the relations that define a point in the tropical Grassmannian
$\text{TropGr}(k,n)$. Therefore, we realize that $\text{trop}(f)$ has
coefficients that satisfy the tropical Plücker relations, which defines a
point $\mu\in\text{TropGr}(k,n)\subseteq Dr(\mathcal{M})$. ∎

We also note that this relation between stable polynomials and positroids does
not extend to the class of matroids, for example in [6] it is shown that if we
consider the matroid $\mathcal{M}$ as the Fano plane then there is no stable
polynomial whose support is $B$. However, the existence of a homogenous multi
affine stable polynomial over the support of a matroid is known for matroids
representable over $\mathbb{C}$ [11, Corollary 8.2]. We want to highlight that
Theorem 16 emphasizes the fact that for the class of positroids, we can
actually provide the exact form of the homogenous multi affine stable
polynomial whose support is the basis of the positroid and whose
tropicalization resides as a point in the Dressian.

Another result from [8] that we are interested in studying is the conjecture
concerning the generating function of matroids, which the authors state as
follows

######  Conjecture 17 (Conjecture 3.12 [8]).

The following conditions are equivalent for any non-empty
$J\subseteq\\{0,1\\}^{n}$ :

  1. (i)

$J$ is the set of bases of a matroid on $[n]$.

  2. (ii)

The generating function $f_{J}$ is a homogeneous $\frac{8}{7}$ -Rayleigh
polynomial.

The constant $\frac{8}{7}$ appearing in this conjecture is strict in the sense
that for any positive real number $c<\frac{8}{7}$, there is a matroid whose
basis generating function is not $c$-Rayleigh [14].

We now highlight that this conjecture is true for the class of Rayleigh
matroids which strictly contain the class of positroids [18] and moreover, a
stronger statement is true for them in the form of Theorem 18.

######  Corollary 18.

The following conditions are equivalent for any non-empty
$J\subseteq\\{0,1\\}^{n}$ :

  1. (i)

$J$ is the set of bases of a Rayleigh matroid on $[n]$.

  2. (ii)

The generating function $f_{J}$ is a homogeneous $1-$Rayleigh polynomial.

Essentially, Conjecture 17 is true for the class of Rayleigh matroids.

###### Proof.

For the implication (i) $\implies$ (ii) by the definition of Rayleigh matroids
in [10], the basis generating function of a Rayleigh matroid is 1-Rayleigh.
Also, if $f$ is a 1-Rayleigh polynomial, then $f$ is $c$-Rayleigh polynomial
for all $c\geq 1$. Additionally, for the opposite implication i.e., (ii)
$\implies$ (i) is straightforward in this case by [8, Theorem 3.10], which
implies that $J$ is the basis of a matroid, which in turn is Rayleigh by our
assumption in this implication. Therefore, this helps us to see that
Conjecture 17 is true for the class of Rayleigh matroids. ∎

An interesting observation is that the statement of Corollary 18 when
considered over the class of positroids, fails to hold true which is confirmed
from the example of the _Vamos_ matroid, which is not representable over any
field and hence is not a positroid, even though it is 1-Rayleigh. This also
explains the strict inclusion of positroids in the class of Rayleigh matroids.

##  4\. Future Work

Motivated by Theorem 11, we now consider the case of the positive Dressian
$Dr^{+}(k,n)$, which is based on the definition of the positive part of a
tropical variety introduced in [27]. A very significant result pertaining to
the positive Dressian is the following [28],

######  Theorem 19 (Theorem 3.9 [28]).

The positive tropical Grassmannian $\text{Trop}^{+}Gr(k,n)$ equals the
positive Dressian $Dr^{+}(k,n)$.

A version of this theorem in terms of the local Dressian is proven in [2]

######  Theorem 20 (Theorem 9.2 [2]).

Let $\mathcal{M}$ be a positroid. Then the positive Dressian
$Dr^{+}(\mathcal{M})$ equals the positive tropical positroidal cell
$\text{Trop}^{+}\Pi_{\mathcal{M}}$.

Theorem 22 collects all previously known equivalent notions [28, 32, 2] of a
point residing in the positive Dressian and captures the essence of Theorem
11,

######  Theorem 21.

Let $\mu$ be a point in $Dr^{+}(d,n)$. Then the following are equivalent

  1. (i)

$\mu$ is a realizable valuated matroid with the underlying positroid
$\mathcal{U}_{d,n}$.

  2. (ii)

$\mu$ defines a realizable $M-convex$ function.

  3. (iii)

$\mu$ satisfies the positive tropical three-term Plücker relations.

  4. (iv)

$\mu$ as a weight vector induces a regular positroidal subdivision of
$\Delta(d,n)$.

  5. (v)

$\mu=\text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on
the bases set $\mathcal{B}$ of the positroid $\mathcal{U}_{d,n}$.

We now present an extension of this theorem to the case of an arbitrary
positroid in Theorem 22, which mostly builds on previously known results in
[2, 28].

######  Theorem 22.

Let $\mathcal{M}$ be a positroid and $\mu$ be a point in
$Dr^{+}(\mathcal{M})$. Then the following are equivalent

  1. (i)

$\mu$ is a realizable valuated matroid with the underlying positroid
$\mathcal{M}$.

  2. (ii)

$\mu$ defines a $M-convex$ function on the positroid $\mathcal{M}$.

  3. (iii)

$\mu$ satisfies the positive tropical three-term Plücker relations.

  4. (iv)

$\mu$ as a weight vector induces a regular positroidal subdivision of
$\mathcal{P}_{\mathcal{M}}$.

  5. (v)

$\mu=\text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on
the bases set $\mathcal{B}$ of the positroid $\mathcal{M}$.

