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# Korn’s inequality in anisotropic Sobolev spaces ††thanks: This work was
partially supported by the Pacific Institute for the Mathematical Sciences
(PIMS).

Gonzalo A. Benavides Sebastián A. Domínguez-Rivera Department of Mathematics,
University of Maryland, College Park, USA.Department of Mathematics and
Statistics, University of Saskatchewan, Saskatoon, Canada.Corresponding
author<EMAIL_ADDRESS>
###### Abstract

Korn’s inequality has been at the heart of much exciting research since its
first appearance in the beginning of the 20th century. Many are the
applications of this inequality to the analysis and construction of
discretizations of a large variety of problems in continuum mechanics. In this
paper, we prove that the classical Korn inequality holds true in anisotropic
Sobolev spaces. We also prove that an extension of Korn’s inequality,
involving non-linear continuous maps, is valid in such spaces. Finally, we
point out that another classical inequality, Poincare’s inequality, also holds
in anisotropic Sobolev spaces.

This paper is dedicated to prof. Nilima Nigam on the occasion of her 50th
birthday.

Keywords: Korn’s inequality, Poincare’s inequality, anisotropic Sobolev
spaces, continuum mechanics

MSC: 26D10, 35A23, 74B05, 74B20.

##  1 Introduction

Many linear and non-linear problems in continuum mechanics are of great
interest in mathematics, engineering, and physics [12]. As part of the
analysis of such problems, Korn’s inequality [38, 39, 40] has been intensively
investigated, extended, and used to establish their well-posedness; see, e.g.
[32, 28]. It has particularly become an important part in the investigation,
both theoretically and numerically speaking, of problems in linear elasticity,
fluid-solid interaction, Newtonian fluids, fluids interface problems,
eigenproblems in continuum mechanics, among others; see, for instance [1, 34,
29, 20, 21].

The classical Korn inequality, named after A. Korn who first proved it in the
beginning of the 1900s [38, 39, 40], reads, in the usual notation for Sobolev
spaces, as follows:

$\displaystyle\|{\bf u}\|_{1,2,\Omega}\leq\,C\big{(}\|{\bm{\epsilon}}({\bf
u})\|_{0,2,\Omega}+\|{\bf u}\|_{0,2,\Omega}\big{)},\quad\forall\,{\bf
u}\in{\bf H}^{1}(\Omega),$ (1)

where $C>0$ is some positive constant, $\Omega$ is an open and bounded domain,
and ${\bf H}^{1}(\Omega)$ denotes the usual Hilbert space $H^{1}$ for vector
fields defined on $\Omega$. The term ${\bm{\epsilon}}$ stands for the usual
strain stress tensor, defined as the symmetric part of the gradient of a
vector field in ${\bf H}^{1}(\Omega)$. One can also obtain a sharper
inequality as follows

$\displaystyle\inf_{{\bf r}\in\ker({\bm{\epsilon}})}\|{\bf u}-{\bf
r}\|_{1,2,\Omega}\leq\,C\|{\bm{\epsilon}}({\bf
u})\|_{0,2,\Omega},\quad\forall\,{\bf u}\in{\bf H}^{1}(\Omega),$ (2)

where $\ker({\bm{\epsilon}})=\\{{\bf v}\in{\bf
H}^{1}(\Omega):{\bm{\epsilon}}({\bf v})={\bf 0}\\}$ denotes the kernel of the
strain tensor. Many versions and extensions of these inequalities have been
proven over the last decades [26, 55, 51, 56, 7, 13, 37, 17]. Some recent work
regarding Korn’s inequalities has extended this inequality to vector fields
belonging to the Sobolev space ${\bf W}^{1,p}(\Omega)$ for $p\in(1,\infty)$,
has studied it on general domains such as John domains and
$(\epsilon,\delta)$-domains [22, 2, 3, 4, 35, 36, 25, 45, 44]. There has also
been development in proving that Korn’s inequality in (1) is valid even when
the ${\bf L}^{2}$-norm of the vector fields in ${\bf H}^{1}(\Omega)$ is
replaced by more general continuous mappings [8, 14, 18, 20, 10]. Others have
studied the necessary conditions needed for Korn’s inequality in (2) may hold
true. In this case we see that the essential part is to restrict the space
${\bf H}^{1}(\Omega)$ to subspaces in which the kernel of the strain stress
tensor is excluded. In particular, if, for example, tangential or normal
components of the vector fields are zero on the boundary of the domain, then
certain domains still support infinitesimal rigid motions; see, e.g. [19, 6,
21]. There has also been motivation to prove that Korn’s inequality extends to
incompatible tensor fields [47, 48, 49, 41, 42, 43]. On the other hand, Korn’s
inequality may fail in certain situations; for example, we note that Korn’s
inequality, at least in some of its forms, fails for vector fields in ${\bf
W}^{1,1}(\Omega)$; see e.g. [52, 16]. See [50] for some counterexamples
regarding some some types of generalized Korn’s inequality.

In this paper we prove that the classical Korn inequalities in (1) and (2)
hold true in more general Sobolev spaces than the classical ${\bf
W}^{1,p}(\Omega)$ spaces. In fact, we prove that the Korn inequalities in (1)
and (2) are valid in anisotropic Sobolev spaces, which we denote by ${\bf
W}^{1,p,q}(\Omega)$. In such spaces, the vector fields considered belong to
${\bf L}^{p}(\Omega)$ while their partial derivatives, in all possible
directions, belong to $L^{q}(\Omega)$, with $1\leq p,q\leq\infty$. With such a
definition, we note that ${\bf W}^{1,p,p}(\Omega)$ reduces to the usual
Sobolev spaces ${\bf W}^{1,p}(\Omega)$. These more general Sobolev spaces were
introduced (in an even more general way) and extensively studied in the works
of Rákosník [53, 54]. More recently, Han utilized them as the appropriate
setting for the study of existence of positive solutions of differential
equations, see for instance [30, 31]. Additionally, we extend the
contributions in [27, 18, 20], regarding a non-linear version of Korn’s
inequality, to these anisotropic Sobolev spaces. As a key factor in our
proofs, we use that anisotropic Sobolev spaces are compactly embedded into the
space ${\bf L}^{p}(\Omega)$, for some values of $p$ and $q$; see, e.g. [30].
Furthermore, as a direct consequence of the results mentioned above, we show
that a generalized version of Poincare’s inequality is also valid in these
anisotropic Sobolev spaces.

The rest of the paper is organized as follows: in section 2 we introduce the
notation to be used throughout this paper and provide an overview of
anisotropic Sobolev spaces. In section 3, we state and prove that Korn’s
inequality, and an extension of it, holds in anisotropic Sobolev spaces.
Finally, we draw some conclusions and point out some future work in section 4

##  2 Sobolev spaces

In this section we introduce some notation and give a brief overview of
anisotropic Sobolev spaces, including the results needed for our study.

###  2.1 Notation

We begin this section by introducing the notation to be used throughout this
paper. Vector fields will be denoted with lower-case bold alphabet letters
whereas tensor fields are denoted with bold Greek letters. When using
matrices, these will be denoted with bold upper-case alphabet letters. Between
vector fields, the operation ${\bf a}\cdot{\bf b}$ is the standard dot product
with induced norm $\|\cdot\|$. In turn, for tensors ${\bm{\sigma}}$ and
${\bm{\tau}}$, the double dot product ${\bm{\sigma}}:{\bm{\tau}}={\rm
tr}({\bm{\tau}}^{\texttt{t}}{\bm{\sigma}})$ is the usual inner product for
matrices, which induces the Frobenius norm $\|\cdot\|_{F}$.

For an open domain $\Omega$ of $\mathbb{R}^{d}$, $d\in\mathbb{N}$, the space
$\mathcal{D}(\Omega)$ denotes all $C^{\infty}$ functions with compact support
in $\Omega$, and $\mathcal{D}^{\prime}(\Omega)$ denotes the space of
distributions over $\Omega$. The space $W^{s,p}(\Omega)$ denotes the usual
Sobolev space of scalar fields, for $s\in\mathbb{R}$ and $1\leq p\leq\infty$,
with norm $\|\cdot\|_{s,p,\Omega}$. For vector fields, we use the notation
${\bf W}^{s,p}(\Omega)$ with the corresponding norm simply denoted by
$\|\cdot\|_{s,p,\Omega}$. If $p=2$, the Sobolev space $W^{s,2}(\Omega)$ is the
usual Hilbert space $H^{s}(\Omega)$ [46]. The same notation is used for the
Hilbert space for vector fields ${\bf H}^{s}(\Omega):={\bf W}^{s,2}(\Omega)$.
We use the convention $W^{0,p}(\Omega)=L^{p}(\Omega)$ and ${\bf
W}^{0,p}(\Omega)={\bf L}^{p}(\Omega)$, for $1\leq p\leq\infty$. Additionally,
the space $\mathbb{L}^{p}(\Omega)$ is the space of all tensor fields with
entries in $L^{p}(\Omega)$. The norm of $L^{p}(\Omega)$, ${\bf
L}^{p}(\Omega)$, and $\mathbb{L}^{p}(\Omega)$ are all denoted in the same way,
as $\|\cdot\|_{0,p,\Omega}$.

For differential operators, $\nabla$ denotes the usual gradient operator
acting on either a scalar field or a vector field. Finally, we employ ${\bf
0}$ to denote the zero vector, tensor, or operator, depending on the context.

###  2.2 Anisotropic Sobolev spaces

The content of this section corresponds to an overview of the most important
results (important for this study) obtained in [5, 53, 30, 15]. Let $\Omega$
be an open subset of $\mathbb{R}^{d}$. For $1\leq p\leq\infty$, a function
$v\in L^{p}(\Omega)$ satisfies

$v\in W^{-1,p}(\Omega)\quad\text{and}\quad\nabla v\in{\bf W}^{-1,p}(\Omega).$
(3)

This is because $v$ can be identified with a distribution $T_{v}$, defined as

$|T_{v}(\varphi)|=\left|\int_{\Omega}v\varphi\right|\leq\|v\|_{0,p,\Omega}\|\varphi\|_{1,q,\Omega}$

and

$|\partial_{i}T_{v}(\varphi)|=|-T_{v}(\partial_{i}\varphi)|=\left|\int_{\Omega}v\partial_{i}\varphi\right|\leq\|v\|_{0,p,\Omega}\|\varphi\|_{1,q,\Omega},\quad\forall\,1\leq
i\leq d,$

for each $\varphi\in\mathcal{D}(\Omega)$, where $q\in[1,+\infty]$ is such that
$\frac{1}{p}+\frac{1}{q}=1$. Here
$\partial_{i}(\cdot)=\frac{\partial(\cdot)}{\partial x_{i}}$, for $1\leq i\leq
d$.

We can now formally define an anisotropic Sobolev space. Let $1\leq
p,q\leq+\infty$. We introduce the anisotropic Sobolev space
$W^{1,p,q}(\Omega)$ as

$W^{1,p,q}(\Omega):=\big{\\{}v\in L^{p}(\Omega):\nabla v\in{\bf
L}^{q}(\Omega)\big{\\}}.$ (4)

From [53, Section 3, Theorem 1] we know that if $p,q\in[1,+\infty)$ the space
$W^{1,p,q}(\Omega)$ is a separable Banach space with respect to the norm

$\|v\|_{1,p,q,\Omega}:=\|v\|_{0,p,\Omega}+\|\nabla v\|_{0,q,\Omega},$ (5)

and in addition, it is reflexive if $p,q\in(1,+\infty)$.

