# Low-rank matrix recovery via regularized nuclear norm minimization

Wendong Wang<EMAIL_ADDRESS>Feng Zhang<EMAIL_ADDRESS>Jianjun
Wang<EMAIL_ADDRESS>School of Mathematics and Statistics, Southwest
University, Chongqing, 400715, China

###### Abstract

In this paper, we theoretically investigate the low-rank matrix recovery
problem in the context of the unconstrained Regularized Nuclear Norm
Minimization (RNNM) framework. Our theoretical findings show that, one can
robustly recover any matrix $X$ from its few noisy measurements
$\bm{b}=\mathcal{A}(X)+\bm{n}$ with a bounded constraint
$\|\bm{n}\|_{2}\leq\epsilon$ through the RNNM, if the linear map $\mathcal{A}$
satisfies Restricted Isometry Property (RIP) with

$\displaystyle\delta_{tk}<\sqrt{\frac{t-1}{t}}$

for certain fixed $t>1$. Recently, this condition for $t\geq 4/3$ has been
proved by Cai and Zhang (2014) to be sharp for exactly recovering any rank-$k$
matrices via the constrained Nuclear Norm Minimization (NNM). To the best of
our knowledge, our work first extends nontrivially this recovery condition for
the constrained NNM to that for its unconstrained counterpart. Furthermore, it
will be shown that similar recovery condition also holds for regularized
$\ell_{1}$-norm minimization, which sometimes is also called Basis Pursuit
DeNoising (BPDN).

###### keywords:

Low-rank matrix recovery , regularized nuclear norm minimization , restricted
isometry property , basis pursuit denoising

††journal: Journal of LaTeX Templates

##  1 Introduction

Over the past decade, Low-Rank Matrix Recovery (LRMR) problem has attracted
considerable interest of researchers in many fields, including computer vision
[1], recommender systems [2], and machine learning [3] to name a few.
Mathematically, this problem aims to recover an unknown low-rank matrix
$X\in\mathbb{R}^{n_{1}\times n_{2}}$ from

$\displaystyle\bm{b}=\mathcal{A}(X)+\bm{n},$

where $\bm{b}\in\mathbb{R}^{m}(m\ll n_{1}n_{2})$ is an observed vector,
$\bm{n}\in\mathbb{R}^{m}$ is the unknown noise, and
$\mathcal{A}:\mathbb{R}^{n_{1}\times n_{2}}\rightarrow\mathbb{R}^{m}$ is a
known linear map defined as

$\displaystyle\mathcal{A}(X)=[\text{tr}(X^{T}A^{(1)}),\text{tr}(X^{T}A^{(2)}),\cdots,\text{tr}(X^{T}A^{(m)})]^{T}.$
(1)

Here, $\text{tr}(\cdot)$ is the trace function and
$A^{(i)}\in\mathbb{R}^{n_{1}\times n_{2}}$ is the $i$th measurement matrix.

A popular approach for the LRMR problem is to solve the Nuclear Norm
Minimization (NNM)

$\displaystyle\min_{X\in\mathbb{R}^{n_{1}\times
n_{2}}}~{}\|X\|_{*},~{}~{}s.t.~{}~{}\|\bm{b}-\mathcal{A}(X)\|_{2}\leq\epsilon,$
(2)

So far, much work has been done to find the explicit conditions under which
the exact/robust recovery of any low-rank matrices can be guaranteed [4, 5, 6,
7]. As one of the most powerful and widely used theoretical tools, Restricted
Isometry Property (RIP) captures particular attention.

######  Definition 1 ([5]).

A linear map $\mathcal{A}$ defined by (1) is said to satisfy the RIP with
Restricted Isometry Constant (RIC) of order $k$, denoted by
$\delta_{k}$111When $k$ is not an integer, we define $\delta_{k}$ as
$\delta_{\lceil k\rceil}$., if $\delta_{k}$ is the smallest value
$\delta\in(0,1)$ such

$\displaystyle(1-\delta)\|X\|_{F}^{2}\leq\|\mathcal{A}(X)\|_{2}^{2}\leq(1+\delta)\|X\|_{F}^{2}$

for every rank-$k$ matrix $X\in\mathbb{R}^{n_{1}\times n_{2}}$, i.e., the
signal whose rank is at most $k$.

There exist many RIP-based sufficient conditions for the exact recovery (i.e.,
the case when $\bm{n}=0$ and $\epsilon=0$) of any rank-$k$ matrices through
(2). These include $\delta_{4k}<\sqrt{2}-1$ [5], $\delta_{4k}<0.558$, and
$\delta_{3k}<0.4721$ [8], $\delta_{2k}<0.4931$ [9], $\delta_{2k}<1/2$ and
$\delta_{k}<1/3$ [10]. In particular, the sharpest conditions with the form of
$\delta_{tk}<\delta^{*}$ for $t>0$ have been completely given by Cai and Zhang
[11] and Zhang and Li [12], where $\delta^{*}=\sqrt{(t-1)/t}$ for $t\geq 4/3$
and $\delta^{*}=t/(4-t)$ otherwise, and they have also proved that under these
conditions, one can still robustly reconstruct any (low-rank) matrices.

An alternative approach to the constrained NNM (2) is to solve its
unconstrained counterpart, i.e., the following Regularized NNM (RNNM):

$\displaystyle\min_{X\in\mathbb{R}^{n_{1}\times
n_{2}}}~{}\|X\|_{*}+\frac{1}{2\lambda}\|\bm{b}-\mathcal{A}(X)\|_{2}^{2}.$ (3)

Compared to the constrained problem (2), this unconstrained problem is much
more suitable for noisy measurements and approximately low-rank matrix
recovery [13]. Currently, almost all the researches are focus on the
algorithms induced by (3), see, e.g., [13, 14, 15]. To the best of our
knowledge, Candès and Plan [5] gave the first RIP-based performance guarantee
for (3), and their results show that, when the noise $\bm{n}$ obeys
$\|\mathcal{A}^{*}(\bm{n})\|\triangleq\|\sum_{i=1}^{m}\bm{n}_{i}\cdot
A^{(i)}\|\leq\lambda/2$, and the map $\mathcal{A}$ satisfies
$\delta_{4k}<(3\sqrt{2}-1)/17$, the robust recovery of any rank-$k$ matrices
can be guaranteed through (3). However, after their initial work, the
theoretical investigation of (3) is rarely reported. Note that their noise
setting is based on the Dantzig selector rather than the often used
$\ell_{2}$-norm setting (i.e., $\|\bm{n}\|_{2}\leq\epsilon$), and the obtained
sufficient condition still has room to improve.

In this paper, we theoretically investigate the RIP-based performance
guarantee of the constrained problem (3) when the noise $\bm{n}$ obeys
$\|\bm{n}\|_{2}\leq\epsilon$. We show that if $\mathcal{A}$ satisfies
$\delta_{tk}<\sqrt{(t-1)/t}$ for certain $t>1$, one can robustly recover any
(low-rank) matrices from (3). The obtained results first extend the recovery
condition recently obtained by Cai and Zhang [11] for the constrained problem
(2) to that for its unconstrained counterpart. It should be also noted that
similar condition also holds for the well-known Basis Pursuit DeNoising (BPDN)
[16] to guarantee the robust recovery of any (sparse) signals.

The remainder of the paper is organized as follows. Section II introduces some
notations and useful lemmas. Section III presents the main results. Section IV
gives the related proofs. Finally, conclusion and future work are given in
Section V.