###### Proof.

The equivalence between (i) $\iff$ (ii) is by definition. The equivalences for
(ii) $\iff$ (iii) are established in [2, Proposition 8.2]. We concentrate on
proving the equivalence of (i). We first concentrate on proving the
equivalence (i) $\iff$ (iii). We already know that a point $\mu\in
Dr^{+}(\mathcal{M})\implies\mu\in Dr(\mathcal{M})$, hence the implication that
$\mu$ defines a valuated matroid follows from the inclusion in the Dressian
$Dr(\mathcal{M})$, the only nontrivial part being the realizability of $\mu$.
The stratification of the tropical positive Grasmmanian for a positroid is
discussed in [2] and it is defined in terms of positroidal cells as

(4)
$\text{Trop}^{+}\Pi_{\mathcal{M}}=\text{val}(\tilde{\Pi}_{\mathcal{M}}(\mathcal{R}_{\geq
0}))=Dr^{+}(\mathcal{M})$

where $\mathcal{R}_{\geq 0}$ represents the positive part of the generalized
Puiseux series and $val()$ represents the _valuaton_ map. The equality in 4
implies that all positive Plücker vectors are realizable therefore by [2,
Theorem 9.2], [9, Remark 3.1.7] implies that the valuated matroid $\mu$ on the
positroid $\mathcal{M}$ is realizable. So (i) $\iff$ (iii). The implication in
(v) is straightforward with the inclusion of $\mu\in Dr(\mathcal{M})$
restricted to a positroid.

∎

We also look at a counterpart of the Dressian, defined in the case of flag
matroids, the _Flag Dressian_ introduced in [9] $FlDr(M)$, where
$M:=(M_{1},\ldots M_{n})$ is a flag matroid, and it is the tropical variety
cut out by the incidence-Plücker relations and the Grassmann-Plücker
relations. A classification result similar to Theorem 11 is proven in [9]

######  Theorem 23 (Theorem A [9]).

Let $\mu=(\mu_{1},\ldots,\mu_{k})$ be a sequence of valuated matroids such
that its sequence of underlying matroids $M=(M_{1},\ldots,M_{k})$ is a flag
matroid. Then the following are equivalent:

  1. (i)

$\mu$ is a point on $FlDr(M)$, i.e. it satisfies tropical incidence-Plücker
relations and Grassmann-Plücker relations,

  2. (ii)

$\mu$ is a valuated flag matroid with underlying flag matroid $M$.

  3. (iii)

$\mu$ induces a subdivision of the base polytope of $M$ into base polytopes of
flag matroids,

  4. (iv)

the projective tropical linear spaces $\overline{trop(\mu_{i})}$ form a flag
$\overline{trop(\mu_{1})}\subseteq\ldots\subseteq\overline{trop(\mu_{k})}$

Our first move concerning the Flag Dressian is establishing the equivalence
between points in the Flag Dressian and a sequence of Lorentzian polynomials.

######  Theorem 24.

Let $\mu=(\mu_{1},\ldots,\mu_{k})$ be a sequence of valuated matroids such
that its sequence of underlying matroids $M=(M_{1},\ldots,M_{k})$ is a flag
matroid. Then the following are equivalent

  1. (i)

$\mu$ is a point on $FlDr(M)$,

  2. (ii)

there exist a sequence of Lorentzian polynomials $f_{1},\ldots f_{k}$ such
that $\mu$ = $(\text{trop}(f_{1})=\mu_{1},\ldots,\text{trop}(f_{k})=\mu_{k})$,
where $(\mu_{1},\ldots\mu_{k})$ forms a valuated flag matroid and $f_{i}$ is a
Lorentzial polynomial defined on the bases set $B_{i}$ of $M_{i}$.

###### Proof.

We first consider the implication (i) $\implies$ (ii). Each valuated matroid
$\mu_{i}$ corresponds to a M-convex function on $M_{i}$ by definition, and by
[8, Theorem 3.20] this further corresponds to a Lorentzian polynomial $f_{i}$
such that $\text{trop}(f_{i})=\mu_{i}$. Therefore, for $\mu\in FlDr(M)$, there
exists a sequence of Lorentzian polynomials $f_{1},\ldots f_{k}$, such that
$\mu$ = $(\text{trop}(f_{1})=\mu_{1},\ldots,\text{trop}(f_{k})=\mu_{k})$,
where $f_{i}$ is a Lorentzial polynomial defined on the bases set $B_{i}$ of
$M_{i}$. We now try to prove the implication (ii) $\implies$ (i). Given a
sequence of Lorentzian polynomials such that $\mu$ =
$(\text{trop}(f_{1})=\mu_{1},\ldots,\text{trop}(f_{k})=\mu_{k})$ is a valuated
flag matroid on the underlying matroid $M=(M_{1},\ldots,M_{k})$, which is
equivalent to saying that $\mu\in FlDr(M)$ by 23. ∎

We acknowledge that stipulating that a sequence of flag matroids form a
valuated flag matroid is a very strong condition and it might be worthwhile to
explore if a refinement of the statement above is possible with weaker
conditions.

There has been recent work on extending the equivalent conditions for points
residing in the Flag Dressian $FlDr(M)$ to the case of the positive Flag
Dressian $FlDr^{+}(M)$, specifically in the case of the _complete tropical
flag variety_ [5, 15, Theorem A] and we wish to explore the relation between
the positive Flag Dressian and Lorentzian polynomials in this case in future
work.

Another important aspect closely related to stable polynomials that we want to
highlight is their connection with _hyperbolic polynomials_ and _positively
hyperbolic_ varieties, studied in detail in [4] and [26] respectively. We
first recall the definitions of a _hyperbolic polynomial_[4, 1] and
_positively hyperbolic_ variety [26].

######  Definition 25.

A homogeneous polynomial $h(x)\in\mathbb{R}[x_{1},\cdots,x_{n}]$ is hyperbolic
with respect to a vector $e\in\mathbb{R}^{n}$ if $h(e)\neq 0$, and if for all
$x\in\mathbb{R}^{n}$ the univariate polynomial $t\rightarrow h(te-x)$ has only
real zeros.

If h is a hyperbolic polynomial of degree d, then we may write

$h(te-x)=h(e)\prod_{j=1}^{d}(t-\lambda_{j}(x))$

where

$\lambda_{\text{max}}(x)=\lambda_{1}(x)\geq\lambda_{2}(x)\geq\cdots\geq\lambda_{d}(x)=\lambda_{\text{min}}(x)$

are called the _eigenvalues_ of $x$ (with respect to $e$). The _hyperbolicity
cone_ of $h$ with respect to $e$ is the set
$\Lambda_{+}(h,e)=\\{x\in\mathbb{R}^{n}:\lambda_{\text{min}}(x)\geq 0\\}$.

We now relate the connection between stable and hyperbolic polynomials [1,
Lemma 2.7],

######  Lemma 26.

Let $P\in\mathbb{R}[x_{1},\cdots,x_{n}]$ be a homogeneous polynomial. Then $P$
is stable if and only if $P$ is hyperbolic with
$\mathbb{R}^{n}_{+}\subseteq\Lambda_{+}$.

