# Generalized binomial edge ideals of complete $r$-partite graphs

Yi-Huang Shen  CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of
Mathematical Sciences, University of Science and Technology of China, Hefei,
Anhui, 230026, P.R. China<EMAIL_ADDRESS>and  Guangjun Zhu∗ School of
Mathematical Sciences, Soochow University, Suzhou, Jiangsu, 215006, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

This paper analyzes the cohomological dimension of the generalized binomial
edge ideal $\mathcal{J}_{K_{m},G}$ for a complete $r$-partite graph $G$.
Additionally, the Krull dimension, the depth, the Castelnuovo–Mumford
regularity, the Hilbert series, and the multiplicity of its quotient ring are
explicitly determined.

∗ Corresponding author

2020 Mathematics Subject Classification. Primary 13C15, 13C70; Secondary
05E40, 13F65

Keywords: Regularity, depth, cohomological dimension, generalized binomial
edge ideal, complete $r$-partite graph

##  1\. Introduction

When $n$ is a positive integer, $[n]$ represents the set
$\\{1,2,\dots,n-1,n\\}$ according to convention. Suppose that $m$ and $n$ are
positive integers. Let
$S={\mathbb{K}}[\bm{X}]\coloneqq{\mathbb{K}}[x_{ij}:(i,j)\in[m]\times[n]]$ be
the polynomial ring in $mn$ variables over a field ${\mathbb{K}}$. In their
paper [MR3290687], Ene et al. introduced the binomial edge ideal of a pair of
graphs. Specifically, let $G_{1}$ and $G_{2}$ be two simple graphs on the
vertex sets $[m]$ and $[n]$ with the edge sets $E(G_{1})$ and $E(G_{2})$,
respectively. Given $e=\\{i,j\\}\in E(G_{1})$ and $f=\\{k,l\\}\in E(G_{2})$,
where $i<j$ and $k<l$, we can assign a $2$-minor
$p_{(e,f)}=[i,j\,|\,k,l]\coloneqq x_{ik}x_{jl}-x_{il}x_{jk}$ to the pair
$(e,f)$. The _binomial edge ideal of the pair $(G_{1},G_{2})$_ is defined as
the ideal

$\mathcal{J}_{G_{1},G_{2}}\coloneqq(p_{(e,f)}:e\in E(G_{1}),f\in E(G_{2}))$

in $S$.

Let $G$ be a finite simple graph with the vertex set $[n]$. The _generalized
binomial edge ideal_ of $G$ is the binomial edge ideal of the pair
$(K_{m},G)$, where $K_{m}$ is a complete graph with $m$ vertices. Generalized
binomial edge ideals have numerous applications in algebraic statistics for
studying conditional independence, as noted in [MR3011436]. These ideals are a
generalization of the classical binomial edge ideals studied in [MR2669070,
MR2782571]. It is not surprising that certain properties of the classical
binomial edge ideal $J_{G}$ can be naturally extended to
$\mathcal{J}_{K_{m},G}$. For instance, $\mathcal{J}_{K_{m},G}$ is a radical
ideal, and the subsets of $G$ with the cut-point property determine the
minimal primes of $\mathcal{J}_{K_{m},G}$, as proven in [MR3011436].

However, for $m\geq 3$, the generalized binomial edge ideal
$\mathcal{J}_{K_{m},G}$ cannot be generated by maximal minors of the matrix
$\bm{X}$. This gives rise to new phenomena arise. For example, Chaudhry and
Irfan in [MR4233116] proved that, for a block graph $G$, the ideal
$\mathcal{J}_{K_{m},G}$ is Cohen–Macaulay if and only if
$\mathcal{J}_{K_{m},G}$ is unmixed, if and only if $G$ is a complete graph.
Kumar [MR4033090] demonstrated that the upper bound of the regularity
$\operatorname{reg}(S/\mathcal{J}_{K_{m},G})$ is $n-1$, which is achieved if
$m\geq n$.

Building on their work, we explored the algebraic properties of generalized
binomial edge ideals for specific graphs. In [arXiv:2305.05365], we provided
precise formulas for the Krull dimension, depth and regularity of
$S/\mathcal{J}_{K_{m},G}$, where $G$ is either a bipartite graph $F_{p}$ or a
$k$-fan graph $F_{k}^{W}(K_{n})$. These graphs serve as the foundation for
constructing Cohen–Macaulay binomial edge ideals of bipartite graphs. In a
recent study [arXiv:2310.20235], we analyzed the regularity and depth of the
powers of the generalized binomial edge ideal $\mathcal{J}_{K_{m},P_{n}}$ of a
path graph $P_{n}$ using the Sagbi basis theory. Furthermore, we showed that
the symbolic powers and ordinary powers of $\mathcal{J}_{K_{m},P_{n}}$ are
identical.

This paper focuses on the generalized binomial edge ideals of complete
$r$-partite graphs. In their work [MR3195706], Schenzel and Zafar computed the
Krull dimension, depth, regularity, Hilbert function and multiplicity of the
binomial edge ideal of a complete bipartite graph. Wang and Tang also studied
the regularity and depth of the powers of the binomial edge ideal of a
complete bipartite graph in their papers [MR4563443, MR4632858]. Meanwhile,
Ohtani [MR3169495] demonstrated that the symbolic powers and ordinary powers
of $J_{G}$ coincide when $G$ is a complete $r$-partite graph. However, there
is limited understanding of the generalized binomial edge ideal of a complete
$r$-partite graph.

The cohomological dimension of binomial edge ideals is also an intriguing
topic. In their work [MR1082012], Bruns and Schwänzl computed the
cohomological dimension of $\mathcal{J}_{K_{m},K_{n}}$. Katsabekis
[arXiv:2207.02256] provided a lower bound on the cohomological dimension of
$\mathcal{J}_{K_{m},G}$ for any non-complete graph $G$, based on its vertex
connectivity. Particularly, he obtained interesting results for path graphs
and star graphs. What can be said about the cohomological dimension of the
generalized binomial edge ideal of complete $r$-partite graphs?

The article is organized as follows. Section 2 presents essential definitions
and terminology that will be necessary later. In Section 3, we begin by
analyzing the primary decomposition of $\mathcal{J}_{K_{m},G}$, where $G$ is a
complete $r$-partite graph. This enables us to quickly obtain the Krull
dimension and the depth of $S/\mathcal{J}_{K_{m},G}$. As a corollary, we
generalize a recent result of Katsabekis on the cohomological dimension. We
then compute the Castelnuovo–Mumford regularity, Hilbert series, and
multiplicity of $S/\mathcal{J}_{K_{m},G}$. It is important to note that
Schenzel and Zafar computed the regularity of binomial edge ideals of complete
bipartite graphs as a by-product of a long and meticulous analysis of the
modules of deficiencies. In contrast, we present here an exact formula for the
regularity of the generalized binomial edge ideal of a complete $r$-partite
graph, using the comparison-by-induced-subgraph strategy, which significantly
shortens the proof. Finally, we conclude this paper by demonstrating that the
binomial edge ideal of a complete $r$-partite graph is of König type,
following the definition of Herzog et al. in [MR4358668]. This generalizes a
recent result of Williams in [arXiv:2310.14410] for complete bipartite graphs.

##  2\. Preliminaries

This section provides the necessary definitions and basic facts for the paper.
For further details, please refer to [MR1613627, MR1251956, MR3838370].

###  2.1. Notions of simple graphs

To begin, let us review some fundamental concepts regarding graphs. Let $G$ be
a simple graph with the vertex set $V(G)$ and the edge set $E(G)$. For any
subset $A$ of $V(G)$, the _induced subgraph_ of $G$ on the vertex set $A$,
denoted by $G|_{A}$, satisfies that $V(G|_{A})=A$ and for any $i,j\in A$,
$\\{i,j\\}\in E(G|_{A})$ if and only if $\\{i,j\\}\in E(G)$. At the same time,
the induced subgraph of $G$ on the set $V(G)\setminus A$ will be denoted by
$G\setminus A$. In particular, if $A=\\{v\\}$ then we will write $G\setminus
v$ instead of $G\setminus\\{v\\}$ for simplicity.

Given positive integers $n_{1}$ and $n_{2}$, where $n_{1}\leq n_{2}$, the set
$[n_{1},n_{2}]$ is defined as $\\{n_{1},n_{1}+1,\dots,n_{2}-1,n_{2}\\}$. If
$n_{1}=1$, we will use the simplified notation $[n_{2}]$.

Let $r$ be an integer greater than or equal to $2$. A simple graph $G$ is _$r$
-partite_ if its vertex set $V(G)$ can be partitioned into $r$ pairwise
disjoint subsets, such that no two vertices in the same subset are adjacent in
$G$. A _complete $r$-partite_ graph with $\sum_{i=1}^{r}n_{i}$ vertices,
denoted by $K_{n_{1},n_{2},\ldots,n_{r}}$, is a graph with the vertex set
$V(K_{n_{1},n_{2},\ldots,n_{r}})=[n_{1}]\sqcup[n_{1}+1,n_{1}+n_{2}]\sqcup\cdots\sqcup[1+\sum_{i=1}^{r-1}n_{i},\sum_{i=1}^{r}n_{i}]$
and the edge set $E(K_{n_{1},n_{2},\ldots,n_{r}})=\\{\\{u,v\\}:u\in V_{i},v\in
V_{j}\text{\ for different\ }i,j\\}$. A graph with $r=2$ is called a _complete
bipartite_ graph. In contrast, a graph with $n_{1}=n_{2}=\cdots=n_{r}=1$ is
known as a _complete graph_ , conventionally denoted by $K_{r}$.

A _walk_ $W$ of length $n-1$ in a graph $G$ is a sequence of vertices $w_{1}$
through $w_{n}$, where each consecutive pair of vertices $\\{w_{i},w_{i+1}\\}$
is connected by an edge in $G$. A _path_ is a walk where all vertices are
distinct. To simplify notation, a path of length $n-1$ is denoted by $P_{n}$.

###  2.2. Primary decompositions

Rauh [MR3011436] has already studied the primary decomposition of generalized
binomial edge ideals. Specifically, let $m,n\geq 2$ be two positive integers
and $G$ be a simple graph with the vertex set $[n]$. For each $T\subseteq[n]$,
let $\overline{T}=V(G)\setminus T$, and $G_{1},\ldots,G_{c(T)}$ be the
connected components of $G|_{\overline{T}}$. In addition, let
$\widetilde{G_{i}}$ be the complete graph on the set $V(G_{i})$ for
$i=1,\dots,c(T)$. Then, we have the ideal

$P_{T}(K_{m},G)\coloneqq(x_{ij}:(i,j)\in[m]\times
T)+\mathcal{J}_{K_{m},\widetilde{G}_{1}}+\cdots+\mathcal{J}_{K_{m},\widetilde{G}_{c(T)}}$

in the polynomial ring $S={\mathbb{K}}[x_{ij}:(i,j)\in[m]\times[n]]$. Note
that $P_{T}(K_{m},G)$ is a prime ideal containing $\mathcal{J}_{K_{m},G}$.

