node style sp/.style=draw,circle,minimum size=,
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On nonemptiness of Newton strata in the B_^+-Grassmannian for _2n

S. Hong]Serin Hong
Simons Laufer Mathematical Sciences Institute, 17 Gauss Way, Berkeley, CA 94720

We build upon our previous work [10] to study the Newton stratification on the $\BdR^+$-Grassmannian for $\GSp_{2n}$. Our main result gives an explicit classification of all nonempty Newton strata associated to a Frobenius-conjugacy class satisfying certain conditions on the Newton polygon.

§ INTRODUCTION

The $\BdR^+$-Grassmannian is a $p$-adic analogue of the complex affine Grassmannian. It has played a fundamental role in a number of major developments in $p$-adic geometry, including the work of Caraiani-Scholze [4] on the cohomology of unitary Shimura varieties, the work of Scholze-Weinstein [21] on the general construction of local Shimura varieties, and the work of Fargues-Scholze [9] on the geometrization of the local Langlands correspondence. It has also been used as a central tool for studying the $p$-adic period domain by many authors, such as Chen-Fargues-Shen [2], Shen [20], Chen [3], Viehmann [22], Nguyen-Viehmann [16], and Chen-Tong [5].

In order to describe the geometry of the $\BdR^+$-Grassmannian, we consider a natural stratification called the Newton stratification.

Let $G$ be a connected reductive group over a finite extension $\genlocfield$ of $\Q_p$. We denote by $\Gr_G$ the $\BdR^+$-Grassmannian associated to $G$, and by $\Gr_{G, \gencocharacter}$ the Schubert cell associated to a dominant cocharacter $\gencocharacter$ of $G$. In addition, we write $\schff = \schff_\genlocfield$ for the Fargues-Fontaine curve over $\genlocfield$ and $\Bun_G$ for the stack of $G$-bundles on $\schff$. The work of Fargues [6] gives a natural bijection between the topological space $\topspace{\Bun_G}$ of $\Bun_G$ and the set $B(G)$ of Frobenius-conjugacy classes of $G(\compmaxunram{\genlocfield})$, where $\compmaxunram{\genlocfield}$ denotes the $p$-adic completion of the maximal unramified extension of $\genlocfield$. If we fix an element $b \in B(G)$ and a complete algebraically closed extension $\FFuntilt$ of $\genlocfield$, we get a natural map

\[ \Newt_b: \Gr_G(\FFuntilt) \longrightarrow |\Bun_G| \simeq B(G)\]

induced by the theorem of Beauville-Laszlo [1].
For each Schubert cell $\Gr_{G, \gencocharacter}$, the Newton stratification associated to $b$ is a decomposition

\[ \Gr_{G, \gencocharacter} = \bigsqcup_{\modification{b} \in B(G)} \Gr_{G, \gencocharacter, b}^{\modification{b}}\]

where $\Gr_{G, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$ is the preimage of $\modification{b}$ in $\Gr_{G, \gencocharacter}(\FFuntilt)$ under the map $\Newt_b$.

In this paper, we study the Newton stratification of minuscule Schubert cells in the $\BdR^+$-Grassmannian for $\GSp_{2n}$. For a precise statement of our main result, let us recall some basic facts regarding the group $\GSp_{2n}$. As observed by Kottwitz [13], an element in $B(\GSp_{2n})$ is determined by an invariant called the Newton polygon, which is a concave polygon on the interval $[0, 2n]$ with integer breakpoints. In addition, we may represent each dominant cocharacter of $\GSp_{2n}$ as a $2n$-tuple of descending integers, or equivalently as a concave polygon on the interval $[0, 2n]$ where the slope on $[i-1, i]$ is given by the $i$-th entry of the $2n$-tuple. After multiplication by a central cocharacter, a minuscule dominant cocharacter of $\GSp_{2n}$ is either trivial or represented by the tuple $(\lineseg{1}{n}, \lineseg{0}{n})$, where we write $\lineseg{d}{n} := (d, \cdots, d) \in \Z^n$ for each $d \in \Z$. Since multiplication by a central cocharacter induces a natural bijection between Newton strata on Schubert cells (as noted in Proposition <ref>), it suffices to consider the minuscule cocharacter represented by $(1^{(n)}, 0^{(n)})$.

Let $b$ and $\modification{b}$ be elements in $B(\GSp_{2n})$ with their Newton polygons respectively denoted by $\newtonmap[b]$ and $\newtonmap[\modification{b}]$. Assume that the difference between any two distinct slopes in $\newtonmap[b]$ is greater than $1$. For the cocharacter $\gencocharacter$ of $\GSp_{2n}$ represented by the tuple $(\lineseg{1}{n}, \lineseg{0}{n})$, the Newton stratum $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$ is nonempty if and only if the following conditions are satisfied:

*  The polygon $\newtonmap[\modification{b}]$ lies below the polygon $\newtonmap[b] + \dualpolygon{\gencocharacter}$ with the same endpoints, where $\dualpolygon{\gencocharacter}$ denotes the cocharacter of $\GSp_{2n}$ represented by the tuple $(\lineseg{0}{n}, \lineseg{-1}{n})$.

*  For $i = 1, 2, \cdots, 2n$, we have an inequality

\[\HNslope{\newtonmap[\modification{b}]}{i} \leq \HNslope{\newtonmap[b]}{i} \leq \HNslope{\newtonmap[\modification{b}]}{i} +1\]

where $\HNslope{\newtonmap[b]}{i}$ and $\HNslope{\newtonmap[\modification{b}]}{i}$ respectively denote the slopes of $\newtonmap[b]$ and $\newtonmap[\modification{b}]$ on $[i-1, i]$.

*  For each breakpoint of $\newtonmap[b]$, there exists

a breakpoint of $\newtonmap[\modification{b}]$ with the same $x$-coordinate.

(p00) at (1.5, 4);
(p11) at (2.8, 6.3);
(r11) at (4, 7.8);
(p22) at (5.2, 8.7);
(p33) at (6.5, 9);
(p44) at (8, 8);

(left) at (0, 0);
(q0) at (1.5, 3.2);
(q1) at (4, 5.2);

(q2) at (6.5, 5.7);
(q3) at (8, 4);

(r1) at (4, 3.8);
(r2) at (6.5, 3.5);

(p0) at (1.5, 2.5);
(p1) at (2.8, 3.5);
(p2) at (5.2, 3.5);
(p3) at (6.5, 2.5);
(p4) at (8, 0);

[step=1cm,thick, color=red] (left) – (q0) –  (q1) – (q2) – (q3);
[step=1cm,thick, color=green] (left) – (p0) –  (p1) – (r1) – (p2) – (p3) – (p4);
[step=1cm,thick, color=teal] (left) – (p00) –  (p11) – (r11) – (p22) – (p33) – (p44);
[step=1cm,thick, color=blue] (q1) – (r2) – (p4);

[fill] (q0) circle [radius=0.05];
[fill] (q1) circle [radius=0.05];
[fill] (q2) circle [radius=0.05];
[fill] (q3) circle [radius=0.05];
[fill] (left) circle [radius=0.05];

[fill] (r1) circle [radius=0.05];
[fill] (r2) circle [radius=0.05];

[fill] (p0) circle [radius=0.05];
[fill] (p1) circle [radius=0.05];
[fill] (p2) circle [radius=0.05];
[fill] (p3) circle [radius=0.05];
[fill] (p4) circle [radius=0.05];

[fill] (p00) circle [radius=0.05];
[fill] (p11) circle [radius=0.05];
[fill] (r11) circle [radius=0.05];
[fill] (p22) circle [radius=0.05];
[fill] (p33) circle [radius=0.05];
[fill] (p44) circle [radius=0.05];

[step=1cm,dotted] (1.5, -0.4) – (1.5, 9.2);
[step=1cm,dotted] (4, -0.4) – (4, 9.2);
[step=1cm,dotted] (6.5, -0.4) – (6.5, 9.2);

(q1) ++(0.8, 0.6) node $\newtonmap[b]$;
(r2) ++(1.8, -0.8) node $\newtonmap[b] + \dualpolygon{\gencocharacter}$;
(p00) ++(-1.8, 0) node $\newtonmap[\modification{b}] + \mathrm{id}_{[0, 2n]}$;
(p0) ++(0.7, -0.3) node $\newtonmap[\modification{b}]$;

Illustration of the conditions in Theorem <ref>

Let us briefly sketch our proof of Theorem <ref>. We consider the natural embedding $\GSp_{2n} \inj \GL_{2n}$ and identify $b, \modification{b} \in B(\GSp_{2n})$ with their images under the induced map $B(\GSp_{2n}) \longrightarrow B(\GL_{2n})$. Then we obtain a natural map

\[ \Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}} \longrightarrow \Gr_{\GL_{2n}, \gencocharacter, b}^{\modification{b}},\]

and consequently deduce the necessity part of Theorem <ref> from a previous result of the author [10]. For the sufficiency part of Theorem <ref>, we regard the Newton polygons $\newtonmap[b]$ and $\newtonmap[\modification{b}]$ as rational cocharacters of $\GSp_{2n}$. Let $\genlevi$ denote the centralizer of $\newtonmap[b]$, which is a Levi subgroup of $\GSp_{2n}$. By construction, the element $b \in B(\GSp_{2n})$ admits a reduction $b_\genlevi \in B(\genlevi)$; in other words, $b$ is the image of $b_\genlevi$ under the natural map $B(\genlevi) \longrightarrow B(\GSp_{2n})$. Moreover, the condition <ref> implies that $\modification{b} \in B(\GSp_{2n})$ admits a reduction $\modification{b}_\genlevi \in B(\genlevi)$. Hence we obtain a natural map

\[ \Gr_{\genlevi, \gencocharacter, b_\genlevi}^{\modification{b}_\genlevi} \longrightarrow \Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}.\]

Let us identify the Newton polygons of $b_\genlevi$ and $\modification{b}_\genlevi$ respectively with $\newtonmap[b]$ and $\newtonmap[\modification{b}]$. Since the Newton polygon of $b_\genlevi$ is central in $\genlevi$, we deduce the nonemptiness of $\Gr_{\genlevi, \gencocharacter, b_\genlevi}^{\modification{b}_\genlevi}$ from the conditions in Theorem <ref> by the work of Rapoport [17], Chen-Fargues-Shen [2], Shen [20], and Viehmann [22], thereby establishing the nonemptiness of $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$.

We expect that our strategy can be used to establish an analogous statement of Theorem <ref> for other classical groups. We plan to update this manuscript as we make progress on other groups.

§.§ Acknowledgments

The author would like to thank Linus Hamann and Tasho Kaletha for interesting discussions.

This work was partially supported by the Simons Foundation under Grant Number 814268.

§ NOTATIONS AND CONVENTIONS

Throughout the paper, we consider the following data:

* $\genlocfield$ is a finite extension of $\Q_p$ with uniformizer $\uniformizer$.

* $\FFuntilt$ is a complete and algebraically closed extension of $\genlocfield$.

* $G$ is a reductive group over $\genlocfield$ with Borel subgroup $\genborel$ and maximal torus $\genmaxtorus \subseteq \genborel$.

Given such data, we set up the following notations:

* $\absgalgrp$ is the absolute Galois group of $\genlocfield$.

* $\compmaxunram{\genlocfield}$ is the $p$-adic completion of the maximal unramified extension of $\genlocfield$.

* $\frobfieldaut$ is the Frobenius automorphism of $\compmaxunram{\genlocfield}$

* $B(G)$ is the set of $\frobfieldaut$-conjugacy classes of $G(\compmaxunram{\genlocfield})$.

* $(\charlatt, \roots, \cocharlatt, \coroots)$ is the absolute root datum of $G$.

* $\weylgrp$ is the Weyl group of $G$.

In addition, we use the following general notations:

* Given a valued field $\genpadicfield$, we denote its valuation ring by $\integerring{\genpadicfield}$.

* Given a perfectoid ring $\genperfdring$, we denote its tilt by $\tilt{\genperfdring}$.

* Given a perfect $\F_p$-algebra $\genperfalg$, we write $\witt(\genperfalg)$ for the ring of Witt vectors over $\genperfalg$.

* Given an abelian group $\genabgrp$ equipped with an action of a group $\gengrp$, we write $\genabgrp^\gengrp$ and $\genabgrp_\gengrp$ respectively for the subgroups of $\gengrp$-invariants and $\gengrp$-coinvariants.