We wish to explore these connections between hyperbolic polynomials and stable
polynomials in the context of the Dressian in future work.

######  Definition 27.

Let $X\subset\mathbb{C}^{n}$ be a variety which is equidimensional of
codimension $c\leq n-1$. X is called _positively hyperbolic_ if for every
linear subspace $L$ in the positive Grassmannian $Gr^{+}(c,n)$ and every $x\in
X$, the imaginary part Im(x) does not belong to $L\setminus\\{0\\}$. A
projective variety in $\mathbb{P}^{n-1}$ is positively hyperbolic if its
affine cone in $\mathbb{C}^{n}$ is.

A straightforward conclusion from [26, Proposition 2.2] and [24, Theorem 1.1]
is that the variety defined by the homogenous multi-affine polynomial in
Equation 2, which _represents_ $V$ in $Gr(k,n)$, is positively hyperbolic if
$V$ corresponds to a positroid.

Another aspect of our discussion is the classification of various classes of
matroids on whether they are Rayleigh, strongly Rayleigh, or if they satisfy
the half-plane property. Before moving on, we also briefly discuss the notion
of _balanced matroids_ [13] and how they are related to our discussion.

######  Definition 28.

Let $\mathcal{M}$ be a matroid on the ground set $E$ with basis $\mathcal{B}$.
Let $\text{Pr}(e)$ denote the probability of an element $e$ being present in
basis element $B$, which is chosen uniformly at random. The matroid
$\mathcal{M}(E,\mathcal{B})$ is said to satisfy the _negatively correlated_
property if

$\text{Pr}(ef)\leq\text{Pr}(e)\text{Pr}(f)$

for all pair of distinct $e,f\in E$.

######  Definition 29.

A matroid $\mathcal{M}$ is said to be _balanced_ if all its minors including
itself satisfy the negative correlation property.

It is clear from the definition that Rayleigh matroids are balanced. In [10]
the authors also provide an explicit example of a transversal matroid of rank
4 [11, Proposition 5.9] (also updated in [14]), which is not balanced and
hence is also not Rayleigh. We recall the example here. Let $\mathcal{L}$ be a
matroid on the ground set $E=\\{1,2,\cdots,10,e,f\\}$ and $\mathcal{B}$ denote
its bases, where the bases are the four transversals to the sets
$\\{1,2,3,4,f\\},\\{5,6,7,f\\},\\{8,9,10,f\\}$ and $\\{1,2,3,5,6,8,9,e,f\\}$

Equivalently the Rayleigh condition can be restated in terms of probability
distribution in the following way [14] for $B\in\mathcal{B}$ chosen uniformly
at random,

$\text{Pr}(i\in B,j\in B)\>\>\text{Pr}(i\not\in B,j\not\in
B)\leq\text{Pr}(i\in B,j\not\in B)\>\>\text{Pr}(j\in B,i\not\in B)$

But,

$\text{Pr}(i\in B,j\in B)\>\>\text{Pr}(i\not\in B,j\not\in B)=0.04355$

and

$\text{Pr}(i\in B,j\not\in B)\>\>\text{Pr}(j\in B,i\not\in B)=0.04298$

which shows that $L$ is not Rayleigh. However, we want to point out the nuance
here that this still does not provide any counterexample to Conjecture 17, as
we see clearly that $L$ is still $\frac{8}{7}$\- Rayleigh. Therefore, one of
the problems that we want to work on is

######  Problem 30.

Verify Conjecture 17 for the class of transversal matroids.

In the class of transversal matroids, it is already known that lattice path
matroids are Rayleigh [12]. Hence, we initially aim to consider families of
transversal matroids that are not necessarily lattice path matroids, and try
to verify if Conjecture 17 holds true for them.

Also as previously mentioned the authors in [11] suggested that it might be
the case that a large subclass of transversal matroids satisfy the half-plane
property. However, since all matroids that satisfy the half-plane property are
Rayleigh and now we know that the class of transversal matroids is not
Rayleigh, therefore it also does not satisfy the half-plane property. But we
wish to modify the problem in the following way,

######  Problem 31.

Find the largest class of transversal matroids for which it satisfies the
half-plane property.

Pertaining to positroids, we wish to work on the following problem

######  Problem 32.

Are positroids strongly Rayleigh?

With the complete classification of matroids on at most eight elements in
[16], one is tempted to search for a counterexample for Problem 32 by looking
for a positroid in the list of 22 matroids in [16, Theorem 5.13] which do not
satisfy the half-plane property, and hence are not strongly Rayleigh. We did
verify that none of these 22 matroids are positroids and hence no
counterexample comes from this list. However, we want to highlight that this
does not guarantee that a counterexample of Problem 32 does not exist in the
form of a matroid on eight elements, because the property of being a positroid
is not invariant under matroid isomorphisms, and the classification in [16]
relies on a listing of matroids up to isomorphisms.

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# Photoinduced collective mode, inhomogeneity, and melting in a charge-order
system

Hitoshi Seo1,2, Yasuhiro Tanaka3, and Sumio Ishihara4 1Condensed Matter Theory
Laboratory, RIKEN, Wako 351-0198, Japan  2Center for Emergent Matter Science
(CEMS), RIKEN, Wako 351-0198, Japan  3Department of Physics, Chuo University,
Tokyo 112-8551, Japan  4Department of Physics, Tohoku University, Sendai
980-8578, Japan

###### Abstract

We theoretically investigate photoresponses of a correlated electron system
upon stimuli of a pulsed laser light. Real-time dynamics of an interacting
spinless fermion model on a one-dimensional chain, as a model of charge order
(CO), are numerically simulated using the time-dependent Hartree-Fock method.
In particular, we discuss the differences between two situations as the
initial state: the homogeneous order and the presence of a domain wall, i.e.,
a kink structure embedded in the CO bulk. Coherent dynamics are seen in the
former case: When the frequency of the pump light $\omega_{p}$ is varied,
along with single particle excitations across the CO gap
($\Delta_{\textrm{CO}}$), the resonantly-excited collective phase mode near
$\omega_{p}\simeq\Delta_{\textrm{CO}}/2$ efficiently destabilizes CO. In clear
contrast, in the latter case, when $\omega_{p}$ is tuned at such in-gap
frequencies and the intensity of light is sufficiently large, inhomogeneity
spreads out from the kink to the bulk region through kink creations. Moreover,
even stronger intensity induces the inhomogeneous melting of CO where the CO
gap is destroyed.