To describe the minimal primes of $\mathcal{J}_{K_{m},G}$, we need to impose
restrictions on $T$. We define a _cut point_ of $G$ as a vertex $v\in V(G)$
such that $c(G)<c(G\setminus v)$, where $c(G)$ denotes the number of connected
components of $G$. Let $T$ be a subset of $V(G)$, and let $c(T)$ denote the
number of connected components of $G\setminus T$. If $v$ is a cut point of the
induced subgraph $G\setminus(T\setminus\\{v\\})$ for any $v\in T$, then we say
that $T$ has the _cut point property_. We define $\mathcal{C}(G)$ as the set
of all subsets of $V(G)$ that have the cut point property, and we let
$\overline{\mathcal{C}}(G)\coloneqq\mathcal{C}(G)\setminus\\{\emptyset\\}$.

Now, we can describe the minimal prime ideals of $\mathcal{J}_{K_{m},G}$.

######  Lemma 2.1 ([MR3011436, Theorem 7]).

Let $H$ be a finite simple graph. Then, the generalized binomial edge ideal
$\mathcal{J}_{K_{m},H}$ is reduced and

$\mathcal{J}_{K_{m},H}=\bigcap_{T\in\mathcal{C}(H)}P_{T}(K_{m},H)$

is a minimal primary decomposition of this ideal.

###  2.3. Depth and regularity

In the sequel, let $\mathfrak{m}$ be the unique graded maximal ideal of $S$
with respect to the standard grading. The local cohomology modules of a
finitely generated graded $S$-module $M$ with respect to $\mathfrak{m}$ are
denoted by $H_{\mathfrak{m}}^{i}(M)$ for $i\in{\mathbb{Z}}$.

######  Definition 2.2.

Let $M$ be a finitely generated graded $S$-module.

  1. a

For each $i$ from $0$ to $\operatorname{dim}(M)$, the _$a_{i}$ -invariant_ of
$M$ is defined as

$a_{i}(M)\coloneqq\max\\{t:(H_{\mathfrak{m}}^{i}(M))_{t}\neq 0\\},$

with the convention that $\max\emptyset=-\infty$.

  2. b

The _Castelnuovo–Mumford regularity_ of $M$ is defined as

$\operatorname{reg}(M)\coloneqq\max\\{a_{i}(M)+i:0\leq
i\leq\operatorname{dim}(M)\\}.$

######  Remark 2.3.

For a finitely generated graded $S$-module $M$, with $M\neq 0$, the local
cohomology module $H_{\mathfrak{m}}^{i}(M)=0$ when $i<\operatorname{depth}(M)$
or $i>\operatorname{dim}(M)$. However,
$H_{\mathfrak{m}}^{\operatorname{depth}(M)}(M)\neq 0$ and
$H_{\mathfrak{m}}^{\operatorname{dim}(M)}(M)\neq 0$ ([MR1251956, Theorem
3.5.7]). In particular, if $M$ is Cohen–Macaulay, then
$\operatorname{reg}(M)=a_{d}(M)+d$ for $d=\operatorname{dim}(M)$.

The lemma below is frequently utilized to calculate a module’s depth and
regularity.

######  Lemma 2.4 ([MR2643966, Lemmas 2.1, 3.1]).

Let $0\rightarrow M\rightarrow N\rightarrow P\rightarrow 0$ be a short exact
sequence of finitely generated graded $S$-modules.

  1. a

One has
$\operatorname{depth}(M)\geq\min\\{\operatorname{depth}(N),\operatorname{depth}(P)+1\\}$.
The equality holds if $\operatorname{depth}(N)\neq\operatorname{depth}(P)$.

  2. b

One has
$\operatorname{reg}(M)\leq\max\\{\operatorname{reg}(N),\operatorname{reg}(P)+1\\}$.
The equality holds if $\operatorname{reg}(N)\neq\operatorname{reg}(P)$.

The following two lemmas are highly valuable for analyzing the regularity of
generalized binomial ideals.

######  Lemma 2.5 ([MR3040610, Proposition 8]).

For a simple graph $G$ and its induced subgraph $H$,
$\operatorname{reg}(\mathcal{J}_{K_{m},H})$ is less than or equal to
$\operatorname{reg}(\mathcal{J}_{K_{m},G})$.

######  Lemma 2.6 ([MR4033090, Theorems 3.6 and 3.7]).

For a connected graph $G$ on the vertex set $[n]$ and $m\geq 2$,
$\operatorname{reg}(S/\mathcal{J}_{K_{m},G})$ is less than or equal to $n-1$.
If $m\geq n\geq 2$, then $\operatorname{reg}(S/\mathcal{J}_{K_{m},G})$ equals
$n-1$.

##  3\. Generalized binomial edge ideals of a complete $r$-partite graph

This section analyzes the generalized binomial edge ideal
$\mathcal{J}_{K_{m},G}$ associated with a complete $r$-partite graph $G$ where
$r\geq 2$. The primary decomposition of $\mathcal{J}_{K_{m},G}$ is examined to
determine the Krull dimension and the depth of $S/\mathcal{J}_{K_{m},G}$. A
recent result of Katsabekis on the cohomological dimension is generalized as a
corollary. Additionally, the Castelnuovo–Mumford regularity, the Hilbert
series, and the multiplicity of the quotient ring $S/\mathcal{J}_{K_{m},G}$
are computed.

Assuming $G$ is a complete $r$-partite graph, its vertices can be rearranged
to meet the following conditions:

######  Setting 3.1.

Let $m,r,n_{1},n_{2},\dots,n_{r}$ be positive integers such that $m\geq 2$,
$r\geq 2$, $n_{r}\geq 2$, and
$1=n_{1}=n_{2}=\cdots=n_{s-1}<n_{s}\leq\cdots\leq n_{r}$ for some $s\in[r]$.
Suppose that $G$ is a complete $r$-partite graph with a partition
$V_{1}\sqcup\cdots\sqcup V_{r}$, where
$V_{k}\coloneqq\left[1+\sum\limits_{j=1}^{k-1}n_{j},\sum\limits_{j=1}^{k}n_{j}\right]$
for each $k\in[r]$. At the same time, let $\widetilde{G}$ be the complete
graph on $[n]$ with $n\coloneqq\sum_{j=1}^{r}n_{j}$. As usual, let
$\mathcal{J}_{K_{m},G}$ and $\mathcal{J}_{K_{m},\widetilde{G}}$ be the
generalized binomial edge ideals of the graphs $G$ and $\widetilde{G}$,
respectively, in the polynomial ring $S={\mathbb{K}}[x_{ij}:i\in[m],j\in[n]]$.
Furthermore, let

$A_{i}\coloneqq\begin{cases}S,&\text{if $n_{i}=1$,}\\\ (x_{ij}:i\in[m],j\in
T_{i}),&\text{if $n_{i}\geq 2$,}\end{cases}$

where $T_{i}\coloneqq\bigsqcup_{j\neq i}V_{j}$.

######  Remark 3.2.

If $n_{1}=\cdots=n_{r}=1$, then $G$ is a complete graph $K_{r}$. In this
scenario, $S/\mathcal{J}_{K_{m},G}$ is Cohen–Macaulay with
$\operatorname{dim}(S/\mathcal{J}_{K_{m},G})=m+r-1$ and
$\operatorname{reg}(S/\mathcal{J}_{K_{m},G})=\min\\{m-1,r-1\\}$, as proven by
Lemmas 3.3 and 3.4 below.

######  Lemma 3.3 ([MR266912, Corollary 4]).

Let $\bm{X}=(x_{ij})_{1\leq i\leq m,1\leq j\leq n}$ be a generic matrix over
${\mathbb{K}}$. Then the ideal $I_{t}(\bm{X})$ in $S$, generated by all
$t\times t$ minors of $\bm{X}$, is perfect of height $(m-t+1)(n-t+1)$. In
particular,
$\operatorname{depth}({\mathbb{K}}[\bm{X}]/I_{t}(\bm{X}))=(m+n-1)(t-1)$.

######  Lemma 3.4 ([MR4033090, Proposition 3.3] or [MR3764061, Theorem 5.1]).

For the complete graph $K_{n}$ with $n$ vertices, we have
$\operatorname{reg}(S/\mathcal{J}_{K_{m},K_{n}})=\min\\{m-1,n-1\\}$.

Now, we will discuss the primary decomposition and the dimension of
$\mathcal{J}_{K_{m},G}$.

######  Proposition 3.5.

Under 3.1, the minimal primary decomposition is

$\mathcal{J}_{K_{m},G}=\mathcal{J}_{K_{m},\widetilde{G}}\cap A_{s}\cap
A_{s+1}\cap\cdots\cap A_{r},$

and the dimension is

$\operatorname{dim}(S/\mathcal{J}_{K_{m},G})=\max\\{m+n-1,mn_{r}\\}.$

###### Proof.

Notice that $\operatorname{dim}(S/\mathcal{J}_{K_{m},\widetilde{G}})=m+n-1$ by
Lemma 3.3 and $\operatorname{dim}(S/A_{i})=mn_{i}$ for each $i\in[r]$. Thus,
by Lemma 2.1, it suffices to show that

$\mathcal{C}(G)=\Set{\emptyset,T_{s},T_{s+1},\dots,T_{r}}.$ (1)

Let $G_{i}$ be the empty graph on the vertex set $V_{i}$ for $i\in[r]$. Then,
$G$ is the join graph $G_{1}*\cdots*G_{r}$ in the sense of [MR3395714, Section
4]. Therefore, the equality of (1) follows from [MR3395714, Propositions 4.1,
4.5 and 4.14]. ∎

Using the primary decomposition of $\mathcal{J}_{K_{m},G}$, we can calculate
the depth of $S/\mathcal{J}_{K_{m},G}$.

######  Theorem 3.6.

Under 3.1, we have $\operatorname{depth}(S/\mathcal{J}_{K_{m},{G}})=m+n_{s}$.

###### Proof.

To utilize the primary decomposition
$\mathcal{J}_{K_{m},G}=\mathcal{J}_{K_{m},\widetilde{G}}\cap
A_{s}\cap\cdots\cap A_{r}$ from 3.5, we introduce
$\mathfrak{a}_{s-1}\coloneqq\mathcal{J}_{K_{m},\widetilde{G}}$ and
$\mathfrak{a}_{k}\coloneqq\mathcal{J}_{K_{m},\widetilde{G}}\cap
A_{s}\cap\cdots\cap A_{k}$ for $s\leq k\leq r$. With these preparations, for
each $k\in[s,r]$, we have

$0\to S/(\mathfrak{a}_{s-1}\cap A_{k})\to S/\mathfrak{a}_{s-1}\oplus
S/A_{k}\to S/(\mathfrak{a}_{s-1}+A_{k})\to 0.$

Note that $\operatorname{depth}(S/\mathfrak{a}_{s-1})=m+n-1$,
$\operatorname{depth}(S/A_{k})=mn_{k}$ and
$\operatorname{depth}(S/(\mathfrak{a}_{s-1}+A_{k}))=m+n_{k}-1$. Since $m\geq
2$, $r\geq 2$ and $n_{k}\geq 2$, we have $\min\\{m+n-1,mn_{k}\\}>m+n_{k}-1$.
So $\operatorname{depth}(S/(\mathfrak{a}_{s-1}\cap A_{k}))=m+n_{k}$ by Lemma
2.4a.