Analogously to the vector Sobolev spaces ${\bf W}^{1,p}(\Omega)$, we can
define the vector anisotropic Sobolev space ${\bf W}^{1,p,q}(\Omega)$, endowed
with the norm as defined as in (5) but applied to vector fields. The norm for
vector fields in ${\bf W}^{1,p,q}(\Omega)$ will be simply denoted as for the
scalar case, that is $\|\cdot\|_{1,p,q,\Omega}$.

For $1\leq q<d$, we let $q^{*}:=\frac{dq}{d-q}$ denote the Sobolev critical
index. We have the following result from [30, Lemma 2.1], which we have
rewritten according to our notation.

######  Theorem 1.

Let $\Omega$ be an open and bounded domain with a compact Lipschitz boundary.
Assume $1\leq p\leq\infty$. When $1\leq q<d$, then the embedding
$\iota:W^{1,p,q}(\Omega)\rightarrow L^{s}(\Omega)$ is continuous if $1\leq
s\leq\max\\{q^{*},p\\}$ and compact if $1\leq s<\max\\{q^{*},p\\}$. When
$q=d$, then the embedding $\iota:W^{1,p,q}(\Omega)\hookrightarrow
L^{s}(\Omega)$ is compact if $1\leq s<\infty$. When $d<q\leq\infty$, then the
embedding $\iota:W^{1,p,q}(\Omega)\rightarrow L^{s}(\Omega)$ is continuous if
$1\leq s\leq\infty$ and compact if $1\leq s<\infty$.

##  3 Korn’s inequality

In this section we prove that Korn’s inequality holds in anisotropic Sobolev
spaces and show that an extension of it is also valid in these spaces.

###  3.1 Extension to anisotropic Sobolev spaces

The first result we present in this section extends the classical Korn
inequality in (1) to vector fields in ${\bf W}^{1,p,q}(\Omega)$. To that end
we recall the following result from [5, Prop. 2.10(i)].

######  Theorem 2.

Let $\Omega$ be an open and bounded domain with Lipschitz boundary of
$\mathbb{R}^{d}$. Aso let $m$ be any integer and $r$ be any real number with
$1<r<\infty$. If $p\in\mathcal{D}^{\prime}(\Omega)$ is a distribution such
that $\partial_{i}p\in W^{m-1,r}(\Omega)$, $1\leq i\leq d$, then $p\in
W^{m,r}(\Omega)$.

######  Theorem 3 (Korn’s inequality in ${\bf W}^{1,p,q}(\Omega)$).

Let $\Omega$ be a bounded domain of $\mathbb{R}^{d}$ with a compact and
Lipschitz continuous boundary. Let $1\leq p,q\leq\infty$, $q\neq 1$. Then,
there exists a constant $C=C(\Omega,p,q)$ such that for all ${\bf v}\in{\bf
W}^{1,p,q}(\Omega)$

$\|{\bf v}\|_{1,p,q,\Omega}\leq C\left\\{\|{\bf
v}\|_{0,p,\Omega}+\|{\bm{\epsilon}}({\bf v})\|_{0,q,\Omega}\right\\}$ (6)

###### Proof.

The proof is a generalization of the one of Theorem 2.1 of [11], which in turn
follows the one of Theorem 3.3 in Chapter 3 of [23].

Let us define the space

${\bf E}^{p,q}(\Omega):=\\{{\bf v}\in{\bf L}^{p}(\Omega):{\bm{\epsilon}}({\bf
v})\in\mathbb{L}_{s}^{q}(\Omega)\\},$ (7)

where $\mathbb{L}^{q}_{s}(\Omega)$ is the space of symmetric tensor fields in
$\mathbb{L}^{q}(\Omega)$. Then, ${\bf E}^{p,q}(\Omega)$ endowed with the norm
$\|{\bf v}\|:=\|{\bf v}\|_{0,p,\Omega}+\|{\bm{\epsilon}}({\bf
v})\|_{0,q,\Omega}$ is a Banach space. Indeed, the condition
${\bm{\epsilon}}({\bf v})\in\mathbb{L}_{s}^{q}(\Omega)$ defining ${\bf
E}^{p,q}(\Omega)$ is understood in the sense of distributions, that is, it
means that there exist symmetrical functions $e_{ij}({\bf v})$ in
$L^{q}(\Omega)$, such that

$\int_{\Omega}e_{ij}({\bf
v})\varphi=-\frac{1}{2}\int_{\Omega}\left(v_{i}\partial_{j}\varphi+v_{j}\partial_{i}\varphi\right),\quad\forall\varphi\in\mathcal{D}(\Omega).$

Now, let $\\{{\bf v}^{k}\\}_{k\in\mathbb{N}}$ be a Cauchy sequence with
respect to $\|\cdot\|$ of elements ${\bf
v}^{k}=(v^{k}_{1},\ldots,v^{k}_{d})\in{\bf E}^{p,q}(\Omega)$. The definition
of $\|\cdot\|$ implies that for each $i,j=1,\dots,d$, the sequences
$\\{v^{k}_{i}\\}_{k\in\mathbb{N}}$ and $\\{e_{ij}({\bf
v}^{k})\\}_{k\in\mathbb{N}}$ are Cauchy in $L^{p}(\Omega)$ and
$L^{q}(\Omega)$, respectively. Being these two spaces complete, there exist
functions $v_{i}\in L^{p}(\Omega)$ and $e_{ij}\in L^{q}(\Omega)$ such that

$\|v^{k}_{i}-v_{i}\|_{0,p,\Omega}\xrightarrow{k\to\infty}0\quad\text{and}\quad\|e_{ij}({\bf
v}^{k})-e_{ij}\|_{0,q,\Omega}\xrightarrow{k\to\infty}0.$

Moreover, since $e_{ij}({\bf v}^{k})=e_{ji}({\bf v}^{k})$ for each natural
$k$, it is clear that $e_{ij}=e_{ji}$.

Given a function $\varphi\in\mathcal{D}(\Omega)$, we let $k\to\infty$ in the
relations

$\int_{\Omega}e_{ij}({\bf
v}^{k})\varphi=-\frac{1}{2}\int_{\Omega}\left(v^{k}_{i}\partial_{j}\varphi+v^{k}_{j}\partial_{i}\varphi\right),\quad\forall\varphi\in\mathcal{D}(\Omega).$

to deduce $e_{ij}=e_{ij}({\bf v})$. Thus ${\bf v}\in{\bf E}^{p,q}(\Omega)$,
and $({\bf v}^{k})_{k\in\mathbb{N}}$ converges to ${\bf v}$ in $\|\cdot\|$.

Furthermore, ${\bf E}^{p,q}(\Omega)$ and ${\bf W}^{1,p,q}(\Omega)$ coincide.
Indeed, since $\|{\bm{\epsilon}}(\cdot)\|_{0,q,\Omega}\leq
c\|\nabla(\cdot)\|_{0,q,\Omega}$ for some positive constant $c$, we clearly
have that ${\bf W}^{1,p,q}(\Omega)\subseteq{\bf E}^{p,q}(\Omega)$. For the
reverse inclusion, consider ${\bf v}=(v_{1},\ldots,v_{d})\in{\bf
E}^{p,q}(\Omega)$. Then for $1\leq i,j,k\leq d$, duality arguments (3) give

$\partial_{k}v_{i}\in
W^{-1,p}(\Omega)\subseteq\mathcal{D}^{\prime}(\Omega),\quad\partial_{j}(\partial_{k}v_{i})=\\{\partial_{j}e_{ik}({\bf
v})+\partial_{k}{\bf e}_{ij}({\bf v})-\partial_{i}e_{jk}({\bf v})\\}\in
W^{-1,q}(\Omega).$

Hence, by Theorem 2 with $m=0$ we know that $\partial_{k}v_{i}\in
L^{q}(\Omega)$, and therefore ${\bf v}\in{\bf W}^{1,p,q}(\Omega)$.

The identity mapping $\iota$ from ${\bf W}^{1,p,q}(\Omega)$ equipped with
$\|\cdot\|_{1,p,q,\Omega}$ into ${\bf E}^{p,q}(\Omega)$ equipped with
$\|\cdot\|$ is bounded (clearly $\|\cdot\|\leq c\|\cdot\|_{1,p,q,\Omega}$, for
some positive constant $c$ and surjective as ${\bf E}^{p,q}(\Omega)={\bf
W}^{1,p,q}(\Omega)$. In this way, the bounded inverse Theorem [9, Corollary
2.7] then shows that the inverse mapping $\iota^{-1}$ is also bounded, which
is exactly what we were aiming to prove. ∎

######  Remark 1.

We stress the fact that the previous result, unlike the forthcoming Theorem 5
and Theorem 6, holds regardless of the embedding
$\iota:W^{1,p,q}(\Omega)\hookrightarrow L^{q}(\Omega)$ being compact or not.

Now, let $\mathbb{A}^{d}$ denote the space of all real $d\times d$ skew-
symmetric matrices. We define the space of rigid motions of $\Omega$, denoted
by ${\bf R}{\bf M}(\Omega)$, as follows

${\bf R}{\bf M}(\Omega):=\big{\\{}{\bf v}\in{\bf W}^{1,p,q}(\Omega):{\bf
v}({\bf x})={\bf A}{\bf x}+{\bf c}\quad\text{a.e. ${\bf
x}\in\Omega$},\quad{\bf A}\in\mathbb{A}^{d},\,\,{\bf
c}\in\mathbb{R}^{d}\big{\\}}.$ (8)

The following result characterizes the kernel of the strain stress tensor in
${\bf W}^{1,p,q}(\Omega)$.

######  Theorem 4.

Let $\Omega$ be an open, and connected domain in $\mathbb{R}^{d}$. Then, the
kernel of the strain stress tensor, $\ker({\bm{\epsilon}}):=\\{{\bf v}\in{\bf
W}^{1,p,q}(\Omega):\,{\bm{\epsilon}}({\bf
v})=\mathbf{0}\,\,\text{in}\,\,\Omega\\}$, coincides with the space of rigid
motions, that is

$\ker({\bm{\epsilon}})={\bf R}{\bf M}(\Omega).$ (9)

In addition, the kernel of the strain stress tensor is finite-dimensional,
with dimension $\frac{d(d+1)}{2}$.

###### Proof.

The proof is a generalization of the one of [11, Theorem 2.2]. Note that the
inclusion ${\bf R}{\bf M}(\Omega)\subseteq\ker({\bm{\epsilon}})$ is
straightforward.