§ $G$-BUNDLES ON THE FARGUES-FONTAINE CURVE

Let us fix a pseudouniformizer $\pseudounif$ of $\tilt{\FFuntilt}$. In addition, we write $q$ for the number of elements in the residue field of $\genlocfield$ and set $\witt_{\integerring{\genlocfield}}(\integerring{\tilt{\FFuntilt}}): = \witt(\integerring{\tilt{\FFuntilt}}) \otimes_{\witt(\F_q)} \integerring{\genlocfield}$.

* We define the perfectoid punctured unit disk associated to the pair $(\genlocfield, \tilt{\FFuntilt})$ by

\[ \Ycal:= \Spa(\witt_{\integerring{\genlocfield}}(\integerring{\tilt{\FFuntilt}})\setminus\{|\uniformizer [\pseudounif]|=0\}\]

where $[\pseudounif]$ denotes the Teichmuller lift of $\pseudounif$ in $\witt_{\integerring{\genlocfield}}(\integerring{\tilt{\FFuntilt}})$.

* We write $\genfrob$ for the automorphism of $\Ycal$ induced by the $q$-Frobenius automorphism on $\witt_{\integerring{\genlocfield}}(\integerring{\tilt{\FFuntilt}})$, and define the adic Fargues-Fontaine curve associated to the pair $(\genlocfield, \tilt{\FFuntilt})$ by

\[ \adicff:= \Ycal/\genfrob^\Z.\]

* We define the schematic Fargues-Fontaine curve associated to the pair $(\genlocfield, \tilt{\FFuntilt})$ by

\[
%\schff_{\genlocfield, \FFfield} := \Proj \left( \bigoplus_{n \geq 0 } H^0(\adicff_{\genlocfield,\FFfield}, \trivbundle(n) ) \right).
%\schff:= \Proj \left( \bigoplus_{n \geq 0 } H^0(\Ycal, \strsheaf[\Ycal])^{\genfrob = \uniformizer^n} \right).
\schff:= \Proj \left( \bigoplus_{n \geq 0 } \FFring^{\genfrob = \uniformizer^n} \right)
\]

where $\FFring$ denotes the ring of global sections on $\Ycal$.

The construction of the adic Fargues-Fontaine curve relies on the fact that the action of $\genfrob$ on $\Ycal$ is properly discontinuous.

We have the following facts regarding $\adicff$ and $\schff$:

* $\adicff$ is a Noetherian adic space over $\genlocfield$.

* $\schff$ is a Dedekind scheme over $\genlocfield$.

*  The the categories of vector bundles on $\adicff$ and $\schff$ are equivalent via the pullback along a natural map $\adicff \longrightarrow \schff$ of locally ringed spaces.

The statement <ref> in Theorem <ref> allows us to identify $G$-bundles on $\adicff$ with $G$-bundles on $\schff$.

Given an element $b \in B(G)$, we define the associated $G$-bundle $\genvb_b$ on $\adicff$ (or on $\schff)$ to be the descent along the map $\Ycal \longrightarrow \Ycal/\genfrob^\Z = \adicff$ of the trial $G$-bundle on $\Ycal$ equipped with the $\genfrob$-linear automorphism induced by $b$.

There is a natural bijection

\[ B(G) \stackrel{\sim}{\longrightarrow} H_{\et}^1(\adicff, G)\]

which maps each $b \in B(G)$ to the isomorphism class of $\genvb_b$.

We define the Newton set of $G$ by

\[\newtonset{G}:= (\cocharlatt_\Q/\weylgrp)^\absgalgrp,\]

where we write $\cocharlatt_\Q:= \cocharlatt \otimes_{\Z} \Q$ for the space of rational cocharacters of $G$, and the algebraic fundamental group of $G$ by

\[\pi_1(G):= \cocharlatt/\langle\coroots\rangle,\]

where $\langle\coroots\rangle$ denotes the subgroup of $\cocharlatt$ generated by $\coroots$.

There exist unique natural transformations

\[\newtonmap: B(\cdot) \longrightarrow \newtonset{\cdot} \quad \text{ and } \quad \kottwitzmap: B(\cdot) \longrightarrow \pi_1(\cdot)_\absgalgrp\]

of set-valued functors on the category of connected reductive groups over $\genlocfield$ such that the following statements hold:

* Under the map $\newtonmap: B(G) \longrightarrow \newtonset{G}$, the image of $b \in B(G)$ with a representative $\tilde{b}$ is given by a unique rational cocharacter $\newtoncochar{\tilde{b}}$ of $G$ for which there exist an integer $m>0$ and an element $c \in G(\compmaxunram{\genlocfield})$ with the following properties:

* $m\newtoncochar{\tilde{b}}$ lies in $\cocharlatt$.

* $\mathrm{Int}(c) \circ m\newtoncochar{\tilde{b}}$ is defined over the fixed field of $\frobfieldaut^m$ in $\compmaxunram{\genlocfield}$, where $\mathrm{Int}(c)$ denotes the inner automorphism mapping each $g \in G(\compmaxunram{\genlocfield})$ to $cgc^{-1}$.

* We have $c \cdot \tilde{b} \cdot \frobfieldaut(\tilde{b}) \cdots \frobfieldaut^m(\tilde{b}) \cdot \frobfieldaut^m(c)^{-1} = c \cdot (m\newtoncochar{\tilde{b}})(\uniformizer) \cdot c^{-1}$.

* We have a commutative diagram

\[
\begin{tikzcd}
\compmaxunram{\genlocfield}^\times \arrow[d, twoheadrightarrow] \arrow[r, "\mathrm{val}"] & \Z \arrow[d, -, double line with arrow={-,-}]\\
%\arrow[d, shift left, dash] \arrow[d, shift right, dash] \\
B(\Gm) \arrow[r, "\kottwitzmap"] & \pi_1(\Gm)_\absgalgrp
\end{tikzcd}
\]

where $\mathrm{val}: \compmaxunram{\genlocfield}^\times  \longrightarrow \Z$ is the $\uniformizer$-adic valuation map on $\compmaxunram{\genlocfield}$.

If $G$ is unramified, there is an explicit description of the map $\kottwitzmap: B(G) \longrightarrow \pi_1(G)_\absgalgrp$ given by Rapoport-Viehmann <cit.>.

We refer to the maps $\newtonmap : B(G) \longrightarrow \newtonset{G}$ and $\kottwitzmap: B(G) \longrightarrow \pi_1(G)_\absgalgrp$ from Proposition <ref> respectively as the Newton map and the Kottwitz map of $G$.

In line with Theorem <ref>, Fargues <cit.> provides a geometric description of $\newtonmap$ and $\kottwitzmap$ in terms of $G$-bundles on the Fargues-Fontaine curve. We will briefly sketch this description for $G = \GL_n$ after Example <ref>.

Every element of $B(G)$ is uniquely determined by its image under the Newton map and the Kottwitz map. In other words, the map

\[ \newtonmap \times \kottwitzmap : B(G) \longrightarrow \newtonset{G} \times \pi_1(G)_\absgalgrp\]

is injective.

The maps $\newtonmap$ and $\kottwitzmap$ fit into a natural commutative diagram

\[
\begin{tikzcd}
B(G) \arrow[d, "\newtonmap", swap] \arrow[rr, "\kottwitzmap"] & & \pi_1(G)_\absgalgrp \arrow[d] \\[0.5em]
\newtonset{G} \arrow[r, hookrightarrow] & \cocharlatt_\Q \arrow[r, twoheadrightarrow] & \pi_1(G)_\absgalgrp \otimes_{\Z} \Q
\end{tikzcd}
\]

where the inclusion $\newtonset{G} \inj \cocharlatt_\Q$ is obtained by identifying $\cocharlatt_\Q/\weylgrp$ with the set of dominant rational cocharacters.

If $\pi_1(G)_\absgalgrp$ is torsion free, the Newton map of $G$ is injective.

When $\pi_1(G)_\absgalgrp$ is torsion free, the natural map $\pi_1(G)_\absgalgrp \longrightarrow \pi_1(G)_\absgalgrp \otimes_\Z \Q$ is injective. Hence the assertion follows from Propositions <ref> and <ref>.

We say that an element $b \in B(G)$ is basic if $\newtonmap[b]$ is represented by a central rational cocharacter.

Let $B(G)_\mathrm{basic}$ denote the set of basic elements in $B(G)$. The Kottwitz map induces a bijection

\[ B(G)_\mathrm{basic} \simeq \pi_1(G)_\absgalgrp.\]

Let us discuss the Newton map and the Kottwitz map for some groups.

* For $\GL_n$, we have natural identifications

\[ \newtonset{\GL_n} \cong \{(\genslope_1, \cdots, \genslope_n) \in \Q^n: \genslope_1 \geq \cdots \geq \genslope_n\} \quad \text{ and } \quad \pi_1(\GL_n)_\absgalgrp \cong \Z.\]

We regard elements of $\newtonset{\GL_n}$ as concave polygons on the interval $[0, n]$ with rational slopes. Then we have the following facts:

* The image of the Newton map for $\GL_n$ consists of all concave polygons on $[0, n]$ with integer breakpoints, essentially by the classical result of Manin [15].

* An element $b \in B(\GL_n)$ is basic if and only if its image under the Newton map is a line segment.

* The Newton map of $\GL_n$ is injective by Corollary <ref>.

* Proposition <ref> yields a commutative diagram

\begin{equation*}\label{comm diagram for newton map and kottwitz map of GLn}
\begin{tikzcd}
B(\GL_n) \arrow[d, "\newtonmap", swap] \arrow[r, "\kottwitzmap"] &[2em] \pi_1(\GL_n)_\absgalgrp \cong \Z \arrow[d, hookrightarrow] \\[0.5em]
\newtonset{\GL_n} \arrow[r, "\sum"] & \Q
\end{tikzcd}
%G \times_\genring G \arrow[rr, "{(x, y) \mapsto (y, x)}"] \arrow[dr, "m", swap] & & G \times_\genring G \arrow[dl, "m"]\\
%& G &
\end{equation*}

where $\sum$ maps each element in $\newtonset{\GL_n}$ to the sum of its entries.

* For $\GSp_{2n}$, we consider the natural embedding into $\GL_{2n}$ and obtain an identification

\[
%\newtonset{\GSp_{2n}} \cong \{(\genslope_1, \cdots, \genslope_{2n}) \in \Q^{2n}: \genslope_1 \geq \cdots \geq \genslope_{2n}
\newtonset{\GSp_{2n}} \cong \{(\genslope_1, \cdots, \genslope_{2n}) \in \newtonset{\GL_{2n}}: \genslope_i + \genslope_{2n+1-i} \text{ constant}\}.
%\quad \text{ and } \quad \pi_1(\GSp_{2n})_\absgalgrp \cong \Z.
\]

Moreover, we have $\pi_1(\GSp_{2n})_\absgalgrp \cong \Z$.
Then we find the following facts:

* An element $(\genslope_1, \cdots, \genslope_{2n}) \in \newtonset{\GSp_{2n}}$ lies in the image of the Newton map if and only if it has integer breakpoints with $\genslope_i + \genslope_{2n+1-i} \in \Z$ for $i = 1, \cdots, n$.

* An element $b \in B(\GSp_{2n})$ is basic if and only if its image under the Newton map is a line segment.

* The Newton map of $\GSp_{2n}$ is injective by Corollary <ref>.

* Proposition <ref> yields a commutative diagram

\begin{equation*}\label{comm diagram for newton map and kottwitz map of GLn}
\begin{tikzcd}
B(\GSp_{2n}) \arrow[d, "\newtonmap", swap] \arrow[r, "\kottwitzmap"] &[2em] \pi_1(\GSp_{2n})_\absgalgrp \cong \Z \arrow[d, hookrightarrow] \\[0.5em]
\newtonset{\GSp_{2n}} \arrow[r, "\frac{1}{n}\sum"] & \Q
\end{tikzcd}
%G \times_\genring G \arrow[rr, "{(x, y) \mapsto (y, x)}"] \arrow[dr, "m", swap] & & G \times_\genring G \arrow[dl, "m"]\\
%& G &
\end{equation*}

where $\sum$ maps each element in $\newtonset{\GSp_{2n}}$ to the sum of its entries.

Let us provide an explicit description of the Newton map and the Kottwitz map for $\GL_n$ in terms of vector bundles on $\schff$.