Next, we prove by induction on $k\in[s,r]$ that
$\operatorname{depth}(S/\mathfrak{a}_{k})=m+n_{s}$. The base case when $k=s$
is clear from the above calculation, since
$\mathfrak{a}_{s}=\mathfrak{a}_{s-1}\cap A_{s}$. Now, suppose that $k\geq s+1$
and we consider the short exact sequence

$0\to S/\mathfrak{a}_{k}\to S/\mathfrak{a}_{k-1}\oplus
S/(\mathfrak{a}_{s-1}\cap A_{k})\to
S/(\mathfrak{a}_{k-1}+\mathfrak{a}_{s-1}\cap A_{k})\to 0.$

By induction, we have $\operatorname{depth}(S/\mathfrak{a}_{k-1})=m+n_{s}$. By
the calculation in the previous part, we have
$\operatorname{depth}(S/(\mathfrak{a}_{s-1}\cap A_{k}))=m+n_{k}$. Let
$A_{k}^{\complement}\coloneqq(x_{ij}:i\in[m],j\in V_{k})\subset S$. Then, we
have

$\mathfrak{a}_{s-1}\supseteq\mathfrak{a}_{k-1}+\mathfrak{a}_{s-1}\cap
A_{k}\supseteq\mathfrak{a}_{s-1}\cap
A_{k}^{\complement}+\mathfrak{a}_{s-1}\cap A_{k}\supseteq\mathfrak{a}_{s-1}.$

So $\mathfrak{a}_{s-1}=\mathfrak{a}_{k-1}+\mathfrak{a}_{s-1}\cap A_{k}$ and
$\operatorname{depth}(S/(\mathfrak{a}_{k-1}+\mathfrak{a}_{s-1}\cap
A_{k}))=\operatorname{depth}(S/\mathfrak{a}_{s-1})=m+n-1$. Since $k\geq s+1$,
we have

$\min\\{m+n_{s},m+n_{k}\\}=m+n_{s}<m+n_{s}+n_{s+1}-1\leq m+n-1.$

So we obtain $\operatorname{depth}(S/\mathfrak{a}_{k})=m+n_{s}$ by Lemma 2.4a.

In particular, we can take $k=r$ and obtain that

$\operatorname{depth}(S/\mathcal{J}_{K_{m},G})=\operatorname{depth}(S/\mathfrak{a}_{r})=m+n_{s}.\qed$

As a corollary to Theorem 3.6, we will consider the cohomological dimension of
$\mathcal{J}_{K_{m},G}$. Recall that in a Noetherian ring $R$, the
_cohomological dimension_ $\operatorname{cd}_{R}(I)$ of an ideal $I$ is
defined as follows:

$\operatorname{cd}_{R}(I)\coloneqq\max\\{i\in{\mathbb{Z}}:H_{I}^{i}(R)\neq
0\\}.$

Here, $H_{I}^{i}(R)$ represents the $i$-th local cohomology module of $R$ with
support in $I$. It is worth noting that the star graph is a special complete
bipartite graph, and the following result is a generalization of
[arXiv:2207.02256, Proposition 2.12].

######  Corollary 3.7.

Under 3.1, we have the following results:

  1. a

If the characteristic of the field ${\mathbb{K}}$ is positive, then

$\operatorname{cd}(\mathcal{J}_{K_{m},G})=mn-m-n_{s}.$

  2. b

If the characteristic of the field ${\mathbb{K}}$ is zero, then

$mn-m-n_{s}\leq\operatorname{cd}(\mathcal{J}_{K_{m},G})\leq mn-3.$

###### Proof.

Based on [MR3011436, Theorem 2], $\mathcal{J}_{K_{m},G}$ has a squarefree
initial ideal with respect to some monomial order on $S$. Therefore, by
applying Theorem 3.6 and [MR4132955, Proposition 3.6], we can infer that
$\operatorname{cd}(\mathcal{J}_{K_{m},G})\geq mn-m-n_{s}$, with equality when
the characteristic of ${\mathbb{K}}$ is positive. Additionally, since
$\operatorname{depth}(R/\mathcal{J}_{K_{m},G})=m+n_{s}\geq 3$, as per Theorem
3.6, we can conclude that $\operatorname{cd}(\mathcal{J}_{K_{m},G})\leq mn-3$,
according to [MR3078644, Theorem 3.5]. ∎

Next, we will compute the regularity of $S/\mathcal{J}_{K_{m},G}$ under 3.1.
Before proceeding, let us examine a special case.

######  Lemma 3.8.

For $m\geq 2$, we have $\operatorname{reg}(S/\mathcal{J}_{K_{m},K_{1,m}})=m$.

###### Proof.

We will treat $K_{1,m}$ here as the complete $r$-partite graph $G$ in 3.1, by
taking $r=2$, $n_{1}=1$, and $n_{2}=m$. When $m=2$, the result can be derived
from [MR3195706, Theorem 4.1]. In the following, we will assume that $m\geq
3$. It follows from 3.5 that there exists a short exact sequence

$0\rightarrow S/\mathcal{J}_{K_{m},G}\rightarrow
S/\mathcal{J}_{K_{m},\widetilde{G}}\oplus S/A_{2}\rightarrow
S/(\mathcal{J}_{K_{m},\widetilde{G}}+A_{2})\rightarrow 0.$

Note that $S/\mathcal{J}_{K_{m},\widetilde{G}}$, $S/A_{2}$ and
$S/(\mathcal{J}_{K_{m},\widetilde{G}}+A_{2})$ are Cohen–Macaulay of dimensions
$2m$, $m^{2}$ and $2m-1$, respectively, by Lemma 3.3. Since $m\geq 3$, we have
$m^{2}>2m$. Applying the local cohomology functor with respect to
$\mathfrak{m}$ to the above exact sequence, we obtain from 2.3 that

$\displaystyle 0\to
H_{\mathfrak{m}}^{2m-1}(S/(\mathcal{J}_{K_{m},\widetilde{G}}+A_{2}))\to
H_{\mathfrak{m}}^{2m}(S/\mathcal{J}_{K_{m},G})\to
H_{\mathfrak{m}}^{2m}(S/\mathcal{J}_{K_{m},\widetilde{G}})\to 0,$ (2)
$\displaystyle H_{\mathfrak{m}}^{m^{2}}(S/\mathcal{J}_{K_{m},G})\cong
H_{\mathfrak{m}}^{m^{2}}(S/A_{2}),$ (3) and $\displaystyle
H_{\mathfrak{m}}^{i}(S/\mathcal{J}_{K_{m},G})=0\qquad\text{for all
$i\notin\\{2m,m^{2}\\}$.}$ (4)

Note that

$\displaystyle\operatorname{reg}(S/\mathcal{J}_{K_{m},G})$
$\displaystyle=\max\\{a_{i}(S/\mathcal{J}_{K_{m},G})+i:\operatorname{depth}(S/\mathcal{J}_{K_{m},G})\leq
i\leq\operatorname{dim}(S/\mathcal{J}_{K_{m},G})\\}$
$\displaystyle\xlongequal{\eqref{eqn:123-3}}\max\\{a_{2m}(S/\mathcal{J}_{K_{m},G})+2m,a_{m^{2}}(S/\mathcal{J}_{K_{m},G})+m^{2}\\}.$

Since $S/A_{2}$ is a polynomial ring of dimension $m^{2}$ with regularity $0$,
it follows from (3) that

$a_{m^{2}}(S/\mathcal{J}_{K_{m},G})=-m^{2}.$

Meanwhile, $S/\mathcal{J}_{K_{m},\widetilde{G}}$ is Cohen–Macaulay of
dimension $2m$ and
$\operatorname{reg}(S/\mathcal{J}_{K_{m},\widetilde{G}})=m-1$ by Lemma 3.4. So
$a_{2m}(S/\mathcal{J}_{K_{m},\widetilde{G}})=(m-1)-2m=-m-1$. Similarly, we get
$a_{2m-1}(S/(\mathcal{J}_{K_{m},\widetilde{G}}+A_{2}))=(m-1)-(2m-1)=-m$. Since
the short exact sequence (2) is graded, we get

$\displaystyle a_{2m}(S/\mathcal{J}_{K_{m},G})$
$\displaystyle=\max\\{-m-1,-m\\}=-m.$

Consequently, $\operatorname{reg}(S/\mathcal{J}_{K_{m},G})=m$. ∎

######  Theorem 3.9.

Under 3.1, we have

$\operatorname{reg}(S/\mathcal{J}_{K_{m},{G}})=\begin{cases}n-1,&\text{if
$m\geq n$,}\\\ m-1,&\text{if $n>m>n_{r}$,}\\\ m,&\text{if $n_{r}\geq m\geq
2$.}\end{cases}$

###### Proof.

By 3.5, we have

$\mathcal{J}_{K_{m},G}=A_{s}\cap\cdots\cap
A_{r}\cap\mathcal{J}_{K_{m},\widetilde{G}}.$

Let $\mathfrak{b}_{k}\coloneqq A_{k}\cap\cdots\cap
A_{r}\cap\mathcal{J}_{K_{m},\widetilde{G}}$ for any $k\in[s,r]$ and
$\mathfrak{b}_{r+1}\coloneqq\mathcal{J}_{K_{m},\widetilde{G}}$. Then,
$\mathfrak{b}_{k}=\mathfrak{b}_{k+1}\cap A_{k}$ and
$\mathfrak{b}_{k+1}+A_{k}=\mathcal{J}_{K_{m},\widetilde{G}|_{V_{k}}}+A_{k}$
for all $k\in[s,r]$.

We have three cases.

  1. a

If $m\geq n$, then $\operatorname{reg}(S/\mathcal{J}_{K_{m},{G}})=n-1$, by
Lemma 2.6.

  2. b

Suppose that $n>m>n_{r}$. In this case, we can see that for any $k\in[s,r]$,

$\displaystyle\operatorname{reg}(S/\mathfrak{b}_{r+1})=\operatorname{reg}(S/\mathcal{J}_{K_{m},\widetilde{G}})=\min\\{m-1,n-1\\}=m-1,$
$\displaystyle\operatorname{reg}(S/(\mathfrak{b}_{k+1}+A_{k}))=\operatorname{reg}(S/(\mathcal{J}_{K_{m},\widetilde{G}|_{V_{k}}}+A_{k}))=\min\\{m-1,n_{k}-1\\}=n_{k}-1,$
and $\displaystyle\operatorname{reg}(S/A_{k})=0,$

as per Lemma 3.4. Consider the following short exact sequence

$0\to S/\mathfrak{b}_{k}\to S/\mathfrak{b}_{k+1}\oplus S/A_{k}\to
S/(\mathfrak{b}_{k+1}+A_{k})\to 0$ (5)

for such $k$. If we argue by induction on $k$ from $r+1$ down to $s$, it
follows from Lemma 2.4b that $\operatorname{reg}(S/\mathfrak{b}_{k})=m-1$. In
particular,
$\operatorname{reg}(S/\mathcal{J}_{K_{m},{G}})=\operatorname{reg}(S/\mathfrak{b}_{s})=m-1$.