##  I Introduction

Photoresponses of condensed matter are now increasingly upon intense research
thanks to rapid advancements in technologies Orenstein ; Kampfrath . Ultrafast
spectroscopies have been applied to a wide variety of correlated systems,
whose displayed physical phenomena include Mott metal-insulator transition
Iwai , charge order (CO) Fiebig , charge-density-wave Chollet ; Tomaljak ,
superconductivity Matsunaga , and magnetism Ogasawara . The understanding of
their dynamical responses upon pulsed laser light emission is of fundamental
interest owing to its nonequilibrium and nonstationary nature, which is mostly
unexplored for correlated matters. Elucidating their mechanisms is crucial
also for possible future applications using phase control. In fact, in certain
materials the light irradiation induces macroscopic conversion of phases; the
phenomenon called photoinduced phase transition (PIPT) Nasu ; Tokura  occurs.
Among them, molecular crystals have been intensively studied, taking advantage
of their lower energy scale which in turn results in slower time scale
compared to other materials, for example, transition metal oxides. Such a
feature enables us to access the early stages leading to PIPT, and even the
initial processes governed by electronic degrees of freedom have started to
become visible.
Experimentally, the existence of spatial inhomogeneity is often indicated to
emerge through the PIPT process, and is discussed to be an essential
ingredient for the realization of PIPT, referred to as the domino effect
Koshihara . A well known example is the case of quasi-one-dimensional mixed
stack compound TTF-CA [tetrathiafulvalene-p-chloranil] which shows a valence
transition by the collective charge transfer between two kinds of molecules
TTF and CA. The light irradiation to the low temperature ionic phase leads to
a transient neutral phase, whose time-resolved measurements revealed domain
wall excitations and growth dynamics Iwai_TTFCA . Such inhomogeneous dynamics
have been discussed also in quasi-two-dimensional CO materials based on the
BEDT-TTF [bis(ethylenedithio)tetrathiafulvalene] molecule. Depending on the
nature of their equilibrium CO transition, the dynamic domain formation and
how they evolve turn out to be material dependent and reflect the ability of
CO melting Iwai_BEDTTTF .
Theoretical efforts to elucidate the early stage dynamics of PIPT phenomena
have been made using microscopic models of correlated electrons, often
including electron-phonon couplings which eventually lead to structural
changes at longer time scales. In such calculations, coherent (spatially
homogeneous) dynamics have been mainly studied, while theoretical treatments
of interactions fully taking into account quantum fluctuation limit the system
size we can treat Yonemitsu ; Takahashi ; Dagotto ; Hashimoto ; Hashimoto2 .
Then, to discuss the role of inhomogeneity mentioned above, different
approaches have been used, e.g., implementing by hand the existence of domain
structures, randomness, or local excitation Nagaosa ; Miyashita ; Iwano ;
Tanaka . The inter-relationship between the two different ways for phase
conversion, i.e., coherent excitation and inhomogeneity, in the initial
processes remains to be clarified, which is indispensable for the microscopic
understanding of PIPT.
In this paper, we propose a unified picture treating the above-mentioned two
processes on equal footing by means of a microscopic model of interacting
particles. For this purpose, we choose a minimal model of a correlated
electron system for CO and apply the time-dependent Hartree-Fock (HF) method,
whose combination enables us to treat system sizes large enough for a
systematic survey including inhomogeneity in the real-time dynamics. We find
that the coherent dynamics can be largely enhanced by tuning the pump-light
frequency at the in-gap region of the CO gap. This is interpreted as
collective modes associated with the CO, bringing about large oscillations of
the order parameter. When a kink structure, i.e., a domain wall of CO phases,
is present in the initial state, we show that the inhomogeneity emerges
starting from this kink when the collective excitations are activated and
eventually results in CO melting at large enough intensity of the light.

##  II Model and method

We treat an interacting spinless fermion model on a one-dimensional chain,
whose time ($\tau$) dependent Hamiltonian is written as

$\displaystyle\mathcal{H}(\tau)=$
$\displaystyle\sum_{i}\left[t_{\textrm{nn}}(\tau)\
c_{i}^{\dagger}c_{i+1}+\text{h.c.}\right]$
$\displaystyle+\sum_{i}\left[t_{\textrm{nnn}}(\tau)\
c_{i}^{\dagger}c_{i+2}+\text{h.c.}\right]+V\sum_{i}n_{i}n_{i+1},$ (1)

where $c_{i}(c_{i}^{\dagger})$ is the annihilation (creation) operator of a
spinless fermion at site $i$, and $n_{i}\equiv c_{i}^{\dagger}c_{i}$ is the
fermion number operator. The first two terms are the kinetic energy with
hopping integrals between site pairs of nearest neighbor and next nearest
neighbor, and the last term is the nearest neighbor Coulomb repulsion which
induces CO at half filling. The effect of external laser pulse polarized along
the chain direction is incorporated by the Peierls substitution in the hopping
integrals as

$\displaystyle t_{\textrm{nn}}(\tau)=t\ e^{iA(\tau)},\hskip
5.69054ptt_{\textrm{nnn}}(\tau)=t^{\prime}e^{2iA(\tau)},$ (2)

where the lattice constant, the light velocity, the elementary charge, and
$\hbar$ are taken as unity. We consider the vector potential for the pump
light centered at $\tau=0$ with a gaussian envelope of frequency $\omega_{p}$

$\displaystyle
A(\tau)=\frac{A_{p}}{\sqrt{2\pi}\tau_{p}}\exp{\left(-\frac{\tau^{2}}{2\tau_{p}^{2}}\right)}\cos(\omega_{p}\tau),$
(3)

where $A_{p}$ and $\tau_{p}$ control the amplitude and the pulse width,
respectively. In the following we set $t=1$ as the unit of energy (and time
$1/t$), and fix $t^{\prime}=0.1$ and $V=4$; we also fix $\tau_{p}=5$. Then the
parameters we vary are $\omega_{p}$ and $A_{p}$.