Observe that for any ${\bf v}\in{\bf W}^{1,p,q}(\Omega)$ and
$i,j,k=1,\dotsc,d$, it holds

$\int_{\Omega}(\partial_{j}v_{i})\partial_{k}\varphi=\int_{\Omega}\left(e_{ij}({\bf
v})\partial_{k}\varphi+e_{ik}({\bf v})\partial_{j}\varphi-e_{jk}({\bf
v})\partial_{i}\varphi\right)\quad\forall\varphi\in\mathcal{D}(\Omega),$

because both sides of the equality coincide with
$\displaystyle-\int_{\Omega}v_{i}\partial_{kj}\varphi$. Therefore, if ${\bf
v}\in{\bf W}^{1,p,q}(\Omega)$ is such that ${\bm{\epsilon}}({\bf v})={\bf 0}$,
then

$\partial_{k}T_{\partial_{j}v_{i}}(\varphi)=-\int_{\Omega}(\partial_{j}v_{i})\partial_{k}\varphi=0,\quad\forall\varphi\in\mathcal{D}(\Omega),\quad\forall
j,k=1,\dotsc,d.$

Thus, given that $\Omega$ is connected, we can use [33, Corollary 3.1.6] in
combination with [24, Theorem 13.17] to deduce the existence of constants
$a_{ij}$, $i,j=1,\dotsc,d$, such that $\partial_{j}v_{i}=a_{ij}$ a.e. in
$\Omega$. Moreover, the condition $e_{ij}({\bf v})=0$ implies that
$a_{ij}=-a_{ji}$.

Let $w_{i}({\bf x}):=\sum_{j=1}^{d}a_{ij}x_{j}$ for ${\bf x}\in\Omega$,
$i=1,\dotsc,d$. Then for each $\varphi\in\mathcal{D}(\Omega)$

$\int_{\Omega}v_{i}\partial_{j}\varphi=-\int_{\Omega}(\partial_{j}v_{i})\varphi=-a_{ij}\int_{\Omega}\varphi=-\int_{\Omega}(\partial_{j}w_{i})\varphi=\int_{\Omega}w_{i}\partial_{j}\varphi,$

that is $\partial_{j}T_{v_{i}-w_{i}}=0$ for each $j=1,\dotsc,d$. In this way,
using [33, Corollary 3.1.6] and [24, Theorem 13.17] there exist constants
$c_{i}$ such that $v_{i}({\bf x})=w_{i}({\bf x})+c_{i}$ a.e. in $\Omega$.
Thus, we have shown that ${\bf v}({\bf x})={\bf A}{\bf x}+{\bf c}$ a.e. in
$\Omega$ with ${\bf A}:=(a_{ij})$ and ${\bf c}:=(c_{i})$. ∎

######  Theorem 5 (Korn’s inequality in the quotient space ${\bf
W}^{1,p,q}(\Omega)/\ker({\bm{\epsilon}})$).

Let $\Omega$ be a bounded domain of $\mathbb{R}^{d}$ with compact, Lipschitz
boundary such that $\ker({\bm{\epsilon}})$ is finite-dimensional. Define the
quotient space

$\dot{\bf W}^{1,p,q}(\Omega):={\bf W}^{1,p,q}(\Omega)/\ker({\bm{\epsilon}}),$
(10)

equipped with the norm

$\|\dot{\bf v}\|_{1,p,q,\Omega}:=\inf_{{\bf r}\in\ker({\bm{\epsilon}})}\|{\bf
v}-{\bf r}\|_{1,p,q,\Omega}\quad\text{for all}\quad\dot{\bf v}\in\dot{\bf
W}^{1,p,q}(\Omega),$

the space $\dot{\bf W}^{1,p,q}(\Omega)$ is a Banach space. Then:

  1. 1.

If $1\leq p,q\leq\infty$, $q\neq 1$, are such that the embedding
$\iota:W^{1,p,q}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact (see Lemma
1), then there exists a constant $\dot{C}:=\dot{C}(\Omega,p,q)$ such that
Korn’s inequality in $\dot{\bf W}^{1,p,q}(\Omega)$ holds, that is,

$\|\dot{\bf v}\|_{1,p,q,\Omega}\leq\dot{C}\|{\bm{\epsilon}}(\dot{\bf
v})\|_{0,q,\Omega}\quad\forall\dot{\bf v}\in\dot{\bf W}^{1,p,q}(\Omega),$ (11)

where ${\bm{\epsilon}}(\dot{\bf v}):={\bm{\epsilon}}({\bf w})$ for any ${\bf
w}\in\dot{\bf v}$.

  2. 2.

Conversely, Korn’s inequality in $\dot{\bf W}^{1,p,q}(\Omega)$ implies Korn’s
inequality in ${\bf W}^{1,p,q}(\Omega)$ (see Theorem 3).

###### Proof.

  1. 1.

Let $M$ be the dimension of $\ker({\bm{\epsilon}})$. A corollary of Hahn-
Banach theorem in normed vector spaces [9] ensures the existence of $M$ linear
and bounded functionals $\ell_{\alpha}$ on ${\bf W}^{1,p,q}(\Omega)$,
$1\leq\alpha\leq M$, such that: given an element ${\bf
r}\in\ker({\bm{\epsilon}})$, ${\bf r}$ is the null vector if and only if
$\ell_{\alpha}({\bf r})=0$, $1\leq\alpha\leq M$. Indeed, if
$\\{r_{1},\dotsc,r_{M}\\}$ is a basis of $\ker({\bm{\epsilon}})$, the
canonical basis of $\ker({\bm{\epsilon}})^{\prime}$ (the dual space of
$\ker({\bm{\epsilon}})$ with respect to the $\|\cdot\|_{1,p,q,\Omega}$) is
$\\{f_{1},\dotsc,f_{M}\\}$, which satisfies
$f_{\alpha}(r_{\beta})=\delta_{\alpha\beta}$ for each $1\leq\alpha,\beta\leq
M$. We extend, via Hahn-Banach Theorem [9, Corollary 1.2] each $f_{\alpha}$ to
a linear and bounded functional $\ell_{\alpha}$ on ${\bf W}^{1,p,q}(\Omega)$.

We claim the existence of a positive constant $D=D(p,q,\Omega)$, such that

$\|{\bf v}\|_{1,p,q,\Omega}\leq D\left\\{\|{\bm{\epsilon}}({\bf
v})\|_{0,q,\Omega}+\sum_{\alpha=1}^{M}|\ell_{\alpha}({\bf
v})|\right\\}\quad\forall{\bf v}\in{\bf W}^{1,p,q}(\Omega).$ (12)

Indeed, let’s assume the contrary. Then, there exists a sequence $\\{{\bf
v}^{k}\\}_{k\in\mathbb{N}}$ on ${\bf W}^{1,p,q}(\Omega)$, such that $\|{\bf
v}^{k}\|_{1,p,q,\Omega}=1$ for all $k\in\mathbb{N}$

$\|{\bm{\epsilon}}({\bf
v}^{k})\|_{0,q,\Omega}+\sum_{\alpha=1}^{M}|\ell_{\alpha}({\bf
v}^{k})|\xrightarrow{k\to\infty}0.$ (13)

Being $\\{{\bf v}^{k}\\}_{k\in\mathbb{N}}$ bounded in ${\bf
W}^{1,p,q}(\Omega)$ by compactness of the embedding $\iota$ there exists a
subsequence $\\{{\bf v}^{l}\\}_{l\in\mathbb{N}}$ that converges in
$L^{p}(\Omega)$. Since the sequence $\\{{\bm{\epsilon}}({\bf
v}^{l})\\}_{l\in\mathbb{N}}$ is convergent in $\mathbb{L}_{s}^{q}$, the
subsequence $\\{{\bf v}^{l}\\}_{l\in\mathbb{N}}$ is a Cauchy sequence with
respect to the norm
$\|\cdot\|:=\|\cdot\|_{0,p,\Omega}+\|{\bm{\epsilon}}(\cdot)\|_{0,q,\Omega}$,
hence also with respect to the norm $\|\cdot\|_{1,p,q,\Omega}$ by Korn’s
inequality in ${\bf W}^{1,p,q}(\Omega)$ (see Theorem 3). As a consequence,
there exists ${\bf v}\in{\bf W}^{1,p,q}(\Omega)$ such that

$\|{\bf v}^{l}-{\bf v}\|_{1,p,q,\Omega}\xrightarrow{l\to\infty}0.$

But, by continuity, it follows from (13) that ${\bm{\epsilon}}({\bf v})={\bf
0}$ (that is, ${\bf v}\in\ker({\bm{\epsilon}})$) and $\ell_{\alpha}({\bf
v})=0$, $1\leq\alpha\leq M$, and therefore ${\bf v}=\bf{0}$; in contradiction
with the fact that $\|{\bf v}^{l}\|_{1,p,q,\Omega}=1$ for all
$l\in\mathbb{N}$. Thus, (12) holds true.

In this way, given any ${\bf v}\in{\bf W}^{1,p,q}(\Omega)$, we can choose
${\bf r}({\bf v})\in\ker({\bm{\epsilon}})$ such that $\ell_{\alpha}({\bf
v}+{\bf r}({\bf v}))=0$, for each $1\leq\alpha\leq M$ (it is equivalent to
solve a uniquely determined $M\times M$ system of equations). Then, by (12)

$\|\dot{\bf v}\|_{1,p,q,\Omega}=\inf_{{\bf r}\in\ker({\bm{\epsilon}})}\|{\bf
v}+{\bf r}\|_{1,p,q,\Omega}\leq\|{\bf v}+{\bf r}({\bf v})\|_{1,p,q,\Omega}\leq
D\|{\bm{\epsilon}}({\bf v})\|_{0,q,\Omega}=D\|{\bm{\epsilon}}(\dot{\bf
v})\|_{0,q,\Omega},$

which completes the proof.

  2. 2.

By means of contradiction, assume that there exists a sequence $\\{{\bf
v}^{k}\\}_{k\in\mathbb{N}}$ in ${\bf W}^{1,p,q}(\Omega)$ such that $\|{\bf
v}^{k}\|_{1,p,q,\Omega}=1$ for each $k\in\mathbb{N}$ and

$\|{\bf v}^{k}\|_{0,p,\Omega}+\|{\bm{\epsilon}}({\bf
v}^{k})\|_{0,q,\Omega}\xrightarrow{k\to\infty}0.$ (14)

By definition of the norm $\|\cdot\|_{1,p,q,\Omega}$ in $\dot{\bf
W}^{1,p,q}(\Omega)$, there exists a sequence $\\{{\bf
r}^{k}\\}_{k\in\mathbb{N}}$ in $\ker({\bm{\epsilon}})$ that satisfies

$\displaystyle\|{\bf v}^{k}-{\bf r}^{k}\|_{1,p,q,\Omega}$
$\displaystyle<\|\dot{\bf v}^{k}\|_{1,p,q,\Omega}+\frac{1}{k}$
$\displaystyle\leq\|{\bf v}^{k}\|_{1,p,q,\Omega}+\frac{1}{k}\leq 2.$

The space $\ker({\bm{\epsilon}})$ being finite-dimensional, the inequality
$\|{\bf r}^{k}\|_{1,p,q,\Omega}\leq\|{\bf v}^{k}-{\bf
r}^{k}\|_{1,p,q,\Omega}+\|{\bf v}^{k}\|_{1,p,q,\Omega}\leq 3$ for all
$k\in\mathbb{N}$ implies the existence of a subsequence $\\{{\bf
r}^{l}\\}_{l\in\mathbb{N}}$ that converges in ${\bf W}^{1,p,q}(\Omega)$ to an
element ${\bf r}\in\ker({\bm{\epsilon}})$. Besides, by Korn’s inequality in
$\dot{\bf W}^{1,p,q}(\Omega)$ and noting that ${\bf v}^{l}-{\bf
r}^{l}\in\dot{\bf v}^{l}$,