  3. c

Suppose that $n_{r}\geq m\geq 2$. Likewise, we have

$\displaystyle\operatorname{reg}(S/\mathfrak{b}_{r+1})=\operatorname{reg}(S/\mathcal{J}_{K_{m},\widetilde{G}})=\min\\{m-1,n-1\\}=m-1,$
$\displaystyle\operatorname{reg}(S/(\mathfrak{b}_{k+1}+A_{k}))=\operatorname{reg}(S/(\mathcal{J}_{K_{m},\widetilde{G}|_{V_{k}}}+A_{k}))=\min\\{m-1,n_{k}-1\\}\leq
m-1,$ and $\displaystyle\operatorname{reg}(S/A_{k})=0$

for each $k\in[s,r]$. Using a similar technique, we can show that
$\operatorname{reg}(S/\mathfrak{b}_{k})\leq m$ for $k=r+1,r,\dots,s$. In
particular,
$\operatorname{reg}(S/\mathcal{J}_{K_{m},{G}})=\operatorname{reg}(S/\mathfrak{b}_{s})\leq
m$.

On the other hand, the complete $r$-partite graph $G$ contains an induced
subgraph $K_{1,m}$, as $n_{r}\geq m$ and $r\geq 2$. Therefore,
$\operatorname{reg}(S/\mathcal{J}_{K_{m},{G}})\geq\operatorname{reg}(S/\mathcal{J}_{K_{m},K_{1,m}})=m$
by Lemmas 2.5 and 3.8. Thus,
$\operatorname{reg}(S/\mathcal{J}_{K_{m},{G}})=m$, which completes the proof.
∎

######  Proposition 3.10.

Under 3.1, the Hilbert series of $S/\mathcal{J}_{K_{m},{G}}$ is given by

$\displaystyle\operatorname{Hilb}(S/\mathcal{J}_{K_{m},G})=\frac{\sum_{i\geq
0}{m-1\choose i}{n-1\choose
i}t^{i}}{(1-t)^{m+n-1}}+\sum_{k=s}^{r}\left(\frac{1}{(1-t)^{mn_{k}}}-\frac{\sum_{i\geq
0}{m-1\choose i}{n_{k}-1\choose i}t^{i}}{(1-t)^{m+n_{k}-1}}\right).$

###### Proof.

We adopt the notation from the proof of Theorem 3.9. It follows from the short
exact sequence in (5) that

$\operatorname{Hilb}(S/\mathfrak{b}_{k},t)=\operatorname{Hilb}(S/\mathfrak{b}_{k+1},t)+\operatorname{Hilb}(S/A_{k},t)-\operatorname{Hilb}(S/(\mathcal{J}_{K_{m},\widetilde{G}|_{V_{k}}}+A_{k}),t).$

Since $\mathcal{J}_{K_{m},G}=\mathfrak{b}_{s}$ and
$\mathcal{J}_{K_{m},\widetilde{G}}=\mathfrak{b}_{r+1}$, we obtain that

$\displaystyle\operatorname{Hilb}(S/\mathcal{J}_{K_{m},G})=\operatorname{Hilb}(S/\mathcal{J}_{K_{m},\widetilde{G}})+\sum_{k=s}^{r}\left(\operatorname{Hilb}(S/A_{k},t)-\operatorname{Hilb}(S/(\mathcal{J}_{K_{m},\widetilde{G}|_{V_{k}}}+A_{k}),t)\right).$

It follows from [MR1213858, Corollary 1] that

$\displaystyle\operatorname{Hilb}(S/\mathcal{J}_{K_{m},\widetilde{G}},t)$
$\displaystyle=\frac{\sum_{i\geq 0}{m-1\choose i}{n-1\choose
i}t^{i}}{(1-t)^{m+n-1}}$ and
$\displaystyle\operatorname{Hilb}(S/(\mathcal{J}_{K_{m},\widetilde{G}|_{V_{k}}}+A_{k}),t)$
$\displaystyle=\frac{\sum_{i\geq 0}{m-1\choose i}{n_{k}-1\choose
i}t^{i}}{(1-t)^{m+n_{k}-1}}.$

Meanwhile, the Hilbert series of the polynomial rings $S/A_{k}$ is
$\frac{1}{(1-t)^{mn_{k}}}$. Thus, the desired formula follows. ∎

######  Corollary 3.11.

Under 3.1, we assume additionally that $r=2$. Then the multiplicity of
$S/\mathcal{J}_{K_{m},{G}}$ is given by

$\operatorname{e}(S/\mathcal{J}_{K_{m},{G}})=\begin{cases}1,&\text{if
$\max\\{m+n_{1}+n_{2}-1,mn_{1}\\}<mn_{2}$},\\\ 2,&\text{if
$m+n_{1}+n_{2}-1<mn_{1}=mn_{2}$},\\\ 2n_{2},&\text{if
$mn_{2}<m+n_{1}+n_{2}-1$},\\\ 12,&\text{if
$m+n_{1}+n_{2}-1=mn_{1}=mn_{2}$},\\\ \sum_{k\geq 0}{m-1\choose
k}{n_{1}+n_{2}-1\choose k}+1,&\text{if
$mn_{1}<m+n_{1}+n_{2}-1=mn_{2}$}.\end{cases}$

###### Proof.

It is a direct consequence of 3.10. We only remark that if
$mn_{2}<m+n_{1}+n_{2}-1$, then $m=2$ and $n_{1}=n_{2}$. Meanwhile, if
$m+n_{1}+n_{2}-1=mn_{1}=mn_{2}$, then $m=3$ and $n_{1}=n_{2}=2$. ∎

In the following, we summarize the applications of the above results to
binomial edge ideals of complete bipartite graphs. Specifically, we recover
the corresponding parts of Corollary 3.4 and Theorem 5.4 (a) and (b) from
Schenzel and Zafar’s research [MR3195706].

######  Corollary 3.12.

Let $G=K_{n_{1},n_{2}}$ be a complete bipartite graph with $n_{1}\leq n_{2}$.
We have the following formulas for its binomial edge ideal $J_{G}$:

  1. a

$\operatorname{depth}(S/J_{G})=\begin{cases}n_{2}+2,&\text{\ if\
$n_{1}=1$,}\\\ n_{1}+2,&\text{\ if\ $n_{1}\geq 2$.}\end{cases}$

  2. b

The Hilbert series of $S/J_{G}$ is given by

$\operatorname{Hilb}(S/J_{G},t)=\frac{1+(n_{1}+n_{2}-1)t}{(1-t)^{n_{1}+n_{2}+1}}+\frac{1}{(1-t)^{2n_{1}}}+\frac{1}{(1-t)^{2n_{2}}}-\frac{1+(n_{1}-1)t}{(1-t)^{n_{1}+1}}-\frac{1+(n_{2}-1)t}{(1-t)^{n_{2}+1}}.$

  3. c

The multiplicity of $S/J_{G}$ is given by

$\operatorname{e}(S/J_{G})=\begin{cases}1,&\text{if $n_{2}>n_{1}+1$},\\\
2n_{2},&\text{otherwise.}\end{cases}$

We conclude this paper by showing that the binomial edge ideal $J_{G}$ of a
complete $r$-partite graph $G$ is of König type. This builds on the work of
Herzog et al. [MR4358668], who introduced the concept of _graded ideals of
König type_ with respect to a monomial order $<$, which generalizes the _edge
ideals of König graphs_. They proved that the Cohen–Macaulay property of the
initial ideal $\operatorname{in}_{<}(I)$ of $I$ is independent of the
characteristic of the base field for any graded ideal $I$ of König type.
Williams [arXiv:2310.14410, Theorem 2.4 and Proposition 2.24] recently
demonstrated that the binomial edge ideal of a complete bipartite graph is of
König type. In fact, this result also applies to complete $r$-partite graphs.

######  Proposition 3.13.

If $G$ is a complete $r$-partite graph, the binomial edge ideal $J_{G}$ is of
König type.

###### Proof.

If $G$ is a complete graph, it is _traceable_ , meaning it has a spanning path
as a subgraph. As a result, $J_{G}$ is of König type according to [MR4358668,
Proposition 3.6].

In the following, we may assume that $G$ is a graph in 3.1. Let $h$ be the
height of $J_{G}$. 3.5 states that $h=2n-\max\\{n+1,2n_{r}\\}$. To show that
$J_{G}$ is of König type, it suffices to find a path of length $h$ in $G$ by
[MR4358668, Theorem 3.5]. Let $n^{\prime}\coloneqq
n_{1}+n_{2}+\cdots+n_{r-1}$. There are two cases to consider:

  1. a

Suppose that $n_{r}>n^{\prime}$. Notice that $V_{1}\sqcup\cdots\sqcup
V_{r-1}=[n^{\prime}]$. The path we choose is:

$n^{\prime}+1,1,n^{\prime}+2,2,n^{\prime}+3,3,\dots,n^{\prime}+n^{\prime},n^{\prime},n^{\prime}+(n^{\prime}+1).$

  2. b

Suppose that $n_{r}\leq n^{\prime}$. We can show that $G$ is traceable. For
notational simplicity, we rewrite the vertices of $V_{i}$ as
$V_{i}=\\{v_{i,1},v_{i,2},\dots,v_{i,n_{i}}\\}$ for $i\in[r]$. Thus, we obtain
a total order $\succ$ on $V_{1}\sqcup\cdots\sqcup V_{r-1}$ by setting
$v_{i,j}\succ v_{i^{\prime},j^{\prime}}$ if and only if $j<j^{\prime}$, or
$j=j^{\prime}$ and $i<i^{\prime}$. With respect to the order $\succ$, we can
rewrite $V_{1}\sqcup\cdots\sqcup V_{r-1}$ as $\\{u_{1}\succ
u_{2}\succ\cdots\succ u_{n^{\prime}}\\}$. Furthermore, set $\delta\coloneqq
n^{\prime}-n_{r}+2$. The path in $G$ we take will be:

$n^{\prime}+1,u_{1},u_{2},\dots,u_{\delta},n^{\prime}+2,u_{\delta+1},n^{\prime}+3,u_{\delta+2},\dots,n^{\prime}+n_{r}-1,u_{\delta+n_{r}-2}=u_{n^{\prime}},n^{\prime}+n_{r}=n.$

∎

######  Remark 3.14.

When $G$ is a complete $r$-partite graph, the _generalized_ binomial edge
ideal $\mathcal{J}_{K_{m},G}$ is not necessarily of König type. For instance,
the height of the ideal $\mathcal{J}_{K_{3},K_{2,2}}$ is $6$, while the
maximal length of the regular sequence in
$\\{\operatorname{in}_{{\operatorname{lex}}}(p_{(e,f)}):e\in E(K_{3}),f\in
E(K_{2,2})\\}$ is $5$.