By Theorem <ref>, the Harder-Narasimhan formalism applies to the category of vector bundles on $\schff$. Under the identifications for $\newtonset{\GL_n}$ and $\pi_1(\GL_n)_\absgalgrp$ in Example <ref>, $\newtonmap[b]$ and $\kottwitzmap[b]$ for each $b \in B(\GL_n)$ are respectively given by the Harder-Narasimhan polygon and degree of $\genvb_b$.
In addition, an element $b \in B(\GL_n)$ is basic if and only if $\genvb_b$ is semistable.
The injectivity of the Newton map for $\GL_n$ means that vector bundles on $\schff$ are determined by its Harder-Narasimhan polygon.

§ THE NEWTON STRATIFICATION OF THE $\BDR^+$-GRASSMANNIAN

Every perfectoid affinoid algebra $(\genperfdring, \genperfdring^+)$ over $\genlocfield$ induces a natural surjective homomorphism $\fontainemap_{\genperfdring, \genperfdring^+}: \witt(\inttilt{\genperfdring}) \surj \genperfdring^+$ whose kernel is a principal ideal of $\witt(\inttilt{\genperfdring})$.

Let $(\genperfdring, \genperfdring^+)$ be a perfectoid affinoid algebra over $\genlocfield$.

* We refer to the map $\fontainemap_{\genperfdring, \genperfdring^+}$ in Proposition <ref> as the Fontaine map for $(\genperfdring, \genperfdring^+)$

* We define the positive de Rham ring associated to $\genperfdring$, denoted by $\BdR^+(\genperfdring)$, to be the completion of $\witt(\inttilt{\genperfdring})[1/p]$ with respect to the kernel of $\fontainemap_{\genperfdring, \genperfdring^+}$.

* We fix an element $\gendRunif_{\genperfdring, \genperfdring^+} \in \witt(\inttilt{\genperfdring})$ which generates the kernel of $\fontainemap_{\genperfdring, \genperfdring^+}$, and define the de Rham ring associated to $\genperfdring$ by $\BdR(\genperfdring):= \BdR^+(\genperfdring)[1/\gendRunif_{\genperfdring, \genperfdring^+}]$.

* For $\genperfdring = \FFuntilt$, we refer to $\BdR:= \BdR(\FFuntilt)$ and $\BdR^+:= \BdR^+(\FFuntilt)$ respectively as the de Rham period ring and the positive de Rham period ring, and write $\gendRunif:= \gendRunif_{\FFuntilt, \integerring{\FFuntilt}}$.

As the notation suggests, the construction of $\BdR^+(\genperfdring)$ and $\BdR(\genperfdring)$ does not depend on $\genperfdring^+$ or $\gendRunif_{\genperfdring, \genperfdring^+}$.

The ring $\BdR$ is a discretely valued field with $\BdR^+$ as the valuation ring and $\FFuntilt$ as the residue field.

The $\BdR^+$-Grassmannian is the functor $\Gr_G$ which associates to each perfectoid affinoid $\genlocfield$-algebra $(\genperfdring, \genperfdring^+)$ the set of pairs $(\gentorsor, \gentrivialization)$ consisting of a $G$-torsor $\gentorsor$ on $\Spec(\BdR^+(\genperfdring))$ and a trivialization $\gentrivialization$ of $\gentorsor$ over $\Spec(\BdR(\genperfdring))$. For simplicity, we write $\Gr_G(\FFuntilt):= \Gr_G(\FFuntilt, \integerring{\FFuntilt})$.

There exists a canonical identification

\[\Gr_G(\FFuntilt) \cong G(\BdR)/G(\BdR^+).\]

Let $\gencocharacter$ be a dominant cocharacter of $G$. There exists a locally spatial diamond $\Gr_{G, \gencocharacter}$ with

\[ \Gr_{G, \gencocharacter}(\FFuntilt) =  G(\BdR^+)\gencocharacter(\gendRunif)^{-1}G(\BdR^+)/G(\BdR^+).\]

We will not use the language of diamonds in a significant way as we will be mainly interested in the $\FFuntilt$-valued points of $\Gr_G$ and $\Gr_{G, \gencocharacter}$.

Given a dominant cocharacter $\gencocharacter$ of $G$, we refer to the locally spatial diamond $\Gr_{G, \gencocharacter}$ given by Proposition <ref> as the Schubert cell of $\Gr_G$ associated to $\gencocharacter$.

There exists a closed point $\FFclosedpt$ on $\schff$ satisfying the following properties:

* $\schff - \FFclosedpt$ is isomorphic to $\Spec(\Be)$ for some principal ideal domain $\Be \subseteq \BdR$.

* The completed local ring at $\FFclosedpt$ is naturally isomorphic to $\BdR^+$.

* Every $G$-bundle on $\schff$ pulls back to a trivial $G$-bundle via the map $\Spec(\BdR^+) \to \schff$ induced by $\FFclosedpt$.

We fix a closed point $\FFclosedpt$ on $\schff$ given by Proposition <ref>, and refer to $\FFclosedpt$ as the distinguished closed point on $\schff$.

Our terminology is justified by the fact that there is a canonical choice of $\FFclosedpt$. The key technical fact is that closed points on $\schff$ are naturally in bijection with the equivalence classes of untilts of $\tilt{\FFuntilt}$, as shown by Kedlaya-Liu <cit.>. Our canonical choice of $\FFclosedpt$ corresponds to the identity map on $\tilt{\FFuntilt}$.

The set $H_{\et}^1(\schff, G)$ is canonically in bijection with the set of isomorphism classes of triples $(\punctuation{\genvb}, \completion{\genvb}, \gentrivialization)$ where

* $\punctuation{\genvb}$ is a $G$-bundle on $\schff - \FFclosedpt$,

* $\completion{\genvb}$ is a trivial $G$-bundle on $\Spec(\BdR^+)$, and

* $\gentrivialization$ is a gluing map of $\punctuation{\genvb}$ and $\completion{\genvb}$ over $\Spec(\BdR)$.

The assertion is evident by the theorem of Beauville-Laszlo [1].

Let $\genvb$ be a $G$-bundle on $\schff$. By a modification of $\genvb$ at $\FFclosedpt$, we mean a $G$-bundle $\modification{\genvb}$ on $\schff$ together with an isomorphism between the restrictions of $\genvb$ and $\modification{\genvb}$ on $\schff - \FFclosedpt$.

Let us consider an element $b \in B(G)$ and a point $x \in \Gr_G(\FFuntilt)$. We have $x = gG(\BdR^+)$ for some $g \in G(\BdR)$ under the identification $\Gr_G(\FFuntilt) \cong G(\BdR)/G(\BdR^+)$ given by Proposition <ref>. In light of Proposition <ref> we take a triple $(\punctuation{\genvb}, \completion{\genvb}, \gentrivialization)$ corresponding to $\genvb_b$ and a $G$-bundle $\genvb_{b, x}$ on $\schff$ corresponding to $(\punctuation{\genvb}, \completion{\genvb}, g\gentrivialization)$.

By construction,

$\genvb_{b, x}$ is a modification of $\genvb_b$ at $\FFclosedpt$.

It is not hard to see that the isomorphism type of $\genvb_{b, x}$ does not depend on the choice of $g \in G(\BdR)$.

Consider an element $b \in B(G)$ and a dominant cocharacter $\gencocharacter$ of $G$.

* For every $x \in \Gr_G(\FFuntilt)$, we refer to the $G$-bundle $\genvb_{b, x}$ in Example <ref> as the modification of $\genvb_b$ at $\FFclosedpt$ induced by $x$.

* For every $\modification{b} \in B(G)$, we define the associated Newton stratum with respect to $b$ in $\Gr_{G, \gencocharacter}$ to be the subdiamond $\Gr_{G, \gencocharacter, b}^{\modification{b}}$ of $\Gr_{G, \gencocharacter}$ with

\[\Gr_{G, \gencocharacter, b}^{\modification{b}}(\FFuntilt) = \{x \in \Gr_{G, \gencocharacter}(\FFuntilt): \genvb_{b, x} \simeq \genvb_{\modification{b}}\}.\]

The subdiamond $\Gr_{G, \gencocharacter, b}^{\modification{b}}$ of $\Gr_{G, \gencocharacter}$ is uniquely determined by its set of $\FFuntilt$-points since $\Gr_{G, \gencocharacter}$ is a locally spatial diamond.

Let $\gencocharacter$ be a dominant cocharacter of $G$. Given two elements $b, \modification{b} \in B(G)$, the Newton stratum $\Gr_{G, \gencocharacter, b}^{\modification{b}}$ is not empty if and only if it has a $\FFuntilt$-valued point.

The assertion is evident by definition.

Let $f: G \inj \genlift{G}$ be an embedding of reductive groups over $\genlocfield$. We take elements $b, \modification{b} \in B(G)$ and write $\genlift{b}, \genlift{\modification{b}}$ for their images under the map $B(G) \longrightarrow B(\genlift{G})$ induced by $f$. In addition, we consider a dominant cocharacter $\gencocharacter$ of $G$ and denote by $\genlift{\gencocharacter}$ the dominant cocharacter of $\genlift{G}$ associated to $f \circ \gencocharacter$. There exists a natural map

\[ \Gr_{G, \gencocharacter, b}^{\modification{b}}(\FFuntilt) \longrightarrow \Gr_{\genlift{G}, \genlift{\gencocharacter}, \genlift{b}}^{\modification{\genlift{b}}}(\FFuntilt).\]

Let us consider the natural map $\Gr_{G, \gencocharacter}(\FFuntilt) \longrightarrow \Gr_{\genlift{G}, \genlift{\gencocharacter}}(\FFuntilt)$ induced by $f$. It suffices to show that the image of $\Gr_{G, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$ lies in $\Gr_{\genlift{G}, \genlift{\gencocharacter}, \genlift{b}}^{\modification{\genlift{b}}}(\FFuntilt)$. For every $G$-bundle $\genvb$ on $\schff$, we write $\genlift{\genvb}$ for the corresponding $\genlift{G}$-bundle on $\schff$. Given an arbitrary point $x \in \Gr_{G, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$, we denote by $\genlift{x}$ its image in $\Gr_{\genlift{G}}(\FFuntilt)$ and find

\[ \genvb_{\genlift{\modification{b}}} \cong \genlift{\genvb}_{\modification{b}} \simeq \genlift{\genvb}_{b, x} \cong \genvb_{\genlift{b}, \genlift{x}},\]

thereby deducing that $\genlift{x}$ lies in $\Gr_{\genlift{G}, \genlift{\gencocharacter}, \genlift{b}}^{\modification{\genlift{b}}}(\FFuntilt)$ as desired.

Let us regard elements of $\newtonset{G}$ as dominant rational cocharacters. We define the Bruhat order $\leq$ on $\newtonset{G}$ by writing $\gencocharacter \leq \gencocharacter'$ if $\gencocharacter' - \gencocharacter$ is a linear combination of positive coroots with nonnegative coefficients.

Given a closed embedding $G \inj \genlift{G}$ of reductive groups over $\genlocfield$, the induced map $\newtonset{G} \longrightarrow \newtonset{\genlift{G}}$ is compatible with the Bruhat order.

The assertion is straightforward to verify by definition.

Let us describe the Bruhat order for groups considered in Example <ref>.

* Given two elements $\gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_n)$ and $\gencocharacter' = (\gencocharacter_1', \cdots, \gencocharacter_n')$ of $\newtonset{\GL_n}$, we have $\gencocharacter \leq \gencocharacter'$ if and only if the following equivalent conditions are satisfied:

* We have inequalities

\[ \sum_{i = 1}^j \gencocharacter_i \leq \sum_{i = 1}^j \gencocharacter_i' \quad \text{ for } j = 1, \cdots, n\]

with equality for $j = n$.

* If $\gencocharacter$ and $\gencocharacter'$ are regarded as polygons on the interval $[0, n]$, then $\gencocharacter$ lies below $\gencocharacter'$ with the same endpoints.

* Since $\newtonset{\GSp_{2n}}$ is a subset of $\newtonset{\GL_{2n}}$, Proposition <ref> implies that the Bruhat order on $\newtonset{\GL_n}$ restricts to the Bruhat order on $\newtonset{\GSp_{2n}}$.

Let $\gencocharacter$ be a dominant cocharacter of $G$.

* The dual of $\gencocharacter$ is the unique dominant cocharacter $\dualpolygon{\gencocharacter}$ in the conjugacy class of $-\gencocharacter$.