For the time evolution, a time-dependent HF method at absolute zero
temperature developed in Refs. Terai ; Kuwabara ; Tanaka  is adopted. In this
method the interaction term is treated within the HF approximation:
$n_{i}n_{i+1}\rightarrow\langle n_{i}\rangle n_{i+1}+n_{i}\langle
n_{i+1}\rangle-\langle n_{i}\rangle\langle n_{i+1}\rangle-\langle
c_{i}^{\dagger}c_{i+1}\rangle
c_{i+1}^{\dagger}c_{i}-c_{i}^{\dagger}c_{i+1}\langle
c_{i+1}^{\dagger}c_{i}\rangle+\langle c_{i}^{\dagger}c_{i+1}\rangle\langle
c_{i+1}^{\dagger}c_{i}\rangle$. After evaluating the mean-fields for all sites
and bonds in a self-consistent manner at equilibrium, the time evolution of
the wave vector $\left|\Psi(\tau)\right>$ is calculated by the discretized
Schr$\rm\ddot{o}$dinger equation as

$\displaystyle\left|\Psi(\tau+\Delta\tau)\right>=\exp{\left[-i\ {\cal
H}^{HF}(\tau+\frac{\Delta\tau}{2})\Delta\tau\right]}\
\left|\Psi(\tau)\right>.$ (4)

The transient Hamiltonian ${\cal H}^{HF}(\tau+\Delta\tau/2)$ is calculated by
a linear interpolation

$\displaystyle{\cal H}^{HF}(\tau+\frac{\Delta\tau}{2})=\frac{1}{2}\left[{\cal
H}^{HF}_{0}(\tau+\Delta\tau)+{\cal H}^{HF}(\tau)\right],$ (5)

where ${\cal H}^{HF}_{0}(\tau+\Delta\tau)$ is the “extrapolated” HF
Hamiltonian with mean fields calculated by the wave vector

$\displaystyle\left|\Psi^{0}(\tau+\Delta\tau)\right>=\exp{\left[-i\ {\cal
H}^{HF}(\tau)\Delta\tau\right]}\ \left|\Psi(\tau)\right>.$ (6)

This scheme gives the accuracy in $\left|\Psi(\tau+\Delta\tau)\right>$ with
errors of the order $(\Delta\tau)^{3}$, and considerably lightens the
numerical calculations. Note that by using HF approximation, we are implying
higher dimension (quasi-one-dimensional) systems in mind, rather than trying
to investigate a purely one-dimensional system, as studied in Ref. Hashimoto3
.

##  III results

In the following, we show results for the system sizes $N=500$ at half filling
(Sec. III.1) and $N=501$ with one fermion added to that (Sec. III.2), using
the periodic boundary condition. These sizes are large enough to neglect
finite-size effect within the time domain we consider, typically $\tau\leq
100$.

###  III.1 Homogeneous order

First, we show results for even number of sites ($N=500$). In this case the
twofold periodicity $\langle n_{i}\rangle=1/2+(-1)^{i}\delta(\tau)$ is always
maintained throughout the simulation, namely, coherent dynamics are seen
note_randomness . In the initial state $\tau=\tau_{\textrm{in}}$, our
parameter set gives the order parameter $\delta(\tau_{\textrm{in}})=0.40$
$(\equiv\delta_{0})$ and the one-particle gap $\Delta_{\textrm{CO}}=6.45$. In
Fig. 1(a), we show the $\tau$ dependence of $\delta(\tau)$ for two pump-light
frequencies: $\omega_{p}=\Delta_{\textrm{CO}}$ and
$\omega_{p}=\Delta_{\textrm{CO}}/2$. When
$\omega_{p}\gtrsim\Delta_{\textrm{CO}}$, during the pump light is shined
$\delta$ decreases and stays constant afterwards. This is interpreted as the
one-particle excitation where the energy of light is absorbed by the fermion
system through photocarrier generation Hashimoto3 ; Ishihara .

Figure 1:  (a) $\tau$ dependence of order parameter $\delta(\tau)$ for an even
number of sites, with two pump-light frequencies:
$\omega_{p}=\Delta_{\textrm{CO}}=6.45$ and
$\omega_{p}=\Delta_{\textrm{CO}}/2$, for different intensity $A_{p}$. (b) A
closer look at the latter case for $A_{p}=1$ together with real and imaginary
parts of the bond orders $\chi_{i}(\tau)$ for even and odd $i$. The light and
dark shaded areas show the time domain where the pump light is applied and one
period of oscillation, respectively.

On the other hand, the response is larger for
$\omega_{p}\simeq\Delta_{\textrm{CO}}/2$, where the decrease in $\delta(\tau)$
is large and coherent oscillations are seen even after the light irradiation
is finished. This is depicted in more detail in Fig. 1(b), where
$\delta(\tau)$ together with the real and imaginary parts of the bond orders
$\chi_{i}(\tau)=\langle c_{i}^{\dagger}c_{i+1}\rangle$, for even and odd $i$,
are plotted for $\omega_{p}=\Delta_{\textrm{CO}}/2$, $A_{p}=1$. Since the
gauge field induces electric current with frequency $\omega_{p}$, the
imaginary part of $\chi_{i}(\tau)$ follows its time profile, in the in-phase
way for even and odd $i$. This directly generates an oscillation in the charge
density in a coherent way.

We can write the modulation in $\langle n_{i}\rangle$ as
$-(-1)^{i}\Delta\delta(\tau)$ where the decrease in the order parameter
$\Delta\delta(\tau)$ is approximately written as
$\Delta\delta(\tau)\propto\cos{\left[\alpha\cos{\omega_{p}\left(\tau-\tau_{0}\right)}\right]}$,
with $\alpha$ being a coefficient representing the oscillation in time. Then
the modulation is rewritten as
$-(-1)^{i}\Delta\delta(\tau)\propto-\cos{\left[\pi
i+\alpha\cos{\omega_{p}\left(\tau-\tau_{0}\right)}\right]}$. This is the
collective phase mode out of CO ground stateFukuyama . As shown in Fig. 1 (b),
within one period $T=2\pi/\omega_{p}$, the induced modulation in $\langle
n_{i}\rangle$ undergoes two oscillations, namely, it has the frequency
$2\omega_{p}=\Delta_{\textrm{CO}}$.