##  2 Notations and Preliminaries

###  2.1 Notations

We assume w.l.o.g. that $n_{1}\leq n_{2}$ and the SVD of
$X\in\mathbb{R}^{n_{1}\times n_{2}}$ is
$X=\sum_{i=1}^{n_{1}}\sigma_{i}(X)\cdot\bm{u}_{X}^{(i)}\cdot(\bm{v}_{X}^{(i)})^{T}$,
where $\bm{u}_{X}^{(i)}$ and $\bm{v}_{X}^{(i)}$ are the left and right
singular value vectors of $X$, respectively, and $\sigma_{i}(X)$ is the $i$th
largest singular value of $X$. For any positive integer $s$, we denote
$[s]=\\{1,2,\cdots,s\\}$, and $E^{c}=[n_{1}]\setminus E$ for any
$E\subset[n_{1}]$. We also denote $\sigma_{E}(X)$ as a vector whose element
$(\sigma_{E}(X))_{i}=\sigma_{i}(X)$ for $i\in E$ and $(\sigma_{E}(X))_{i}=0$
otherwise, and $X_{E}=\sum_{i\in
E}\sigma_{i}(X)\cdot\bm{u}_{X}^{(i)}\cdot(\bm{v}_{X}^{(i)})^{T}$ and
$X_{[s]}=\sum_{i=1}^{s}\sigma_{i}(X)\bm{u}_{X}^{(i)}(\bm{v}_{X}^{(i)})^{T}$.
Besides, we denote $\|\cdot\|_{\alpha}^{\beta}=(\|\cdot\|_{\alpha})^{\beta}$
where $\|\cdot\|_{\alpha}$ is certain (quasi-)norm. Then clearly
$\|\sigma_{E}(X)\|_{1}=\|X_{E}\|_{*}$. In the end, $\|\bm{x}\|_{0}$ is defined
to be the number of the nonzero elements in $\bm{x}$.

###  2.2 Three key lemmas

Before presenting our main results, we need some lemmas.

######  Lemma 1 ([11]).

For a positive number $\alpha$ and a positive integer $k$, define the polytope
$T(\alpha,k)\subset\mathbb{R}^{n}$ by
$T(\alpha,k)=\\{\bm{v}\in\mathbb{R}^{n}:\|\bm{v}\|_{\infty}\leq\alpha,\|\bm{v}\|_{1}\leq
k\alpha\\}$. For any $\bm{v}\in\mathbb{R}^{n}$, define the set
$U(\alpha,k,\bm{v})\subset\mathbb{R}^{n}$ by
$U(\alpha,k,\bm{v})=\\{\bm{u}\in\mathbb{R}^{n}:\text{supp}(\bm{u})\subseteq\text{supp}(\bm{v}),\|\bm{u}\|_{0}\leq
k,\|\bm{u}\|_{1}=\|\bm{v}\|_{1},\|\bm{u}\|_{\infty}\leq\alpha\\}$. Then
$\bm{v}\in T(\alpha,k)$ iff $\bm{v}$ is in the convex hull of
$U(\alpha,k,\bm{v})$. In particular, any $\bm{v}\in T(\alpha,k)$ can be
expressed as

$\displaystyle\bm{v}=\sum_{i=1}^{c}\gamma_{i}\bm{u}_{i}$

where $\bm{u}_{i}\in U(\alpha,k,\bm{v})$ and $0\leq\gamma_{i}\leq 1$,
$\sum_{i=1}^{c}\gamma_{i}=1$.

######  Lemma 2.

If the map $\mathcal{A}$ obeys the RIP of order $tk(t>1)$ with RIC
$\delta_{tk}\in(0,1)$, then for any matrix $H\in\mathbb{R}^{n_{1}\times
n_{2}}$ and $E\subset[n_{1}]$ with $|E|=k$, it holds that

$\displaystyle\|H_{E}\|_{F}\leq\beta_{1}\|\mathcal{A}(H)\|_{2}+\beta_{2}\frac{\|H_{E^{c}}\|_{*}}{\sqrt{k}},$
(4)

where

$\displaystyle\beta_{1}\triangleq\frac{2}{(1-\delta_{tk})\sqrt{1+\delta_{tk}}},\text{~{}and~{}}\beta_{2}\triangleq\frac{\delta_{tk}}{\sqrt{(1-(\delta_{tk})^{2})(t-1)}}.$

######  Lemma 3.

Assume that $X^{\sharp}$ is the solution of (3) and $H=X^{\sharp}-X$. If the
noisy measurements $\bm{b}=\mathcal{A}(X)+\bm{n}$ are observed with the noise
level $\|\bm{n}\|_{2}\leq\epsilon$, then for any subset $E\subset[n_{1}]$ with
$|E|=k$, we have

$\displaystyle\begin{split}\|\mathcal{A}(H)\|_{2}^{2}-2\epsilon\|\mathcal{A}(H)\|_{2}\leq&2\lambda(\|H_{E}\|_{*}-\|H_{E^{c}}\|_{*}+2\|X_{E^{c}}\|_{*})\end{split}$
(5)

and

$\displaystyle\|H_{E^{c}}\|_{*}\leq\|H_{E}\|_{*}+2\|X_{E^{c}}\|_{*}+\frac{\epsilon}{\lambda}\|\mathcal{A}(H)\|_{2}.$
(6)

##  3 Main results

With previous preparations in mind, we now present our main results.

######  Theorem 4.

For any observed vector $\bm{b}=\mathcal{A}(X)+\bm{n}$ with a bounded
constraint $\|\bm{n}\|_{2}\leq\lambda/2$, if the map $\mathcal{A}$ satisfies
RIP with

$\displaystyle\delta_{tk}<\sqrt{\frac{t-1}{t}}$ (7)

for certain fixed $t>1$, then we have

$\displaystyle\|\mathcal{A}(X^{\sharp}-X)\|_{2}$ $\displaystyle\leq
C_{1}\|X-X_{[k]}\|_{*}+C_{2},$ (8) $\displaystyle\|X^{\sharp}-X\|_{F}$
$\displaystyle\leq C_{3}\|X-X_{[k]}\|_{*}+C_{4},$ (9)

where $X^{\sharp}$ is the optimal solution of (3), and

$\displaystyle C_{1}$
$\displaystyle=\frac{2\lambda}{\sqrt{k}\beta_{1}\lambda+\epsilon},~{}~{}C_{2}=2\sqrt{k}\beta_{1}\lambda+2\epsilon,$
$\displaystyle C_{3}$
$\displaystyle=\frac{2\sqrt{k}\beta_{1}(2\sqrt{k}+1+\beta_{2})\lambda+2(\sqrt{k}\beta_{2}+2\beta_{2}+\sqrt{k})\epsilon}{k\beta_{1}(1-\beta_{2})\lambda},$
$\displaystyle C_{4}$
$\displaystyle=\frac{2(k+\sqrt{k})\beta_{1}\lambda+(\beta_{2}+2\sqrt{k}-\sqrt{k}\beta_{2})\epsilon}{\sqrt{k}(1-\beta_{2})\lambda(\sqrt{k}\beta_{1}\lambda+\epsilon)^{-1}}.$

######  Remark 1.

The condition (7) has been obtained recently by Cai and Zhang in [11] for
exact/robust signal recovery from (2), and it was proved to be sharp for the
exact rank-$k$ matrix recovery when $t>4/3$. To the best of our knowledge, we
first extend nontrivially this condition from the constrained problem (2) to
its unconstrained counterpart. When compared to some existing results, e.g,
[17], our upper bound estimate for $\|X^{\sharp}-X\|_{F}$ seems relatively
loose. However it can be further improved by using the skills in [17].