* The Galois average of $\gencocharacter$ is defined by

\[\galavg{\gencocharacter}:= \dfrac{1}{[\absgalgrp: \absgalgrp_\gencocharacter]} \sum_{\tau \in \absgalgrp/\absgalgrp_\gencocharacter} \tau(\gencocharacter) \in \newtonset{G}\]

where $\absgalgrp_\gencocharacter$ denotes the stabilizer of $\gencocharacter$ in $\absgalgrp$.

* The degree of $\gencocharacter$, denoted by $\cochardeg{\gencocharacter}$, is the image of $\gencocharacter$ under the natural projection map $\cocharlatt \surj \pi_1(G)_\absgalgrp$.

Let us illustrate Definition <ref> for groups considered in Example <ref>.

* For $\GL_n$, the Galois group $\absgalgrp$ acts trivially on cocharacters as $\GL_n$ is split. Hence all dominant cocharacters of $\GL_n$ are elements of $\newtonset{\GL_n}$. Given a dominant cocharacter $\gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_n)$ of $\GL_n$, we have

\[ \dualpolygon{\gencocharacter} = (-\gencocharacter_n, \cdots, -\gencocharacter_1), \quad \galavg{\gencocharacter} = \gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_n), \quad \cochardeg{\gencocharacter} = \gencocharacter_1 + \cdots + \gencocharacter_n.\]

* For $\GSp_{2n}$, the Galois group $\absgalgrp$ acts trivially on cocharacters as $\GSp_{2n}$ is split. Hence all dominant cocharacters of $\GSp_{2n}$ are elements of $\newtonset{\GSp_{2n}}$. Given a dominant cocharacter $\gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_{2n})$ of $\GSp_{2n}$, we have

\[ \dualpolygon{\gencocharacter} = (-\gencocharacter_{2n}, \cdots, -\gencocharacter_1), \quad \galavg{\gencocharacter} = \gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_{2n}), \quad \cochardeg{\gencocharacter} = \dfrac{1}{n} (\gencocharacter_1 + \cdots + \gencocharacter_{2n}).\]

Let $\gencocharacter$ be a dominant cocharacter of $G$. Take elements $b, \modification{b} \in B(G)$ such that $b$ is basic. The Newton stratum $\Gr_{G, \gencocharacter, b}^{\modification{b}}$ is not empty if and only if we have $\kottwitzmap[\modification{b}] = \kottwitzmap[b] - \cochardeg{\gencocharacter}$ and $\newtonmap[\modification{b}] \leq \newtonmap[b] + \galavg{(\dualpolygon{\gencocharacter})}$.

§ NONEMPTY NEWTON STRATA IN MINUSCULE SCHUBERT CELLS FOR $\GSP_{2N}$

Given a rational tuple $\gentuple$, we denote its $i$-th entry by $\HNslope{\gentuple}{i}$.

The identity cocharacter of $\GSp_{2n}$ is the element $\idcochar \in \newtonset{\GSp_{2n}}$ with

\[\HNslope{\idcochar}{i} = 1 \quad \text{ for } i = 1, \cdots, 2n.\]

The ordinary cocharacter of $\GSp_{2n}$ is the element $\ordcochar \in \newtonset{\GSp_{2n}}$ with

\[ \HNslope{\ordcochar}{i} = \begin{cases} 1 & \text{ for } i = 1, \cdots, n, \\ 0 & \text{ for } i = n+1, \cdots, 2n. \end{cases}\]

* We define the slopewise dominance order $\slopedom$ on $\newtonset{\GSp_{2n}}$ by writing $\gencocharacter \slopedom \gencocharacter'$ if we have $\HNslope{\gencocharacter}{i} \leq \HNslope{\gencocharacter'}{i}$ for $i = 1, \cdots, 2n$.

Our terminologies for $\idcochar$ and $\ordcochar$ come from the following facts:

* As a polygon, $\idcochar$ coincides with the graph of the identity function on $[0, 2n]$.

* As a polygon, $\ordcochar$ coincides with the Newton polygon of the ordinary polarized abelian variety of dimension $2n$.

The natural map $B(\GSp_{2n}) \longrightarrow B(\GL_{2n})$ is injective.

The assertion is evident by the commutative diagram

\[
\begin{tikzcd}
B(\GL_{2n}) \arrow[r, "\newtonmap", hookrightarrow] &[2em] \newtonset{\GL_{2n}} \\[0.5em]
B(\GSp_{2n}) \arrow[r, "\newtonmap", hookrightarrow] \arrow[u] & \newtonset{\GSp_{2n}} \arrow[u, hookrightarrow]
\end{tikzcd}
\]

where the Newton maps are injective as noted in Example <ref>.

While Proposition <ref> is an interesting fact, it is not essential for our argument as we will use it only for notational convenience. Every part in our argument that invokes Proposition <ref> will rely only on the injectivity of the map $\newtonset{\GSp_{2n}} \inj \newtonset{\GL_{2n}}$.

Let us regard $\idcochar$ and $\ordcochar$ as dominant cocharacters of $\GL_{2n}$. Consider elements $b, \modification{b} \in B(\GL_{2n})$.

* If the Newton stratum $\Gr_{\GL_{2n}, \idcochar, b}^{\modification{b}}$ contains $\genlift{x}:= \idcochar(\gendRunif) \GL_{2n}(\BdR^+) \in \Gr_{\GL_{2n}, \idcochar}(\FFuntilt)$, we have $\newtonmap[\modification{b}] = \newtonmap[b] - \idcochar$.

* If the Newton stratum $\Gr_{\GL_{2n}, \ordcochar, b}^{\modification{b}}$ is not empty, we have

\[\newtonmap[\modification{b}] \leq \newtonmap[b] + \dualpolygon{\ordcochar} \quad\text{ and } \quad
\newtonmap[\modification{b}] \slopedom \newtonmap[b] \slopedom \newtonmap[\modification{b}] + \idcochar.\]

In fact, the second statement holds for any dominant cocharacter $\gencocharacter$ of $\GL_{2n}$ whose entries are either $0$ or $1$.

Given elements $b, \modification{b} \in B(\GSp_{2n})$ such that the Newton stratum $\Gr_{\GSp_{2n}, \idcochar, b}^{\modification{b}}$ contains $x:= \idcochar(\gendRunif) \GSp_{2n}(\BdR^+) \in \Gr_{\GSp_{2n}, \idcochar}(\FFuntilt)$, we have $\newtonmap[\modification{b}] = \newtonmap[b] - \idcochar$.

We may regard $\idcochar$ as a dominant cocharacter of $\GL_{2n}$. In addition, in light of Proposition <ref>, we may regard $b$ and $\modification{b}$ as elemnts of $B(\GL_{2n})$. By Proposition <ref>, we have a natural map

\[ \Gr_{\GSp_{2n}, \idcochar, b}^{\modification{b}}(\FFuntilt) \longrightarrow \Gr_{\GL_{2n}, \idcochar, b}^{\modification{b}}(\FFuntilt)\]

which sends $x$ to $\genlift{x}:= \idcochar(\gendRunif) \GL_{2n}(\BdR^+)$. Hence the desired assertion is evident by Proposition <ref>.

Let $\gencocharacter$ be a dominant cocharacter of $\GSp_{2n}$ with nonnegative entries.

Given arbitrary elements $b, \modification{b} \in B(\GSp_{2n})$, we have the following equivalent conditions:

*  $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$ is nonempty.

*  $\Gr_{\GSp_{2n}, \dualpolygon{\gencocharacter}, \modification{b}}^{b}$ is nonempty.

*  $\Gr_{\GSp_{2n}, \gencocharacter+\idcochar, b}^{\genlift{\modification{b}}}$ is nonempty for $\genlift{\modification{b}} \in B(\GSp_{2n})$ with $\newtonmap[\genlift{\modification{b}}] = \newtonmap[\modification{b}] - \idcochar$.

For an arbitrary point $x = g\gencocharacter(\gendRunif)\GSp_{2n}(\BdR^+) \in \Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$, we take

\[\dualpolygon{x} := g^{-1}\dualpolygon{\gencocharacter}(\gendRunif)\GSp_{2n}(\BdR^+) \in \Gr_{\GSp_{2n}, \dualpolygon{\gencocharacter}}(\FFuntilt)\]

and find $\genvb_{\modification{b}, \dualpolygon{x}} \simeq \genvb_b$, thereby deducing that $\dualpolygon{x}$ lies in $\Gr_{\GSp_{2n}, \dualpolygon{\gencocharacter}, \modification{b}}^{b}(\FFuntilt)$. Similarly, every point in $\Gr_{\GSp_{2n}, \dualpolygon{\gencocharacter}, \modification{b}}^{b}(\FFuntilt)$ yields a point in $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$. Hence we establish the equivalence of the conditions <ref> and <ref> by Proposition <ref>.

It remains to verify the equivalence of the conditions <ref> and <ref>. For an arbitrary point $x = g\gencocharacter(\gendRunif)\GSp_{2n}(\BdR^+) \in \Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$, we take

\[\genlift{x} := g\gencocharacter(\gendRunif)\idcochar(\gendRunif) \GSp_{2n}(\BdR^+) \in \Gr_{\GSp_{2n}, \gencocharacter+\idcochar}(\FFuntilt)\]

and find $\genvb_{b, \genlift{x}} \simeq \genvb_{\genlift{\modification{b}}}$ by Lemma <ref>, thereby deducing that $\genlift{x}$ lies in $\Gr_{\GSp_{2n}, \gencocharacter+\idcochar, b}^{\genlift{\modification{b}}}(\FFuntilt)$. Similarly, every point in $\Gr_{\GSp_{2n}, \gencocharacter+\idcochar}(\FFuntilt)$ gives rise to a point in $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}(\FFuntilt)$. Hence we complete the proof by Proposition <ref>.

For a minuscule dominant cocharacter $\gencocharacter$ of $\GSp_{2n}$, we have $\gencocharacter = d \cdot \idcochar$ or $\gencocharacter = d \cdot \idcochar + \ordcochar$ for some $d \in \Z$.

Let us identify the character lattice of $\GL_{2n}$ with $\Z^{2n}$ and write $e_1, \cdots, e_{2n}$ for its standard basis elements. Then we can identify the character lattice of $\GSp_{2n}$ as the lattice generated by

\[ e_0' := e_{n+1} + \cdots + e_{2n} \quad \text{ and } \quad e_i':= e_i - e_{2n+1-i} \text{ for } i = 1, \cdots, n. \]

The set of positive roots for $\GL_{2n}$ and $\GSp_{2n}$ are given by

\begin{align*}
\roots_{\GL_{2n}}^+ &= \{ e_i - e_j: 1 \leq i < j \leq 2n\}, \\
\roots_{\GSp_{2n}}^+ &= \{ e_i' - e_j': 1 \leq i < j \leq n\} \cup \{ e_i' + e_j' - e_0': 1 \leq i \leq j \leq n\}.
\end{align*}

Now the assertion is straightforward to verify.

Let $\gencocharacter$ be a minuscule dominant cocharacter of $\GSp_{2n}$. Assume that all entries of $\gencocharacter$ are either $d$ or $d+1$ for some $d \in \Z$. For elements $b, \modification{b} \in B(\GSp_{2n})$ such that the Newton stratum $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$ is not empty, we have

\[\newtonmap[\modification{b}] \leq \newtonmap[b] + \dualpolygon{\gencocharacter} \quad\text{ and } \quad
\newtonmap[\modification{b}] + d \cdot \idcochar \slopedom \newtonmap[b] \slopedom \newtonmap[\modification{b}] + (d+1) \cdot \idcochar.\]

By Propositions <ref> and <ref>, we may take $d = 0$ and assume that $\gencocharacter$ is either $0$ or $\ordcochar$. For $\gencocharacter = 0$, the assertion is evident since the only nonempty Newton stratum in $\Gr_{\GSp_{2n}, 0}$ with respect to $b$ is $\Gr_{\GSp_{2n}, 0, b}^b$. For $\gencocharacter = \ordcochar$, we have a natural map

\[ \Gr_{\GSp_{2n}, \ordcochar, b}^{\modification{b}}(\FFuntilt) \longrightarrow \Gr_{\GL_{2n}, \ordcochar, b}^{\modification{b}}(\FFuntilt)\]

given by Proposition <ref>, where for $\Gr_{\GL_{2n}, \ordcochar, b}^{\modification{b}}$ we regard $\ordcochar$ as a dominant cocharacter of $\GL_{2n}$ and $b, \modification{b}$ as elements of $B(\GL_{2n})$ in light of Proposition <ref>, and thus deduce the desired assertion by Propositions <ref> and <ref>.