In such a picture, the real part of $\chi_{i}(\tau)$ shows a modulation
proportional to $\cos{\left[\pi\left(i+1/2\right)+\beta\
\cos{\omega_{p}\left(\tau-\tau_{0}\right)}\right]}$, where $\beta$ is a
coefficient as above; $i+1/2$ is the midpoint coordinate between neighboring
sites. This is equal to $-\sin{\left[\pi i+\beta\
\cos{\omega_{p}\left(\tau-\tau_{0}\right)}\right]}$ with period $T$ in time,
consistent with the numerical results in Fig. 1(b). We note that such in-gap
modes are observed in other systems such as 1/2-filled models Lu  and models
for BCS superconductors Matsunaga ; Krull  whose origins are discussed to be
two-photon processes. In clear contrast, in our case the excitation is an
optically-allowed process seen in the linear absorption spectra in the small
$A_{p}$ limit (see Appendix A).

Figure 2: (Color online) Two-dimensional plots on the {$\omega_{p},A_{p}$}
plane of the time averaged photoresponses for the $N=500$ sites case: (a)
decrease in the order parameter $\Delta\delta$ renormalized by $\delta_{0}$,
(b) standard deviation $\sigma$ in $\delta(\tau)$, (c) effective absorbed
photon number $N_{ph}$, and (d) in-gap density of states (dos). Their
definitions are depicted in the text.

We can see the resonant character in the two-dimensional plots on the
{$\omega_{p},A_{p}$} plane. In Fig. 2, we plot (a) the time averaged decrease
in the order parameter
$\Delta\delta=\langle\Delta\delta(\tau)\rangle_{\textrm{av}}$ renormalized by
$\delta_{0}$, (b) the standard deviation
$\sigma=\langle\sqrt{(\delta(\tau)-\bar{\delta})^{2}}\rangle_{\textrm{av}}$
where $\bar{\delta}=\langle\delta(\tau)\rangle_{\textrm{av}}$, a measure for
the oscillatory part, (c) the effective absorbed photon number $N_{ph}$
defined as $\Delta E/\omega_{p}$ where $\Delta E$ is the increase in the
expectation value of energy, and (d) the in-gap density of states (dos)
measured within $\pm 0.1t$ range of the Fermi level, compared to that in the
$V=0$ equilibrium metallic state. $\Delta\delta/\delta_{0}$, $\sigma$,
$\bar{\delta}$, and the in-gap density of states are the values averaged over
$\tau=20$ to 60, and $\Delta E$ is converged after the light emission in our
simulation. We note that negative values of $\delta(\tau)$ are observed in
some cases for large $A_{p}$ as seen in Fig. 1(a). This indicates that the
phase of CO is reverted. For the cases where the time average $\bar{\delta}$
is negative, we take its absolute value $|\bar{\delta}|$ to calculate the time
averaged decrease, as $\Delta\delta=\delta_{0}-|\bar{\delta}|$.

The excitation across the gap is seen for
$\omega_{p}\gtrsim\Delta_{\textrm{CO}}$ in $\Delta\delta$ and $N_{ph}$ in
Figs. 2(a) and 2(c), respectively, while $\sigma$ is negligibly small there,
as seen in Fig 2(b). On the other hand, the large decrease $\Delta\delta$
occurs resonantly as in-gap excitations, together with appreciable $\sigma$
indicating its oscillating nature. In the small $A_{p}$ region, the resonance
frequency approaches $\omega_{p}\simeq\Delta_{\textrm{CO}}/2$, whereas the
mode broadens and enlarges as $A_{p}$ is increased. This is because the pulsed
laser light includes Fourier components of different frequencies and also the
nonlinear effects become prominent. We note the existence of another mode at
$\omega_{p}\simeq 2$, separated from the phase mode in the small $A_{p}$
region of Fig. 2, which is the amplitude mode excitation of CO. This is more
clearly seen in $\sigma$ than in $\Delta\delta$, indicating that it does not
efficiently acts to decrease $\delta$, compared to the phase mode (see
Appendix B for details).

Figure 3: (Color online) Time evolution of $\langle n_{i}\rangle$ on each
site, plotted for $i=$ 76 to 426, among $N=$ 501 sites. (a) The initial state
having a kink structure centered at $i=$ 250-251. (b) $A_{p}=$ 4 and (c)
$A_{p}=$ 8, for time $\tau=$ 20 and 40 [$\omega_{p}=\Delta_{\textrm{CO}}/2$].
On the right of each plot, the temporary density of states (dos) is plotted,
where the Fermi energy is set to the origin.

These modes become mixed in the large $A_{p}$ regime and therefore the
photoresponses show complex $\tau$ dependence with different frequency
components difficult to resolve [see Fig. 1(a)]. The in-gap density of states
shown in Fig. 2(d) is negligibly small in all parameter space, despite the
fact that resonant excitation sometimes accompanies phase inversion of CO as
mentioned above, leading to instantaneous disappearance of the order
[$\delta(\tau)=0$], for example seen in Fig. 1 (a) for $A_{p}=5$ (see Appendix
C for examples of temporary density of states). This will be contrasted with
the case of inhomogeneous dynamics in the next subsection.