######  Remark 2.

BPDN is closely related to (3), and there are some recovery conditions for
this BPDN, see, e.g., [17, 18, 19]. However, most of these conditions are
unsatisfactory. In fact, by combing Lemma 2 (with setting $D$ be an identity
matrix) in [20] and also using the techniques in proof of our Theorem 4, one
will obtain a new and much weaker recovery condition for the BPDN. Besides,
our theoretical results can still be extended to deal with the noise under
Dantzig Selector settings for low-rank matrix recovery. Due to the limited
space (up to 10 printed pages), we omit them here.

######  Remark 3.

There are some special cases of Theorem 4 which can be used to cope with
several different LRMR tasks. For examples, one can set $\bm{n}=0$ and
$\epsilon=0$ for the noiseless recovery. In this case, the error will almost
disappear if one chooses the parameter $\lambda$ as small as possible, and
this result is also coincident with the results obtained in [17, 20]; one can
consider the rank-$k$ matrix recovery in presence of noise; similar with [5,
17, 20], one can also associate $\epsilon$ with $\lambda$, and set
$\epsilon=\lambda/2$. Due to the limited space, we also omit these potential
conclusions.

##  4 Proofs

###  4.1 Proof of Lemma 2

######  Proof 1.

The proof mainly follows from [20]. When $tk$ is not an integer, let
$t^{\prime}=\lceil tk\rceil/k$, then $t^{\prime}>t$ and $t^{\prime}k$ is an
integer. In view of this, we here only need to prove Lemma 2 when $tk$ is a
positive integer for a given $t>1$. To do so, we first denote the SVD of $H$
as

$\displaystyle
H=\sum_{i=1}^{n_{1}}\sigma_{i}(H)\cdot\bm{u}_{H}^{(i)}\cdot(\bm{v}_{H}^{(i)})^{T}.$

We also denote $\alpha=\|H_{E^{c}}\|_{*}/((t-1)k)$, and

$\displaystyle E_{1}=\\{i\in
E^{c}:\sigma_{i}(H)>\alpha\\},~{}~{}~{}E_{2}=\\{i\in
E^{c}:\sigma_{i}(H)\leq\alpha\\}.$

Then clearly $E_{1}\cup E_{2}=E^{c}$ and $E_{1}\cap E_{2}=\emptyset$. We will
begin with proving

$\displaystyle\|H_{E\cup
E_{1}}\|_{F}\leq\beta_{1}\|\mathcal{A}(H)\|_{2}+\frac{\beta_{2}}{\sqrt{k}}\|H_{E^{c}}\|_{*}$
(10)

Before this, we will show that $s\triangleq|E_{1}|<(t-1)k$. In fact it holds
naturally for $E_{1}=\emptyset$. When $E_{1}\neq\emptyset$, we know that

$\displaystyle\|\sigma_{E_{1}}(H)\|_{1}=\|H_{E_{1}}\|_{*}>s\alpha=s\frac{\|H_{E^{c}}\|_{*}}{(t-1)k}\geq\frac{s}{(t-1)k}\|H_{E_{1}}\|_{*}=\frac{s}{(t-1)k}\|\sigma_{E_{1}}(H)\|_{1}.$

Thus a quick simplification of the above inequality yields the desired result.

On the other hand, in terms of $\sigma_{E_{2}}(H)$, we have

$\displaystyle\|\sigma_{E_{2}}(H)\|_{1}=\|H_{E_{2}}\|_{*}=\|H_{E^{c}}\|_{*}-\|H_{E_{1}}\|_{*}\leq(t-1)k\alpha-s\alpha=((t-1)k-s)\alpha,$

and $\|\sigma_{E_{2}}(H)\|_{\infty}=\max_{i\in E_{2}}\sigma_{i}(H)\leq\alpha$.
Then using Lemma 1, we have

$\displaystyle\sigma_{E_{2}}(H)=\sum_{i=1}^{l}\gamma_{i}\bm{z}^{(i)},$

where $l$ is a certain positive integer, $\bm{z}^{(i)}\in
U(\alpha,(t-1)k-s,\sigma_{E_{2}}(H))$ and $0\leq\gamma_{i}\leq 1$,
$\sum_{i=1}^{l}\gamma_{i}=1$. By further defining

$\displaystyle\bm{b}^{(i)}$ $\displaystyle=(1+\delta_{tk})\sigma_{E\cup
E_{1}}(H)+\delta_{tk}\bm{z}^{(i)},~{}~{}~{}\bm{d}^{(i)}=(1-\delta_{tk})\sigma_{E\cup
E_{1}}(H)-\delta_{tk}\bm{z}^{(i)},$ $\displaystyle Z^{(i)}$
$\displaystyle=\sum_{j=1}^{n_{1}}(\bm{z}^{(i)})_{j}\cdot\bm{u}_{H}^{(j)}\cdot(\bm{v}_{H}^{(j)})^{T},~{}~{}~{}B^{(i)}=\sum_{j=1}^{n_{1}}(\bm{b}^{(i)})_{j}\cdot\bm{u}_{H}^{(j)}\cdot(\bm{v}_{H}^{(j)})^{T},$
$\displaystyle D^{(i)}$
$\displaystyle=\sum_{j=1}^{n_{1}}(\bm{d}^{(i)})_{j}\cdot\bm{u}_{H}^{(j)}\cdot(\bm{v}_{H}^{(j)})^{T},$

we can easily induce that both $\bm{b}^{(i)}$ and $\bm{d}^{(i)}$ are all
$tk$-sparse, and

$\displaystyle
H_{E_{2}}=\sum_{i=1}^{l}\gamma_{i}Z^{(i)},~{}~{}B^{(i)}=(1+\delta_{tk})H_{E\cup
E_{1}}+\delta_{tk}Z^{(i)},~{}~{}D^{(i)}=(1-\delta_{tk})H_{E\cup
E_{1}}-\delta_{tk}Z^{(i)}.$

Now applying Definition 1, we will estimate the upper and lower bounds of

$\displaystyle\rho\triangleq\sum_{i=1}^{l}\gamma_{i}\bigg{(}\|\mathcal{A}(B^{(i)})\|_{2}^{2}-\|\mathcal{A}(D^{(i)})\|_{2}^{2}\bigg{)}.$

As to the upper bound of $\rho$, we have

$\displaystyle\rho$ $\displaystyle=4\delta_{tk}\langle\mathcal{A}(H_{E\cup
E_{1}}),\mathcal{A}(H_{E\cup E_{1}}+\sum_{i=1}^{l}\gamma_{i}Z^{(i)})\rangle$
$\displaystyle=4\delta_{tk}\langle\mathcal{A}(H_{E\cup
E_{1}}),\mathcal{A}(H)\rangle\leq 4\delta_{tk}\|\mathcal{A}(H_{E\cup
E_{1}})\|_{2}\|\mathcal{A}(H)\|_{2}$ $\displaystyle\leq
4\delta_{tk}\sqrt{1+\delta_{tk}}\|H_{E\cup
E_{1}}\|_{F}\|\mathcal{A}(H)\|_{2}.$ (11)

As to the lower bound of $\rho$, we have

$\displaystyle\rho$
$\displaystyle\geq\sum_{i=1}^{l}\gamma_{i}\left((1-\delta_{tk})\|\bm{b}^{(i)}\|_{2}^{2}-(1+\delta_{tk})\|\bm{d}^{(i)}\|_{2}^{2}\right)$
$\displaystyle=2\delta_{tk}(1-(\delta_{tk})^{2})\|\sigma_{E\cup
E_{1}}(H)\|_{2}^{2}-2(\delta_{tk})^{3}\sum_{i=1}^{l}\gamma_{i}\|\bm{z}_{i}\|_{2}^{2}$
$\displaystyle\geq 2\delta_{tk}(1-(\delta_{tk})^{2})\|H_{E\cup
E_{1}}\|_{F}^{2}-\frac{2(\delta_{tk})^{3}}{(t-1)k}\|H_{E^{c}}\|_{*}^{2},$ (12)

where we used $\langle\sigma_{E\cup E_{1}}(H),\bm{z}^{(i)}\rangle=0$ for the
equation,and