$\|{\bf v}^{l}-{\bf r}^{l}\|_{1,p,q,\Omega}<\|\dot{\bf
v}^{l}\|_{1,p,q,\Omega}+\frac{1}{l}\leq C\|{\bm{\epsilon}}(\dot{\bf
v}^{l})\|_{0,q,\Omega}+\frac{1}{l}\\\ =C\|{\bm{\epsilon}}({\bf v}^{l}-{\bf
r}^{l})\|_{0,q,\Omega}+\frac{1}{l}=C\|{\bm{\epsilon}}({\bf
v}^{l})\|_{0,q,\Omega}+\frac{1}{l}\xrightarrow{l\to\infty}0;$

which readily implies that $\|{\bf v}^{l}-{\bf
r}\|_{1,p,q,\Omega}\xrightarrow{l\to\infty}0$. Hence, $\|{\bf v}^{l}-{\bf
r}\|_{0,p,\Omega}\xrightarrow{l\to\infty}0$, which force ${\bf r}$ to be
$\bf{0}$ since $\|{\bf v}^{l}\|_{0,p,\Omega}\xrightarrow{l\to\infty}0$ (cf.
(14)). Thus, we reach to the conclusion that $\|{\bf
v}^{l}\|_{1,p,q,\Omega}\xrightarrow{l\to\infty}0$, which contradicts $\|{\bf
v}^{l}\|_{1,p,q,\Omega}=1$.

∎

###  3.2 Extension to continuous mappings

Consider a continuous map $F:{\bf W}^{1,p,q}(\Omega)\to\mathbb{R}$, different
from the $\|\cdot\|_{1,p,q,\Omega}$ norm (cf. (5)). Let $N(F)$ be the set of
zeros of $F$, that is

$\displaystyle N(F):=\Big{\\{}{\bf v}\in{\bf W}^{p,q}(\Omega):\,F({\bf
v})=0\Big{\\}}.$

Note that the set $N(F)$ could be the empty set. We assume that this set
satisfies any of the two conditions below

$\displaystyle N(F)\cap\ker({\bm{\epsilon}})=\\{{\bf 0}\\}\quad\text{or}\quad
N(F)\cap\ker({\bm{\epsilon}})=\emptyset.$ (15)

Korn’s inequality in ${\bf W}^{1,p,q}(\Omega)$ for general continuous maps is
stated and proven below.

######  Theorem 6.

Assume $\Omega$ is a bounded domain of $\mathbb{R}^{d}$ with compact,
Lipschitz boundary such that $\ker({\bm{\epsilon}})$ is finite-dimensional.
Let $p,q\in(1,+\infty)$ such that the embedding
$\iota:W^{1,p,q}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact (see Theorem
1). Then, there is a constant $C>0$ such that

$\displaystyle\|{\bf u}\|_{1,p,q,\Omega}\leq C\Big{(}\|{\bm{\epsilon}}({\bf
u})\|_{0,q,\Omega}+|F({\bf u})|\Big{)},\quad\forall\,{\bf u}\in{\bf
W}^{1,p,q}(\Omega).$ (16)

###### Proof.

By contradiction, suppose there is a sequence $\\{{\bf
u}^{k}\\}_{k\in\mathbb{N}}\in{\bf W}^{1,p,q}(\Omega)$ such that

$\displaystyle\|{\bf u}^{k}\|_{1,p,q,\Omega}=1,\quad\|{\bm{\epsilon}}({\bf
u}^{k})\|_{0,q,\Omega}+|F({\bf u}^{k})|\xrightarrow{k\to\infty}0$

Since $\\{{\bf u}^{k}\\}_{k\in\mathbb{N}}$ is a bounded sequence in the
$\|\cdot\|_{1,p,q,\Omega}$ norm and ${\bf W}^{1,p,q}(\Omega)$ is reflexive
(for $1<p,q<+\infty$), we know from [9, Theorem 3.18] that there is a
subsequence $\\{{\bf u}^{k_{l}}\\}_{l\in\mathbb{N}}$ of $\\{{\bf
u}^{k}\\}_{k\in\mathbb{N}}$ and ${\bf u}\in{\bf W}^{1,p,q}(\Omega)$ such that
${\bf u}^{k_{l}}\xrightarrow{l\to\infty}{\bf u}$ weakly in ${\bf
W}^{1,p,q}(\Omega)$. The embedding $\iota:W^{1,p,q}(\Omega)\hookrightarrow
L^{p}(\Omega)$ being compact, it further implies that ${\bf u}^{k_{l}}\to{\bf
u}$ strongly in ${\bf L}^{p}(\Omega)$.

In turn, we see that $|F({\bf u}^{k})|\xrightarrow{k\to\infty}0$ and
$\|{\bm{\epsilon}}({\bf u}^{k})\|_{0,q,\Omega}\xrightarrow{k\to\infty}0$.
Besides, from Theorem 3,

$\displaystyle\|{\bf u}^{k_{l}}-{\bf u}^{k_{m}}\|_{1,p,q,\Omega}\leq
C\,\Big{(}\|{\bm{\epsilon}}({\bf u}^{k_{l}})-{\bm{\epsilon}}({\bf
u}^{k_{m}})\|_{0,q,\Omega}+\|{\bf u}^{k_{l}}-{\bf
u}^{k_{m}}\|_{0,p,\Omega}\Big{)}\quad\forall l,m\in\mathbb{N}$

From the observations above we have that $\\{{\bf
u}^{k_{l}}\\}_{l\in\mathbb{N}}$ is a Cauchy sequence in ${\bf
W}^{1,p,q}(\Omega)$ and thus ${\bf u}^{k_{l}}\xrightarrow{l\to\infty}{\bf u}$
strongly in ${\bf W}^{1,p,q}(\Omega)$. The continuity of $F$ then gives
$|F({\bf u}^{k_{l}})|\xrightarrow{l\to\infty}|F({\bf u})|$ and therefore ${\bf
u}\in N(F)$. Also, the fact that $\|{\bm{\epsilon}}({\bf
u}^{k})\|_{0,q,\Omega}\xrightarrow{k\to\infty}0$ and the continuity of
${\bm{\epsilon}}$ in ${\bf W}^{1,p,q}(\Omega)$ implies that ${\bf
u}\in\ker({\bm{\epsilon}})$ that is, ${\bf u}$ belongs to
$N(F)\cap\ker({\bm{\epsilon}})$. That is, if the second condition of (15)
holds, we have reached a contradiction. Otherwise, the first condition of (15)
would imply that ${\bf u}={\bf 0}$, condition that crashes with ${\bf
u}^{k_{l}}\to{\bf u}$ strongly in ${\bf W}^{1,p,q}(\Omega)$, because from it
it follows that $1=\|{\bf u}^{k_{l}}\|_{1,p,q,\Omega}\to\|{\bf
u}\|_{1,p,q,\Omega}=1$. ∎

######  Remark 2.

Note that the functional $F$ is meant to be different from the
$\|\cdot\|_{1,p,q,\Omega}$ norm as this case corresponds to the usual Korn’s
inequality in Theorem 3 and it is used in the proof of the result above.

###  3.3 A note on Poincare’s inequality

The results already obtained for Korn’s inequality in the spaces ${\bf
W}^{1,p,q}(\Omega)$ can also be proved, with no much more work, for the
classical Poincare inequality. In fact, define

$\ker(\nabla):=\\{{\bf v}\in{\bf W}^{1,p,q}(\Omega):\nabla{\bf
v}=\mathbf{0}\quad\text{in}\quad\Omega\\}.$ (17)

Using both [33, Corollary 3.1.6] and [24, Theorem 13.17] it readily follows
that if $\Omega$ is an open connected set, then

$\ker(\nabla)=\big{\\{}{\bf v}\in{\bf W}^{1,p,q}(\Omega):{\bf v}={\bf
c}\quad\text{a.e.~{}$\in\Omega$},\quad{\bf c}\in\mathbb{R}^{d}\big{\\}},$

and in particular, it is finite-dimensional with dimension equal to $d$.

######  Theorem 7 (Poincare’s inequality in the quotient space ${\bf
W}^{1,p,q}(\Omega)/\ker(\nabla)$).

Let $\Omega$ be a bounded domain of $\mathbb{R}^{d}$ with compact, Lipschitz
boundary, such that $\ker(\nabla)$ is finite-dimensional. Define the quotient
space

$\dot{\bf W}^{1,p,q}(\Omega):={\bf W}^{1,p,q}(\Omega)/\ker(\nabla),$ (18)

equipped with the norm

$\|\dot{\bf v}\|_{1,p,q,\Omega}:=\inf_{{\bf r}\in\ker(\nabla)}\|{\bf v}-{\bf
r}\|_{1,p,q,\Omega},\quad\forall\,\dot{\bf v}\in\dot{\bf W}^{1,p,q}(\Omega),$

the space $\dot{\bf W}^{1,p,q}(\Omega)$ is a Banach space. Then: If $1\leq
p,q\leq+\infty$ are such that the embedding
$\iota:W^{1,p,q}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact (see Lemma
1), then there exists a constant $\dot{C}:=\dot{C}(\Omega,p,q)$ such that
Poincare’s inequality in $\dot{\bf W}^{1,p,q}(\Omega)$ holds, that is,

$\|\dot{\bf v}\|_{1,p,q,\Omega}\leq\dot{C}\|\nabla(\dot{\bf
v})\|_{0,q,\Omega},\quad\forall\dot{\bf v}\in\dot{\bf W}^{1,p,q}(\Omega),$

where $\nabla(\dot{\bf v}):=\nabla{\bf w}$ for any ${\bf w}\in\dot{\bf v}$.

###### Proof.

The proof follows almost verbatim as the proof of item 1 of Theorem 5 by
replacing the role of ${\bm{\epsilon}}$ by $\nabla$, and without utilizing
Theorem 3 and hence it is allowed for $q$ to take the value $1$. ∎

Consider a continuous map $F:{\bf W}^{1,p,q}(\Omega)\to\mathbb{R}$, different
from the $\|\cdot\|_{1,p,q,\Omega}$ norm (cf. (5)). Let $N(F)$ be the set of
zeros of $F$, that is

$\displaystyle N(F):=\Big{\\{}{\bf v}\in{\bf W}^{p,q}(\Omega):\,F({\bf
v})=0\Big{\\}}.$

Note that the set $N(F)$ could be the empty set. We assume that this set
satisfies any of the two conditions below

$\displaystyle N(F)\cap\ker(\nabla)=\\{{\bf 0}\\}\quad\text{or}\quad
N(F)\cap\ker(\nabla)=\emptyset.$ (19)

Poincare’s inequality in ${\bf W}^{1,p,q}(\Omega)$ for general continuous maps
is stated and proven below.

