---
title: Polar Vertices Overview
url: https://www.emergentmind.com/topics/polar-vertices
type: topic
---

# Polar Vertices Overview

The cited literature suggests that **polar vertices** is a polysemous research term rather than a single standardized object. In convex geometry it refers to vertices of a shifted polar polytope and to a polarity construction used in a quantitative Steinitz theorem [2403.14761]. In discrete polarization on the sphere it denotes the points of $\pm w_*$ at which the $f$-potential of a regular simplex attains its minima [2003.02165]. In optics, a narrative exposition applies the label to polarization-singularity C-points in tailored beams [1402.5442]. In computer vision, it designates polygon vertices represented by angle–distance pairs in polar coordinates [2304.02250]. In finite geometry, the vertices of a polar graph are isotropic one-dimensional subspaces of a vector space with respect to a non-degenerate form [1706.08175]. In oxide ferroelectrics, isolated three-fold polar vertices are electrically manipulable topological structures [2109.14173]. Related planetary-atmospheric usage concerns the six vertices of Saturn’s north-polar hexagon [1412.6416].

## 1. Terminological scope and unifying motifs

Across these literatures, the common element is not a single ontology but a recurring reliance on **polarity**, **polar coordinates**, or **polar topology**. The convex-geometric usage is duality-theoretic: given a polytope $P$ and an interior point $c$, one studies the vertices of $(P-c)^\circ$ [2403.14761]. The spherical-polarization usage is potential-theoretic: for a regular simplex $w_*=\{v_0,\dots,v_d\}\subset S^{d-1}$, the minima of the $f$-potential occur at $\pm w_*$, called “polar vertices” [2003.02165]. The computer-vision usage is coordinate-parametric: a polygon is represented by an origin $O=(o_x,o_y)$ and vertices $(r_i,\theta_i)$ in Polar form [2304.02250].

A different cluster of meanings is topological or structural. In the optics exposition, polarization-singularity C-points are described in terms of local Stokes fields, half-integer index, and lemon/star/monstar morphology [1402.5442]. In Pantangi–Sin’s graph-theoretic setting, “polar vertices” are the vertices of polar graphs, namely isotropic lines in finite vector spaces endowed with symplectic, orthogonal, or Hermitian forms [1706.08175]. In Li et al., isolated three-fold polar vertices arise at the PbTiO$_3$/SrRuO$_3$ interface under incomplete screening and are characterized by a winding number $n=+\tfrac12$ [2109.14173]. This suggests that the phrase functions as a domain-specific shorthand whose precise meaning is determined by the ambient formalism.

## 2. Convex-geometric polarity and the quantitative Steinitz theorem

In Ivanov’s formulation, let $P$ be a convex polytope in $\mathbb{R}^d$ with nonempty interior and let $c\in \operatorname{Int} P$. The shifted polytope is
$$
P-c=\{\,y-c:y\in P\},
$$
and for a convex set $K\subset \mathbb{R}^d$ containing the origin, the polar is
$$
K^\circ=\{\,x\in\mathbb{R}^d:\langle x,y\rangle\le 1\ \text{for all}\ y\in K\}.
$$
Hence
$$
(P-c)^\circ=\{\,x\in\mathbb{R}^d:\langle x,y-c\rangle\le 1\ \text{for all}\ y\in P\}.
$$
The central statement is Theorem 3.1: if $P\subset \mathbb{R}^d$ has facets $F_1,\dots,F_n$ and $\alpha_i>0$, then there exists a unique point
$$
c\in \operatorname{Int} P
\quad\text{such that}\quad
\sum_{i=1}^n \alpha_i w_i=0,
$$
where $w_i$ is the vertex of $(P-c)^\circ$ dual to the facet $F_i$ of $P$. Setting all $\alpha_i=1$ yields the corollary
$$
\sum_{w\in \operatorname{Vert}((P-c)^\circ)} w=0.
$$
The proof is based on the strictly concave maximization problem
$$
F(x)=\prod_{i=1}^n \bigl(1-\langle x,v_i\rangle\bigr)^{\alpha_i},
$$
with
$$
P=\{\,x\in\mathbb{R}^d:\langle x,v_i\rangle\le 1\ \text{for all}\ i\},
$$
and on the gradient identity
$$
\nabla F(x)=F(x)\sum_{i=1}^n \frac{-\alpha_i v_i}{1-\langle x,v_i\rangle}.
$$
At the unique maximizer $c$,
$$
\sum_{i=1}^n \frac{\alpha_i v_i}{1-\langle c,v_i\rangle}=0,
$$
and the facet–vertex correspondence gives
$$
w_i=\frac{v_i}{1-\langle c,v_i\rangle}.
$$
These are the “balanced vertices” of the shifted polar polytope [2403.14761].