$\displaystyle\|\bm{z}^{(i)}\|_{2}^{2}\leq\|\bm{z}^{(i)}\|_{0}\|\bm{z}^{(i)}\|_{\infty}^{2}\leq((t-1)k-s)\alpha^{2}=\frac{\|H_{E^{c}}\|_{*}^{2}}{(t-1)k}$

for the last inequality. Combing (1) and (1) yields

$\displaystyle(1-(\delta_{tk})^{2})\|H_{E\cup
E_{1}}\|_{F}^{2}-2\sqrt{1+\delta_{tk}}\|\mathcal{A}(H)\|_{2}\|H_{E\cup
E_{1}}\|_{F}-\frac{(\delta_{tk})^{2}}{(t-1)k}\|H_{E^{c}}\|_{*}^{2}\leq 0.$

Therefore,

$\displaystyle\|H_{E\cup E_{1}}\|_{F}\leq$
$\displaystyle\frac{2\sqrt{1+\delta_{tk}}\|\mathcal{A}(H)\|_{2}}{2(1-(\delta_{tk})^{2})}+\frac{\sqrt{(2\sqrt{1+\delta_{tk}}\|\mathcal{A}(H)\|_{2})^{2}+4(1-(\delta_{tk})^{2})\frac{(\delta_{tk})^{2}}{(t-1)k}\|H_{E^{c}}\|_{*}^{2}}}{2(1-(\delta_{tk})^{2})}$
$\displaystyle\leq$
$\displaystyle\frac{2(1-\delta_{tk})^{-1}}{\sqrt{1+\delta_{tk}}}\|\mathcal{A}(H)\|_{2}+\frac{\delta_{tk}}{\sqrt{(1-(\delta_{tk})^{2})(t-1)}}\frac{\|H_{E^{c}}\|_{*}}{\sqrt{k}},$

where we used $\sqrt{x^{2}+y^{2}}\leq|x|+|y|$ for the last inequality. Then
combing (10) and $\|H_{E}\|_{F}\leq\|H_{E\cup E_{1}}\|_{F}$ directly leads to
(4), which completes the proof.

###  4.2 Proof of Lemma 3

######  Proof 2.

Since $X^{\sharp}$ is the optimal solution of (3), we have

$\displaystyle\|X^{\sharp}\|_{*}+\frac{1}{2\lambda}\|\bm{b}-\mathcal{A}(X^{\sharp})\|_{2}^{2}\leq\|X\|_{*}+\frac{1}{2\lambda}\|\bm{b}-\mathcal{A}(X)\|_{2}^{2},$

which is equivalent to

$\displaystyle\|\mathcal{A}(H)\|_{2}^{2}-2\langle\bm{n},\mathcal{A}(H)\rangle\leq
2\lambda(\|X\|_{*}-\|X^{\sharp}\|_{*}).$ (13)

As to the left-hand side of (13), we have

$\displaystyle\|\mathcal{A}(H)\|_{2}^{2}-2\langle\bm{n},\mathcal{A}(H)\rangle\geq\|\mathcal{A}(H)\|_{2}^{2}-2\epsilon\|\mathcal{A}(H)\|_{2}.$
(14)

As to the right-hand side of (13), we know

$\displaystyle\|X^{\sharp}\|_{*}-\|X\|_{*}=$
$\displaystyle\sum_{i=1}^{n_{1}}\sigma_{i}(X+H)-(\|X_{E}\|_{*}+\|X_{E^{c}}\|_{*})$
$\displaystyle\geq$
$\displaystyle\sum_{i=1}^{n_{1}}|\sigma_{i}(X)-\sigma_{i}(-H)|-(\|X_{E}\|_{*}+\|X_{E^{c}}\|_{*})$
$\displaystyle\geq$ $\displaystyle\sum_{i\in
E}(\sigma_{i}(X)-\sigma_{i}(H))+\sum_{i\in
E^{c}}(\sigma_{i}(H)-\sigma_{i}(X))-(\|X_{E}\|_{*}+\|X_{E^{c}}\|_{*})$
$\displaystyle=$
$\displaystyle-\|H_{E}\|_{*}+\|H_{E^{c}}\|_{*}-2\|X_{E^{c}}\|_{*},$ (15)

where we used Theorem 1 in [21] for the first inequality. Then combing (13),
(14), and (2) leads to the desired result (5), and (6) follows trivially from
(5).

###  4.3 Proof of Theorem 4

######  Proof 3.

We start with Denoting $E=[k]$ and $H=X^{\sharp}-X$. Then by Lemma 2 and Lemma
3, we have

$\displaystyle\|\mathcal{A}(H)\|_{2}^{2}-2\epsilon\|\mathcal{A}(H)\|_{2}\leq$
$\displaystyle
2\lambda(\sqrt{k}\|H_{E}\|_{F}-\|H_{E^{c}}\|_{*}+2\|X_{E^{c}}\|_{*})$
$\displaystyle\leq$ $\displaystyle
2\sqrt{k}\lambda(\beta_{1}\|\mathcal{A}(H)\|_{2}+\frac{\beta_{2}}{\sqrt{k}}\|H_{E^{c}}\|_{*})-2\lambda\|H_{E^{c}}\|_{*}+4\lambda\|X_{E^{c}}\|_{*}$
$\displaystyle=$ $\displaystyle
2\sqrt{k}\beta_{1}\lambda\|\mathcal{A}(H)\|_{2}-2(1-\beta_{2})\lambda\|H_{E^{c}}\|_{*}+4\lambda\|X_{E^{c}}\|_{*}$
(16)

According to the condition (7), we know

$\displaystyle
1-\beta_{2}=1-\frac{\delta_{tk}}{\sqrt{(1-(\delta_{tk})^{2})(t-1)}}>1-\frac{\sqrt{(t-1)/t}}{\sqrt{(1-(t-1)/t)(t-1)}}=0.$

Therefore we can further know from (3) that

$\displaystyle\|\mathcal{A}(H)\|_{2}^{2}-2(\sqrt{k}\beta_{1}\lambda+\epsilon)\|\mathcal{A}(H)\|_{2}-4\lambda\|X_{E^{c}}\|_{*}\leq
0,$

which implies that

$\displaystyle\|\mathcal{A}(H)\|_{2}\leq$
$\displaystyle(\sqrt{k}\beta_{1}\lambda+\epsilon)+\sqrt{(\sqrt{k}\beta_{1}\lambda+\epsilon)^{2}+4\lambda\|X_{E^{c}}\|_{*}}$
$\displaystyle\leq$
$\displaystyle(\sqrt{k}\beta_{1}\lambda+\epsilon)+(\sqrt{k}\beta_{1}\lambda+\epsilon)+\frac{2\lambda\|X_{E^{c}}\|_{*}}{(\sqrt{k}\beta_{1}\lambda+\epsilon)}$
$\displaystyle\leq$
$\displaystyle\frac{2\lambda}{\sqrt{k}\beta_{1}\lambda+\epsilon}\|X_{E^{c}}\|_{*}+2\sqrt{k}\beta_{1}\lambda+2\epsilon.$

This completes (8). Based on (6) and (8), we now give a new upper bound
estimate for $\|H_{E^{c}}\|_{*}$, i.e.,

$\displaystyle\|H_{E^{c}}\|_{*}\leq$
$\displaystyle\sqrt{k}\|H_{E}\|_{F}+\frac{2(\sqrt{k}\beta_{1}\lambda+2\epsilon)}{\sqrt{k}\beta_{1}\lambda+\epsilon}\|X_{E^{c}}\|_{*}+\frac{2\epsilon}{\lambda}(\sqrt{k}\beta_{1}\lambda+\epsilon),$
(17)

where we used $\|H_{E}\|_{*}\leq\sqrt{k}\|H_{E}\|_{F}$.