Let $\genpartition = (\genpartition_1, \cdots, \genpartition_l)$ be an ordered partition of an integer $m \leq n$.

* We define the associated Levi subgroup $\genlevi_\genpartition$ of $\GSp_{2n}$ to be the group of block diagonal matrices

\[
\quad\quad\quad
\begin{pmatrix}
g_1 & & & & & & \\
& \ddots & & & & & \\
& & g_{l} & & & & \\
& & & h & & &\\
& & & & \simcochar(h) (g_{l}^T)^{-1} & & \\
& & & & & \ddots & \\
& & & & & & \simcochar(h) (g_{1}^T)^{-1}
\end{pmatrix}
\text{ with } g_i \in \GL_{\genpartition_i}, h \in \GSp_{2(n-m)},
\]

where $\simcochar$ denotes the similitude character of $\GSp_{2(n-m)}$.

* Given an element $b \in B(\GSp_{2n})$, its reduction to $\genlevi_\genpartition$ is an element $\genred{b} \in B(\genlevi_\genpartition)$ whose image under the natural map $B(\genlevi_\genpartition) \longrightarrow B(\GSp_{2n})$ is equal to $b$.

It is not hard to show that every standard Levi subgroup of $\GSp_{2n}$ is equal to $\genlevi_\genpartition$ for some ordered partition $\genpartition$ of an integer $m \leq n$.

Let $\genpartition = (\genpartition_1, \cdots, \genpartition_l)$ be an ordered partition of an integer $m \leq n$. We write $n_i := \genpartition_1 + \cdots + \genpartition_i$ for $i = 1, \cdots, l$. In addition, we set $n_0 := 0$ and $n_{l+1}:= n$.

* The Newton set of $\genlevi_\genpartition$ is the set of dominant rational cocharacters of $\genlevi_\genpartition$, which is also identified with the set of tuples $(\genslope_1, \cdots, \genslope_{2n}) \in \Q^{2n}$ satisfying the following properties:

* We have $\genslope_{n_{i-1}+1} \geq \cdots \geq \genslope_{n_i}$ for $i = 1, \cdots, l+1$.

* There exists a rational number $d$ with $\genslope_i + \genslope_{2n+1-i} = d$ for $i = 1, \cdots, n$.

* Given two elements $\gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_{2n})$ and $\gencocharacter' = (\gencocharacter_1', \cdots, \gencocharacter_{2n}')$ of $\newtonset{\genlevi_\genpartition}$, the inequality $\gencocharacter \leq \gencocharacter'$ holds if and only if we have inequalities

\[ \sum_{i = 1}^j \gencocharacter_i \leq \sum_{i = 1}^j \gencocharacter_i' \quad \text{ for } j = 1, \cdots, 2n\]

with equalities for $j = n_1, \cdots, n_{l+1}$.

* There exists a natural isomorphism $\pi_1(\genlevi_\genpartition)_\absgalgrp \cong \Z^{l+1}$.

* Given a dominant cocharacter $\gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_{2n})$ of $\genlevi_\genpartition$, its dual $\dualpolygon{\gencocharacter} = (\dualpolygon{\gencocharacter}_1, \cdots, \dualpolygon{\gencocharacter}_{2n})$ is specified by the following identities:

* $\dualpolygon{\gencocharacter}_{n_{i-1}+j} = -{\gencocharacter}_{n_i+1-j}$ for $i = 1, \cdots, l$ and $j = 1, \cdots, \genpartition_i$.

* $\dualpolygon{\gencocharacter}_{m+i} = -{\gencocharacter}_{2n+1-m-i}$ for $i = 1, \cdots, n-m$.

* $\dualpolygon{\gencocharacter}_i + \dualpolygon{\gencocharacter}_{2n+1-i} = - (\gencocharacter_i + \gencocharacter_{2n+1-i})$ for $i = 1, \cdots, n$.

* Given a dominant cocharacter $\gencocharacter = (\gencocharacter_1, \cdots, \gencocharacter_{2n})$ of $\genlevi_\genpartition$, its degree $\cochardeg{\gencocharacter} = (\cochardeg{\gencocharacter}_1, \cdots, \cochardeg{\gencocharacter}_{l+1})$ is given by

\[ \cochardeg{\gencocharacter}_i = \begin{cases} \gencocharacter_{n_{i-1}+1} + \cdots + \gencocharacter_{n_i} & \text{ for } i = 1, \cdots, l\\[1em]  \dfrac{\gencocharacter_{m+1} + \cdots \gencocharacter_{2n-m}}{n-m} & \text{ for } i = l+1\end{cases}.\]

All statements are straightforward to verify by our discussion in Examples <ref>, <ref>, and <ref>.

Let $\genpartition = (\genpartition_1, \cdots, \genpartition_l)$ be an ordered partition of an integer $m \leq n$. We write $n_i = \genpartition_1 + \cdots + \genpartition_i$ for $i = 1, \cdots, l$. In addition, we set $n_0: = 0$ and $n_{l+1}:=n$.

Given an element $b \in B(\GSp_{2n})$ such that $\newtonmap[b]$ has integer points with $n_1, \cdots, n_l$ as $x$-coordinates, there exists a reduction of $b$ to $\genlevi_\genpartition$.

For a rational tuple $\gentuple$, we denote its $i$-th entry by $\HNslope{\gentuple}{i}$.
Our discussion in Example <ref> implies that there exists an integer $d$ with

\begin{equation}\label{symmetry for slopes of newton polygon, levi reduction lemma}
\HNslope{\newtonmap[b]}{i} + \HNslope{\newtonmap[b]}{2n+1-i} = d \quad \text{ for } i = 1, \cdots, n.
\end{equation}

For each $i = 1, \cdots, l$, we define $\genpartition_i$-tuples $\newtonmaplevipart{i}$ and $\newtonmaplevipartdual{i}$ by

\[ \HNslope{\newtonmaplevipart{i}}{j} = \HNslope{\newtonmap[b]}{n_{i-1}+j} \quad \text{ and } \quad \HNslope{\newtonmaplevipartdual{i}}{j} = \HNslope{\newtonmap[b]}{2n-n_i+j} \quad \text{ for } j = 1, \cdots, \genpartition_j.\]

In addition, we define a $2(n-m)$-tuple $\newtonmaplevipartmid$ by

\[ \HNslope{\newtonmaplevipartmid}{i} = \HNslope{\newtonmap[b]}{m+i} \quad \text{ for } i = 1, \cdots, 2(n-m).\]

As polygons, these tuples form a partition of $\newtonmap[b]$ as illustrated in Figure <ref>.

(left) at (0, 0);
(p1) at (1, 2);
(p2) at (2, 3);
(p3) at (4, 5);
(p4) at (6, 6);
(p5) at (8, 6);
(p6) at (9, 6);
(right) at (10, 5);

[step=1cm,thick, color=blue] (left) – (p1) –  (p2) – (p3) – (p4) – (p5) – (p6) – (right);

[fill] (left) circle [radius=0.08];
[fill] (right) circle [radius=0.08];

[fill] (p1) circle [radius=0.08];
[fill] (p2) circle [radius=0.08];
[fill] (p3) circle [radius=0.08];
[fill] (p4) circle [radius=0.08];
[fill] (p5) circle [radius=0.08];
[fill] (p6) circle [radius=0.08];

[step=1cm,dotted] (2, -0.4) – (2, 6.8);
[step=1cm,dotted] (4, -0.4) – (4, 6.8);
[step=1cm,dotted] (6, -0.4) – (6, 6.8);
[step=1cm,dotted] (8, -0.4) – (8, 6.8);

at (2,-0.8) $n_1$;
at (4,-0.8) $n_2$;

(p6) ++(1.2, 0) node $\newtonmap[b]$;

[->, line width=0.6pt] (0, ) – (1.5,);
(0,0) circle [radius=0.00];

(left) at (0, 0);
(p1) at (1, 2);
(p2) at (2, 3);
(p3) at (4, 5);
(p4) at (6, 6);
(p5) at (8, 6);
(p6) at (9, 6);
(right) at (10, 5);

[step=1cm,thick, color=red] (left) – (p1) –  (p2);
[step=1cm,thick, color=orange] (p2) – (p3);
[step=1cm,thick, color=green] (p3) –  (p4);
[step=1cm,thick, color=magenta] (p4) –  (p5);
[step=1cm,thick, color=violet] (p5) – (p6) –  (right);

[fill] (left) circle [radius=0.08];
[fill] (right) circle [radius=0.08];

[fill] (p1) circle [radius=0.08];
[fill] (p2) circle [radius=0.08];
[fill] (p3) circle [radius=0.08];
[fill] (p4) circle [radius=0.08];
[fill] (p5) circle [radius=0.08];
[fill] (p6) circle [radius=0.08];

[step=1cm,dotted] (2, -0.4) – (2, 6.8);
[step=1cm,dotted] (4, -0.4) – (4, 6.8);
[step=1cm,dotted] (6, -0.4) – (6, 6.8);
[step=1cm,dotted] (8, -0.4) – (8, 6.8);

at (2,-0.8) $n_1$;
at (4,-0.8) $n_2$;

(p1) ++(-1, 0) node $\newtonmaplevipart{1}$;
(p2) ++(1, 1.8) node $\newtonmaplevipart{2}$;
(p3) ++(1, 1) node $\newtonmaplevipartmid$;
(p4) ++(1.1, 0.6) node $\newtonmaplevipartdual{2}$;
(p6) ++(1.2, 0) node $\newtonmaplevipartdual{1}$;

Partition of $\newtonmap[b]$ by integer points

By construction, these polygons all have integer breakpoints. In addition, by (<ref>) we find

\begin{equation}\label{symmetry for slopes of newton polygon, levi reduction lemma mid part}
\HNslope{\newtonmaplevipartmid}{i} + \HNslope{\newtonmaplevipartmid}{2n-2m+1-i} = d \quad \text{ for } i = 1, \cdots, 2(n-m).
\end{equation}

Hence by our discussion in Example <ref>, we find an element $b_i \in B(\GL_{\genpartition_i})$ with $\newtonmap[b_i] = \newtonmaplevipart{i}$ for $i = 1, \cdots, l$ and also an element $c \in B(\GSp_{2(n-m)})$ with $\newtonmap[c] = \newtonmaplevipartmid$. Let us choose a representative $g_i \in \GL_{\genpartition_i}(\compmaxunram{\genlocfield})$ of $b_i$ for each $i = 1, \cdots, l$ and  a representative $h \in \GSp_{2(n-m)}(\compmaxunram{\genlocfield})$ of $c$. We take $\genred{b} \in B(\genlevi_\genpartition)$ to be the $\frobfieldaut$-conjugacy class of

\[
\begin{pmatrix}
g_1 & & & & & & \\
& \ddots & & & & & \\
& & g_{l} & & & & \\
& & & h & & &\\
& & & & \simcochar(h) (g_{l}^T)^{-1} & & \\
& & & & & \ddots & \\
& & & & & & \simcochar(h) (g_{1}^T)^{-1}
\end{pmatrix}
\in \genlevi_{\genpartition}(\compmaxunram{\genlocfield})
\]

where $\simcochar$ denotes the similitude character of $\GSp_{2(n-m)}$, and denote by $\levilift{b}$ the image of $\genred{b}$ under the natural map $B(\genlevi_\genpartition) \longrightarrow B(\GSp_{2n})$. By the functoriality of the Newton map, $\newtonmap[\levilift{b}]$ is given by the image of $\newtonmap[\genred{b}]$ under the natural map $\newtonset{\genlevi_\genpartition} \longrightarrow \newtonset{\GSp_{2n}}$. It is then not hard to see by (<ref>) and (<ref>) that $\newtonmap[\levilift{b}]$ coincides with $\newtonmap[b]$. Since the Newton map for $\GSp_{2n}$ is injective as noted in Example <ref>, we find $b = \levilift{b}$ and consequently complete the proof.

Consider an arbitrary element $b \in B(\GSp_{2n})$. Let us denote the centralizer of $\newtonmap[b]$ by $\genlevi_b$.

* $\genlevi_b$ is given by $\genlevi_\genpartition$ for some ordered partition $\genpartition = (\genpartition_1, \cdots, \genpartition_l)$ of an integer $m \leq n$.