###  III.2 Presence of a domain wall

Next, we show results for odd number of sites ($N=501$). Since the twofold
periodicity of CO does not fit to the system, the equilibrium self-consistent
HF solution shows a kink structure OnoTerai  as a domain wall embedded in the
CO bulk. This is shown in Fig. 3(a), where the real-space plot of charge
density on each site $\langle n_{i}\rangle$ is shown. For
$\omega_{p}\gtrsim\Delta_{\textrm{CO}}$, the response is unchanged from the
$N$ even case: The bulk region shows coherent dynamics irrespective of the
presence of the kink.
When the pump light frequency is tuned in the in-gap region, especially near
$\omega_{p}\simeq\Delta_{\textrm{CO}}/2$ as in the $N$ even case, a large
response is seen again. When $A_{p}$ is small the response is similar to the
$N$ even case, namely, the collective phase mode is resonantly excited and the
system remains coherent. However, appreciable differences are observed in the
large $A_{p}$ regime: Two distinct types of inhomogeneous dynamics are seen,
as represented in the time evolution of $\langle n_{i}\rangle$ plotted for
$\tau=20$ and 40 (“snapshots”), in Figs. 3(b) and 3(c) for $A_{p}=4$ and 8,
respectively Suppl . At first sight, as a common feature, one can see the
generation of inhomogeneous regions in both cases, and its growth as time
evolves originated from the kink in the initial state. In the following we
discuss qualitatively different features of the two cases.
In the former case of $A_{p}=4$ [Fig. 3 (b)], the inhomogeneity is accompanied
by kink generations. This is reflected in the $\tau$ dependence of the
averaged “order parameter” $\delta_{\textrm{av}}(\tau)$, defined as
$\delta_{\textrm{av}}(\tau)=(2N)^{-1}\sum_{i}|\langle n_{i}\rangle-\langle
n_{i+1}\rangle|$ shown in Fig. 4; $\delta_{\textrm{av}}(\tau)$ coincides with
$\delta(\tau)$ in the homogeneous case. As $A_{p}$ is increased, first the
curve is almost identical to the $N=$ 500 case [see data for $A_{p}=1$ in Fig.
2 (a)]. In the data for $A_{p}=$ 3 and 4, the large oscillations in
$\delta_{\textrm{av}}(\tau)$ are seen in the initial stage and then they decay
gradually. These behaviors come from the phase mode excitation in the bulk CO
region and the growth of inhomogeneous region reducing such oscillating CO,
respectively. On the other hand, for the latter case with a larger value of
$A_{p}=8$ [Fig. 3(c)], in the inhomogeneous region the CO structure is nearly
lost; a CO melting occurs. In this case, $\tau$ dependence of
$\delta_{\textrm{av}}(\tau)$ shows a very different curve as seen in Fig. 4;
the oscillatory behavior is almost diminished at some time after the laser
radiation and the value of $\delta_{\textrm{av}}(\tau)$ approaches to a very
small value ($\simeq 0.04$ in this case).
The difference of the two cases is highlighted in the temporal evolution of
the density of states, plotted in the right panels in Fig. 3 (see also Ref.
Suppl ). One can see that for $A_{p}=4$ the system essentially maintains the
gap structure due to CO, reflecting the fact that the CO structure in real
space is basically kept even in the presence of multiple kinks which give rise
to in-gap states. For $A_{p}=8$, in contrast, the gap structure itself gets
diminished. The shape of the density of states becomes similar to that for the
metallic state in equilibrium at $V=0$ but with heavy influence of disorder.
The transfer integrals are renormalized in the case of $V\neq 0$, owing to the
Fock term, as $t\rightarrow\tilde{t}\equiv t-V\chi_{i}(\tau)$. The calculated
$\chi_{i}(\tau)$ typically take values such as $-0.1\sim-0.4$, therefore
$\tilde{t}\approx 1.2t\sim 2.6t$. Moreover, the Hartree term gives rise to
modulations in the on-site potentials of $V(\langle
n_{i}\rangle-0.5)\approx\pm 0.8t$ at maximum. These two randomness effects
result in the broad and blurred shapes of the density of states in Fig. 3(c).
In Fig. 5, we show the two-dimensional plots on the {$\omega_{p},A_{p}$}
plane, for the same quantities as in Fig. 2, also averaged over $\tau=20$ to
60. One can see the response is large in the same in-gap region as in the $N$
even case, indicating that the resonant collective mode excitation is an
essential ingredient. The difference from the $N$ even case is prominent in
the in-gap density of states in Fig. 5(d): There is a threshold behavior for
the raise, around $A_{p}=3$ \- $4$ for the resonant condition
$\omega_{p}=\Delta_{\textrm{CO}}/2\simeq 3.2$, where the kink generation and
inhomogeneous growth start to occur. The transformation to the “CO melting”
behavior is rather obscure and not well defined but at $A_{p}=7$ \- $8$ and
larger, the in-gap density of states becomes large and a metallic like state
is achieved. These are in clear contrast with the $N$ even case, where such
in-gap density of states are always negligibly small [Fig. 2(d)].

We note that, in both cases, the inhomogeneity is generated by the cooperative
effect of the phase mode excitation and the existence of a kink in the middle
working as a seed. As $A_{p}$ increases, the former brings about the reduction
of CO order parameter together with the large oscillation in the bulk CO
region, sometimes even accompanying its sign change. This makes instantaneous
disappearances of CO and enables the kink generations and melting possible. In
fact, the growth of inhomogeneity is promoted when CO amplitude in the bulk is
small, and slower when the CO amplitude is large. We note that the generated
kinks themselves show little motions Suppl . This picture is different from
the case of the purely one-dimensional model where a CO decrease is ascribed
to solitonic motions of kinks Hashimoto3 . In any case, the phenomena occur in
the highly nonlinear regime, as seen, for example, from the threshold behavior
of the induced in-gap density of states.

Figure 4: (Color online) $\tau$ dependence of the averaged “order parameter”
$\delta_{\textrm{av}}(\tau)$ for $N=501$; The frequency of the pump light is
$\omega_{p}=\Delta_{\textrm{CO}}/2$, for different intensity $A_{p}$. The
shaded area show the time domain where the pump light is applied.  Figure 5:
(Color online) Two-dimensional plots on the {$\omega_{p},A_{p}$} plane of the
time averaged photoresponses for the $N=501$ sites case: (a) decrease in the
order parameter $\Delta\delta$ renormalized by $\delta_{0}$, (b) standard
deviation $\sigma$ in $\delta(\tau)$, (c) effective absorbed photon number
$N_{ph}$, and (d) in-gap density of states (dos). Their definitions are
depicted in the text.

##  IV discussion and summary

Finally we briefly discuss implications of our results to the experiments on
PIPT, although our one-dimensional model here containing only the electronic
degree of freedom is too simple. Nevertheless, the inhomogeneity addressed
above is reminiscent of the phenomena observed in molecular materials. As
mentioned in the introduction, domain wall excitations and growth of the
domains have been discussed in TTF-CA and CO melting with domain growth in
BEDT-TTF based salts. One discrepancy between our results here and the
experimental situation is that the pump light frequency needed to be tuned in
the in-gap region in the simulation, whereas the above-mentioned experiments
use off resonance photon energy even larger than the gap. However, we have
presented the results here with a relatively large value of $V=4$ which gives
large $\Delta_{\textrm{CO}}$ to resolve the in-gap region. When $V$ gets
smaller, the excitations across and in the gap get considerably mixed; we show
results for $V=2$ in Appendix D. In such cases it is possible to trigger the
inhomogeneity even at large $\omega_{p}$. Another important factor is the
dimensionality; higher dimensional models are known to stabilize the metallic
phase at equilibrium, whose dynamics are left for future problems.