######  Theorem 8.

Assume $\Omega$ is a bounded domain of $\mathbb{R}^{d}$ with compact,
Lipschitz boundary. Let $1<p,q<\infty$ be such that the embedding
$\iota:W^{1,p,q}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact (see Theorem
1). Then, there is a constant $C>0$ such that

$\displaystyle\|{\bf u}\|_{1,p,q,\Omega}\leq C\Big{(}\|\nabla{\bf
u}\|_{0,q,\Omega}+|F({\bf u})|\Big{)},\quad\forall\,{\bf u}\in{\bf
W}^{1,p,q}(\Omega).$ (20)

###### Proof.

Its proof follows the same ideas of the one of Theorem 6 by replacing the role
of ${\bm{\epsilon}}$ by $\nabla$. It does not require the use of Theorem 3. ∎

##  4 Conclusions

In this paper we proved that the classical Korn inequality, usually posed for
vector fields in ${\bf H}^{1}(\Omega)$, is valid in the anisotropic Sobolev
spaces ${\bf W}^{1,p,q}(\Omega)$, under different assumptions on the Sobolev
parameters $p$ and $q$. In particular, we point out that Theorem 3 and Theorem
5 imply that Korn’s inequality (cf. Equation 6 and Equation 11) holds for
vector fields in ${\bf W}^{1,1,q}(\Omega)$ as long as $q$ is strictly larger
than 1. In fact, Korn’s inequality for vector fields in ${\bf
W}^{1,1,1}(\Omega)$ cannot hold as previously proved in [52, 16]. This follows
because ${\bf W}^{1,1,1}(\Omega)$ reduces to the classical Sobolev space ${\bf
W}^{1,1}(\Omega)$.

Furthermore, we were also able to extend the results from [27, 18, 20] and
proved that a non-linear version of Korn’s inequality (cf. (16)) holds true in
these anisotropic Sobolev spaces (cf. Theorem 6). Similarly, we also proved
that Poincare’s inequality, and a more general non-linear version of it, hold
in this more general Sobolev spaces (cf. results in subsection 3.3). Note that
to be able to extend Korn’s and Poincare’s inequality to a non-linear version,
the continuous mappings considered as in Theorem 6 and Theorem 8 cannot share
any elements (or only share the zero vector) with the kernels of the strain
stress tensor or the gradient, respectively.

In [53, 54], the author studied anisotropic Sobolev spaces in a more general
way, in which for example, each distributional derivative lies in a possibly
different $L^{p}$-space. A natural question question would be, then, whether
the Korn inequality also holds for this more general setting. Nevertheless, it
is not yet clear in what $L^{p}$-spaces we should measure each component of
the symmetric part of the gradient, and hence an appropriate expression for a
Korn-type inequality in this general setting is unknown to the authors. This
is left as subject of future work.

## Acknowledgments

Sebastián A. Domínguez-Rivera thanks the support of the Pacific Institute for
the Mathematical Sciences (PIMS) through a PIMS postdoctoral fellowship. The
authors are very grateful to prof. J. Rákosník (Institute of Mathematics CAS)
for providing us with digital copies of [53, 54], and to prof. Gabriel N.
Gatica (Universidad de Concepción) for his comments which helped improving
this manuscript.

#### Note for the reader

Please note that this manuscript represents a preprint only and has not been
(or is in the process of being) peer-reviewed. A DOI link will be made
available for this ArXiv preprint as soon as the peer-reviewed version is
published online.

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#  $N$-Band Photonic Hopf Insulators Based on 2D Microring Lattices

Bo Leng, Vien Van
Department of Electrical and Computer Engineering
University of Alberta
Edmonton, Alberta, T6G 2W3, Canada
{Bo<EMAIL_ADDRESS>

###### Abstract

Hopf insulators are topological insulators whose topological behavior arises
from the nontrivial mapping from a 3D sphere to a 2D sphere, known as the Hopf
map. The Hopf map, typically encountered in the study of spinor and skyrmion
systems, is classified topologically by an integer invariant called the Hopf
index. Here we show that due to the periodic circulation of light inside each
microring, a 2D lattice of microring resonators can emulate an $N$-band
photonic Hopf insulator with nontrivial Hopf index. In particular, we show by
numerical computation and direct analytical proof that the $N$-band Hopf index
of the microring lattice is identical to its winding number. The result shows
that the Hopf index is an alternative topological invariant for classifying 2D
microring photonic lattices and establishes a correspondence between the Hopf
insulator phase and the anomalous Floquet insulator phase of the lattice. More
generally, our work shows that 2D microring lattices can provide a versatile
nanophotonic platform for studying non-Abelian topological photonic systems.

_K_ eywords Hopf insulator $\cdot$ Topological photonics

## Introduction

Topological insulators are band insulators whose energy bands or band gaps are
characterized by global invariants that remain unchanged in the presence of
disorders. In addition to their importance in the study of solid-state
systems, these materials can potentially have applications in realizing
electronic [1, 2, 3] and photonic devices [4, 5, 6, 7] that are robust to
imperfections. The "ten-fold" classification has successfully provided a
systematic way to classify a large number of topological insulators based on
lattice symmetry [8]. However, there also exist topological systems beyond the
ten-fold classification which exhibit topologically protected edge states even
though the relevant topological invariants (e.g. Chern numbers) of all the
energy bands are trivial. One prominent example is the anomalous Floquet
insulator (AFI) phase of periodically driven systems [9, 10, 11, 12, 13, 14,
15, 16, 17], whose band gaps can have nontrivial winding number even when the
Chern invariants of all the quasi-energy bands vanish. Another topological
insulator which exists beyond the ten-fold classification is the Hopf
insulator, first investigated in static 3D magnetic topological insulator with
two energy bands [18]. When all relevant weak invariants vanish, the system
can still exhibit nontrivial behavior and is classified by the Hopf index,
which is related to the linking number of the Hopf map $S^{3}\rightarrow
S^{2}$ [19, 20, 21, 22] typically encountered in spinor and skyrmion systems
[23, 24, 25, 26]. Interestingly, it was later shown [27] that this Hopf index
exactly corresponds to the winding number of a 2D+1 Floquet insulator with two
quasi-energy bands, thus establishing a connection between the Hopf phase and
the AFI phase. The two-band Hopf insulator was generalized to the multi-band
case in [28], where it was shown that a 3D system with $N>2$ distinct bands
and vanishing weak invariants can be classified by an $N$-band Hopf index
associated with the third homotopy group $\pi_{3}(SU(N))$. It should be
stressed that a defining feature of Hopf insulators is that they do not
possess any symmetry of the "ten-fold" classification so that their nontrivial
topological behavior arises solely from the Hopf map.

The Hopf invariant has previously been used to characterize dynamical
quenching processes of 2D cold atom topological systems, enabling the Chern
number of the system under equilibrium to be experimentally measured [29, 30].
In the photonics domain, there have so far been no reported realizations of
Hopf insulators, although AFI insulators have been demonstrated using coupled
waveguide arrays[15, 17, 16] and microring lattices [10, 6, 7]. However, not
all AFIs are governed by Hamiltonians that are homotopic to the Hopf map and
are thus Hopf insulators. In this paper, we show that 2D square lattices of
coupled microrings can emulate an $N$-band Hopf insulator with $N=3$ and
$N=4$. The direction of light circulation in each microring serves as the
third periodic dimension so that our system is a map from $T^{2}\times
S^{1}\rightarrow SU(N)$. In particular, we show by numerical simulations as
well as by direct analytical proof that the $N$-band Hopf index of the
microring lattice is identical to the winding number of each band gap and can
thus provide a complete topological classification of the insulator. Both
$N=3$ and $N=4$ band microring lattices have been experimentally demonstrated
before as AFIs [10, 7, 6]. Here we show that these systems can also be
regarded as photonic Hopf insulators, thereby providing further insight into
the nontrivial topological behaviors of 2D microring lattices and
demonstrating their versatility as a nanophotonic platform for studying non-
Abelian topological photonic systems.

## Hopf insulator and its multi-band generalization

We begin by first giving a brief review of Hopf insulator and its multi-band
generalization. Given any 2D insulator with two energy bands separated by a
band gap, we can express its Hamiltonian as
$H(k_{x},k_{y})=U(k_{x},k_{y})\sigma_{3}U^{\dagger}(k_{x},k_{y})$, where
$\sigma_{3}=\mathrm{diag}[1,-1]$ is a Pauli matrix and $U(k_{x},k_{y})\in
SU(2)$ is a $2\times 2$ unitary transformation matrix which flattens the bands
to have constant energy spectra of $\pm 1$. We can regard $H(k_{x},k_{y})$ as
a map of each point $(k_{x},k_{y})$ in the 2-torus ($T^{2}$) Brillouin zone
(BZ) to the quotient group $SU(2)/U(1)$, where $U(1)$ accounts for the fact
that only the relative phase between the two Bloch eigenstates is fixed [31].
Since the group $SU(2)/U(1)$ is isomorphic to a 2-sphere ($S^{2}$),
$H(k_{x},k_{y})$ is homotopic to the map $T^{2}\rightarrow S^{2}$. The
topological invariant classifying this map is the Chern number,
$C_{z}=\int_{BZ}dk_{x}dk_{y}B_{z}$, where $B_{z}$ is the $z$-component of the
Berry curvature. This corresponds to the well-known classification of 2D Chern
insulators. For a 3D insulator with the same two-band constraint, the
Hamiltonian $H(\vec{k})$, $\vec{k}=(k_{x},k_{y},k_{z})$, becomes a map from
the 3-torus ($T^{3}$) BZ to the group $SU(2)/U(1)$, or $T^{3}\rightarrow
S^{2}$. A 3-torus has three submanifolds, each being a cross-section $T^{2}$
perpendicular to the axis $k_{x}$, $k_{y}$, or $k_{z}$. As in the 2D case,
each submanifold is classified by a Chern number corresponding to a component
$\mu=x,y,z$ of the Berry curvature,
$C_{\mu}=\int_{BZ}dk_{\rho}dk_{\lambda}B_{\mu},$ where $dk_{\rho}dk_{\lambda}$
describes a cross-section of the $T^{3}$ BZ perpendicular to the $\mu$
direction [32]. When all three invariants vanish, $H(\vec{k})$ becomes
homotopic to the Hopf map, $S^{3}\rightarrow S^{2}$, which can be nontrivial
since it is classified by the third homotopy group $\pi_{3}(S^{2})=\mathbb{Z}$
[33]. Thus, even though the Chern numbers of the submanifolds are zero, the
insulator can still be topologically nontrivial. Such a 3D two-band
topological insulator is called a Hopf insulator [18, 32]. Under the
assumption of vanishing weak invariants ($C_{x}=C_{y}=C_{z}=0$), the
topological invariant of this map is given by the Hopf index,
$n_{H}=2P_{3}^{1}\in\mathbb{Z}$, where $P_{3}^{n}$ is the Abelian third Chern-
Simons form of band $n$,

$P_{3}^{n}=\int_{BZ}\frac{d^{3}\vec{k}}{8\pi^{2}}\vec{A}_{n}\cdot\nabla_{\vec{k}}\times\vec{A}_{n}$
(1)

with $\vec{A}_{n}=i\bra{\psi_{n}}\nabla_{\vec{k}}\ket{\psi_{n}}$ being the
Berry connection associated with the $n^{\mathrm{th}}$ Bloch state
$|\psi_{n}(\vec{k})\rangle$.