* Given an element $\modification{b} \in B(\GSp_{2n})$ such that for each breakpoint of $\newtonmap[b]$ there exists an integer point of $\newtonmap[\modification{b}]$ with the same $x$-coordinate, we have a reduction of $\modification{b}$ to $\genlevi_b$.

We consider the breakpoints of $\newtonmap[b]$ on the interval $[0, n]$ and denote their $x$-coordinates by $n_1, \cdots, n_l$ in ascending order. It is straightforward to verify the first statement by taking an ordered partition $\genpartition = (\genpartition_1, \cdots, \genpartition_l)$ of $m = n_l$ with $\genpartition_i:= n_1 + \cdots + n_i$ for $i = 1, \cdots, l$. The second statement then immediately follows from Proposition <ref>.

Consider elements $\gencocharacter, \modification{\gencocharacter} \in \newtonset{\GSp_{2n}}$ with the following properties:

*  Any two distinct slopes in $\gencocharacter$ differ by more than 1.

*  We have $\modification{\gencocharacter} \slopedom \gencocharacter \slopedom \gencocharacter + \idcochar$.

Then for each breakpoint of $\gencocharacter$, there exists a break point of $\modification{\gencocharacter}$ with the same $x$-coordinate.

For a rational tuple $\gentuple$, we denote its $i$-th entry by $\HNslope{\gentuple}{i}$. Take an arbitrary breakpoint of $\gencocharacter$ and denote its $x$-coordinate by $m$. By the properties <ref> and <ref>, we find

\begin{equation*}
\HNslope{\modification{\gencocharacter}}{m} \leq \HNslope{\gencocharacter}{m} < \HNslope{\gencocharacter}{m+1} -1 \leq \HNslope{\modification{\gencocharacter}}{m+1}.
\end{equation*}

Hence we deduce that $\modification{\gencocharacter}$ has a breakpoint with $x$-coordinate $m$, thereby completing the proof.

Let $\gencocharacter$ be a minuscule dominant cocharacter of $\GSp_{2n}$ whose entries are either $d$ or $d+1$ for some $d \in \Z$. Consider elements $b, \modification{b} \in B(\GSp_{2n})$ such that any two distinct slopes in $\newtonmap[b]$ differ by more than $1$.
The Newton stratum $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$ is nonempty if and only if the following conditions are satisfied:

*  We have
$\newtonmap[\modification{b}] \leq \newtonmap[b] + \dualpolygon{\gencocharacter}$ and $\newtonmap[\modification{b}] + d \cdot \idcochar \slopedom \newtonmap[b] \slopedom \newtonmap[\modification{b}] + (d+1) \cdot \idcochar$.

*  For each breakpoint of $\newtonmap[b]$, there exists

a breakpoint of $\newtonmap[\modification{b}]$ with the same $x$-coordinate.

(p00) at (1.5, 4);
(p11) at (2.8, 6.3);
(r11) at (4, 7.8);
(p22) at (5.2, 8.7);
(p33) at (6.5, 9);
(p44) at (8, 8);

(left) at (0, 0);
(q0) at (1.5, 3.2);
(q1) at (4, 5.2);

(q2) at (6.5, 5.7);
(q3) at (8, 4);

(r1) at (4, 3.8);
(r2) at (6.5, 3.5);

(p0) at (1.5, 2.5);
(p1) at (2.8, 3.5);
(p2) at (5.2, 3.5);
(p3) at (6.5, 2.5);
(p4) at (8, 0);

[step=1cm,thick, color=red] (left) – (q0) –  (q1) – (q2) – (q3);
[step=1cm,thick, color=green] (left) – (p0) –  (p1) – (r1) – (p2) – (p3) – (p4);
[step=1cm,thick, color=teal] (left) – (p00) –  (p11) – (r11) – (p22) – (p33) – (p44);
[step=1cm,thick, color=blue] (q1) – (r2) – (p4);

[fill] (q0) circle [radius=0.05];
[fill] (q1) circle [radius=0.05];
[fill] (q2) circle [radius=0.05];
[fill] (q3) circle [radius=0.05];
[fill] (left) circle [radius=0.05];

[fill] (r1) circle [radius=0.05];
[fill] (r2) circle [radius=0.05];

[fill] (p0) circle [radius=0.05];
[fill] (p1) circle [radius=0.05];
[fill] (p2) circle [radius=0.05];
[fill] (p3) circle [radius=0.05];
[fill] (p4) circle [radius=0.05];

[fill] (p00) circle [radius=0.05];
[fill] (p11) circle [radius=0.05];
[fill] (r11) circle [radius=0.05];
[fill] (p22) circle [radius=0.05];
[fill] (p33) circle [radius=0.05];
[fill] (p44) circle [radius=0.05];

[step=1cm,dotted] (1.5, -0.4) – (1.5, 9.2);
[step=1cm,dotted] (4, -0.4) – (4, 9.2);
[step=1cm,dotted] (6.5, -0.4) – (6.5, 9.2);

(q1) ++(0.8, 0.6) node $\newtonmap[b]$;
(r2) ++(1.8, -0.8) node $\newtonmap[b] + \dualpolygon{\gencocharacter}$;
(p00) ++(-1.3, 0) node $\newtonmap[\modification{b}] + \idcochar$;
(p0) ++(0.7, -0.3) node $\newtonmap[\modification{b}]$;

Illustration of the conditions in Theorem <ref> for $d =0$

For a rational tuple $\gentuple$, we denote its $i$-th entry by $\HNslope{\gentuple}{i}$. By Propositions <ref> and <ref>, we may set $d = 0$ and assume that $\gencocharacter$ is either $0$ or $\ordcochar$. For $\gencocharacter = 0$, the assertion is evident since the only nonempty Newton stratum in $\Gr_{\GSp_{2n}, 0}$ with respect to $b$ is $\Gr_{\GSp_{2n}, 0, b}^b$. Therefore we may henceforth take $\gencocharacter = \ordcochar$.

Let us first assume that $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$ is nonempty. The necessity of the condition <ref> immediately follows from Proposition <ref>. Now we take an arbitrary breakpoint of $\newtonmap[b]$ and denote its $x$-coordinate by $m$. By the condition <ref> and the assumption on slopes of $\newtonmap[b]$, we find

\begin{equation*}
\HNslope{\newtonmap[\modification{b}]}{m+1} \leq \HNslope{\newtonmap[b]}{m+1} < \HNslope{\newtonmap[b]}{m} -1 \leq \HNslope{\newtonmap[\modification{b}]}{m}
\end{equation*}

and consequently deduce that $\modification{\gencocharacter}$ has a breakpoint with $x$-coordinate $m$, thereby establishing the necessity of the condition <ref>.

For the converse, we now assume that the conditions <ref> and <ref> are satisfied. We consider the breakpoints of $\newtonmap[b]$ on the interval $[0, n]$ and denote their $x$-coordinates by $n_1, \cdots, n_l$ in ascending order. We also set $n_0: = 0$ and take $\genpartition = (\genpartition_1, \cdots, \genpartition_l)$ to be an ordered partition of $m = n_l$ with

\[\genpartition_i:= n_i - n_{i-1} \quad \text{ for } i = 1, \cdots, l.\]

It is not hard to see that the centralizer of $\newtonmap[b]$ is $\genlevi_\genpartition$. Moreover, Proposition <ref> and the condition <ref> together imply that $b$ and $\modification{b}$ admit reductions to $\genlevi_\genpartition$, which we respectively denote by $\genred{b}$ and $\genred{\modification{b}}$. Let us now set

\[d_i:= \sum_{j = n_{i-1}+1}^{n_i} \HNslope{\newtonmap[b]}{j} - \sum_{j=n_{i-1}+1}^{n_i} \HNslope{\newtonmap[\modification{b}]}{j} \quad \text{ for } i = 1, \cdots, l.\]

By the condition <ref>, we have $0 \leq d_i \leq \genpartition_i$ for $i = 1, \cdots, l$. Take $\genred{\gencocharacter}$ to be the $2n$-tuple specified by the following properties:

* For $i = 1, \cdots, l+1$ and $j = 1, \cdots, \genpartition_i$, we have

\[\HNslope{\genred{\gencocharacter}}{n_{i-1}+j} = \begin{cases} 1 & \text{ if } j \leq d_i,\\ 0 & \text { if } j > d_i,\end{cases}\]

where we set $d_{l+1}:= n- n_l = n-m$.

* For $j = 1, \cdots, n$, we have $\HNslope{\genred{\gencocharacter}}{j} + \HNslope{\genred{\gencocharacter}}{2n+1-j} = 1$.

It is not hard to see by Proposition <ref> that $\genred{\gencocharacter}$ is a dominant cocharacter of $\genlevi_\genpartition$ with

\[\kottwitzmap[\genred{\modification{b}}] = \kottwitzmap[\genred{b}] - \cochardeg{\genred{\gencocharacter}} \quad \text{ and } \quad \newtonmap[\genred{\modification{b}}] \leq \newtonmap[\genred{b}] - \galavg{(\dualpolygon{\genred{\gencocharacter}})}.\]

Moreover, $\genred{b}$ is basic in $B(\genlevi_\genpartition)$ by construction. Hence Proposition <ref> implies that the Newton stratum $\Gr_{\genlevi_\genpartition, \genred{\gencocharacter}, \genred{b}}^{\genred{\modification{b}}}$ is not empty. Since we have a natural map

\[ \Gr_{\genlevi_\genpartition, \genred{\gencocharacter}, \genred{b}}^{\genred{\modification{b}}}(\FFuntilt) \longrightarrow \Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}(\FFuntilt)\]

by Proposition <ref>, we deduce by Proposition <ref> that $\Gr_{\GSp_{2n}, \gencocharacter, b}^{\modification{b}}$ is not empty as desired, thereby completing the proof.

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# Octupolar edge state in an $e_{g}$ orbital system on a square lattice

Katsunori Kubo  Advanced Science Research Center, Japan Atomic Energy Agency,
Tokai, Ibaraki 319-1195, Japan

###### Abstract

A tight-binding model for $e_{g}$ orbitals on a square lattice is
investigated. We consider only the nearest-neighbor hopping and the model is
characterized by two hopping parameters, $t_{1}$ and $t_{2}$. There are Dirac
points in the electronic band structure and the type of the Dirac points
(type-I or type-II) depends on the ratio $t_{2}/t_{1}$. For the case of the
type-I Dirac points, edge states appear for a lattice with edges perpendicular
to the $[11]$ direction. The edge states at a certain momentum along the edges
have octupole moments with opposite signs between the right and left edges.
Thus, these edge states can be called helical octupolar edge states. This
study bridges the research fields of topological phenomena and multipole
physics.

##  I Introduction

Topological semimetals, characterized by the presence of Dirac or Weyl points
(topological nodal points) in their electronic band structures, have garnered
significant attention due to their intriguing physical properties [1, 2, 3,
4]. A topological nodal point is defined as a point where the upper and lower
bands touch each other, possessing topological protection. A Dirac point
exhibits spin degeneracy, while a Weyl point does not. Realizing such
topological nodal points necessitates at least two bands. In systems with two
sites in a unit cell, two bands are present. Graphene is a prime example of
such a system, exhibiting Dirac points [1, 2]. Alternatively, the spin degrees
of freedom can be used to achieve two bands. For instance, the Rashba spin-
orbit coupling [5] lifts the spin degeneracy of a band, except at certain
$\bm{k}$ points where the degeneracy is preserved by time-reversal symmetry,
resulting in Weyl points [6].

In this study, we explore an alternative approach to achieve two bands and
Dirac points by exploiting the orbital degrees of freedom. As a representative
two-orbital system, we investigate a model for the $e_{g}$ orbitals of $d$
electrons. Notably, type-II Dirac points have been observed in $e_{g}$ orbital
systems, including an $e_{g}$ orbital model [7], a cuprate superconductor [8],
and a LaAlO2/LaNiO3/LaAlO3 quantum well [9]. A type-II Dirac point is
characterized by the touching of upper and lower bands, with electron and hole
pockets extending from it even when the Fermi level is situated at this point
[10] [see, Fig.1(f)]. A type-I Dirac point also involves the touching of upper
and lower bands, but in this case, the Fermi surface vanishes around it when
the Fermi level is located at this point [see, Fig.1(h)]. Traditionally, only
type-I Dirac points were referred to as Dirac points until the term “type-II
Dirac point” was introduced [10]. Despite the anticipation of several
intriguing features, phenomena arising from topological signatures might be
obscured by the presence of finite density states at the Fermi level in type-
II Dirac point cases.