The polarity construction is then combined with an “Atlantis” lemma. If $P\subset B_d\subset \mathbb{R}^d$, $0\in P$, $c\in \operatorname{Int} P$, $K_1=P^\circ$, and $K_2=(P-c)^\circ$, then for any subset of vertices $\{w_{i_1},\dots,w_{i_k}\}\subset \operatorname{Vert}(K_2)$ satisfying
$$
\operatorname{conv}\{w_{i_1},\dots,w_{i_k}\}\supset A\,B_d,
$$
the corresponding vertices $\{v_{i_1},\dots,v_{i_k}\}\subset \operatorname{Vert}(K_1)$ satisfy
$$
\operatorname{conv}\{v_{i_1},\dots,v_{i_k}\}\supset (A/(1+\|c\|))\,B_d\supset A'B_d,
$$
and since $c\in B_d$ one takes $A'=A/2$ [2403.14761].

This machinery yields the quantitative Steinitz bound. If $Q\subset \mathbb{R}^d$ is finite, $\operatorname{conv} Q\supset B_d$, and $|Q|=m$, then there are at most $2d$ points of $Q$ whose convex hull contains
$$
rB_d\quad\text{with}\quad r=\frac{1}{4(m+d)}.
$$
If $|Q|=m=a\,d$, then
$$
r=\frac{1}{4(a\,d+d)}=\frac{1}{4d(a+1)}\ge \frac{1}{5ad}\qquad (a>1),
$$
so one can remove all but $2d$ vertices from $Q$ while still containing $\tfrac{1}{5ad}B_d$. In the abstract, the corresponding formulation is: if the number of vertices of $Q$ scales linearly with the dimension, i.e. $\alpha d$, then one can select $2d$ vertices such that $r\ge \frac{1}{5\alpha d}$ [2403.14761]. The significance of the result lies in how polarity turns a sparsification problem for $Q$ into a centroid-type balancing problem on $(P-c)^\circ$, extending the classical Steinitz theorem and the earlier quantitative result of Bárány, Katchalski, and Pach.

## 3. Polar vertices in maximal discrete polarization on the sphere

For the maximal discrete polarization problem, let $S^{d-1}\subset \mathbb{R}^d$, let $X=\{x_0,\dots,x_d\}\subset S^{d-1}$, and let
$$
P_f(X;x)=\sum_{i=0}^d f(\|x-x_i\|^2),
$$
where $f:[0,4]\to (-\infty,\infty]$ is continuous in the extended sense on $[0,4]$, monotone non-increasing on $[0,4]$, and convex on $(0,4]$, with either concave or convex derivative $f'$. The quantity of interest is
$$
P_f(X)=\min_{x\in S^{d-1}} P_f(X;x),
$$
and the $(d+1)$-point $f$-polarization constant is
$$
P_f(S^{d-1})=\sup_{X\subset S^{d-1},\,|X|=d+1} P_f(X).
$$
Borodachov proves that the configuration of the vertices of a regular $d$-simplex inscribed in $S^{d-1}$, denoted $w_*=\{v_0,\dots,v_d\}$, is optimal [2003.02165].