The two-band Hopf insulator can be generalized to multi-band systems in 3D,
which are called $N$-band Hopf insulators [28]. The Hamiltonian of an
insulator with $N$ distinct flattened bands can be expressed as
$H(\vec{k})=U\text{diag}[1,\ldots,N]U^{\dagger}$, where $U(\vec{k})\in SU(N)$
is an $N\times N$ unitary matrix. The Hamiltonian in this case is a map from
the 3-torus BZ to the quotient group $SU(N)/U(1)^{N-1}$ [28]. Under the same
assumption of vanishing weak invariants, this map is classified by the third
homotopy group $\pi_{3}(SU(N)/U(1)^{N-1})\cong\pi_{3}(SU(N))$. The topological
invariant is also given by the Hopf index but in this case computed from the
non-Abelian third Chern-Simons form,

$n_{H}=\int_{BZ}\frac{d^{3}\vec{k}}{8\pi^{2}}\text{Tr}\bigg{\\{}\vec{\hat{A}}\cdot\nabla_{\vec{k}}\times\vec{\hat{A}}-\frac{2i}{3}\vec{\hat{A}}\cdot\vec{\hat{A}}\times\vec{\hat{A}}\bigg{\\}}$
(2)

Here $\vec{\hat{A}}$ is the matrix-valued connection,
$\vec{\hat{A}}=\hat{A}_{k_{x}}\vec{\textbf{k}}_{x}+\hat{A}_{k_{y}}\vec{\textbf{k}}_{y}+\hat{A}_{k_{z}}\vec{\textbf{k}}_{z}$,
where the $\vec{\textbf{k}}_{\mu}$-component $\hat{A}_{k_{\mu}}$ is an
$N\times N$ matrix with elements
$(\hat{A}_{k_{\mu}})_{mn}=i\expectationvalue{\psi_{m}|\partial_{k_{\mu}}\psi_{n}}$.

Since the Hamiltonian of a 3D insulator can be emulated by a 2D periodically-
driven system with the substitution $k_{z}\rightarrow 2\pi t/T$, where $T$ is
the driving period, there is a correspondence between a Hopf insulator and the
AFI phase of a Floquet insulator in 2D+1 dimensions [27, 28]. In particular,
the AFI phase exists when the Chern numbers of all the quasi-energy bands of a
Floquet insulator vanish and is classified by the third winding number of the
evolution operator $\mathcal{U}$, $W[\mathcal{U}]$. For a 2D+1 Floquet
insulator, it can be shown that the winding number is identical to the Hopf
invariant, $W=n_{H}$ [27, 28].

## Photonic Hopf Insulator Based on 2D Microring Lattice

The photonic Hopf insulator we consider is a 2D square lattice of
evanescently-coupled microring resonators with identical resonance
frequencies, as shown in Fig. 1(a). Each unit cell consists of four microrings
labeled $\\{A,B,C,D\\}$ characterized by two unequal coupling angles:
$\theta_{a}$ between microring $A$ and its neighbors, and $\theta_{b}$ between
microring $D$ and its neighbors. The coupling angles correspond to power
coupling coefficients $\kappa_{(a,b)}^{2}=\sin^{2}\theta_{(a,b)}$ between
adjacent resonators. We assume that each microring in the lattice supports
only a clockwise or counterclockwise propagating mode. As light circulates
around each microring, it couples periodically to its neighbors, with a period
$L$ equal to the microring circumference. The direction of light propagation
in each microring, denoted by $z$, thus takes the role of time in our
periodically-driven system. Using waveguide coupled mode theory, the evolution
of the system can be described by a Schrodinger-like equation (see
Supplementary Materials for derivation),

$-i\frac{\partial}{\partial z}|\psi(\vec{k},z)\rangle=[\beta
I+H_{FB}(\vec{k},z)]|\psi(\vec{k},z)\rangle$ (3)

where $I$ is the $4\times 4$ identity matrix, $\beta$ is the propagation
constant of the microring waveguides, $\vec{k}=(k_{x},k_{y})$ is the crystal
momentum, and
$|\psi(\vec{k},z)\rangle=[\psi_{A},\psi_{B},\psi_{C},\psi_{D}]^{\mathrm{T}}$
is the wave function representing the fields in the four microrings in each
unit cell. The Floquet-Bloch Hamiltonian $H_{FB}$ describes a hopping sequence
of four coupling steps, $H_{FB}(\vec{k},z)=\sum_{j=1}^{4}H_{j}(\vec{k})$,
where $H_{j}$ in step $j$ is given by

$\begin{array}[]{ll}H_{1}(\vec{k})=K\otimes\sigma_{1},&0\leq z<L/4\\\
H_{2}(\vec{k})=\sigma_{1}\otimes K,&L/4\leq z<L/2\\\
H_{3}(\vec{k})=K\otimes(\sigma_{1}\cos{k_{x}}+\sigma_{2}\sin{k_{x}}),&L/2\leq
z<3L/4\\\ H_{4}(\vec{k})=(\sigma_{1}\cos{k_{y}}+\sigma_{2}\sin{k_{y}})\otimes
K,&3L/4\leq z<L\end{array}$ (4)

with $K=\mathrm{diag}[\theta_{a}L/4,\theta_{b}L/4]$ and $\sigma_{i}$ being the
Pauli matrices. The Hamiltonian is periodic in $z$ with periodicity $L$,
$H_{FB}(\vec{k},z)=H_{FB}(\vec{k},z+L)$. Neglecting the dynamic phase term
($e^{i\beta z}$), the evolution operator of the lattice is given by
$\mathcal{U}(\vec{k},z)=\mathcal{T}e^{i\int_{0}^{z}H_{FB}(\vec{k},z^{\prime})dz^{\prime}}$,
where $\mathcal{T}$ is the time-order operator. Over one period $L$, the
system evolution is captured by the Floquet operator
$U_{F}(\vec{k})=\mathcal{U}(\vec{k},L)$, whose eigenstates are the Floquet
modes $\Phi_{n}(\vec{k})$,

$U_{F}(\vec{k})|\Phi_{n}(\vec{k})\rangle=e^{i\epsilon_{n}(\vec{k})L}|\Phi_{n}(\vec{k})\rangle$
(5)

with associated quasi-energy bands $\epsilon_{n}(\vec{k})$. These bands can be
regarded as the energy bands of a 2D static system with an effective
Hamiltonian $H_{eff}(\vec{k})$ defined such that
$U_{F}(\vec{k})=e^{iH_{eff}(\vec{k})L}$. Over each Floquet-Brillouin zone, the
microring lattice has $N=4$ quasi-energy bands separated by 3 gaps, labeled I,
II and III, as shown in Fig. 1(b) for the case
($\theta_{a},\theta_{b})=(0.48\pi,0.05\pi)$. Band gap II is centered at quasi-
energy $\epsilon=\pi/L$ and is always open except for $\theta_{a}=\theta_{b}$
or $\theta^{2}_{a}+\theta^{2}_{b}\approx\pi/8$, while band gaps I and III are
symmetric about $\epsilon=0$ and are only open for certain range of
($\theta_{a},\theta_{b}$) values (red and blue areas in Fig. 2(a)). When
$\theta_{b}$ is set to 0, which corresponds to removal of microring $D$ from
each unit cell, the two bands around quasi-energy $\epsilon=0$ collapse to
form a single flat band. In this case the microring lattice forms a Lieb
lattice [34] with $N=3$ quasi-energy bands, with a flat band pinned at
$\epsilon=0$, as shown in Fig. 1(c). For each quasi-energy band, we can
compute the Chern number from the Floquet state as[9]

$C_{z}^{n}=\frac{1}{2\pi
i}\int_{BZ}\text{Tr}\bigg{\\{}P_{n}\left[\partial_{k_{x}}P_{n},\partial_{k_{y}}P_{n}\right]\bigg{\\}}dk_{x}dk_{y}$
(6)

where $P_{n}(\vec{k})=|\Phi_{n}(\vec{k})\rangle\langle\Phi_{n}(\vec{k})|$ is
the projection operator onto the Floquet state $\Phi_{n}(\vec{k})$. In
general, the Chern number is zero for quasi-energy bands $n=1$ and 4 but
nontrivial for bands $n=2$ and 3 over the range of $(\theta_{a},\theta_{b})$
values shown by the red areas in Fig. 2(a). In this region the microring
lattice exhibits Chern insulator behavior in band gaps I and III [11].

Figure 1: (a) Schematic of a 2D microring lattice characterized by dissimilar
coupling angles $\theta_{a}\neq\theta_{b}$ in each 4-ring unit cell
$\\{A,B,C,D\\}$. In each microring, light propagates in either the clockwise
or counterclockwise direction, which is denoted by $z$. (b) and (c) Quasi-
energy bands and band gaps of (b) $N=4$ band lattice with
$\theta_{a}=0.48\pi,\theta_{b}=0.05\pi$ and (c) $N=3$ band lattice with
$\theta_{a}=0.46\pi,\theta_{b}=0$.

Due to the periodic circulation of light in each microring, the Floquet-Bloch
Hamiltonian $H_{FB}(\vec{k},z)$ of the $N$-band microring lattice can be
regarded as a map from $T^{2}\times S^{1}$ to the group $SU(N)/U(1)^{(N-1)}$.
The Chern number computed in Eq.(6) classifies only the 2-torus sub-manifold
$k_{x}-k_{y}$ in each Floquet-Brillouin zone and thus does not completely
capture the full topological properties of the lattice. In particular, when
the Chern numbers of all the bands vanish, $H_{FB}$ is homotopic to the map
$S^{2}\times S^{1}\rightarrow SU(N)/U(1)^{(N-1)}\cong\pi_{3}(SU(N))$, which is
classified by the $N$-band Hopf index in Eq.(2). Computation of the Hopf index
requires us to first construct the connection matrix $\vec{\hat{A}}$, which is
a non-Abelian gauge potential since the gauge transformation matrices $U\in
SU(N)$ acting on the states of the $N$-band microring lattice are non-
commutative. For our Floquet system, we can construct the connection matrix
$\vec{\hat{A}}$ using the $z$-evolved Floquet state $\Psi_{n,\xi}(\vec{k},z)$
defined as

$|\Psi_{n,\xi}(\vec{k},z)\rangle=\mathcal{U}_{\xi}(\vec{k},z)|\Phi_{n}(\vec{k})\rangle$
(7)

where
$\mathcal{U}_{\xi}(\vec{k},z)=\mathcal{U}(\vec{k},z)e^{-iH_{eff,\xi}(\vec{k})z}$
is the periodized evolution operator with the property
$\mathcal{U}_{\xi}(\vec{k},0)=\mathcal{U}_{\xi}(\vec{k},L)=I$. Here the
effective Hamiltonian $H_{eff,\xi}(\vec{k})=-(i/L)ln(U_{F}(\vec{k}))$ is
defined for the band gap at energy $\xi$ by choosing its eigenvalues to be
between $\xi$ and $\xi+2\pi/L$ [9]. The $z$-evolved Floquet states are
periodic, $|\Psi_{n,\xi}(\vec{k},L)\rangle=|\Psi_{n,\xi}(\vec{k},0)\rangle$,
and orthogonal
$\langle\Psi_{m,\xi}(\vec{k},z)|\Psi_{n,\xi}(\vec{k},z)\rangle=\delta_{mn}$,
since the periodic evolution operator preserves the orthogonality of the
initial Floquet states during evolution. The elements of the connection matrix
are computed as
$(\hat{A}_{\mu})_{mn}=i\langle\Psi_{m,\xi}(\vec{k},z)|\partial_{\mu}\Psi_{n,\xi}(\vec{k},z)\rangle$,
where $\mu\in\\{k_{x},k_{y},z\\}$. The Hopf index can then be written
explicitly in terms of the $z$-evolved Floquet states as

