# Submanifolds of Generalized Sasakian-space-forms with respect to certain
connections

Pradip Mandal, Shyam Kishor and Shyamal Kumar Hui∗

###### Abstract.

The present paper deals with some results of submanifolds of generalized
Sasakian-space-forms in [3] with respect to semisymmetric metric connection,
semisymmetric non-metric connection, Schouten-van Kampen connection and
Tanaka-webster connection.

###### Key words and phrases:

generalized Sasakian-space-forms, semisymmetric metric connection,
semisymmetric non-metric connection, Schouten-van Kampen Connection, Tanaka-
Webster connection.
* corresponding author

###### 2010 Mathematics Subject Classification:

53C15, 53C40

##  1\. Introduction

As a generalization of Sasakian-space-form, Alegre et al. [2] introduced the
notion of generalized Sasakian-space-form as that an almost contact metric
manifold $\bar{M}(\phi,\xi,\eta,g)$ whose curvature tensor $\bar{R}$ of
$\bar{M}$ satisfies

(1.1) $\displaystyle\bar{R}(X,Y)Z$
$\displaystyle=f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y$ $\displaystyle-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi
Z\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$

for all vector fields $X$, $Y$, $Z$ on $\bar{M}$ and $f_{1},f_{2},f_{3}$ are
certain smooth functions on $\bar{M}$. Such a manifold of dimension $(2n+1)$,
$n>1$ (the condition $n>1$ is assumed throughout the paper), is denoted by
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ [2]. Many authors studied this space form
with different aspects. For this, we may refer ([11], [12], [13], [14], [15],
[17], [18] and [23]). It reduces to Sasakian-space-form if
$f_{1}=\frac{c+3}{4}$, $f_{2}=f_{3}=\frac{c-1}{4}$ [2].

After introducing the semisymmetric linear connection by Friedman and Schouten
[7], Hayden [9] gave the idea of metric connection with torsion on a
Riemannian manifold. Later, Yano [29] and many others (see, [21], [22], [24]
and references therein) studied semisymmetric metric connection in different
context. The idea of semisymmetric non-metric connection was introduced by
Agashe and Chafle [1].

The Schouten-van Kampen connection introduced for the study of non-holomorphic
manifolds ([20], [27]). In $2006$, Bejancu [6] studied Schouten-van Kampen
connection on foliated manifolds. Recently Olszak [19] studied Schouten-van
Kampen connection on almost(para) contact metric structure.

The Tanaka-Webster connection ([25], [28]) is the canonical affine connection
defined on a non-degenerate pseudo-Hermitian CR-manifold. Tanno [26] defined
the Tanaka-Webster connection for contact metric manifolds.

The submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ are studied in ([3],
[10], [16]). In [3], Alegre and Carriazo studied submanifolds of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to Levi-Civita connection
$\bar{\nabla}$. The present paper deals with study of such submanifolds of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to semisymmetric metric
connection, semisymmetric non-metric connection, Schouten-van Kampen
connection and Tanaka-webster connection respectively.

##  2\. preliminaries

In an almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$, we have [4]

(2.1) $\displaystyle\phi^{2}(X)=-X+\eta(X)\xi,\ \phi\xi=0,$ (2.2)
$\displaystyle\eta(\xi)=1,\ g(X,\xi)=\eta(X),\ \eta(\phi X)=0,$ (2.3)
$\displaystyle g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y),$ (2.4) $\displaystyle
g(\phi X,Y)=-g(X,\phi Y).$

In $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$, we have [2]

(2.5)
$\displaystyle(\bar{\nabla}_{X}\phi)(Y)=(f_{1}-f_{3})[g(X,Y)\xi-\eta(Y)X],$
(2.6) $\displaystyle\bar{\nabla}_{X}\xi=-(f_{1}-f_{3})\phi X,$

where $\bar{\nabla}$ is the Levi-Civita connection of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$.

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$. If $\nabla$
and $\nabla^{\perp}$ are the induced connections on the tangent bundle $TM$
and the normal bundle $T^{\perp}{M}$ of $M$, respectively then the Gauss and
Weingarten formulae are given by [30]

(2.7) $\displaystyle\bar{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y),\
\bar{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{\perp}V$

for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and $A_{V}$
are second fundamental form and shape operator (corresponding to the normal
vector field V), respectively and they are related by [30]
$g(h(X,Y),V)=g(A_{V}X,Y)$.

For any $X\in\Gamma(TM)$, we may write

(2.8) $\phi X=TX+FX,$

where $TX$ is the tangential component and $FX$ is the normal component of
$\phi X$.

In particular, if $F=0$ then $M$ is invariant [5] and here $\phi(TM)\subset
TM$. Also if $T=0$ then $M$ is anti-invariant [5] and here $\phi(TM)\subset
T^{\bot}M$. Also here we assume that $\xi$ is tangent to $M$.