######  Acknowledgment.

This work is supported by the Natural Science Foundation of Jiangsu Province
(No. BK20221353). In addition, the first author is partially supported by the
Anhui Initiative in Quantum Information Technologies (No. AHY150200) and the
“Innovation Program for Quantum Science and Technology” (2021ZD0302902). And
the second author is supported by the Foundation of the Priority Academic
Program Development of Jiangsu Higher Education Institutions.

## References


eprint Nisho-3-2021

# Detectable Electric Current induced by Dark Matter Axion in a Conductor

Aiichi Iwazaki  International Economics and Politics, Nishogakusha University,
6-16 3-bantyo Chiyoda Tokyo 102-8336, Japan

(Nov. 18, 2021)

###### Abstract

We propose a way of detecting dark matter axion by using two slabs of
conductor. The flat surfaces are put to meet face to face so that they are
parallel to each other. External magnetic field $B$ parallel to the surfaces
is impressed. Radiations converted from the axion arise between two slabs.
When we tune the spacing $l$ between two surfaces such as $l=\pi/m_{a}$ with
axion mass $m_{a}$, a resonance occurs so that the radiations become strong.
Furthermore, electric current flowing on the surface of the slabs is enhanced.
We show that the electric current is large enough to be detectable at the
resonance. It reaches $0.7\times
10^{-9}$A$(10^{-5}\mbox{eV}/m_{a})^{1/2}(B/5\mbox{T})(L/10\mbox{cm})(\sigma/3.3\times
10^{7}\rm eV)$, using $6$N copper of the square slab with side length $L$ and
high electrical conductivity $\sigma$ at temperature $T\sim 1$K. The power of
the Joule heating is $0.3\times
10^{-22}\mbox{W}(B/5\mbox{T})^{2}(10^{-5}\mbox{eV}/m_{a})^{1/2}(L/10\mbox{cm})^{2}(\sigma/3.3\times
10^{7}\rm eV)$. When we amplify the power using LC circuit with $Q_{LC}$
factor, the signal to noise ratio is $4.5\times
10^{4}(Q_{LC}/10^{6})(B/5\mbox{T})^{2}(t_{obs}/1\sec)^{1/2}\,(10^{-5}\mbox{eV}/m_{a})(L/10\mbox{cm})^{2}(\sigma/3.3\times
10^{7}\rm eV)$ with $t_{obs}$ observational time.

The axionaxion ; Wil  is one of the most important candidates of the dark
matter. Especially, it appears that it stands on the top of the candidates
because QCD axion gives natural solutions for strong CP problem as well as the
dark matter problem. Other candidates of the dark matter e.g. WINPs ( weakly
interacting massive particles ) have not been detected in spite of extensive
searches for long terms.

The axion has recently been searched with various ways. Most of the
experimentsadmx ; carrack ; haystac ; abracadabra ; organ ; madmax ; brass ;
cast ; sumico  exploit the Primakoff effect. That is, they detect photon
directly produced by the dark matter axion $a(t)$ under strong magnetic field
$B$. But, the axion photon coupling $g_{a\gamma\gamma}a(t)\sim 10^{-19}$ is so
tiny that the direct photon production is hard to be detected. So we need, for
instance, very sensitive radio receiver using fine technology such as SQUIDs.

In this letter we propose a new way of detection of the dark matter axion. The
detection is performed by observing electric current flowing a conductor with
high electrical conductivity. The current is induced by the axion under strong
magnetic field $\vec{B}$. We use two parallel slabsiwazaki1  Fig.1 which are
put to meet face to face. When we impress magnetic field between the slabs,
the axion is converted to radiation between the slabs. Simultaneously,
electric current $I_{a}$ flows the surfces of the slabs in the direction
parallel to the magnetic field $\vec{B}$. When we appropriately tune a spacing
$l$ between the parallel slabs, a resonance occurs and the radiation is
enhanced. At the resonance, the electric current is also enhanced. We find
that the electric current reaches $\sim
10^{-10}$A$(10^{-5}\mbox{eV}/m_{a})^{1/2}(B/5\mbox{T})(L/10\mbox{cm})^{2}(\sigma/3.3\times
10^{7}\rm eV)$, using high quality $6$N copper of the square slab with the
side length $L=10$cm and high electrical conductivity $\sigma\simeq 3.3\times
10^{7}$eV at $T=1$K. This system resembles the system used in ADMX, where
conductor of cylinder form is used to obtain the resonance. In addition to the
amplification of radiation by the tuning the spacing, we also amplify the
electric current with LC circuitsikivie ; kishimoto  by tuning parameters of
the circuit. Consequently, we can obtain sufficiently large power $\sim
10^{-17}\mbox{W}(Q_{LC}/10^{6})(10^{-5}\mbox{eV}/m_{a})^{1/2}(B/5\mbox{T})^{2}(L/10\mbox{cm})^{2}(\sigma/3.3\times
10^{7}\rm eV)$ to be detected by the circuit with $Q_{LC}$ factor.

Before we discuss radiation in the space between two slabs of normal
conductors, we briefly explain the axion photon coupling. The axion
$a(\vec{x},t)$ couples with both electric $\vec{E}$ and magnetic fields
$\vec{B}$ in the following,

$L_{aEB}=g_{\gamma}\alpha\frac{a(\vec{x},t)\vec{E}\cdot\vec{B}}{f_{a}\pi}\equiv
g_{a\gamma\gamma}a(\vec{x},t)\vec{E}\cdot\vec{B}$ (1)

with the decay constant $f_{a}$ of the axion and the fine structure constant
$\alpha\simeq 1/137$, where the numerical constant $g_{\gamma}$ depends on
axion models; typically it is of the order of one. The standard notation
$g_{a\gamma\gamma}$ is such that
$g_{a\gamma\gamma}=g_{\gamma}\alpha/f_{a}\pi\simeq 0.14(m_{a}/\rm GeV^{2})$
for DFSZ modeldfsz  and $g_{a\gamma\gamma}\simeq-0.39(m_{a}/\rm GeV^{2})$ for
KSVZ modelksvz . In other words, $g_{\gamma}\simeq 0.37$ for DFSZ and
$g_{\gamma}\simeq-0.96$ for KSVZ. The axion decay constant $f_{a}$ is related
with the axion mass $m_{a}$ in the QCD axion; $m_{a}f_{a}\simeq 6\times
10^{-6}\rm eV\times 10^{12}$GeV. We only consider the QCD axion in this paper.

Because de Broglie wave length of the axion inside our galaxy is $10^{3}$
times larger than $m_{a}^{-1}$, we may neglect spatial dependence of the axion
field in experiment at room; $a(\vec{x},t)=a(t)$. We can see that the coupling
parameter $g_{a\gamma\gamma}a(t)$ in the Lagrangian eq(1) is extremely small
for the dark matter axion $a(t)$. We note that the energy density of the dark
matter axion taken by time average is given by

$\rho_{a}=\frac{1}{2}\overline{(\dot{a}^{2}+(\vec{\partial}a)^{2}+m_{a}^{2}a^{2})}\simeq\frac{m_{a}^{2}a_{0}^{2}}{2}$
(2)

where $a(t)=a_{0}\cos(t\sqrt{m_{a}^{2}+k^{2}})\simeq a_{0}\cos(m_{a}t)$,
because the velocity $k/m_{a}$ of the axion is about $10^{-3}$ in our galaxy.

The local energy density $\rho$ of dark matter in our galaxy is supposed such
as $\rho\simeq 0.3\rm GeV\,\rm cm^{-3}\simeq 2.4\times 10^{-42}\rm GeV^{4}$.
Assuming that the density is equal to that of the dark matter axion, i.e.
$\rho=\rho_{a}$, we find extremely small parameter $g_{a\gamma\gamma}a\sim
10^{-19}$. The energy density also gives the large number density of the
axions $\rho_{a}/m_{a}\sim 10^{15}\mbox{cm}^{-3}(10^{-6}\mbox{eV}/m_{a})$,
which give rise to their coherence. This allows us to treat the axions as the
classical axion field $a(t)\propto\cos(m_{a}t)$ spatially smoothly varying.
The parameter $g_{a\gamma\gamma}a(t)$ is extremely small so that perturbative
analysis is valid.

The interaction term in eq(1) between axion and electromagnetic field slightly
modifies Maxwell equations in vacuum between two slabs,

$\displaystyle\vec{\partial}\cdot\vec{E}+g_{a\gamma\gamma}\vec{\partial}\cdot(a(t)\vec{B})$
$\displaystyle=0$
$\displaystyle,\quad\vec{\partial}\times\Big{(}\vec{B}-g_{a\gamma\gamma}a(t)\vec{E}\Big{)}-\partial_{t}\Big{(}\vec{E}+g_{a\gamma\gamma}a(t)\vec{B}\Big{)}=0,$
$\displaystyle\vec{\partial}\cdot\vec{B}$ $\displaystyle=0$
$\displaystyle,\quad\vec{\partial}\times\vec{E}+\partial_{t}\vec{B}=0.$ (3)

From the equations, we approximately obtain the electric field $\vec{E}$
generated by the axion $a(t)$ under static and spatially homogeneous
background magnetic field $\vec{B}$. It is assumed to be of the order of
$g_{a\gamma\gamma}a(t)B$,

$\vec{E}_{a}(t)=-g_{a\gamma\gamma}a(t)\vec{B}.$ (4)

Other solutions are those of standard electromagnetic waves in vacuum.

Figure 1: rectangular flat conductors with size $L\times R$ and spacing $l$.
Electric current $I_{a}$ flows on the surface.

Now, we discussiwazaki1  radiations arising between two slabs and electric
currents flowing the slabs. In order to discuss electric and magnetic fields
in the slabs composed of normal conductor, we suppose conductor with trivial
permeability $\mu=1$, dielectric permittivity $\epsilon=1$ and large
electrical conductivity $\sigma$.

The configurations Fig.1 of the conductors are in the following. The
conductors with flat surfaces are put parallel to each other. They occupy the
regions $x\geq 0$ and $x\leq-l$. Their surfaces are present at $x=0$ and
$x=-l$, respectively. They are uniform in $y$ and $z$ directions. We impose
spatially uniform static magnetic field $\vec{B}=(0,0,B)$ parallel to the
surfaces of the conductors. We denote magnetic field induced in the conductor
as $\vec{B}(\mbox{inside})+\vec{B}$. The term $\vec{B}$ corresponds to the
external magnetic field $\vec{B}$ because the component of the field
$\vec{B}(\mbox{inside})+\vec{B}$ parallel to the surface ( at $x=0$ and $x=-l$
) must be continuous at the surface, Although we assume two slabs infinitely
extended in $y$ and $z$ directions, results obtained below are valid as long
as the wave length of radiation is much smaller than actual length of the
conductors.