$\begin{split}n_{H}&=-\frac{1}{8\pi^{2}}\int_{0}^{L}dz\int_{BZ}dk_{x}dk_{y}(F_{1}+F_{2})\\\
&F_{1}=\sum_{n=1}^{N}\sum_{m=1}^{N}\epsilon_{ijk}\langle\Psi_{m,\xi}|\partial_{i}\Psi_{n,\xi}\rangle\langle\partial_{j}\Psi_{n,\xi}|\partial_{k}\Psi_{m,\xi}\rangle\\\
&F_{2}=\sum_{l=1}^{N}\sum_{m=1}^{N}\sum_{n=1}^{N}\epsilon_{ijk}\langle\Psi_{l,\xi}|\partial_{i}\Psi_{m,\xi}\rangle\langle\Psi_{m,\xi}|\partial_{j}\Psi_{n,\xi}\rangle\langle\Psi_{n,\xi}|\partial_{k}\Psi_{l,\xi}\rangle\end{split}$
(8)

where $\epsilon_{ijk}$ is the Levi-Civita symbol representing summation over
$k_{x},k_{y},z$. From its definition, the $N$-band Hopf index is a band gap
invariant which is summed over the topological structures of all the bands
anchored at quasi-energy $\xi$.

We computed the Hopf invariant for each open band gap of the microring lattice
as a function of the coupling angles $\theta_{a}$ and $\theta_{b}$. The
results are shown in Fig. 2(b). For band gaps I and III, the Hopf invariant is
equal to 1 so the lattice is always nontrivial as long as these gaps are open.
For band gap II, the Hopf index transitions from 0 to 1 when the coupling
angles satisfy the approximate relation
$\theta_{a}^{2}+\theta_{b}^{2}\gtrapprox\pi^{2}/8$. For the $N=3$ band
lattice, which corresponds to the $\theta_{b}=0$ (or $\theta_{a}=0$) axis of
the map, the Hopf index is 1 in band gap I for all values of coupling angle
$\theta_{a}$ while for band gap II, it transitions from 0 to 1 for
$\theta_{a}>\pi/\sqrt{8}$. The Hopf insulator phase is defined as having
nontrivial Hopf invariant when the Chern numbers of all the quasi-energy bands
vanish. This region is highlighted by the red lines in Fig. 2(b) showing the
existence of the nontrivial Hopf phase for the microring lattice with $N=3$
and $N=4$ bands.

Figure 2: (a) Map of the Chern numbers $C_{2}$ and $C_{3}$ of quasi-energy
bands $n=2$ and $n=3$ of the microring lattice. (b) Map of the $N$-band Hopf
index and winding number of each band gap, with the regions of Hopf insulator
phase and AFI phase marked by red lines. The $N=3$ band lattice is indicated
by the green lines along the $\theta_{a}=0$ and $\theta_{b}=0$ axes.

As mentioned above, there is a correspondence between the Hopf phase and the
AFI phase of a Floquet insulator. We have also calculated the winding number
for each band gap of the 2D microring lattice [11]. The winding number
associated with a gap at quasi-energy $\xi$ is computed using the periodized
evolution operator as [9]

$\begin{split}W[\mathcal{U}_{\xi}]=&\frac{1}{8\pi^{2}}\int_{0}^{L}dz\int_{BZ}dk_{x}dk_{y}\\\
&\text{Tr}\bigg{\\{}\mathcal{U}_{\xi}^{\dagger}\partial_{z}\mathcal{U}_{\xi}\left[\mathcal{U}_{\xi}^{\dagger}\partial_{k_{x}}\mathcal{U}_{\xi},\mathcal{U}_{\xi}^{\dagger}\partial_{k_{y}}\mathcal{U}_{\xi}\right]\bigg{\\}}\end{split}$
(9)

For the 2D microring lattice, the map of the winding number vs. coupling
angles is found to be identical to the Hopf invariant, as shown in Fig. 2(b).
Like the Hopf insulator phase, the AFI phase is defined as having nontrivial
winding number when the Chern numbers of all the bands vanish. Fig. 2(b) shows
that the region where the microring lattice exhibits AFI behavior exactly
matches with the region for the Hopf phase, thus confirming the correspondence
between these two phases for the microring lattice. This result shows that the
$N$-band Hopf index can be used as an alternative invariant to the winding
number to classify 2D microring lattices, correctly predicting the existence
of topologically protected edge states in each band gap[10].

The $N$-band Hopf invariant is computed using the $z$-evolved Floquet states
while the winding number is computed from the evolution operator. It is
possible to analytically prove the correspondence of these two invariants for
a general $N$-band Floquet lattice, which we have included in the
Supplementary Materials. While this correspondence has been shown for 3D
static systems with $N$ bands in [28], our proof provides a direct analytical
connection between these two invariants for a 2D+1 Floquet system. It also
shows how the non-Abelian connection matrix $\vec{\hat{A}}$ can also be
constructed using states evolving from the standard bases for the computation
of the Hopf invariant.

The presence of the Hopf insulator phase is attributed to the nontrivial
topological behavior of the entire $N$-band structure which is irreducible to
the topology of the individual bands. A constraint specified in [28] for the
$N$-band Hopf index to be well defined is that the $N$ bands must be distinct
and separated by open gaps; otherwise there may be an integer ambiguity in the
Hopf index given by Eq.(2). On the other hand, the winding number
$W[\mathcal{U}_{\xi}]$ of the evolution operator does not have such ambiguity.
As a result, the Hopf invariant computed using the $z$-evolved Floquet states,
being identical to $W[\mathcal{U}_{\xi}]$, remains well-defined. We found that
this is the case for the $N=4$ band microring lattice, which has well-defined
Hopf index even although bands $n=1$ and $n=4$ are entangled to form a
composite band.

## Conclusions

In summary, 2D microring lattices have previously been shown to emulate 2D+1
Floquet systems which can exhibit AFI phase as classified by the winding
number; here we showed that these systems can also be regarded as $N$-band
Hopf insulators classified by an alternative invariant called the $N$-band
Hopf index. We also showed by numerical computation and analytical proof that
the Hopf invariant is identical to the winding number and established the
correspondence between the AFI phase and Hopf insulator phase of the microring
lattice. These lattices represent the first realization of photonic Hopf
insulators and more generally, can provide a versatile nanophotonic platform
for studying non-Abelian topological photonic systems.

Data Availability. Data underlying the results presented in this paper are
available in the fig.2(b) of this paper.

Funding. Natural Sciences and Engineering Research Council of Canada.

Disclosures. The authors declare no conflicts of interest.

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## Appendix

### Floquet-Bloch Hamiltonian of 2D square microring lattices

Here we derive the Floquet-Bloch Hamiltonian for a 2D square microring lattice
shown in Fig. S1(a). Each unit cell consists of four resonators labeled
$A,B,C,D$. In each microring light propagates in either the counter-clockwise
or clockwise direction, which we denote as the $z$ direction. By cutting each
microring at the point indicated by the open circle in Fig. S3(a) and
unrolling it into a straight waveguide, we can transform the microring lattice
into a 2D array of coupled waveguides as shown in Fig. S3(b) [11]. Each
waveguide has length $L$ equal to the microring’s circumference, which is
divided into a sequence of 4 coupling steps, each step of length $L/4$.
Denoting the fields in the microrings in unit cell $(m,n)$ as
$\psi_{A}^{m,n},\psi_{B}^{m,n},\psi_{C}^{m,n},\psi_{D}^{m,n}$, we can write
the coupled mode equations for the evolution of the fields along the waveguide
array as

Figure S3: (a) Schematic of a unit cell ($m,n$) of the 2D square microring
lattice. (b) Equivalent 2D coupled waveguide array of the unit cell obtained
by cutting each microring at the point indicated by the yellow open circle and
unrolling it into a straight waveguide. The propagation of a light over one
period $L$ is divided into 4 equal-length segments. An example of light
propagating along segment 1 of microring $B$ is shown in red for both the
microring lattice and the waveguide array.

$\displaystyle-i\frac{\partial\psi_{A}^{m,n}}{\partial z}$
$\displaystyle=\beta\psi_{A}^{m,n}+k_{a}(1)\psi_{B}^{m,n}+k_{a}(2)\psi_{C}^{m,n}+k_{a}(3)\psi_{B}^{m-1,n}+k_{a}(4)\psi_{C}^{m,n-1}$
$\displaystyle-i\frac{\partial\psi_{B}^{m,n}}{\partial z}$
$\displaystyle=\beta\psi_{B}^{m,n}+k_{a}(1)\psi_{A}^{m,n}+k_{b}(2)\psi_{D}^{m,n}+k_{a}(3)\psi_{A}^{m+1,n}+k_{b}(4)\psi_{D}^{m,n-1}$
$\displaystyle-i\frac{\partial\psi_{C}^{m,n}}{\partial z}$
$\displaystyle=\beta\psi_{C}^{m,n}+k_{b}(1)\psi_{D}^{m,n}+k_{a}(2)\psi_{A}^{m,n}+k_{b}(3)\psi_{D}^{m-1,n}+k_{a}(4)\psi_{A}^{m,n+1}$
$\displaystyle-i\frac{\partial\psi_{D}^{m,n}}{\partial z}$
$\displaystyle=\beta\psi_{D}^{m,n}+k_{b}(1)\psi_{C}^{m,n}+k_{b}(2)\psi_{B}^{m,n}+k_{b}(3)\psi_{C}^{m+1,n}+k_{b}(4)\psi_{B}^{m,n+1}$
(S10)

where $\beta$ is the propagation constant of the microring waveguides and the
coupling strength $k_{(a,b)}(j)=4\theta_{(a,b)}/L$ in step $j$ and 0
otherwise. Applying Bloch’s theorem in the $x$ and $y$ directions of the
microring lattice, we have

$\psi_{A,B,C,D}^{m\pm\mu,n\pm\nu}=\psi_{A,B,C,D}^{m,n}e^{i(\pm\mu k_{x}\pm\nu
k_{y})\Lambda}$ (S11)

where $\Lambda$ is the lattice constant and $\mu$ and $\nu$ are integers.
Using the above boundary conditions, we can cast eq.(Floquet-Bloch Hamiltonian
of 2D square microring lattices) in the form of a Schrodinger-like equation
for the state vector
$\ket{\psi}=[\psi_{A}^{m,n},\psi_{B}^{m,n},\psi_{C}^{m,n},\psi_{D}^{m,n}]^{\mathrm{T}}$
of each unit cell,