Here, we consider a tight-binding model for the $e_{g}$ orbitals on a square
lattice with nearest-neighbor hopping in a general form. This model contains
two hopping parameters. Such a simple tight-binding model, like the Kane-Mele
model for topological insulators [11], the tight-binding model for graphene
possessing Dirac points [12], and the Rashba-Hubbard model for a Weyl
semimetal [6], is invaluable for exploring and demonstrating topological
phenomena. By tuning the ratio of the two parameters, we demonstrate the
emergence of type-I Dirac points in the $e_{g}$ orbital system. Furthermore,
we explore the emergence of edge states unique to the type-I Dirac point
cases. Specifically, we observe that these edge states have octupolar moments,
with opposite signs between the left and right edges, at a certain momentum
along the edges.

Figure 1:  Energy dispersion for $\Delta=0$. Energy dispersion along symmetric
directions: (a) for $\alpha=0$ ($t_{2}=0$), (b) for $\alpha=0.25\pi$
($t_{2}=t_{1}$), (c) for $\alpha=\tan^{-1}2$ ($=0.3524\pi$, $t_{2}=2t_{1}$),
and (d) for $\alpha=0.5\pi$ ($t_{1}=0$). Lower panels show energy dispersion
in the entire Brillouin zone: (e) for $\alpha=0$, (f) for $\alpha=0.25\pi$,
(g) for $\alpha=\tan^{-1}2$, and (h) for $\alpha=0.5\pi$. The zero-energy
plane is drawn in these figures to distinguish the types of Dirac points
clearly.

##  II Model

In this study, we do not consider a magnetic field or magnetic order. Hence,
we can disregard the spin degrees of freedom and simplify the model to a
spinless one for the $e_{g}$ orbitals. The model Hamiltonian on a square
lattice with nearest-neighbor hopping is represented as follows:

$H=\sum_{\tau\tau^{\prime}}c_{\bm{k}\tau}^{\dagger}\epsilon_{\tau\tau^{\prime}}(\bm{k})c_{\bm{k}\tau^{\prime}},$
(1)

where $c_{\bm{k}\tau}$ is the annihilation operator for an electron with
momentum $\bm{k}$ and orbital $\tau$. Here, $\tau=1$ denotes the $x^{2}-y^{2}$
orbital, and $\tau=2$ denotes the $3z^{2}-r^{2}$ orbital. The matrix elements
of the Hamiltonian are given by [13, 14]

$\displaystyle\epsilon_{11}(\bm{k})$
$\displaystyle=\frac{1}{2}[3(dd\sigma)+(dd\delta)](c_{x}+c_{y})+\frac{\Delta}{2},$
(2) $\displaystyle\epsilon_{22}(\bm{k})$
$\displaystyle=\frac{1}{2}[(dd\sigma)+3(dd\delta)](c_{x}+c_{y})-\frac{\Delta}{2},$
(3) $\displaystyle\epsilon_{12}(\bm{k})$
$\displaystyle=\epsilon_{21}(\bm{k})=-\frac{\sqrt{3}}{2}[(dd\sigma)-(dd\delta)](c_{x}-c_{y}),$
(4)

where $c_{\mu}=\cos k_{\mu}$ with $\mu=x$ or $y$, and $\Delta$ represents the
difference in the onsite energy level between the two orbitals. The lattice
constant is set as unity here. The quantities $(dd\sigma)$ and $(dd\delta)$
denote the two-center integrals [15]. This model can be regarded as a model
for $\Gamma_{8}$ orbitals of $f$ electrons by replacing $(dd\sigma)$ with
$[3(ff\sigma)+4(ff\pi)]/7$ and $(dd\delta)$ with
$[(ff\pi)+5(ff\delta)+15(ff\phi)]/21$. Since it is a two-orbital model, the
matrix $\epsilon(\bm{k})$ can be expressed using the Pauli matrices as

$\begin{split}\epsilon(\bm{k})&=\frac{1}{2}[\epsilon_{11}(\bm{k})+\epsilon_{22}(\bm{k})]\sigma^{0}\\\
&\quad+\frac{1}{2}[\epsilon_{11}(\bm{k})-\epsilon_{22}(\bm{k})]\sigma^{z}+\epsilon_{12}(\bm{k})\sigma^{x}\\\
&=h_{0}(\bm{k})\sigma^{0}+h_{x}(\bm{k})\sigma^{x}+h_{z}(\bm{k})\sigma^{z},\end{split}$
(5)

where $\sigma^{\mu}$ represents the $\mu$ component of the Pauli matrix, and
$\sigma^{0}$ is defined as the unit matrix. The coefficients are given by

$\displaystyle\begin{split}h_{0}(\bm{k})&=[\epsilon_{11}(\bm{k})+\epsilon_{22}(\bm{k})]/2\\\
&=[(dd\sigma)+(dd\delta)](c_{x}+c_{y})\\\ &=2t_{1}(c_{x}+c_{y}),\end{split}$
(6)
$\displaystyle\begin{split}h_{x}(\bm{k})&=\epsilon_{12}(\bm{k})=\epsilon_{21}(\bm{k})\\\
&=-\frac{\sqrt{3}}{2}[(dd\sigma)-(dd\delta)](c_{x}-c_{y})\\\
&=-\sqrt{3}t_{2}(c_{x}-c_{y}),\end{split}$ (7)
$\displaystyle\begin{split}h_{z}(\bm{k})&=[\epsilon_{11}(\bm{k})-\epsilon_{22}(\bm{k})]/2\\\
&=\frac{1}{2}[(dd\sigma)-(dd\delta)](c_{x}+c_{y})+\frac{\Delta}{2}\\\
&=t_{2}(c_{x}+c_{y})+\Delta/2,\end{split}$ (8)

where we have defined

$\displaystyle t_{1}$ $\displaystyle=[(dd\sigma)+(dd\delta)]/2,$ (9)
$\displaystyle t_{2}$ $\displaystyle=[(dd\sigma)-(dd\delta)]/2.$ (10)

We can assume $t_{1}\geq 0$ and $t_{2}\geq 0$ without loss of generality.
Then, we parameterize them as

$\displaystyle t_{1}$ $\displaystyle=t\cos\alpha,$ (11) $\displaystyle t_{2}$
$\displaystyle=t\sin\alpha,$ (12)

with $t>0$ and $0\leq\alpha\leq\pi/2$.

This model has been investigated in various contexts with different values of
$\alpha$. The model at $\alpha=\pi/4$ [$t_{1}=t_{2}$, $(dd\delta)=0$, see Fig.
1(b)] has been used as a model for perovskite manganites [16] and as a model
for $\Gamma_{8}$ orbitals of $f$ electrons, considering only $(ff\sigma)$ [17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. From this model, an effective
Hamiltonian in the strong coupling limit is derived, and frustration arising
from the orbital anisotropy is discussed [29, 30, 31, 32, 33]. A similar
frustrated model is known as the Kitaev model and has been intensively studied
[34, 35]. By using the model at $\alpha=0$ [$t_{2}=0$,
$(dd\delta)=(dd\sigma)$, see Fig. 1(a)], the possible anisotropic
superconducting pairing originating from the orbital anisotropy of $e_{g}$
orbitals was discussed [36]. Such anisotropic superconductivity has been
discussed for the $f$-electron system PrT2X20 (where T denotes a transition
metal element and X denotes Zn or Al) [25, 26, 27, 28]. The model at
$\alpha=\pi/2$ [$t_{1}=0$, $(dd\delta)=-(dd\sigma)$, see Fig. 1(d)] near half-
filling exhibits pocket Fermi surfaces around $(\pm\pi/2,\pm\pi/2)$. Suppose a
superconducting pair is composed of electrons on the same Fermi pocket. In
that case, the superconducting pair has a finite total momentum, resembling
the Fulde-Ferrell-Larkin-Ovchinnikov state [37, 38], even without a magnetic
field similar to the $\eta$-pairing state [39]. This possibility has been
investigated in Refs. [13, 14].

##  III Energy bands

The energy dispersion of the model is given by

$\begin{split}E(\bm{k})&=h_{0}(\bm{k})\pm\sqrt{h_{x}^{2}(\bm{k})+h_{z}^{2}(\bm{k})}\\\
&=h_{0}(\bm{k})\pm h(\bm{k}).\end{split}$ (13)

The two bands touch each other at the Dirac points, where $h(\bm{k})=0$.

In Fig. 1, we show the energy dispersion for $\Delta=0$ for some values of
$\alpha$. The bandwidth $W$ varies non-monotonically with $\alpha$: $W=8t$ at
$\alpha=0$ [Fig. 1(a)], $W$ exceeds $8t$, for example, at $\alpha=0.25\pi$
[Fig. 1(b)], and $W=4\sqrt{3}t$ at $\alpha=0.5\pi$ [Fig. 1(d)]. Note that
hybridization between the $x^{2}-y^{2}$ and $3z^{2}-r^{2}$ orbitals is
forbidden along the $k_{x}=k_{y}$ line due to mirror symmetry [7, 8]. For the
half-filled case with $\Delta=0$, the Fermi level is zero due to electron-hole
symmetry and the Dirac points are always located at the Fermi level.

The two bands are degenerate for $\alpha=0$ [$t_{2}=0$, Figs. 1(a) and 1(e)].
For $\alpha\neq 0$, Dirac points emerge at $(\pm\pi/2,\pm\pi/2)$, and the type
of the Dirac points depends on $\alpha$. For $\alpha<\tan^{-1}2$ [Figs. 1(b)
and 1(f)], they are of type-II, where the two Fermi surfaces touch each other
at the Dirac points for half-filling. For $\alpha>\tan^{-1}2$ [Figs. 1(d) and
1(h)], they are of type-I, and the Fermi surface disappears for half-filling.
The disappearance of the Fermi surface at $\alpha=0.5\pi$ was already pointed
out in Refs. [13, 14]. At the border $\alpha=\tan^{-1}2$ ($=0.3524\pi$,
$t_{2}=2t_{1}$), a flat dispersion is observed along the $(0,0)$–$(\pi,\pi)$
direction [Figs. 1(c) and 1(g)], and the Dirac points are called type-III [40,
41]. For the half-filled case, the change in the type of Dirac points is a
Lifshitz transition [42].

The energy dispersions for finite $\Delta$ are depicted in Fig. 2 for
$\alpha=0.3\pi$ (type-II Dirac point) and for $\alpha=0.4\pi$ (type-I Dirac
point).

Figure 2:  Energy dispersion for $\Delta=-2t$, $0$, and $2t$ along symmetric
directions. Upper panels show energy dispersion for $\alpha=0.3\pi$ (a) for
$\Delta=-2t$, (b) for $\Delta=0$, and (c) for $\Delta=2t$. Lower panels show
energy dispersion for $\alpha=0.4\pi$ (d) for $\Delta=-2t$, (e) for
$\Delta=0$, and (f) for $\Delta=2t$.

The Dirac point is given by $k_{x}=k_{y}=\cos^{-1}[-\Delta/(4t_{2})]$.
Although the position of the Dirac point shifts from $(\pi/2,\pi/2)$ for
finite $\Delta$, the type of the Dirac point remains unchanged. While the
energy at the Dirac point deviates from zero for $\Delta\neq 0$, it still
resides at the Fermi level $E=-\Delta/\tan\alpha$ for half-filling in the
type-I Dirac point cases, maintaining the system semimetallic. On the other
hand, for the type-II Dirac point cases, the Dirac points are not aligned with
the Fermi level for $\Delta\neq 0$. For a large level splitting
$|\Delta|>4t_{2}$ (not shown), a gap opens between the two bands, leading to
the disappearance of the Dirac points.

Figure 3 shows the parameter regions of the type-I and type-II Dirac
semimetals on both the $t_{1}$-$t_{2}$ plane [Fig. 3(a)] and on the
$(dd\sigma)$-$(dd\delta)$ plane [Fig. 3(b)].

Figure 3:  Regions of the type-I and type-II Dirac semimetals: (a) on the
$t_{1}$-$t_{2}$ plane and (b) on the $(dd\sigma)$-$(dd\delta)$ plane. The
energy unit in these figures is arbitrary.