On the other hand, using (4), (8), and (17), we can also give a new upper
bound estimate for $\|H_{E}\|_{F}$, i.e.,

$\displaystyle\|H_{E}\|_{F}\leq\beta_{2}\|H_{E}\|_{F}+\frac{2\sqrt{k}\beta_{1}(1+\beta_{2})\lambda+4\beta_{2}\epsilon}{k\beta_{1}\lambda+\sqrt{k}\epsilon}\|X_{E^{c}}\|_{*}+2(\beta_{1}+\frac{\beta_{2}\epsilon}{\sqrt{k}\lambda})(\sqrt{k}\beta_{1}\lambda+\epsilon),$

which is equivalent to

$\displaystyle\|H_{E}\|_{F}\leq$
$\displaystyle\frac{2\sqrt{k}\beta_{1}(1+\beta_{2})\lambda+4\beta_{2}\epsilon}{(1-\beta_{2})(k\beta_{1}\lambda+\sqrt{k}\epsilon)}\|X_{E^{c}}\|_{*}+\frac{2(\sqrt{k}\beta_{1}\lambda+\beta_{2}\epsilon)(\sqrt{k}\beta_{1}\lambda+\epsilon)}{\sqrt{k}(1-\beta_{2})\lambda}.$
(18)

Combining (17), (18), and $\|H_{E^{c}}\|_{F}\leq\|H_{E^{c}}\|_{*}$, we have

$\displaystyle\|H\|_{F}\leq$ $\displaystyle\|H_{E}\|_{F}+\|H_{E^{c}}\|_{F}$
$\displaystyle\leq$
$\displaystyle(\sqrt{k}+1)\|H_{E}\|_{F}+\frac{2(\sqrt{k}\beta_{1}\lambda+2\epsilon)}{\sqrt{k}\beta_{1}\lambda+\epsilon}\|X_{E^{c}}\|_{*}+\frac{2\epsilon}{\lambda}(\sqrt{k}\beta_{1}\lambda+\epsilon)$
$\displaystyle\leq$ $\displaystyle C_{3}\|X_{E^{c}}\|_{*}+C_{4},$

where $C_{3}$ and $C_{4}$ are defined in Theorem 4. This completes the proof.

##  5 Conclusion and future work

By using the powerful RIP tool, in this paper, we provided a series of RIP-
based sufficient conditions (related to the $\delta_{tk}$) for the
unconstrained RNNM to recover any (low-rank) matrices with the $\ell_{2}$-norm
bounded noise. This result first extended the sharp recovery conditions
obtained recently by Cai and Zhang [11] for constrained NNM to those for the
unconstrained RNNM. Some resulting conclusions were also fully discussed. One
of our future works will focus on deriving the new recovery conditions on the
$\delta_{tk}$ for $0<t\leq 1$. Besides, extending the current theoretical
results to more unconstrained convex/nonconvex models for vector/matrix/tensor
recovery will be another future work.

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# Long-range super-Planckian heat transfer between nanoemitters in a resonant
cavity

Kiryl Asheichyk  Department of Theoretical Physics and Astrophysics,
Belarusian State University, 5 Babruiskaya Street, 220006 Minsk, Belarus
Philippe Ben-Abdallah  Laboratoire Charles Fabry, UMR 8501, Institut
d’Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127
Palaiseau Cedex, France  Matthias Krüger  Institute for Theoretical Physics,
Georg-August-Universität Göttingen, 37073 Göttingen, Germany  Riccardo Messina
<EMAIL_ADDRESS>Laboratoire Charles Fabry, UMR 8501,
Institut d’Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin Fresnel,
91127 Palaiseau Cedex, France

###### Abstract

We study radiative heat transfer between two nanoemitters placed inside
different types of closed cavities by means of a fluctuational-electrodynamics
approach. We highlight a very sharp dependence of this transfer on cavity
width, and connect this to the matching between the material-induced resonance
and the resonant modes of the cavity. In resonant configurations, this allows
for an energy-flux amplification of several orders of magnitude with respect
to the one exchanged between two emitters in vacuum as well as between two
black-bodies, even at separation distances much larger than the thermal
wavelength. On the other hand, variations of the cavity width by a few percent
allow a reduction of the flux by several orders of magnitude and even a
transition to inhibition compared to the vacuum scenario. Our results pave the
way to the design of thermal waveguides for the long-distance transport of
super-Planckian heat flux and selective heat transfer in many-body system.

##  I Introduction

Two bodies kept at different temperatures and separated by vacuum exchange
heat radiatively via the transfer of thermal photons. While this energy flux
is limited at far separation distances by Stefan–Boltzmann’s law Planck10 ,
the pioneering works of Rytov Rytov53 , Polder and van Hove Polder71  showed
that this limit can be overcome in the so-called near field, i.e. when the
separation distance is small compared to the thermal wavelength (around 10
microns at ambient temperature). This amplification can be remarkable for
materials supporting resonant modes of the electromagnetic field Joulain05
(in particular polar materials supporting phonon-polaritonic resonances in the
infrared region of the spectrum) or a continuum of hyperbolic modes Biehs12
(in composite materials made with a combination of dielectric and metals).
Since the first theoretical investigations in the 1970s, several experiments
have verified the predicted near-field heat flux amplification (see Refs.
Song15, ; Bimonte17, ; Cuevas18, ; Biehs21,  and references therein).
Moreover, several applications have been put forward exploiting this strong
flux increase in the near field, ranging from thermal management Latella21a
to solid-state cooling Chen15 ; Zhu19 , heat-assisted data recording and
storage Srituravanich04 ; BenAbdallah19 ; Challener09 ; Stipe20 , infrared
sensing and spectroscopy De Wilde06 ; Jones12 , energy-conversion devices
DiMatteo01 ; Narayanaswamy03 ; Laroche06 ; Park08 ; Latella21b  and
thermotronics BenAbdallah13a ; BenAbdallah15 .

The possibility to transport this energy from a thermal source at distances
larger than its thermal wavelength remains today a challenging problem which
could find broad applications in the fields of thermal management and
information transfer Rustomji19 ; Rustomji21 ; Wei21  (i.e. Förster resonance
energy transfer). Although a first strategy to achieve this transport using
hyperbolic waveguides has been proposed in 2016 Messina16 , this problem
remains today largely unexplored.