The two main formulas depend on the curvature of $f'$. If $f'$ is concave on $(0,4)$, then for every $(d+1)$-point $X\subset S^{d-1}$,
$$
P_f(X)\le P_f(w_*)=f(4)+d\,f(2-2/d).
$$
If $f'$ is convex on $(0,4)$, then
$$
P_f(X)\le P_f(w_*)=f(0)+d\,f(2+2/d).
$$
Strict convexity of $f$ on $(0,4)$ forces uniqueness at $w_*$. The proof introduces the auxiliary function
$$
u(t)=f(2+2dt)+d\,f(2-2t), \qquad t\in[-1/d,1/d],
$$
and shows that $u$ increases on $[0,1/d]$, decreases on $[-1/d,0]$, and satisfies
$$
\min_{t\in[-1/d,1/d]}u(t)=\min\{u(1/d),u(-1/d)\}.
$$

The terminology “polar vertices” enters through the location of the minima of the simplex potential. Proposition 6.1 and Theorem 2.4 show that the $f$-potential of $X=w_*$ attains its minima exactly at the points of $\pm w_*$, called “polar vertices,” with
$$
P_f(w_*)=u(1/d)=f(4)+d\,f(2-2/d)\quad\text{if } f' \text{ is concave},
$$
or
$$
P_f(w_*)=u(-1/d)=f(0)+d\,f(2+2/d)\quad\text{if } f' \text{ is convex}.
$$
The comparison with an arbitrary $X$ splits into the cases where $X$ is contained in a closed hemisphere and where $\operatorname{conv}(X)$ contains $0$ [2003.02165].

Low-dimensional instances recover known extremals. For $d=2$, the optimal configuration on $S^1$ is the equilateral triangle, and
$$
P_f(S^1)=3f(3).
$$
Its polar vertices are the three antipodes of the triangle’s vertices. For $d=3$, the optimal configuration on $S^2$ is the regular tetrahedron, with
$$
P_f(S^2)=4f(8/3).
$$
As a byproduct, the same geometric lemmas imply that among all $(d+1)$-point subsets of $S^{d-1}$, the regular simplex uniquely minimizes the covering radius
$$
\rho(X)=\max_{x\in S^{d-1}}\min_i \|x-x_i\|,
$$
and
$$
\rho(w_*)=\sqrt{2-2/d}.
$$
Here the “polar” designation is antipodal and extremal rather than duality-theoretic.

## 4. Polarization-singularity C-points in optics

In the optics exposition of Galvez et al., polarization-singularity C-points are treated as “polar vertices” of a polarization-spatial field [1402.5442]. At each point in an optical beam the state of polarization is represented by the Stokes vector
$$
S=(S_0,S_1,S_2,S_3),
$$
with
$$
S_0=|E_x|^2+|E_y|^2,\quad
S_1=|E_x|^2-|E_y|^2,\quad
S_2=2\,\operatorname{Re}(E_xE_y^*),\quad
S_3=2\,\operatorname{Im}(E_xE_y^*).
$$
Near a C-point, a very general local expansion for the normalized Jones vector in polar coordinates $(r,\phi)$ is
$$
E_R(r,\phi)=r\,[ \cos\beta\, e^{i\phi}+\sin\beta\, e^{i(\gamma-\phi)} ]\,e^{i\delta},\qquad
E_L(r,\phi)=1.
$$
To first order in $r$,
$$
S_1(r,\phi)\simeq 2r\,[\,\cos\beta\cos(\phi+\delta)+\sin\beta\cos(\gamma-\phi+\delta)\,],
$$
$$
S_2(r,\phi)\simeq 2r\,[\,\cos\beta\sin(\phi+\delta)+\sin\beta\sin(\gamma-\phi+\delta)\,].
$$
The local ellipse-orientation angle and ellipticity angle are
$$
\chi(r,\phi)=\tfrac12\,\arctan\!\bigl[S_2(r,\phi)/S_1(r,\phi)\bigr],\qquad
\psi(r,\phi)=\tfrac12\,\arcsin\!\bigl[S_3(r,\phi)/S_0(r,\phi)\bigr].
$$
In practice one plots the “Stokes-field” $\arg(S_1+iS_2)=2\chi$.