On the $t_{1}$-$t_{2}$ plane, the boundaries are given by $t_{2}=\pm 2t_{1}$.
On the $(dd\sigma)$-$(dd\delta)$ plane, the boundaries are given by
$(dd\delta)=-(dd\sigma)/3$ and $(dd\delta)=-3(dd\sigma)$. For perovskite
manganites, models with $(dd\delta)=0$ are commonly employed, resulting in
type-II Dirac points. However, it is worth noting that the condition
$|(dd\delta)|>|(dd\sigma)|/3$ with opposite signs for $(dd\delta)$ and
$(dd\sigma)$ is not unrealistic, indicating the possibility of type-I Dirac
points emerging in $e_{g}$ orbital systems.

Figure 4 shows the normalized vector field
$\hat{\bm{h}}(\bm{k})=\bm{h}(\bm{k})/h(\bm{k})=[h_{x}(\bm{k}),h_{z}(\bm{k})]/h(\bm{k})=[\hat{h}_{x}(\bm{k}),\hat{h}_{z}(\bm{k})]$
around the Dirac points $\bm{k}=(\pi/2,\pi/2)$ and $\bm{k}=(\pi/2,-\pi/2)$ for
$\Delta=0$.

Figure 4:  $\hat{\bm{h}}(\bm{k})$ (a) around $\bm{k}=(\pi/2,\pi/2)$ and (b)
around $\bm{k}=(\pi/2,-\pi/2)$.

The vector $\hat{\bm{h}}(\bm{k})$ has vortex structures around the Dirac
points. The winding number of a normalized two-component vector field
$\hat{\bm{h}}(\bm{k})$ is [43, 44]

$w=\oint_{C}\frac{d\bm{k}}{2\pi}\cdot\left[\hat{h}_{x}(\bm{k})\bm{\nabla}\hat{h}_{z}(\bm{k})-\hat{h}_{z}(\bm{k})\bm{\nabla}\hat{h}_{x}(\bm{k})\right],$
(14)

where $C$ is a loop enclosing a Dirac point. We obtain $w=-1$ for
$\bm{k}=(\pi/2,\pi/2)$ and $\bm{k}=(-\pi/2,-\pi/2)$ and $w=1$ for
$\bm{k}=(\pi/2,-\pi/2)$ and $(-\pi/2,\pi/2)$.

##  IV Edge states

From the presence of topological defects such as Dirac points, we anticipate
the emergence of edge states similar to those observed in a model of graphene
[12, 45, 46]. We consider two types of edges: those aligned along the $y$
direction [straight edges, Fig. 5(a)], and those perpendicular to the $[11]$
direction [zigzag edges, Fig. 5(b)].

Figure 5:  Lattices with edges. (a) Straight edges. (b) Zigzag edges. $i$
denotes the label representing positions perpendicular to the edges, and $N$
is the number of these positions.

We denote the momentum along the edges as $k$ and the momentum perpendicular
to the edges as $k_{\perp}$. To examine the presence of edge states, the
winding number $w(k)$ for a fixed $k$ is crucial [45, 46]:

$w(k)=\int_{0}^{2\pi}\frac{dk_{\perp}}{2\pi}\left[\hat{h}_{x}(\bm{k})\frac{\partial}{\partial
k_{\perp}}\hat{h}_{z}(\bm{k})-\hat{h}_{z}(\bm{k})\frac{\partial}{\partial
k_{\perp}}\hat{h}_{x}(\bm{k})\right].$ (15)

For the zigzag edges, $h_{x}(\bm{k})=-2\sqrt{3}t_{2}\sin k\sin k_{\perp}$ and
$h_{z}(\bm{k})=2t_{2}\cos k\cos k_{\perp}+\Delta/2$, where we have set
$1/\sqrt{2}$ times the bond length as unity here. For simplicity, we evaluate
the winding number $w(k)$ for $\Delta=0$. For the straight edges, we find
$w(k)=0$ and the edge states will probably be absent. For the zigzag edges, we
find $w(k)=\text{sgn}[\sin(2k)]$ except for $k=0$, $\pm\pi/2$, and $\pm\pi$
(projected Dirac points). Hence, edge states should exist, at least without
$t_{1}$ and $\Delta$, except at the projected Dirac points.

To explicitly demonstrate the existence of the edge states, we calculate the
energy bands for lattices with finite widths. Figure 6 shows the energy bands
for a lattice with straight edges with various values of $\alpha$ for
$\Delta=0$.

Figure 6:  Energy dispersion at $\Delta=0$ for a lattice with straight edges
with $N=71$: (a) for $\alpha=0$, (b) for $\alpha=0.25\pi$, (c) for
$\alpha=0.4\pi$, and (d) for $\alpha=0.5\pi$.

We notice that all bands are bulk states; that is, no edge state isolated from
the bulk states appears. This observation is consistent with $w(k)=0$. We have
also calculated the band dispersions for a lattice with straight edges with
$\Delta\neq 0$ (not shown) and found no edge state. In Fig. 7, we show the
energy bands for a lattice with zigzag edges with various values of $\alpha$
for $\Delta=0$.

Figure 7:  Energy dispersion at $\Delta=0$ for a lattice with zigzag edges
with $N=71$: (a) for $\alpha=0$, (b) for $\alpha=0.25\pi$, (c) for
$\alpha=0.4\pi$, and (d) for $\alpha=0.5\pi$. The solid circles in (c)
indicate edge states in which the multipole density is evaluated in Figs. 9(a)
and 9(c).

We observe the zero-energy states isolated from the bulk states when the Dirac
points become type-I, that is, $\alpha>\tan^{-1}2$. In particular, for
$\alpha=0.5\pi$ ($t_{1}=0$), we find the zero-energy edge states except for
the projected Dirac points, consistent with the non-zero values of the winding
number $w(k)$. For an even number of $N$, the energy of the zero-energy states
deviates from zero when $N$ is small. Thus, we chose $N=71$ for the
calculations presented above.

To examine the stability of the edge states against the inclusion of the level
splitting, we also evaluate the energy bands for finite $\Delta$. In Fig. 8,
we show the energy bands for $\Delta=\pm 2t$ at $\alpha=0.4\pi$ and $0.5\pi$.

Figure 8:  Energy dispersion for a lattice with zigzag edges for finite
$\Delta$ with $N=71$: (a) for $\alpha=0.4\pi$ and $\Delta=-2t$, (b) for
$\alpha=0.4\pi$ and $\Delta=2t$, (c) for $\alpha=0.5\pi$ and $\Delta=-2t$, and
(d) for $\alpha=0.5\pi$ and $\Delta=2t$. The solid circles in (b) indicate
edge states in which the multipole density is evaluated in Figs. 9(b) and
9(d).

While the energy of the edge states deviates from zero for $\Delta\neq 0$ at
$\alpha\neq 0.5\pi$, the edge states persist even for finite $\Delta$. These
edge states are situated at $E=-\Delta/\tan\alpha$, corresponding to the
energy at the Dirac points and the Fermi level for the half-filled case. We
have checked that the edge states appear as long as the type-I Dirac points
exist, that is, $\alpha>\tan^{-1}2$ and $|\Delta|<4t_{2}$.

To gain further insights into the edge states, we evaluate the multipole
density. The multipole operators at site $(i,j)$ are defined as

$\hat{\sigma}^{\mu}(i,j)=\sum_{\tau\tau^{\prime}}c^{\dagger}_{ij\tau}\sigma^{\mu}_{\tau\tau^{\prime}}c_{ij\tau^{\prime}},$
(16)

where $c_{ij\tau}$ denotes the annihilation operator of the electron at site
$(i,j)$ with orbital $\tau$, and $\mu=0$, $x$, $y$, or $z$. The multipole
density operator at position $i$ is

$\begin{split}\hat{\sigma}^{\mu}(i)&=\sum_{j}\hat{\sigma}^{\mu}(i,j)\\\
&=\sum_{k\tau\tau^{\prime}}c^{\dagger}_{ik\tau}\sigma^{\mu}_{\tau\tau^{\prime}}c_{ik\tau^{\prime}}\\\
&=\sum_{k}\hat{\sigma}^{\mu}(i,k),\end{split}$ (17)

where $c_{ik\tau}$ is the Fourier transform of $c_{ij\tau}$ along the edges.
The 0th component is the charge operator $\hat{Q}(i)=\hat{\sigma}^{0}(i)$, the
$z$ and $x$ components are the quadrupole operators
$\hat{O}_{20}(i)=\hat{\sigma}^{z}(i)$ and
$\hat{O}_{22}(i)=\hat{\sigma}^{x}(i)$, respectively, and the $y$ component is
the octupole operator $\hat{T}_{xyz}(i)=\hat{\sigma}^{y}(i)$ [47, 48].

For the lattice with zigzag edges, we can show that at $k=\pi/6$, the
eigenstates of $\hat{\sigma}^{y}(i,k)$ with eigenvalue of $-1$ at $i=-N/2$ and
with eigenvalue of $+1$ at $i=N/2$ do not have matrix elements of the
Hamiltonian proportional to $t_{2}$. Thus, these states are the zero-energy
states for $\alpha=0.5\pi$ and $\Delta=0$ completely localized on the left and
right edges, respectively, possessing opposite octupole moments.

For other values of $k$, $\alpha$, and $\Delta$, we need to numerically
evaluate the multipole density: $Q(i)=\langle\hat{Q}(i)\rangle$,
$O_{20}(i)=\langle\hat{O}_{20}(i)\rangle$,
$O_{22}(i)=\langle\hat{O}_{22}(i)\rangle$, and
$T_{xyz}(i)=\langle\hat{T}_{xyz}(i)\rangle$, where $\langle\cdots\rangle$
denotes the expectation value. In Figs. 9(a)–9(d), we show the multipole
density in the edge states for $\alpha=0.4\pi$ at $k=\pm 0.1\pi$ with
$\Delta=0$ and $2t$.

Figure 9:  Multipole density in the edge states for a lattice with zigzag
edges at $\alpha=0.4\pi$ with $N=71$: (a) for $k=0.1\pi$ and $\Delta=0$, (b)
for $k=0.1\pi$ and $\Delta=2t$, (c) for $k=-0.1\pi$ and $\Delta=0$, and (d)
for $k=-0.1\pi$ and $\Delta=2t$. (e) Schematic view of the edge states with
octupole moments. The shape of the two objects indicates the cubic symmetric
charge distribution in the wavefunctions. The red (blue) color represents the
dipole moment density parallel (antiparallel) to the $z$ direction.

The edge states are doubly degenerate, and we take the average of the two
states. From the charge density $Q$, we recognize these states are well
localized around the edges. We find that the quadrupole moment $O_{22}$ is
always zero. The finite quadrupole moment $O_{20}$ indicates an imbalance in
the orbital occupations between $x^{2}-y^{2}$ and $3z^{2}-r^{2}$. This moment
has the same sign on both edges. The octupole moment $T_{xyz}$ has opposite
signs between the edges. The octupole moment changes its sign when we change
the sign of the momentum $k$ [compare Fig. 9(a) with Fig. 9(c) and Fig. 9(b)
with Fig. 9(d)]. Thus, the edge states are helical regarding the octupole
moment. In Fig.9(e), we show a schematic view of the helical octupolar edge
states.

##  V Summary and Discussion

We have studied an $e_{g}$ orbital model on a square lattice with nearest-
neighbor hopping. This model is characterized by two hopping parameters,
$t_{1}$ and $t_{2}$. The type of the Dirac points depends on their ratio. For
the case of the type-I Dirac point, edge states appear on a lattice with
zigzag edges. We find that these edge states possess helical octupolar
moments.

In $f$ electron systems, octupole order has been identified in certain
materials such as NpO2 [49, 50, 51, 52, 18, 53, 20, 21] and CexLa1-xB6 [54,
55, 56, 57, 58, 59]. For $e_{g}$ orbital systems of $d$ electrons, the
potential for octupole ordering in the bulk perovskite manganites was
previously discussed [47, 60, 61, 62], but a theory incorporating fluctuations
beyond the mean-field approximation concluded it is unlikely [48, 63, 33].

The present study reveals the potential emergence of the octupole moments on
edges of $e_{g}$ orbital systems. If a helical octupolar edge current appears
for an insulator, it could be designated as a quantum octupole Hall system,
analogous to a quantum spin Hall system exhibiting a helical spin edge
current. Such explorations into $e_{g}$ orbital systems hold promise for
fostering deeper connections between topological phenomena and multipole
physics.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number JP23K03330.

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