During the last decade, a remarkable attention has been paid to the study of
heat transport between two closely separated emitters or within a larger set
of emitters, the main goal being to manipulate and tailor the radiative heat
exchanges at the nanoscale Messina13 ; BenAbdallah13b ; Saaskilahti14 ;
Biehs16 ; Dong18 ; Asheichyk18 ; Messina18 ; Deshmukh18 ; Tervo19 ; Zhang19a ;
Zhang19b ; Luo20 ; Ott20 ; Zhang20 ; Ott21 ; Zhang21 ; Chen22 ; Fang22 ;
Asheichyk22 ; Fang23 ; Zhang23 . Some of these studies have focused on the
transfer between emitters in the proximity of a substrate Biehs16 ; Dong18 ;
Messina18 ; Deshmukh18 ; Zhang19a ; Zhang19b ; Ott20 ; Zhang20 ; Ott21 ;
Fang22  or a cylindrical waveguide Asheichyk22 . These works discuss how the
two nanoemitters can couple through the surface resonance existing at the
interface between this external body and vacuum and how this additional
channel contributes to the transfer by amplifying or inhibiting it. Several
works have considered the impact of confinement on the heat transfer (HT)
between two nanoparticles (NP), by placing them inside a planar cavity,
revealing interesting effects Saaskilahti14 ; Asheichyk18 ; Zhang21 ; Chen22 ;
Zhang23 . However, they do not address at all the role played by the cavity
width in the exchanged flux and especially how the presence of the cavity can
make the heat transport strongly super-Planckian (far above the black-body
limit) at large separation distance between the emitters.

Investigating the role of confinement is the main objective of this work,
where we study the near-field radiative HT between two NP placed inside a
cavity (both planar and cylindrical) and highlight the crucial role played by
the cavity width. More specifically, we unveil the major interplay between
cavity resonant modes, whose frequencies are purely governed by the cavity
width and material, and resonances existing at the interface between each NP
and vacuum, giving the leading contribution to near-field radiative HT in
vacuum. We show that this interplay is at the origin of a very sharp
dependence of HT on cavity width, allowing for order-of-magnitude variations
of HT, as well as transitions from amplification to inhibition with respect to
transfer in vacuum, with variations of the cavity width in the micron range.

##  II Nanoparticles inside planar and cylindrical cavities

In order to highlight the effect of confinement and cavity resonances on
radiative HT we consider in the following two NP placed at distance $d$, and
address several different surroundings, taking as a reference the
configuration in vacuum. We start with NP in proximity of a planar surface. In
this configuration, as shown in Messina18 , HT can be both amplified and
inhibited, depending on the distance between NP and on their distance from the
substrate. We then consider two confined regimes, in which the NP are placed
inside either a planar or a cylindrical cavity. The former is geometrically
defined by its width $w$ (the distance between the two plates), the latter by
its radius $R$, corresponding to a cylindrical-cavity width $w=2R$. For
simplicity, the NP are always placed in the middle of the cavity (at equal
distance from the two plates forming the planar cavity and along the axis of
the cylindrical one). For a fair comparison, in the single-plate case the
distance between NP and interface is chosen to be $w/2$. The four scenarios
are depicted in the inset of Fig. 1.

Figure 1: Ratio of heat flux $H$ in the presence of boundary conditions and
the one $H_{0}$ in vacuum, for two NP at distance $d=10\,\mu$m. The vertical
red dashed lines correspond to the planar-cavity resonances
$w=(2n+1)c\pi/\omega_{\mathrm{spp}}$ ($n=0,1,\dots$), whereas the vertical
blue and magenta dashed lines correspond to the cylindrical-cavity resonances
$w=2x^{\prime}_{1m}c/\omega_{\mathrm{spp}}$ and
$w=2x_{0m}c/\omega_{\mathrm{spp}}$ ($m=1,2,\dots$, see text for more details),
respectively. The orange dot-dashed line represents the black-body limit for
two NP having a radius of 100 nm.

The HT between the two NP in an arbitrary geometry, within the dipolar
approximation, can be expressed as follows Messina13 ; Asheichyk17

$\begin{split}H=\frac{32\pi\hbar}{c^{4}}\int_{0}^{\infty}d\omega\,&\omega^{5}[\Theta(\omega,T_{1})-\Theta(\omega,T_{2})]\\\
&\,\times\mathrm{Im}(\alpha_{1})\mathrm{Im}(\alpha_{2})\mathrm{Tr}\left(\mathbb{G}\mathbb{G}^{\dagger}\right),\end{split}$
(1)

where $c$ is the speed of light, $\hbar$ and $k_{\textrm{B}}$ are Planck’s and
Boltzmann’s constants, respectively, and
$\Theta(\omega,T)=[\exp(\hbar\omega/k_{\textrm{B}}T)-1]^{-1}$, $T_{1}=400\,$K
and $T_{2}=300\,$K being the temperatures of the two NP. The temperature of
the cavity walls is fixed at 300 K, so that the colder NP exchanges heat only
with the other NP. This expression contains the imaginary part of the
polarizability of two NP ($i=1,2$), for which we take the Clausius–Mossotti
expression
$\alpha_{i}(\omega)=R_{i}^{3}[\varepsilon_{i}(\omega)-1]/[\varepsilon_{i}(\omega)+2]$,
where $R_{i}$ and $\varepsilon_{i}(\omega)$ are the radius and the frequency-
dependent permittivity, respectively. Assuming that the two NP are made of
silicon carbide (SiC), the latter is well described by the following
Drude–Lorentz model Spitzer59

$\varepsilon_{\textrm{SiC}}(\omega)=\varepsilon_{\infty}\frac{\omega^{2}-\omega_{\textrm{LO}}^{2}+i\omega\gamma}{\omega^{2}-\omega_{\textrm{TO}}^{2}+i\omega\gamma},$
(2)

with $\varepsilon_{\infty}=6.7$, $\omega_{\textrm{LO}}=1.82\times
10^{14}\,\textrm{rad}\,\textrm{s}^{-1}$, $\omega_{\textrm{TO}}=1.48\times
10^{14}\,\textrm{rad}\,\textrm{s}^{-1}$, $\gamma=8.93\times
10^{11}\,\textrm{rad}\,\textrm{s}^{-1}$. Equation (1) also contains the trace
of the product of the dyadic Green’s function $\mathbb{G}$, calculated at the
NP coordinates, by its conjugate transpose $\mathbb{G}^{\dagger}$. The Green’s
function $\mathbb{G}=\mathbb{G}_{0}+\mathbb{G}_{T}$ is written as the sum of
the vacuum Green’s function (known analytically Asheichyk17 ) and the
scattering part $\mathbb{G}_{T}$, describing the effect of the boundary
condition. In the following, we assume that both the planar and cylindrical
boundaries are made of gold, described by the Drude model LambrechtEPJD00
$\varepsilon_{\rm
Au}(\omega)=1-\omega_{p}^{2}/[\omega(\omega+i\omega_{\tau})]$, with
$\omega_{p}=9\,{\rm eV}$ and $\omega_{\tau}=35\,{\rm meV}$.

While the expressions of $\mathbb{G}_{T}$ for a single plate and for a planar
cavity are given in Refs. Messina18,  and Asheichyk18, , respectively, we
present here its expression inside a cylindrical cavity of radius $R$.
Starting from the general expression of the Green’s function given in Ref.
GolykPRE2012,  and recalling our assumption of NP placed along the cylinder
axis, $\mathbb{G}_{T}$ proves to be diagonal with elements

$\begin{split}G_{T11}=G_{T22}&=\frac{i}{8\pi}\int_{0}^{\infty}dk_{z}\Bigl{[}T_{1,k_{z}}^{MM}+2\frac{k_{z}}{k}T_{1,k_{z}}^{MN}\\\
\,&\qquad\qquad+\frac{k_{z}^{2}}{k^{2}}T_{1,k_{z}}^{NN}\Bigr{]}\cos(k_{z}d),\\\
G_{T33}&=\frac{i}{4\pi}\int_{0}^{\infty}dk_{z}\frac{q^{2}}{k^{2}}T_{0,k_{z}}^{NN}\cos(k_{z}d),\end{split}$
(3)

expressed in terms of the scattering coefficients for $N$ (electric) and $M$
(magnetic) polarization modes of the field (for a non-magnetic medium). It is
easy to show that these coefficients are the same as those outside a cylinder
GolykPRE2012  of radius $R$, but with Bessel functions replaced by Hankel
functions, and vice versa, as well as with an additional minus sign for
$T_{n,k_{z}}^{MN}$ (stemming from equality
$J_{n}^{\prime}(qR)H_{n}(qR)-J_{n}(qR)H_{n}^{\prime}(qR)=-2i/\pi qR$), and are
given by