Classification is by half-integer topological index
$$
m=\frac{1}{2\pi}\oint \nabla\chi\cdot d\ell=\pm \tfrac12.
$$
Lemons have $m=+\tfrac12$ and exactly one angle $\phi_0$ at which $\chi(\phi_0)=\phi_0$; stars have $m=-\tfrac12$ and three such radial lines; monstars have $m=+\tfrac12$ like lemons, but three radial lines like stars. The radial-line condition produces a cubic in $t=\tan\phi$, whose number of real roots is $1$ or $3$. At $\gamma=\pi$, the cusp boundary between lemon and monstar occurs at
$$
\tan\beta=1/3,
$$
and more generally the lemon–monstar boundary satisfies
$$
\sin\gamma=\frac{3\tan^2\beta-1}{2\tan\beta}.
$$
A convenient global representation is the “C-point sphere,” with polar coordinates $(2\beta,\gamma)$ on a unit sphere, whose three regions are lemon, monstar, and star [1402.5442].

Galvez et al. describe two complementary generation methods. The first is interferometric creation by superposing two circularly polarized Laguerre–Gaussian modes in a Mach–Zehnder-type set-up, with one arm encoded on a spatial light-modulator with
$$
\cos\beta\,LG_{+1}+\sin\beta\,e^{i\gamma}LG_{-1},
$$
and the relative phase $\delta$ set by a Pancharatnam–Berry phase shifter. The second is an in-line fiber-mode converter obtained by skew-launching into a few-mode optical fiber, so that at the output
$$
E_{\text{out}}\propto (\cos\psi\,HG_{10}+\sin\psi\,e^{i\alpha}HG_{01})e^{i\eta}e_R+e_L.
$$
In both cases imaging polarimetry records six intensity frames through polarizers at $0^\circ,45^\circ,90^\circ,135^\circ$, right- and left-circular, from which the full Stokes vector is reconstructed pixel by pixel. In the LG-based experiment, monstars at $\beta=30^\circ,\gamma=0$ with radial lines at $\phi=0,\pm 34.2^\circ$ and at $\beta=40^\circ,\gamma=0$ with $\phi=0,\pm 42.2^\circ$ were distinguished from the adjacent lemon and star; agreement with theory was within a few degrees. The reported applications include high-sensitivity interferometry and sensing, vector-beam microscopy and imaging, optical trapping and micromanipulation, quantum information, and fundamental studies of natural polarization singularities [1402.5442].

## 5. Polar vertices as polygon parameters in computer vision

In DPPD, “polar vertices” are the sparse set of flexible vertices used to construct a polygon representation of an object [2304.02250]. Each object is represented by a center, or polygon “origin,”
$$
O=(o_x,o_y),
$$
and by $K$ vertices in Polar form
$$
(r_i,\theta_i),\qquad i=0,\dots,K-1.
$$
The Cartesian coordinates are
$$
x_i=o_x+r_i\cos\theta_i,\qquad y_i=o_y+r_i\sin\theta_i,
$$
and conversely,
$$
r_i=\sqrt{(x_i-o_x)^2+(y_i-o_y)^2},\qquad
\theta_i=\operatorname{atan2}(y_i-o_y,x_i-o_x).
$$
The network is anchor-free, NMS-free, and set-prediction style: it produces $N\gg M$ candidate detections, matches them to $M$ ground truths via Hungarian matching, and applies classification and regression loss on the assigned ones [2304.02250].

The regression head outputs a $(2+2K)$-dimensional vector
$$
f=[f_0,f_1,f_2,\dots,f_{2K+1}].
$$
Origin offsets relative to the grid-cell center $(g_x,g_y)$ are decoded as
$$
o_x=g_x+s_x\cdot \sigma(f_0),\qquad
o_y=g_y+s_y\cdot \sigma(f_1),
$$
where $s_x,s_y$ are the cell stride and $\sigma$ is sigmoid. Radial distances are
$$
r_i=\mu\cdot \exp(f_{2+i}),\qquad i=0,\dots,K-1,
$$
where $\mu$ is a scale prior. Angles are predicted by un-normalized “angle deltas”:
$$
\alpha_j=\exp(f_{K+1+j}),\qquad
S=\sum_{j=0}^{K-1}\alpha_j,\qquad
C_i=\sum_{j=0}^i \alpha_j,
$$
then
$$
\theta_i=2\pi\cdot C_i/S,\qquad i=0,\dots,K-1.
$$
This ensures
$$
0=\theta_0<\theta_1<\cdots<\theta_{K-1}<2\pi,
$$
and enforces a valid polygon ordering. The paper explicitly contrasts this with PolarMask, which fixes $\theta_i=2\pi i/M$ and only regresses $r_i$ [2304.02250].