The electromagnetic fields in the conductor are described by the modified
Maxwell equations,

$\displaystyle\vec{\partial}\cdot\vec{E}(\mbox{inside})+g_{a\gamma\gamma}\vec{\partial}\cdot(a(t)\vec{B}(\mbox{inside}))=0,$
(5)
$\displaystyle\vec{\partial}\times\Big{(}\vec{B}(\mbox{inside})-g_{a\gamma\gamma}a(t)\vec{E}(\mbox{inside})\Big{)}-\partial_{t}\Big{(}\vec{E}(\mbox{inside})+g_{a\gamma\gamma}a(t)(\vec{B}(\mbox{inside})+\vec{B})\Big{)}=\vec{J}_{e},$
$\displaystyle\vec{\partial}\cdot\vec{B}(\mbox{inside})=0,\quad\vec{\partial}\times\vec{E}(\mbox{inside})+\partial_{t}\vec{B}(\mbox{inside})=0,$
(6)

where we denote electric field induced in the conductors as
$\vec{E}(\mbox{inside})$.

In eq(5) we have included the current
$\vec{J}_{e}=\sigma\vec{E}(\mbox{inside})$ induced by electric field
$\vec{E}(\mbox{inside})$, but have neglected external current $\vec{J}_{ext}$
generating the background static magnetic field $\vec{B}$ (
$\vec{\partial}\times\vec{B}=\vec{J}_{ext}$ ). The current is outside of the
system. Here we have assumed Ohm’s law
$\vec{J}_{e}=\sigma\vec{E}(\mbox{inside})$. Only the presence of the current
$\vec{J}_{e}$ distinguishes Maxwell equations in conductor from those in
vacuum.

When we neglect axion contribution, we obtain magnetic field $\vec{B}$ inside
the conductor. Obviously, there is no electric field inside the conductor.
Once we take into account the axion contribution, the oscillating electric
field $\vec{E}(\mbox{inside})$ is induced as well as the magnetic field
$\vec{B}(\mbox{inside})$ in addition to $\vec{B}$

Assuming the parameter $g_{a\gamma\gamma}a(t)$ small and noting that the
fields $\vec{E}(\mbox{inside})$ and $\vec{B}(\mbox{inside})$ are of the order
of $g_{a\gamma\gamma}a(t)$, we derive the equations,

$\vec{\partial}\cdot\vec{E}(\mbox{inside})=0,\quad\vec{\partial}\times\vec{B}(\mbox{inside})=\vec{J}_{e}+\partial_{t}(\vec{E}(\mbox{inside})-\vec{E}_{a}),\quad\vec{\partial}\cdot\vec{B}(\mbox{inside})=0,\quad\mbox{and}\quad\vec{\partial}\times\vec{E}(\mbox{inside})+\partial_{t}\vec{B}(\mbox{inside})=0,$
(7)

where we have used the relation $\vec{\partial}\times\vec{E}_{a}=0$ because
$\vec{\partial}\times\vec{B}=0$ inside the conductor.

Using the Ohm’s law $\vec{J}_{e}=\sigma\vec{E}(\mbox{inside})$ in eqs(7), we
derive the equation for $\vec{E}(\mbox{inside})$,

$(\vec{\partial}^{2}-\partial_{t}^{2})\vec{E}(\mbox{inside})=\sigma\partial_{t}\vec{E}(\mbox{inside})-\partial_{t}^{2}\vec{E}_{a}$
(8)

where we note that $\vec{E}_{a}\propto\cos(m_{a}t)$. Then, we find a solution
for $x>0$ ( later we discuss the solution for $x<-l$ ),

$\vec{E}(\mbox{inside})\simeq\vec{E}_{1}\exp(-\frac{x}{\delta_{e}})\Big{(}\cos(\omega
t-\frac{x}{\delta_{e}})+A\sin(\omega
t-\frac{x}{\delta_{e}})\Big{)}+\frac{1}{\sigma}\partial_{t}\vec{E}_{a},$ (9)

with arbitrary field strength $\vec{E}_{1}$, parameter $A$ and frequency
$\omega$. The skin depth $\delta_{e}$ is given by
$\delta_{e}=\sqrt{2/\sigma\omega}$. In the derivation, we have neglected the
term $\partial_{t}^{2}\vec{E}(\mbox{inside})$ in the left hand side of eq(8),
which is much smaller than the term $\sigma\partial_{t}\vec{E}(\mbox{inside})$
in the right hand side. Namely, we have used the inequality
$\sigma\gg\omega\,(\sim m_{a})$ because the electric conductivity $\sigma\sim
10^{4}$eV of copper is much larger than the axion mass $m_{a}=10^{-6}$eV $\sim
10^{-3}$eV under consideration. We note that the parameter
$m_{a}\delta_{e}=\sqrt{2m_{a}/\sigma}$ is much small such as
$m_{a}\delta_{e}\sim 10^{-4}$ for $m_{a}\sim 10^{-5}\rm eV$ and $\sigma\sim
10^{3}$eV.

The first term of the solution represents radiation non propagating in $z$ and
$y$ directions. It is only present in the surface of the conductor due to the
skin effect. It induces electric current $\sigma\vec{E}(\mbox{inside})$ on the
surface. Although $\omega$ is arbitrary at this stage, it turns out from
boundary conditions at $x=0,-l$ that it is equal to $m_{a}$. Hence, we put
$\omega=m_{a}$ hereafter. Furthermore, the boundary conditions make the
electric field $\vec{E}(\mbox{inside})$ be parallel to $\vec{B}$;
$\vec{E}_{1}=(0,0,E_{1})$ just as $\vec{E}_{a}=-g_{a\gamma\gamma}a(t)\vec{B}$.

The second term represents the oscillating electric field
$\frac{1}{\sigma}\partial_{t}\vec{E}_{a}$ present uniformly inside the
conductor. This term is absent when we consider conductor with finite length
in $z$ direction because electric charges on two ends of upper and lower
surfaces induced by the electric field screen the field
$\frac{1}{\sigma}\partial_{t}\vec{E}_{a}$. Hereafter, we neglect the term
$\frac{1}{\sigma}\partial_{t}\vec{E}_{a}$.

We also have magnetic field
$\vec{B}(\mbox{inside})=(0,B_{y}(\mbox{inside}),0)$ corresponding to the
electric field $\vec{E}(\mbox{inside})=(0,0,E_{z}(\mbox{inside}))$ because
$\vec{\partial}\times\vec{E}(\mbox{inside})=-\partial_{t}\vec{B}(\mbox{inside})$,

$\displaystyle B_{y}(\mbox{inside})$ $\displaystyle=$
$\displaystyle-\frac{E_{1}}{\delta_{e}m_{a}}\exp\big{(}-\frac{x}{\delta_{e}}\big{)}\Big{(}\sin(m_{a}t-\frac{x}{\delta_{e}})+\cos(m_{a}t-\frac{x}{\delta_{e}})\Big{)}$
(10) $\displaystyle-$
$\displaystyle\frac{AE_{1}}{\delta_{e}m_{a}}\exp\big{(}-\frac{x}{\delta_{e}}\big{)}\Big{(}\sin(m_{a}t-\frac{x}{\delta_{e}})-\cos(m_{a}t-\frac{x}{\delta_{e}})\Big{)}.$

As a whole, we find the magnetic field $\vec{B}(\mbox{inside})+\vec{B}$ as
well as the electric field $\vec{E}(\mbox{inside})$ inside the conductor (
$x>0$ ).

The strength $\vec{E}_{1}$ ( frequency $\omega$ and parameter $A$ ) is
determined by the boundary conditions at the surfaces $x=0$ and $x=-l$. The
boundary conditions we impose demand that the parallel components of electric
and magnetic fields in the both sides of the boundary must be continuous. That
is, $\vec{E}(\mbox{inside})=\vec{E}_{a}+\vec{E}(\mbox{outside})$ and
$\vec{B}(\mbox{inside})=\vec{B}(\mbox{outside})$, where both of the fields
$\vec{E}(\mbox{outside})$ and $\vec{B}(\mbox{outside})$ represent standing
waves arising in the space between two slabs. It implies that we only consider
the mode of electric field
$\vec{E}(\mbox{outside})=(0,0,E_{z}(\mbox{outside}))$ and magnetic field
$\vec{B}(\mbox{outside})=(0,B_{y}(\mbox{outside}),0)$ arising between two
slabs. They represent the lowest TM mode non propagating in $z$ and $y$
direction. They depends only on the coordinate $x$ and $t$. Then, the
solutions of the Maxwell equations in vacuum representing the mode are

$\displaystyle E_{z}(\mbox{outside})$ $\displaystyle=$ $\displaystyle-
b_{1}\cos(m_{a}x+\delta_{1})\cos(m_{a}t)+b_{2}\cos(m_{a}x+\delta_{2})\sin(m_{a}t)$
(11) $\displaystyle B_{y}(\mbox{outside})$ $\displaystyle=$ $\displaystyle
b_{1}\sin(m_{a}x+\delta_{1})\sin(m_{a}t)+b_{2}\sin(m_{a}x+\delta_{2})\cos(m_{a}t)$
(12)

with constants $b_{1,2}$ and $\delta_{1,2}$. Additionally, we have the
electric field
$\vec{E}_{a}=-g_{a\gamma\gamma}a(t)\vec{B}=\big{(}0,0,-g_{a\gamma\gamma}a_{0}\cos(m_{a}t)B\big{)}\equiv\big{(}0,0,E_{a}\cos(m_{a}t)\big{)}$
induced by axion between the slabs; $E_{a}\equiv-g_{a\gamma\gamma}a_{0}B$.

We determine the constants, $E_{1}$, $A$, $b_{1,2}$ and $\delta_{1,2}$ by
imposing the boundary conditions,
$E_{z}(\mbox{inside})=E_{z}(\mbox{outside})+E_{a}$,
$B_{y}(\mbox{inside})=B_{y}(\mbox{outside})$ at $x=0$.