$\begin{split}T_{n,k_{z}}^{MM}&=-\frac{H_{n}(qR)}{J_{n}(qR)}\frac{\Delta_{1}\Delta_{4}-K^{2}}{\Delta_{1}\Delta_{2}-K^{2}},\\\
T_{n,k_{z}}^{NN}&=-\frac{H_{n}(qR)}{J_{n}(qR)}\frac{\Delta_{2}\Delta_{3}-K^{2}}{\Delta_{1}\Delta_{2}-K^{2}},\\\
T_{n,k_{z}}^{MN}&=T_{n,k_{z}}^{NM}=-\frac{2i}{\pi\sqrt{\varepsilon}\left[qRJ_{n}(qR)\right]^{2}}\frac{K}{\Delta_{1}\Delta_{2}-K^{2}}.\end{split}$
(4)

Here, $J_{n}$ ($H_{n}$) is the Bessel (Hankel) function of order $n$,

$\begin{split}&\Delta_{1}=\frac{H^{\prime}_{n}(q_{\varepsilon}R)}{q_{\varepsilon}RH_{n}(q_{\varepsilon}R)}-\frac{1}{\varepsilon}\frac{J^{\prime}_{n}(qR)}{qRJ_{n}(qR)},\\\
&\Delta_{2}=\frac{H^{\prime}_{n}(q_{\varepsilon}R)}{q_{\varepsilon}RH_{n}(q_{\varepsilon}R)}-\frac{J^{\prime}_{n}(qR)}{qRJ_{n}(qR)},\\\
&\Delta_{3}=\frac{H^{\prime}_{n}(q_{\varepsilon}R)}{q_{\varepsilon}RH_{n}(q_{\varepsilon}R)}-\frac{1}{\varepsilon}\frac{H^{\prime}_{n}(qR)}{qRH_{n}(qR)},\\\
&\Delta_{4}=\frac{H^{\prime}_{n}(q_{\varepsilon}R)}{q_{\varepsilon}RH_{n}(q_{\varepsilon}R)}-\frac{H^{\prime}_{n}(qR)}{qRH_{n}(qR)},\end{split}$
(5)

and

$K=\frac{nk_{z}}{\sqrt{\varepsilon}kR^{2}}\left(\frac{1}{q_{\varepsilon}^{2}}-\frac{1}{q^{2}}\right),$
(6)

where $k=\omega/c$, $q=\sqrt{k^{2}-k_{z}^{2}}$, and
$q_{\varepsilon}=\sqrt{\varepsilon k^{2}-k_{z}^{2}}$ the wave vector
perpendicular to the axis inside the cavity walls. Note that the integrand of
$G_{T11}$ ($G_{T22}$) in Eq. (3) has a pole at $q=0$ (i.e. $k_{z}=k$), which
is integrable in principal value Asheichyk22 , whereas the integrand of
$G_{T33}$ has no poles. It can be shown that the scattering matrices, and
hence the integrands of $G_{Tii}$ ($i=1,2,3$), have no poles at zeros of
$J_{n}(qR)$ [the Bessel functions in denominators of $T_{n,k_{z}}^{MM}$,
$T_{n,k_{z}}^{NN}$ and $T_{n,k_{z}}^{MN}$ in Eq. (4) cancel with those in
$\Delta_{j}$, with $j=1,2,3,4$].

##  III Cavity-induced amplification of heat transfer

We present in Fig. 1 the heat flux $H$ exchanged by the two SiC NP placed at
distance $d=10\,\mu$m in the four scenarios described above (vacuum, single
plate, planar cavity and cylindrical cavity) as a function of the cavity width
$w$. In order to address directly the amplification factor, $H$ is divided by
the heat flux $H_{0}$ exchanged in vacuum. We start by noticing that the three
non-vacuum configurations result in an amplified energy flux with respect to
vacuum. Nevertheless, two crucial features differentiate the single-plate
configuration with respect to the two cavity (confined) ones. First, the
amplification factor for a single-plate (less than 2) is strikingly over-
performed by both cavities, with a flux enhancement going up to four orders of
magnitude. Moreover, while the $w$ dependence in the case of a single plate is
rather flat, the cavity results clearly have a resonant behavior as a function
of the cavity width, suggesting the existence of a geometry-induced resonance.
We notice that the curve associated with a planar cavity has a steep behavior
when the width tends to very small values. However, for such small cavity
widths the validity of the dipolar approximation starts to be questioned, at
least for reasonable values of the NP radius. Furthermore, the design and
fabrication of such cavities is rather unpractical, and, as this amplification
effect when $w$ tends to zero is not connected to the resonant behavior of
interest, we do not analyze it in detail. We also observe that, while the
vacuum result is below the black-body limit (not surprisingly at
$d=10\,\mu$m), shown in Fig. 1 for two NP having radii $R_{1}=R_{2}=100\,$nm,
the cavity scenarios allow to produce a super-Planckian flux, with an
amplification factor going up to more than two orders of magnitude for the
cylindrical cavity.

Figure 2: Spectral flux $H(\omega)$ for two NP, normalized by their volumes
$V_{i}$, at distance $d=10\,\mu$m inside a cylindrical cavity of radius
$3.13\,\mu$m (magenta) and $5.4\,\mu$m (black). The vertical blue and red
lines correspond to the $R$-dependent resonances $\omega=x^{\prime}_{1m}c/R$
and $\omega=x_{0m}c/R$ ($m=1,2,\dots$, see text for more details),
respectively.

To get more insight into this resonant behavior, we start by recalling that
the HT between two NP in vacuum is spectrally dominated by the surface phonon-
polariton supported by each NP. This resonance occurs at the frequency
$\omega_{\mathrm{spp}}\simeq 1.75\times
10^{14}\,\mathrm{rad}\,\mathrm{s}^{-1}$ such that
$\varepsilon_{\textrm{SiC}}(\omega_{\mathrm{spp}})\simeq-2$. We now focus on
the cavities, and discuss their resonant frequencies. The simplest scenario in
which these frequencies can be identified is the one when their walls are
perfectly conducting. In this configuration, it is well known that imposing
the boundary conditions results in an eigenvalue problem giving direct access
to the cutoff frequencies Greiner98 . In the simplest case of a planar cavity
occupying the region $[-w/2,w/2]$, the $z$ dependence of the field modes has
the form $\cos(k_{z}z)$, yielding the cutoff frequencies
$\Omega_{\mathrm{pl},n}=nc\pi/w$ for $n=1,2,\dots$. Based on this analysis we
may expect a matching condition of
$\Omega_{\mathrm{pl},n}=\omega_{\mathrm{spp}}$ between cavity-induced
resonance and the phonon-polariton NP resonance. Nevertheless, we should
notice that for even values of $n$, the field vanishes in the center of the
cavity, making these modes unable to carry energy between the two NP. We thus
expect that the thicknesses $w$ resulting in a resonant amplification have the
form $w=(2n+1)c\pi/\omega_{\mathrm{spp}}$ for $n=0,1,\dots$. This is indeed
confirmed by the vertical red dashed lines in Fig. 1. The reasoning for a
cylindrical cavity is analogous, and starts from the observation [see Eq.
(3)], that only $M$ modes with $n=1$ and $N$ modes with $n=0,1$ can in
principle contribute to the flux amplification. As discussed in Greiner98 ,
$N$ ($M$) modes of order $n$ have cutoff frequencies of the form $x_{nm}c/R$
($x^{\prime}_{nm}c/R$), $x_{nm}$ ($x^{\prime}_{nm}$) being the $m$-th zero of
$J_{n}(x)$ [$J^{\prime}_{n}(x)$]. One should note that for a $N$ mode the $z$
component of the electric field is proportional to $J_{n}(x)$, and thus for
$n=1$, being $J_{1}(0)=0$, the $z$ component of both electric and the magnetic
fields vanish in the cavity axis. For this reason, only two sets of modes ($N$
with $n=0$ and $M$ with $n=1$) are expected to participate, giving the
resonant conditions $R=x_{0m}c/\omega_{\mathrm{spp}}$ and
$R=x^{\prime}_{1m}c/\omega_{\mathrm{spp}}$. These resonant radii are
represented by the vertical blue and magenta dashed lines in Fig. 1 and are in
very good agreement with the resonances of the NP heat flux. We remark that
the smallest resonant radius for a cylindrical cavity is given by
$R=x^{\prime}_{11}c/\omega_{\mathrm{spp}}\simeq 3.13\,\mu$m.