Training uses differentiable dense resampling. One chooses a large $M$ such as $M=360$ of equally spaced rays at angles
$$
\phi_m=2\pi m/M,\qquad m=0,\dots,M-1,
$$
and samples both the ground-truth polygon and the predicted polygon to obtain two $M$-length radial-distance vectors. For each ray $\phi_m$, if $(r_a,\theta_a)$ and $(r_b,\theta_b)$ are neighboring vertices with $\theta_a\le \phi_m\le \theta_b$, then with
$$
t=(\phi_m-\theta_a)/(\theta_b-\theta_a)
$$
the interpolated radius is
$$
r_m=(1-t)\,r_a+t\,r_b.
$$
The computation is fully differentiable, so gradients from any loss on $r_m$ flow back into both the predicted $r_i$ and $\theta_i$. The regression objective is
$$
L_{\text{reg}}=w_1L_o+w_2L_{\text{IoU}}+w_3L_{\text{sm}},
$$
with origin loss
$$
L_o=\bigl[\operatorname{smoothL1}(o_x,\hat o_x)/\hat w+\operatorname{smoothL1}(o_y,\hat o_y)/\hat h\bigr],
$$
polar-IoU loss
$$
L_{\text{IoU}}=-\log\left[\frac{\sum_m \min(r_m,\hat r_m)}{\sum_m \max(r_m,\hat r_m)}\right],
$$
and smoothness loss
$$
\Delta^1 r_m=|r_m-r_{m-1}|,\qquad
\Delta^2 r_m=|\Delta^1 r_m-\Delta^1 r_{m-1}|,
$$
$$
L_{\text{sm}}=\frac{1}{M-1}\left[\sum_m \Delta^1 r_m+\sum_m \Delta^2 r_m\right].
$$

At inference, the dense resampling machinery is dropped. For each surviving candidate, DPPD decodes $f\to o_x,o_y,\{r_i\},\{\theta_i\}$, converts each vertex to Cartesian coordinates, and assembles the $K$-vertex polygon in ascending $\theta_i$ order. The paper states that these $K$ vertices, “typically as few as 12 or 24,” already approximate the object shape very accurately. DPPD is reported on traffic-sign, crosswalk, vehicle, and pedestrian detection for autonomous driving [2304.02250]. Here the adjective “polar” is strictly coordinate-theoretic.

## 6. Vertices of polar graphs in finite geometry

Pantangi and Sin use “polar vertices” in the literal graph-theoretic sense: the vertices of a polar graph are the isotropic one-dimensional subspaces of a finite vector space with respect to a non-degenerate form [1706.08175]. Let $V$ be a finite vector space over $\mathbb{F}_q$ or $\mathbb{F}_{q^2}$ endowed with a symplectic, orthogonal, or Hermitian form $\varphi$. A one-dimensional subspace $\langle v\rangle\subseteq V$ is isotropic, or singular, if
$$
\varphi(v,v)=0.
$$
The vertex set is
$$
P=\{\text{isotropic lines in }V\},
$$
and the notation $[v]$ is used for the vertex $\langle v\rangle$. Two distinct vertices $[v],[u]\in P$ are adjacent precisely when
$$
\varphi(v,u)=0.
$$

The number of vertices is expressed through the Witt index $z$, a shift parameter $h$, and $\tilde q=q$ in the orthogonal and symplectic cases or $\tilde q=q^2$ in the Hermitian case:
$$
|P|=\bigl(\tilde q^{\,z-1+h}+1\bigr)\binom{z}{1}_{\tilde q},
\qquad
\binom{z}{1}_{\tilde q}=\frac{\tilde q^z-1}{\tilde q-1}.
$$
Hence
$$
|P|=(\tilde q^{z-1+h}+1)\frac{\tilde q^z-1}{\tilde q-1}.
$$
The six classical families are listed as
$\Gamma_s(q,m)$, $\Gamma_o(q,m)$, $\Gamma_{o+}(q,m)$, $\Gamma_{o-}(q,m)$, $\Gamma_{ue}(q,m)$, and $\Gamma_{uo}(q,m)$, with the corresponding values of $h$ and $z$ specified in the paper [1706.08175].