$\displaystyle b_{1}\sin\delta_{1}$ $\displaystyle=$
$\displaystyle-\frac{E_{1}(1+A)}{m_{a}\delta_{e}},\quad
b_{2}\sin\delta_{2}=-\frac{E_{1}(1-A)}{m_{a}\delta_{e}}$ $\displaystyle
E_{a}-b_{1}\cos\delta_{1}$ $\displaystyle=$ $\displaystyle E_{1},\quad
b_{2}\cos\delta_{2}=AE_{1}.$ (13)

Additionally, by imposing the boundary condition at $x=-l$, we obtain

$\displaystyle b_{1}\sin(\delta_{1}-m_{a}l)$ $\displaystyle=$
$\displaystyle+\frac{E_{1}(1+A)}{m_{a}\delta_{e}}\quad
b_{2}\sin(\delta_{2}-m_{a}l)=+\frac{E_{1}(1-A)}{m_{a}\delta_{e}}$ (14)
$\displaystyle E_{a}-b_{1}\cos(\delta_{1}-m_{a}l)$ $\displaystyle=$
$\displaystyle E_{1}\quad b_{2}\cos(\delta_{2}-m_{a}l)=AE_{1}$ (15)

In order to obtain the relations from the boundary condition at $x=-l$, we
note that the solutions of the electric and magnetic fields inside the
conductor $x\leq-l$ are given by

$\displaystyle E_{z}(\mbox{inside},l)$ $\displaystyle=$
$\displaystyle+E_{1}\exp\Big{(}\frac{x+l}{\delta_{e}}\Big{)}\Big{(}\cos(\omega
t+\frac{x+l}{\delta_{e}})+A\sin(\omega t+\frac{x+l}{\delta_{e}})\Big{)}$
$\displaystyle B_{y}(\mbox{inside},l)$ $\displaystyle=$
$\displaystyle+\frac{E_{1}}{\delta_{e}m_{a}}\exp\big{(}\frac{x+l}{\delta_{e}}\big{)}\Big{(}\sin(m_{a}t+\frac{x+l}{\delta_{e}})+\cos(m_{a}t+\frac{x+l}{\delta_{e}})\Big{)}$
(16) $\displaystyle+$
$\displaystyle\frac{AE_{1}}{\delta_{e}m_{a}}\exp\big{(}\frac{x+l}{\delta_{e}}\big{)}\Big{(}\sin(m_{a}t+\frac{x+l}{\delta_{e}})-\cos(m_{a}t+\frac{x+l}{\delta_{e}})\Big{)}$
for $\displaystyle\quad x\leq-l$

Therefore, we find that the coefficients, $E_{1}$ and $AE_{1}$ as well as
$b_{1,2}$ are given such that

$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle
E_{a}\frac{(m_{a}\delta_{e}\tan\delta)^{2}-m_{a}\delta_{e}\tan\delta}{1+(m_{a}\delta_{e}\tan\delta-1)^{2}}\quad
E_{1}A=-E_{a}\frac{m_{a}\delta_{e}\tan\delta}{1+(m_{a}\delta_{e}\tan\delta-1)^{2}}$
$\displaystyle\quad b_{1}$ $\displaystyle=$
$\displaystyle\frac{E_{a}}{\cos\delta}\,\frac{2-m_{a}\delta_{e}\tan\delta}{1+(m_{a}\delta_{e}\tan\delta-1)^{2}},\quad
b_{2}=-\frac{E_{a}}{\cos\delta}\,\frac{m_{a}\delta_{e}\tan\delta}{1+(m_{a}\delta_{e}\tan\delta-1)^{2}},$
(17)

and the phases $\delta_{1,2}=\delta$ satisfying the identical equation,

$\tan\delta=\frac{\sin(m_{a}l)}{1+\cos(m_{a}l)}.$ (18)

Therefore, the coefficients behave in the limit of large electrical
conductivity $\sigma$ ( small skin depth $m_{a}\delta_{e}\ll 1$ ) such that

$E_{1}=E_{a}\times O(m_{a}\delta_{e}),\quad A=-1+O(m_{a}\delta_{e}),\quad
b_{1}=\frac{E_{a}}{\cos\delta_{1}}\Big{(}1+O(m_{a}\delta_{e})\Big{)},\quad
b_{2}=-\frac{E_{a}}{\cos\delta_{1}}O(m_{a}\delta_{e}).$ (19)

We find that the standing waves in eq(11) of the electromagnetic fields in the
space between the two slabs are given by in the limit $m_{a}\delta_{e}\to 0$ (
$\delta_{e}=\sqrt{2/m_{a}\sigma}$ ) for $\sigma\to\infty$ ,

$\displaystyle E_{z}(\mbox{outside})$ $\displaystyle=$
$\displaystyle-\frac{E_{a}}{\cos\delta}\cos(m_{a}x+\delta)\cos(m_{a}t)$ (20)
$\displaystyle B_{y}(\mbox{outside})$ $\displaystyle=$
$\displaystyle\frac{E_{a}}{\cos\delta}\sin(m_{a}x+\delta)\sin(m_{a}t).$ (21)

In addition to the standing waves, we have electric field
$E_{z}(\mbox{inside})$ in the conductor,

$\vec{E}(\mbox{inside})=(0,0,E_{z}),\quad
E_{z}=-\frac{E_{a}m_{a}\delta_{e}\tan\delta}{\sqrt{2}}\exp(-\frac{x}{\delta_{e}})\sin(m_{a}t-\frac{x}{\delta_{e}}+\frac{\pi}{4})$
(22)

which generates electric current $J=\sigma E_{z}(\mbox{inside})$ in the
surface of the conductor. These are solutions in the limit of perfect
conductor.

Because $\cos\delta_{1}$ is of the order of $1$, we find that the strengths of
the standing waves in the space between the two slabs are identical to the
electric field $E_{a}$ generated by the axion in vacuum. It is similar to the
case in resonant cavity, as has been pointed out in our previous paperiwazaki1
.

We note that in the limit of perfect conductor with $\sigma\to\infty$, the
electric field $E_{a}+E_{z}(\mbox{outside})$ present between the slabs
vanishes at the boundaries at $x=0$ and $x=-l$. Similarly we find that the
electric field $E(\mbox{inside})\sim E_{a}\times m_{a}\delta_{e}$ inside the
conductors vanishes in the limit of the perfect conductor. The magnetic field
$B_{y}(\mbox{inside})\propto\exp(-x/\delta_{e})$ also vanishes. On the other
hand, the magnetic field $B_{y}(\mbox{outside})$ does not vanish at the
boundaries, $B_{y}(\mbox{outside})=E_{a}\tan\delta_{1}\sin(m_{a}t)$ at $x=0$
and $x=-l$. The discontinuity takes place because of the presence of surface
current. For instance, the surface current at $x=0$ is given by

$J_{a}=\sigma E(\mbox{inside})\to-
E_{a}\tan\delta\sin(m_{a}t)\delta(x)\quad\mbox{for}\quad\sigma\to\infty.$ (23)

This surface current generates the discontinuity $\Delta B\equiv
B_{y}(\mbox{outside})-B_{y}(\mbox{inside})=E_{a}\tan\delta\sin(m_{a}t)$ at the
surface.

We would like to make a comment on the resultsiwazaki  in our previous papers,
in which electromagnetic radiations from a cylindrical conductor converted
from the dark matter axion have been discussed. In the papers we have naively
derived electric current flowing its surface induced by the axion without
appropriately imposing the boundary conditions; the boundary conditions are
carefully treated in the present paper as well as our paperiwazaki1 . As a
result, the radiations from the conductor have been overestimated. It has
already been indicated in the paperiwazaki1 . Their electric fields do not
vanish at the surface even in the limit of perfect conductor. The careful
treatmentiwazaki2  of the boundary conditions lead to physically reasonable
view: The radiations $\vec{E}_{a}$ arising in vacuum from the axion under
magnetic field collides the conductor and reflected radiations are just ones
emitted from the conductor. The reflected radiations are never enhanced
compared with the incident radiations $\vec{E}_{a}$. The identical conclusion
has also been obtained in the paperkishimoto  by Kishimoto and Nakayama.

Now we show that a resonance occurs when we adjust spacing $l$ just as
$l=\pi/m_{a}$ in the case of finite $\sigma$. ( The results shown below hold
even for $l=\pi(2n+1)/m_{a}$ with non negative integer $n$. ) The resonance
implies that electric and magnetic field of radiation is enhanced compared
with the radiation in eq(20). With the choice of $l=\pi/m_{a}$, we obtain
$\delta=\pi/2$ so that $A=0,\quad E_{1}=E_{a},\quad
b_{1,2}=-\frac{E_{a}}{m_{a}\delta_{e}}$. Therefore, the radiation fields
$E_{z}(\mbox{outside})$ and $B_{y}(\mbox{outside})$ are enhanced with the
enhancement factor $1/m_{a}\delta_{e}$,

$\displaystyle E_{z}(\mbox{outside})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}E_{a}}{m_{a}\delta_{e}}\sin(m_{a}x)\sin(m_{a}t-\frac{\pi}{4})$
(24) $\displaystyle B_{y}(\mbox{outside})$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}E_{a}}{m_{a}\delta_{e}}\cos(m_{a}x)\sin(m_{a}t+\frac{\pi}{4}).$
(25)

We also have electric field inside the conductor $x>0$

$E_{z}(\mbox{inside})=E_{a}\exp(-\frac{x}{\delta_{e}})\cos(m_{a}t-\frac{x}{\delta_{e}}),$
(26)

which induces large oscillating electric current $\sigma
E_{a}\exp(-\frac{x}{\delta_{e}})\cos(m_{a}t-\frac{x}{\delta_{e}})$ in the
surface.

We estimate the electric current $I_{a}(m_{a}t)=L\int_{0}^{\infty}dx\sigma
E_{a}\exp(-x/\delta_{e})\cos(m_{a}t-x/\delta_{e})=L\sigma\delta_{e}E_{a}\sin(m_{a}t+\pi/4)/\sqrt{2}\equiv
I_{a}\sin(m_{a}t+\pi/4)$ with the length $L$ of the slab in $y$ direction,

$I_{a}=\frac{\sigma\delta_{e}E_{a}L}{\sqrt{2}}=\frac{\sigma\delta_{e}g_{a\gamma\gamma}a_{0}BL}{\sqrt{2}}=0.7\times
10^{-9}\mbox{A}\Big{(}\frac{10^{-5}\rm
eV}{m_{a}}\Big{)}^{1/2}\Big{(}\frac{B}{5\rm T}\Big{)}\Big{(}\frac{L}{10\rm
cm}\Big{)}\Big{(}\frac{\sigma}{3.3\times 10^{7}\rm eV}\Big{)}$ (27)

where we have adopted high quality $6$N copper with electrical conductivity
$\sigma\simeq 3.3\times 10^{7}$eV at low temperature of the order of $1$K. We
have used the parameters of the dark matter axion energy density
$\rho_{a}=0.3\mbox{GeV}\,\mbox{cm}^{-3}=m_{a}^{2}a_{0}^{2}/2$ and the axion
model $g_{\gamma}=1$ ( $g_{a\gamma\gamma}=g_{\gamma}\alpha/\pi f_{a}$ ). These
values are used throughout the paper.

The power of Joule heating by the electric current $I_{a}$ is given by

$\displaystyle P_{a}$ $\displaystyle=$ $\displaystyle LR\sigma
E_{a}^{2}\int_{0}^{\infty}dx\exp(-\frac{2x}{\delta_{e}})\,\overline{\cos^{2}(m_{a}t-x/\delta_{e})}=\frac{LR\delta_{e}\sigma
E_{a}^{2}}{4}$ (28) $\displaystyle\simeq$ $\displaystyle 0.3\times
10^{-22}\mbox{W}\Big{(}\frac{10^{-5}\rm
eV}{m_{a}}\Big{)}^{1/2}\Big{(}\frac{B}{5\rm
T}\Big{)}^{2}\Big{(}\frac{\sigma}{3.3\times 10^{7}\rm
eV}\Big{)}\Big{(}\frac{L}{10\rm cm}\Big{)}\Big{(}\frac{R}{10\rm cm}\Big{)}$

where $R$ denotes a side length of the slab ( the surface area is $R\times L$
). $LR\delta_{e}$ is the volume of the conductor in which the electric current
flows. $\overline{O}$ is time average of $O$. In order to detect the electric
current, we amplify the power $P_{a}$ using LC circuitsikivie ; kishimoto .