$-i\frac{\partial}{\partial z}|\psi(\vec{k},z)\rangle=[\beta
I+H_{FB}(\vec{k},z)]|\psi(\vec{k},z)\rangle$ (S12)

where $\vec{k}=(k_{x},k_{y})$ is the crystal momentum vector. The Floquet-
Bloch Hamiltonian consists of 4 steps, $H_{FB}=\sum_{j=1}^{4}H_{j}$, with the
Hamiltonian in each step explicitly given by

$\displaystyle H_{1}$ $\displaystyle=\begin{bmatrix}0&k_{a}&0&0\\\
k_{a}&0&0&0\\\ 0&0&0&k_{b}\\\ 0&0&k_{b}&0\\\ \end{bmatrix}$ $\displaystyle
H_{2}$ $\displaystyle=\begin{bmatrix}0&0&k_{a}&0\\\ 0&0&0&k_{b}\\\
k_{a}&0&0&0\\\ 0&k_{b}&0&0\\\ \end{bmatrix}$ $\displaystyle H_{3}$
$\displaystyle=\begin{bmatrix}0&k_{a}e^{-ik_{x}\Lambda}&0&0\\\
k_{a}e^{ik_{x}\Lambda}&0&0&0\\\ 0&0&0&k_{b}e^{-ik_{x}\Lambda}\\\
0&0&k_{b}e^{ik_{x}\Lambda}&0\\\ \end{bmatrix}$ $\displaystyle H_{4}$
$\displaystyle=\begin{bmatrix}0&0&k_{a}e^{-ik_{y}\Lambda}&0\\\
0&0&0&k_{b}e^{-ik_{y}\Lambda}\\\ k_{a}e^{ik_{y}\Lambda}&0&0&0\\\
0&k_{b}e^{ik_{y}\Lambda}&0&0\\\ \end{bmatrix}$

The above expressions can be written in the compact form of Eq.(4) in the main
text with $\Lambda=1$. The $N=3$ band lattice is obtained by setting either
$\theta_{a}=0$ or $\theta_{b}=0$. The explicit form of the Floquet Bloch
Hamiltonian for the $N=3$ band lattice can be found in the Supplementary
Material of [7].

Neglecting the dynamic phase term ($e^{i\beta z}$), the evolution operator of
the lattice is given by
$\mathcal{U}(\vec{k},z)=\mathcal{T}e^{i\int_{0}^{z}H_{FB}(\vec{k},z^{\prime})dz^{\prime}}$,
where $\mathcal{T}$ is the time-order operator. Since the Hamiltonian $H_{j}$
in each step $j$ is independent of $z$, we can define the evolution during
step $j$ as $\mathcal{U}_{j}(\vec{k},z)=e^{iH_{j}(\vec{k})z}$. The evolution
operator over each period can then be explicitly computed as

$\mathcal{U}(\vec{k},z)=\Bigg{\\{}\begin{array}[]{ll}\mathcal{U}_{1}(\vec{k},z),&0\leq
z<L/4\\\ \mathcal{U}_{2}(\vec{k},z-L/4)\mathcal{U}_{1}(\vec{k},L/4),&L/4\leq
z<L/2\\\
\mathcal{U}_{3}(\vec{k},z-L/2)\mathcal{U}_{2}(\vec{k},L/4)\mathcal{U}_{1}(\vec{k},L/4),&L/2\leq
z<3L/4\\\
\mathcal{U}_{4}(\vec{k},z-3L/4)\mathcal{U}_{3}(\vec{k},L/4)\mathcal{U}_{2}(\vec{k},L/4)\mathcal{U}_{1}(\vec{k},L/4),&3L/4\leq
z<L\end{array}$ (S14)

### Proof of the correspondence between the $N$-band Hopf index and the
winding number

Here we provide an analytical proof of the correspondence between the Hopf
index and the winding number of a 2D $N$-band Floquet insulator with
$z$-periodic Hamiltonian $H(k_{x},k_{y},z)$ (for time-periodic systems,
substitute $z\rightarrow t$). The $N$-band Hopf invariant is

$n_{H}=\frac{1}{8\pi^{2}}\int_{0}^{L}dz\int_{BZ}Q_{3}dk_{x}dk_{y}$ (S15)

where $L$ is the evolution period and $Q_{3}$ is the non-Abelian third Chern-
Simons form,

$Q_{3}=\text{Tr}\bigg{\\{}\vec{\hat{A}}\cdot\nabla\times\vec{\hat{A}}-\frac{2i}{3}\vec{\hat{A}}\cdot\vec{\hat{A}}\times\vec{\hat{A}}\bigg{\\}}=\epsilon_{jkl}\text{Tr}\bigg{\\{}\hat{A}_{j}\partial_{k}\hat{A}_{l}-\frac{2i}{3}\hat{A}_{j}\hat{A}_{k}\hat{A}_{l}\bigg{\\}}$
(S16)

with the indices $j,k,l$ running over $k_{x},k_{y},z$. We will first construct
the connection matrices $\hat{A}_{k},k\in\\{k_{x},k_{y},z\\}$, using states
$|u_{m}\rangle$ that evolve from a set of $N$ constant orthogonal states
$|m\rangle$ at $z=0$,

$|u_{m}(k_{x},k_{y},z)\rangle=\mathcal{U}_{\xi}(k_{x},k_{y},z)|m\rangle$ (S17)

where $\mathcal{U}_{\xi}$ is the periodized evolution operator anchored at
band gap quasi-energy $\xi$. For simplicity, we can take $|m\rangle$ to be the
$N$-dimensional standard basis vector, with all zero elements except for the
$m^{\mathrm{th}}$ element set to 1. To obtain the connection matrix computed
from the $z$-evolved Floquet states in Eq.(8) in the main text, we simply
perform the gauge transformation,
$|\Psi_{m}\rangle=\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}|u_{m}\rangle$,
where $Q_{F}$ is the eigenvector matrix of the Floquet operator
$U_{F}(k_{x},k_{y})$. In the basis of the states $|u_{m}\rangle$, the $(m,n)$
element of the connection matrix $\hat{A}_{k}$ is given by
$(\hat{A}_{k})_{mn}=i\langle u_{m}|\partial_{k}u_{n}\rangle$. The matrix
$\hat{A}_{k}$ can thus be expressed as

$\hat{A}_{k}=i\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi}.$ (S18)

Substituting the above expression for $\hat{A}_{k}$ into Eq.(S16), we have

$\displaystyle Q_{3}$
$\displaystyle=\epsilon_{jkl}\text{Tr}\bigg{\\{}-\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi}\partial_{k}(\mathcal{U}_{\xi}^{\dagger}\partial_{l}\mathcal{U}_{\xi})-\frac{2}{3}(\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{l}\mathcal{U}_{\xi})\bigg{\\}}$
$\displaystyle=\epsilon_{jkl}\text{Tr}\bigg{\\{}-\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi}\partial_{k}\mathcal{U}_{\xi}^{\dagger}\partial_{l}\mathcal{U}_{\xi}-(\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi})\mathcal{U}_{\xi}^{\dagger}\partial_{k}\partial_{l}\mathcal{U}_{\xi}$
$\displaystyle-\frac{2}{3}(\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{l}\mathcal{U}_{\xi})\bigg{\\}}$
(S19)

Since $\hat{A}_{k}=\hat{A}_{k}^{\dagger}$ (due to the orthogonality of
$|u_{m}\rangle$), we have
$i\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi}=-i(\partial_{k}\mathcal{U}_{\xi}^{\dagger})\mathcal{U}_{\xi}$,
which gives
$\partial_{k}\mathcal{U}_{\xi}^{\dagger}=-(\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi})\mathcal{U}_{\xi}^{\dagger}$.
Applying this relation to the first term on the right of Eq.(S19), we obtain

$Q_{3}=\epsilon_{jkl}\mathrm{Tr}\bigg{\\{}\frac{1}{3}(\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{l}\mathcal{U}_{\xi})\bigg{\\}}-\epsilon_{jkl}\mathrm{Tr}\bigg{\\{}(\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi})\mathcal{U}_{\xi}^{\dagger}\partial_{k}\partial_{l}\mathcal{U}_{\xi}\bigg{\\}}$
(S20)

The second term on the right hand side vanishes since $\partial_{k}$ commutes
with $\partial_{l}$ but $\epsilon_{jkl}=-\epsilon_{jlk}$, yielding

$Q_{3}=\frac{1}{3}\epsilon_{jkl}\text{Tr}\bigg{\\{}(\mathcal{U}_{\xi}^{\dagger}\partial_{j}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{k}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{l}\mathcal{U}_{\xi})\bigg{\\}}$
(S21)

By writing out the terms of the summation over $i,j,k$ and simplifying using
the invariance property of the trace under cyclic permutation of the matrices,
we obtain

$Q_{3}=\text{Tr}\bigg{\\{}(\mathcal{U}_{\xi}^{\dagger}\partial_{z}\mathcal{U}_{\xi})\left[(\mathcal{U}_{\xi}^{\dagger}\partial_{k_{x}}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{k_{y}}\mathcal{U}_{\xi})-(\mathcal{U}_{\xi}^{\dagger}\partial_{k_{y}}\mathcal{U}_{\xi})(\mathcal{U}_{\xi}^{\dagger}\partial_{k_{x}}\mathcal{U}_{\xi})\right]\bigg{\\}}$
(S22)

Substitution of the above expression for $Q_{3}$ into Eq.(S15) yields the same
expression for the winding number, $W[\mathcal{U}_{\xi}]$, in Eq.(9) in the
main text for a 2D+1 Floquet insulator.

It remains to be shown that when we transform to the basis of the $z$-evolved
Floquet states,
$|\Psi_{m}\rangle=\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}|u_{m}\rangle$,
the non-Abelian third Chern-Simons form remains unchanged. Given the gauge
transformation
$|\Psi_{m}\rangle=\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}|u_{m}\rangle$,
the non-Abelian third Chern-Simons forms $P_{3}$ computed in the bases
$|\Psi_{m}\rangle$ and $|u_{m}\rangle$ are related by [35, 28]

$P_{3}[\Psi_{m}]=P_{3}[u_{m}]+W[\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}]$
(S23)

The winding number $W[\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}]$ can
be expressed as [36]

$W[\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}]=W[\mathcal{U}_{\xi}]+W[Q_{F}]+W[\mathcal{U}_{\xi}^{\dagger}]$
(S24)

Since $\mathcal{U}_{\xi}\mathcal{U}_{\xi}^{\dagger}=I$, we have
$W[\mathcal{U}_{\xi}]+W[\mathcal{U}_{\xi}^{\dagger}]=0$. Furthermore
$W[Q_{F}]=0$ since $Q_{F}$ is independent of $z$. Thus
$W[\mathcal{U}_{\xi}Q_{F}\mathcal{U}_{\xi}^{\dagger}]=0$ so that
$P_{3}[\Psi_{m}]=P_{3}[u_{m}]$. Thus the Hopf index remains unchanged if we
construct the connection matrix $\vec{\hat{A}}$ from the $z$-evolved Floquet
states instead of the states $|u_{m}\rangle$.