Each $\Gamma(V)$ is a strongly regular graph with parameters
$$
v=|P|,
$$
$$
k=\tilde q\binom{z-1}{1}_{\tilde q}(\tilde q^{z-2+h}+1),
$$
$$
\lambda=(\tilde q-1)+\tilde q^2(\tilde q^{z-3+h}+1)\binom{z-2}{1}_{\tilde q},
$$
$$
\mu=\binom{z-1}{1}_{\tilde q}(\tilde q^{z-2+h}+1).
$$
Writing $A$ for the adjacency matrix and $L=D-A$ for the Laplacian, the spectra are
$$
\operatorname{spec}(A)=\{k\ (\text{mult }1),\, r\ (\text{mult }f),\, s\ (\text{mult }g)\},
$$
$$
\operatorname{spec}(L)=\{0\ (\text{mult }1),\, t=k-r\ (\text{mult }f),\, u=k-s\ (\text{mult }g)\},
$$
with
$$
r=\tilde q^{z-1}-1,\qquad
s=-(\tilde q^{z-2+h}+1),
$$
$$
t=\binom{z-1}{1}_{\tilde q}(\tilde q^{z-1+h}+1),\qquad
u=(\tilde q^{z-2+h}+1)\binom{z}{1}_{\tilde q},
$$
and
$$
f=\frac{\tilde q^h(\tilde q^{z-2+h}+1)(\tilde q^z-1)}{(\tilde q-1)(\tilde q^{h-1}+1)},
\qquad
g=\frac{\tilde q(\tilde q^{z-1+h}+1)(\tilde q^{z-1}-1)}{(\tilde q-1)(\tilde q^{h-1}+1)}.
$$

These data feed into the computation of the Smith group
$$
S(\Gamma)=\operatorname{coker}(A:\mathbb{Z}^v\to \mathbb{Z}^v)
$$
and the critical group
$$
K(\Gamma)=\text{finite part of }\operatorname{coker}(L:\mathbb{Z}^v\to \mathbb{Z}^v).
$$
By Kirchhoff’s Matrix-Tree theorem,
$$
|K(\Gamma)|=t^f\cdot u^g/v.
$$
The analysis separates the **non-nilpotent case**, where a prime $p$ divides at most two of the relevant eigenvalues and rank arguments apply, from the **nilpotent case**, where $A\bmod p$ or $L\bmod p$ is nilpotent and the submodule structure of the permutation module $\mathbb{F}_p[P]$ under the form-preserving group $G(V)$ is used. The paper states that the end result is twelve uniform tables—six for $S(\Gamma)$ and six for $K(\Gamma)$—giving a complete determination of the Smith normal form of $A$ and $L$ in each family [1706.08175]. In this setting, “polar vertices” are the combinatorial carriers of orthogonality geometry.

## 7. Planetary and ferroelectric topological structures

In Saturn studies, the relevant objects are the six vertices of the north-polar hexagon rather than a formally defined class called “polar vertices.” Cassini/CIRS thermal-infrared maps show a warm belt encircling the pole at planetographic latitude $77.5^\circ\pm 0.2^\circ$, and the six nearly equally spaced vertices of the hexagon appear as local maxima in brightness temperature $T_B$ when one steps around System III longitude [1412.6416]. Their longitudes are extracted by sampling $T_B(\lambda)$ at fixed latitude $\phi_0\approx 77.5^\circ\text{N}$ and fitting
$$
T_B(\lambda)=T_0+\sum_{k=1}^6 A_k\cos[k(\lambda-\phi_k)].
$$
A representative time series from $2007.20$ to $2013.20$ gives vertex longitudes shifting from $(10^\circ,70^\circ,130^\circ,190^\circ,250^\circ,310^\circ)$ to $(340^\circ,40^\circ,100^\circ,160^\circ,220^\circ,280^\circ)$, corresponding to a westward drift of $\Delta\lambda\approx -30^\circ$ over $\Delta t\approx 6.0\,\text{yr}$:
$$
\omega\equiv \Delta\lambda/\Delta t\approx -5^\circ\,\text{yr}^{-1}\approx -0.013^\circ\,\text{day}^{-1}.
$$
The conversion from observed longitude to System III longitude is
$$
\lambda_{\text{SYSIII}}=\lambda_{\text{obs}}+\Omega_{\text{Sat}}(t-t_0)+C,
$$
with
$$
\Omega_{\text{Sat}}=360^\circ/(10\,\text{h}\,39\,\text{m}\,22.4\,\text{s})\simeq 33.8^\circ\,\text{h}^{-1}\simeq 810.7^\circ\,\text{d}^{-1}.
$$
The paper interprets the drift as evidence that the hexagon is a nearly stationary barotropic Rossby wave trapped in the prograde jet at $\phi\approx 78^\circ\text{N}$ rather than a feature rigidly locked to System III [1412.6416].