Because thermal noise of the system with temperature $T=1$K is given by

$P_{T}=\frac{T\delta\omega}{2\pi}\simeq 3.3\times
10^{-20}\mbox{W}\Big{(}\frac{T}{1\rm
K}\Big{)}\Big{(}\frac{\delta\omega}{10^{-6}m_{a}}\Big{)}\Big{(}\frac{m_{a}}{10^{-5}\rm
eV}\Big{)},$ (29)

the signal to noise ratio is given by

$\displaystyle\frac{S}{N}$ $\displaystyle=$
$\displaystyle\frac{P_{a}}{P_{T}}Q_{LC}\sqrt{\frac{\delta\omega
t_{obs}}{2\pi}}$ (30) $\displaystyle\simeq$ $\displaystyle 4.5\times
10^{-2}Q_{LC}\sqrt{\frac{t_{obs}}{1\rm sec}}\Big{(}\frac{10^{-5}\rm
eV}{m_{a}}\Big{)}\Big{(}\frac{B}{5\rm
T}\Big{)}^{2}\Big{(}\frac{\sigma}{3.3\times 10^{7}\rm
eV}\Big{)}\Big{(}\frac{L}{10\rm cm}\Big{)}\Big{(}\frac{R}{10\rm
cm}\Big{)}\Big{(}\frac{1\rm
K}{T}\Big{)}\Big{(}\frac{10^{-6}m_{a}}{\delta\omega}\Big{)}^{1/2}$

where $Q_{LC}$ denotes $Q$ factor in LC circuit which is used to amplify the
power $P_{a}$. $t_{obs}$ is the observation time. $Q_{LC}$ is given such that
$Q_{LC}=\frac{1}{R_{LC}}\sqrt{L_{i}/C}$ with inductance $L_{i}$ of coil,
capacitance $C$ of condenser and resistance $R_{LC}$ of the circuit.
Especially, $R_{LC}=R/(\sigma L\delta_{e})$. Thus, we find

$Q_{LC}=\sqrt{\frac{2\sigma}{m_{a}}}\times\sqrt{\frac{L_{i}}{C}}\Big{(}\frac{L}{R}\Big{)}\simeq
2.5\times
10^{6}\sqrt{\frac{10^{-5}\mbox{eV}}{m_{a}}}\sqrt{\frac{L_{i}}{C}}\Big{(}\frac{L}{R}\Big{)}.$
(31)

Therefore, we obtain sufficiently large $S/N$ ratio to detect the electric
current induced by the dark matter axion, even if the observational time
$t_{obs}$ is $1$ second.

We would like to point out that $Q$ value associated with the absorption by
the flat conductors is derived using the definition $Q=m_{a}U/2P_{a}$ with the
energy $U$ of radiations in the space between two slabs,

$U\equiv\frac{1}{2}\int(\overline{E_{z}^{2}(\mbox{outside})}+\overline{B_{y}^{2}(\mbox{ouside})})dV=\frac{LRE_{a}^{2}l}{2(m_{a}\delta_{e})^{2}}$
(32)

with $l=\pi/m_{a}$. We find that

$Q=\frac{m_{a}l}{\delta_{e}\sigma(m_{a}\delta_{e})^{2}}=\frac{\pi}{4}\sqrt{\frac{2\sigma}{m_{a}}}\simeq
2\times 10^{6}\sqrt{\frac{10^{-5}\mbox{eV}}{m_{a}}}$ (33)

The formula is almost identical to the $Q_{LC}$ factor in the LC circuit.

In order to obtain the high value of the electric current $I_{a}$, we need to
exactly tune the spacing $l=\pi/m_{a}$ between two slabs. But, such an exact
tuning would be not possible because $\delta_{e}=\sqrt{2/m_{a}\sigma}\simeq
1.6\times 10^{-6}\mbox{cm}\sqrt{10^{-5}\mbox{eV}/m_{a}}$. In actual experiment
we tune it approximately. Thus, we need to see electric current and its power
in the approximate tuning.

We note that in general, electric current $I_{a}(l)$ flowing the slab is given
by

$\displaystyle I_{a}(l)$ $\displaystyle=$ $\displaystyle L\sigma
E_{1}\int_{0}^{\infty}dx\exp(-\frac{x}{\delta_{e}})\Big{(}\cos(m_{a}t-x/\delta_{e})+A\sin(m_{a}t-x/\delta_{e})\Big{)}$
(34) $\displaystyle=$
$\displaystyle\frac{L\sigma\delta_{e}E_{1}}{\sqrt{2}}\Big{(}\sin(m_{a}t+\frac{\pi}{4})+A\sin(m_{a}t-\frac{\pi}{4})\Big{)},$

and the power of Joule heating $P_{a}(l)$ is

$\displaystyle P_{a}(l)$ $\displaystyle=$ $\displaystyle LR\sigma
E_{1}^{2}\int_{0}^{\infty}dx\exp(-\frac{2x}{\delta_{e}})\overline{\Big{(}\cos(m_{a}t-x/\delta_{e})+A\sin(m_{a}t-x/\delta_{e})\Big{)}^{2}}$
(35) $\displaystyle=$
$\displaystyle\frac{LR\delta_{e}\sigma(E_{1}^{2}+(E_{1}A)^{2})}{4},$

where the parameters $E_{1}$ and $A$ depend on the spacing $l$ between two
slabs.

When we take the spacing $l=\pi/m_{a}+\delta l$ with $\delta l\ll\pi/m_{a}$ (
then, $\tan\delta\simeq-2/(m_{a}\delta l)$ ), the parameters $E_{1}$ and
$E_{1}A$ are found such as

$E_{1}=-\frac{E_{a}(y+1)}{2y^{2}+2y+1},\quad
E_{1}A=\frac{E_{a}y}{2y^{2}+2y+1}\quad\mbox{with}\quad y\equiv\delta
l/2\delta_{e}.$ (36)

The tuning $y\ll 1$ would be very difficult because $\delta_{e}\simeq
1.6\times 10^{-6}\mbox{cm}\sqrt{10^{-5}\mbox{eV}/m_{a}}$. But even if we tune
$y$ such as $y\sim 10^{2}$, i.e. $\delta l\sim 10^{-4}$cm, we obtain electric
current $I_{a}(l=\pi/m_{a}+\delta l)=-I_{a}(\delta l)\cos(m_{a}t)$,

$I_{a}(\delta l)=\frac{\sigma\delta_{e}E_{a}L}{2y}\simeq 1.4\times
10^{-11}\mbox{A}\Big{(}\frac{10^{-4}\rm cm}{\delta
l}\Big{)}\Big{(}\frac{10^{-5}\rm eV}{m_{a}}\Big{)}\Big{(}\frac{B}{5\rm
T}\Big{)}\Big{(}\frac{L}{10\rm cm}\Big{)}\Big{(}\frac{\sigma}{3.3\times
10^{7}\rm eV}\Big{)},$ (37)

and the corresponding power $P_{a}(\delta l)\equiv P_{a}(l=\pi/m_{a}+\delta
l)$,

$P_{a}(\delta l)=\frac{LR\delta_{e}\sigma E_{a}^{2}}{8y^{2}}\simeq 1.5\times
10^{-26}\mbox{W}\Big{(}\frac{10^{-4}\rm cm}{\delta
l}\Big{)}^{2}\Big{(}\frac{10^{-5}\rm
eV}{m_{a}}\Big{)}^{3/2}\Big{(}\frac{B}{5\rm
T}\Big{)}^{2}\Big{(}\frac{\sigma}{3.3\times 10^{7}\rm
eV}\Big{)}.\Big{(}\frac{L}{10\rm cm}\Big{)}\Big{(}\frac{R}{10\rm cm}\Big{)}$
(38)

Then, we find the signal to noise ratio $S(\delta l)/N$

$\displaystyle\frac{S(\delta l)}{N}=\frac{P_{a}(\delta
l)}{P_{T}}Q_{LC}\sqrt{\frac{\delta\omega t_{obs}}{2\pi}}$ (39)
$\displaystyle\simeq$ $\displaystyle 2.3\times
10^{-5}Q_{LC}\sqrt{\frac{t_{obs}}{1\rm sec}}\Big{(}\frac{10^{-4}\rm cm}{\delta
l}\Big{)}^{2}\Big{(}\frac{10^{-5}\rm eV}{m_{a}}\Big{)}^{2}\Big{(}\frac{B}{5\rm
T}\Big{)}^{2}\Big{(}\frac{\sigma}{3.3\times 10^{7}\rm
eV}\Big{)}\Big{(}\frac{L}{10\rm cm}\Big{)}\Big{(}\frac{R}{10\rm
cm}\Big{)}\Big{(}\frac{1\rm
K}{T}\Big{)}\Big{(}\frac{10^{-6}m_{a}}{\delta\omega}\Big{)}^{1/2}$

where $Q_{LC}=\frac{1}{R_{LC}}\sqrt{L_{i}/C}\sim
10^{6}\sqrt{L_{i}/C}\sqrt{10^{-5}\mbox{eV}/m_{a}}$ with the use of $L=10$cm
and $R=10$cm.

In the actual experiment with the use of $Q_{LC}=10^{6}$, we can search the
spacing $\Delta l\sim 6$cm within $1$ day, if we tune the spacing with the
speed $\partial_{t}\delta l\sim 10^{-4}\rm cm/s$ ( $t_{obs}=1$sec for $S/N>1$
), It covers the range of the QCD axion mass $m_{a}\sim 10^{-5}\mbox{eV}$
because $\pi/m_{a}\sim 6$cm. On the other hand, in order to cover the range of
the axion mass $\sim 10^{-4}$eV ( $\pi/m_{a}\sim 6\times 10^{-1}$cm ), we can
search it for $10$ days if we tune the spacing with the speed
$\partial_{t}\delta l\sim 10^{-4}\rm cm/10^{2}s$ ( $t_{obs}=10^{2}$sec for
$S/N>1$ ). But, to cover the range of $m_{a}=10^{-3}$eV, we need
$t_{obs}=10^{6}$sec for $S/N>1$ so that it takes more than $10$ years to
search the spacing $\Delta l\sim 0.06$cm. However, if we can tune the spacing
$\delta l=10^{-5}$cm, by taking $\partial_{t}\delta l\sim 10^{-5}\rm
cm/10^{2}s$ ( $t_{obs}=10^{2}$sec for $S/N>1$ ), we can search the spacing
$\Delta l\sim 0.06$cm within $10$ day even for $m_{a}=10^{-3}$eV.

On the other hand, when we search the mass $\sim 3\times 10^{-6}$eV (
$\pi/m_{a}\sim 20$cm ), we need larger slab such as $L=R=40$cm so as for our
approximation of infinitely extending conductor to hold. Then, the power of
Joule heating or $S/N$ ratio is much bigger so that the detection is much
easier because we can take $\delta l=10^{-3}$cm and $t_{obs}=1$sec, as we can
see from eq(39). In this way, it turns out that the new way of the axion
detection is very effective for the QCD axion search.

The author expresses thanks to Yasuhiro Kishimoto and Kazunori Nakayama for
useful comments and discussions. He also expresses thanks to Ngyuen, Le Hoang
for useful suggestions. This work is supported in part by Grant-in-Aid for
Scientific Research ( KAKENHI ), No.19K03832.

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