While these resonances explain most of Fig. 1, there is a difference between
planar and cylindrical cavities. The curve associated with a planar cavity
shows amplification when the width tends to very small values, a behavior
whose investigation we leave for future work. In striking contrast, the
cylindrical cavity displays pronounced inhibition for small $w$: $H$ changes
by six orders of magnitude when changing $w$ by as little as two microns. We
attribute this difference to different shapes of the cavities and to different
degrees of confinement (two directions are allowed for a planar cavity,
whereas one direction is allowed for a cylindrical one). This shows that such
cavities modes lead to strongly selective transfer.

A complementary view can be obtained by looking at the spectral heat flux
$H(\omega)$ (of which the flux $H$ is the integral over frequency) for
different cavity widths. The result is shown in Fig. 2, where the spectral
flux associated with the first resonant radius $R=3.13\,\mu$m is compared to
the one for a non-resonant radius $R=5.4\,\mu$m. We observe that both spectra
peak around $\omega_{\mathrm{spp}}$. This is not surprising since this
resonance stems from purely material properties of NP and represents the
surface resonant mode existing at the interface between each NP and vacuum.
The difference lies, on the contrary, in the additional cavity-induced
resonances. Also in this case, their positions are very well predicted by the
expressions $x_{0m}c/R$ and $x^{\prime}_{1m}c/R$, which depend on the radius.
This curve confirms that the strong flux amplification we obtain at
$R=3.13\,\mu$m results from the matching between the first cavity-induced
resonance (independent of the NP), and the one at $\omega_{\mathrm{spp}}$
associated with NP material properties.

As final aspect, in Fig. 3 we address the dependence of the highlighted flux
amplification mechanism on the distance $d$ between the two NP. Naturally, in
the graphs, we choose the optimal values of $w$ for each case, i.e,
$w=5.39\,\mu$m for the planar cavity (the first resonance in Fig. 1) and
$R=w/2=3.13\,\mu$m for the cylinder. For fair comparison, the associated
single-plate scenario is studied at a distance $z=2.7\,\mu$m.

Figure 3: Ratio of heat flux $H$ in the presence of boundary conditions and
the one $H_{0}$ in vacuum as a function of the NP distance $d$ in three
different scenarios (see legend). The orange dot-dashed line represents the
$d$-dependent black-body limit for two NP having a radius of 100 nm, shown
above $d=1\,\mu$m in order to be coherent with the dipolar approximation.

We first notice that the three curves share a common qualitative behavior,
with a flux amplification starting from 1 (no amplification), increasing up to
a maximum value and then decreasing and switching to a flux inhibition. The
first aspect, namely the absence of amplification for small distances $d$ is
expected, since in this regime, namely when $d$ is much smaller than the
cavity width, the NP are strongly coupled to each other, making the presence
of a boundary condition irrelevant. On the contrary, moving to larger
distances makes the amplification effect manifest, and we stress that the
maximum value strongly depends on the geometry under scrutiny and reaches a
remarkable value of more than 4 orders of magnitude ($\simeq 5\times 10^{4}$)
in the case of a cylindrical cavity, reached around $d\simeq 87\,\mu$m. An
interesting way of restating this result is that the cylindrical cavity is
able to export to $87\,\mu$m, a distance clearly in the far-field regime, the
same flux two NP would exchange in vacuum at a distance of $1.2\,\mu$m, in the
near-field regime. We also notice that, while the vacuum value is below the
black-body limit for distances $d\geq 2\,\mu$m, the cavities allow for a
super-Planckian flux with an amplification factor going beyond 3 orders of
magnitude, which stays above the black-body limit up to distances of some mm.
Finally, we remark that while the three geometries imply rather different
maximum flux-amplification factors, the decay length of the effect they induce
(the distance at which amplification turns into inhibition) also strongly
depends on the geometry. Interestingly, even if for the three geometries we
observe amplification up to relatively high values of the distance, this decay
length actually goes in the opposite order with respect to the maximum
amplification factor. In order to understand this behavior we can recall the
image of heat flux between the two NP conveyed at larger distance by the
surface mode existing at the cavity walls. As a consequence, we expect the
decay length of amplification observed in Fig. 3 to be compatible with the one
of these surface modes at the flux resonance $\omega_{\mathrm{spp}}$. This can
be checked by seeking for the resonance (pole) of the reflection coefficient
in each geometry at $\omega=\omega_{\mathrm{spp}}$: this gives us a parallel
wavevector $k$ such that $\delta=1/\mathrm{Im}(k)$ represents an estimate of
the decay length of the surface mode along the interface. This gives
$\delta\simeq 7\times 10^{4}\,\mu$m for a single plate, $\delta\simeq 5\times
10^{3}\,\mu$m for a planar cavity and $\delta\simeq 35\,\mu$m for a
cylindrical one. These result confirm the trend observed in Fig. 3 and give a
quantitative estimate of the distance at which the flux amplification reaches
its maximum.

##  IV Conclusions

In conclusion, we have shown that confinement of two nanoemitters inside a
closed cavity can induce a strong amplification or inhibition of the radiative
heat flux they exchange. By analyzing two different closed geometries (a
planar and a cylindrical cavity) and comparing the results to exchanges in
vacuum and the simpler case of a planar interface, we have shown that this
amplification stems from the matching between the spectral resonance
associated with material properties (depending only on the nature of materials
and on the shape of emitter) and cavity-induced resonances (depending only on
the cavity geometry), and which may be used to selectively tune NP transfer.
We have also shown that the characteristic distance at which the effect takes
place depends on the geometry, providing a simple tool to choose the most
convenient geometry depending on the desired distance. Our results pave the
way to easier strategies to export and tune near-field super-Planckian
radiative heat flux to distances larger than the thermal wavelength. In future
work, they could be extended to other more complex geometries (curved cavity,
splitter…) and the impact on many-body heat transport inside a cavity could
also be addressed Zhang21 ; Chen22 , also addressing the role of selectivity.
Finally, the role played by multipolar orders in the heat transport between
nanoemitters in strongly confined systems could also be investigated.

###### Acknowledgements.

P. B.-A. and R. M. acknowledge financial support from Labex Nanosaclay,
ANR-10-LABX-0035 (Flagship project MaCaCQu). K.A. is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the Walter
Benjamin fellowship (Project No. 453458207).

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