In ferroelectric oxide thin films, Li et al. study isolated three-fold polar vertices at the PbTiO$_3$/SrRuO$_3$ interface [2109.14173]. The film stack comprises a $(110)$-DyScO$_3$ substrate, a $50\,\text{nm}$ SrRuO$_3$ bottom electrode, a $100\,\text{nm}$ PbTiO$_3$ layer, and a diffused $\sim 0.8\,\text{nm}$ SrTiO$_3$ “transition” layer of approximately $2$ unit cells at the interface. Atomic-resolution EDS/HAADF indicates that Ti diffuses into SrRuO$_3$, substituting Ru and spontaneously forming an insulating SrTiO$_3$ slab; within approximately $6$ unit cells of the interface the out-of-plane Ti–Pb displacements are strongly suppressed, and the transition layer itself shows approximately $0.1\,\text{\AA}$ residual displacement, indicating incomplete screening. Phase-field comparison shows that only with the $0.8\,\text{nm}$ SrTiO$_3$ does a three-fold vertex spontaneously form at the intersection of an incoming $180^\circ$ domain wall and the interface [2109.14173].

Under in situ biasing, the experimental sequence is: initial $a/c$-domains with downward out-of-plane polarization, application of negative tip bias such as $U=-6\,\text{V}$, nucleation of an upward-polarized domain under the tip, propagation of a $180^\circ$ domain wall toward the SrTiO$_3$ layer, insertion of a triangular $90^\circ$ sector when the wall reaches the imperfectly screened interface, and creation of two three-fold vertices. Reversing the bias drives the wall and its attached vertices back and eventually annihilates them. The local free-energy density is written in Landau–Ginzburg–Devonshire form as
$$
f(P,\nabla P,E)=\alpha P^2+\beta P^4+g(\nabla P)^2-E\!\cdot\!P,
$$
and the onset of three-fold vertex nucleation occurs at approximately $U\approx -6.7\,\text{V}$ across $100\,\text{nm}$, giving
$$
E_c\approx \frac{6.7\,\text{V}}{100\,\text{nm}}\approx 6.7\times 10^7\,\text{V/m}.
$$
The maximum observed velocity is approximately $629\,\text{nm/s}$ under $U=-8\,\text{V}$, and over the accessible field range the velocity is approximately linear in field:
$$
v(E)\approx \mu\,(E-E_c),\qquad E>E_c,
$$
with
$$
\mu \simeq 7.9\times 10^{-15}\,\text{m}^2/(\text{V}\cdot \text{s}).
$$
The phase-field dynamics obey
$$
\frac{\partial P_i}{\partial t}=-L\,\frac{\delta F}{\delta P_i},\qquad
\frac{\delta F}{\delta u_j}=0,
$$
and the winding number around an isolated three-fold vertex is
$$
n=\frac{1}{2\pi}\oint_s d\theta=+\tfrac12.
$$
The reported implications include movable topological bits, analogy to racetrack memory, and potential for ultrahigh-density nonvolatile memories if integrated with nanoscale electrodes [2109.14173].

Taken together, these planetary and ferroelectric examples show that “vertex” language at a pole or in a polar medium may denote either geometrically identifiable extrema of a large-scale pattern or nanoscale topological defects whose nucleation and propagation are field controlled.

Source: https://www.emergentmind.com/topics/polar-vertices