In summary, we simulated the time evolution of a CO system applying time-
dependent HF method to a spinless fermion model. We found that inhomogeneity
can emerge out of a homogeneous light emission through cooperative effect of a
collective mode excitation and the triggering by a kink structure.

###### Acknowledgements.

The authors would like to thank S. Iwai, A. Ono, and K. Yonemistu for valuable
discussions. This work was supported by JSPS KAKENHI Grant Nos. 26400377,
15H02100, 16H02393, 17H02916, and 18H05208.

##  Appendix A Absorption spectra

We show the optical absorption spectrum $\alpha(\omega)$, calculated as the
increment in the total energy upon the response to a continuous-wave gauge
field introduced at $\tau=0$ as $A(\tau)=A_{0}\
\theta(\tau)\sin(\omega\tau)\exp(-\gamma\tau)$, where $\gamma$ is the damping
factor. Figure 6 shows the linear spectrum for the parameter set in the main
text ($t=1,t^{\prime}=0.1,V=4$) obtained for a small value of $A_{0}=0.01$ and
$\gamma=0.02$. One can see the continuum approximately above the charge gap
$\Delta_{\textrm{CO}}=6.45$ and the resonant mode located at $\omega\sim 3.5$.

Figure 6: (Color online) The linear optical absorption spectrum
$\alpha(\omega)$ using the time-dependent Hartree-Fock method, for $V=4$.

##  Appendix B Collective mode excitations

Figure 7: $\tau$ dependence of order parameter $\delta(\tau)$ and the real and
imaginary parts of the bond orders $\chi_{i}(\tau)$ for even and odd $i$ for
$N=500$. (a) $\omega_{p}=\Delta_{\textrm{CO}}/2=3.225,A_{p}=1$ as in the main
text [Fig. 1(b)] and (b) $\omega_{p}=2.15,A_{p}=1$. The light and dark shaded
areas show the time domain where the pump light is applied and one period of
the excited oscillation, respectively.

We plot the time evolution of the two resonant modes observed in the
simulation for $N=500$ in Fig. 7. The order parameter $\delta(\tau)$ and the
real and imaginary parts of bond orders $\chi_{i}(\tau)=\langle
c_{i}^{\dagger}c_{i+1}\rangle$, for even and odd $i$, are plotted, for (a)
$\omega_{p}=\Delta_{\textrm{CO}}/2=3.225,A_{p}=1$, and (b)
$\omega_{p}=2.15,A_{p}=1$. Figure 7(a) shows the same data as Fig. 1(b) in the
main text for a longer time domain: the phase mode excitation. Figure 7(b) is
the amplitude modeFukuyama , as seen in the approximately symmetric
oscillation about the average $\delta_{0}$. Therefore it does not efficiently
decrease CO, in contrast with the phase mode. The oscillation frequency is
near twice of $\omega_{p}$, as two oscillations emerge during the light pulse;
the real and imaginary parts of $\chi_{i}(\tau)$, the latter representing the
local current, show in-phase and out-of-phase alternating behavior for even
and odd $i$, respectively, in contrast with the phase mode. The phase mode
excitation can be attributed to the exciton band in the linear optical spectra
(see Appendix A) discussed in the literature Bruinsma ; Gagliano ; Lorenzana ,
whereas the amplitude mode excitation is a two-photon process.

##  Appendix C Density of states for $N=500$

In Fig. 8, we plot the temporary density of states for the $N=500$ case,
corresponding to the case plotted in Fig. 1(a) for
$\omega_{p}=\Delta_{\textrm{CO}}/2$ and $A_{p}=5$. We can see that the density
of states in the coherent state essentially maintains its gap structure,
except when the order parameter $\delta(\tau)$ coincidentally becomes 0. This
only happens instantaneously when $\delta(\tau)$ changes sign. Nevertheless,
in the case of $\delta(\tau)<0$ the CO just reverts it phase, therefore its
gap structure is common to the case of $\delta(\tau)>0$, as seen in Fig. 8.

Figure 8: Examples of temporary density of states for the $N=500$ case,
corresponding to the case in Fig. 1(a), $\omega_{p}=\Delta_{\textrm{CO}}/2$.
The data are for $\tau=10,20,$ and 30.

##  Appendix D Data for V=2

We show two-dimensional plots of the time averaged photoresponses for the case
of $V=2$, as shown in the main text for $V=4$ [Figs. 2 and 5], in Fig. 9 for
(a) $N=500$ and (b) $N=501$. The plotted values are: (i) the decrease in the
order parameter $\Delta\delta$ renormalized by $\delta_{0}$, (ii) the standard
deviation
$\sigma=\langle\sqrt{(\delta(\tau)-\bar{\delta})^{2}}\rangle_{\textrm{av}}$, a
measure for the oscillatory part, (iii) the effective absorbed photon number
$N_{ph}$ defined as $\Delta E/\omega_{p}$ where $\Delta E$ is the increase in
the expectation value of energy, and (iv) the in-gap density of states (dos)
measured within $\pm 0.1t$ range of the Fermi level, compared to that in the
$V=0$ equilibrium metallic state. $\Delta\delta/\delta_{0}$, $\sigma$,
$\bar{\delta}$, and the in-gap density of states are the values averaged over
$\tau=20$ to 60, and $\Delta E$ is converged after the light emission in our
simulation.
Similarly to the case of $V=4$, the large values of $\Delta\delta/\delta_{0}$
are seen in the in-gap region, at small $A_{p}$ toward
$\omega_{p}\simeq\Delta_{\textrm{CO}}/2$; $\Delta_{\textrm{CO}}=2.05$ for
$V=2$. The large values of in-gap dos are seen only for $N=501$ with a
threshold behavior for $A_{p}$. One can see that, in contrast with the $V=4$
data in the main text, the gap structure and the in-gap regime are mixed at
large $A_{p}$ and become difficult to resolve.

Figure 9: (Color online) Two-dimensional plots on the {$\omega_{p},A_{p}$}
plane of the photo-esponses for $V=2$, with (a) $N=500$ and (b) $N=501$ sites.
The plotted quantities are: (i) decrease in the order parameter $\Delta\delta$
renormalized by $\delta_{0}$, (ii) standard deviation $\sigma$ in $\delta$,
(iii) effective absorbed photon number $N_{ph}$, and (iv) in-gap density of
states (dos). Their definitions are depicted in the main text.

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