The semisymmetric metric connection $\widetilde{\bar{\nabla}}$ and the
Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$
are related by [29]

(2.9)
$\displaystyle\widetilde{\bar{\nabla}}_{X}Y=\bar{\nabla}_{X}Y+\eta(Y)X-g(X,Y)\xi.$

The Riemannian curvature tensor $\widetilde{\bar{R}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$
is

(2.10) $\displaystyle\widetilde{\bar{R}}(X,Y)Z$
$\displaystyle=(f_{1}-1)\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y-g(Y,\phi Z)\phi X$ $\displaystyle+2g(X,\phi Y)\phi
Z\big{\\}}+(f_{3}-1)\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi$
$\displaystyle-g(Y,Z)\eta(X)\xi\big{\\}}+(f_{1}-f_{3})\\{g(X,\phi Z)Y-g(Y,\phi
Z)X+g(Y,Z)\phi X$ $\displaystyle-g(X,Z)\phi Y\\}.$

The semisymmetric non-metric connection ${\bar{\nabla}}^{{}^{\prime}}$ and the
Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$
are related by [1]

(2.11)
$\displaystyle\bar{\nabla}^{{}^{\prime}}_{X}Y=\bar{\nabla}_{X}Y+\eta(Y)X.$

The Riemannian curvature tensor ${\bar{R}}^{{}^{\prime}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
${\bar{\nabla}}^{{}^{\prime}}$ is

$\displaystyle{\bar{R}}^{{}^{\prime}}(X,Y)Z$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi Z)\phi Y$
$\displaystyle-$ $\displaystyle g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi
Z\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(Z)Y$ $\displaystyle-$
$\displaystyle\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})[g(X,\phi Z)Y-g(Y,\phi Z)X]$
$\displaystyle+$ $\displaystyle\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y.$

The Schouten-van Kampen connection $\hat{\bar{\nabla}}$ and the Riemannian
connection $\bar{\nabla}$ of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related
by [19]

(2.13)
$\displaystyle\hat{\bar{\nabla}}_{X}Y=\bar{\nabla}_{X}Y+(f_{1}-f_{3})\eta(Y)\phi
X-(f_{1}-f_{3})g(\phi X,Y)\xi.$

The Riemannian curvature tensor $\hat{\bar{R}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is

(2.14) $\displaystyle\hat{\bar{R}}(X,Y)Z$
$\displaystyle=f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y$ $\displaystyle-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi
Z\big{\\}}+\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
$\displaystyle+(f_{1}-f_{3})^{2}\big{[}g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi
X\big{]}.$

The Tanaka-Webster connection $\stackrel{{\scriptstyle\ast}}{{\bar{\nabla}}}$
and the Riemannian connection $\bar{\nabla}$ of
${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [8]

(2.15)
$\displaystyle\stackrel{{\scriptstyle\ast}}{{\bar{\nabla}}}_{X}Y=\bar{\nabla}_{X}Y+\eta(X)\phi
Y+(f_{1}-f_{3})\eta(Y)\phi X-(f_{1}-f_{3})g(\phi X,Y)\xi.$

The Riemannian curvature tensor $\stackrel{{\scriptstyle*}}{{\bar{R}}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ is

(2.16) $\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$
$\displaystyle=f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y-g(Y,\phi Z)\phi X$ $\displaystyle+2g(X,\phi Y)\phi
Z\big{\\}}+\\{f_{3}+{(f_{1}-f_{3})^{2}}\\}\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$
$\displaystyle+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}+{(f_{1}-f_{3})^{2}}\big{[}g(X,\phi
Z)\phi Y$ $\displaystyle-g(Y,\phi Z)\phi X\big{]}+2(f_{1}-f_{3})g(X,\phi
Y)\phi Z.$

##  3\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
$\widetilde{\bar{\nabla}}$

######  Lemma 3.1.

If $M$ is invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)Z$ is
tangent to $M$, for any $X,Y,Z\in\Gamma(TM)$.

###### Proof.

If $M$ is invariant then from (2.10) we say that $\widetilde{\bar{R}}(X,Y)Z$
is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$. This
proves the lemma. ∎

######  Lemma 3.2.

If $M$ is anti-invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$
with respect to $\widetilde{\bar{\nabla}}$, then

(3.1) $\displaystyle tan(\widetilde{\bar{R}}(X,Y)Z)$
$\displaystyle=(f_{1}-1)\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+(f_{3}-1)\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}},$
(3.2) $\displaystyle nor(\widetilde{\bar{R}}(X,Y)Z)$ $\displaystyle=$
$\displaystyle(f_{1}-f_{3})\\{g(Y,Z)\phi X-g(X,Z)\phi Y\\}$

for any $X,Y,Z\in\Gamma(TM)$.

###### Proof.

Since $M$ is anti-invariant, we have $\phi X,\phi Y\in\Gamma(T^{\bot}M)$. Then
equating tangent and normal component of (2.10) we get the result. ∎

######  Lemma 3.3.

If $f_{1}(p)=f_{3}(p)$ and $M$ is either invariant or anti-invariant
submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$
for any $X,Y,Z\in\Gamma(TM)$.

###### Proof.

Using Lemma $3.1$ and Lemma $3.2$ we get the result. ∎

######  Lemma 3.4.

If $M$ is invariant or anti-invariant submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)V$ is normal to $M$,
for any $X,Y,\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

###### Proof.

If $M$ is invariant from (2.10) we have $\widetilde{\bar{R}}(X,Y)V$ normal to
$M$, and if $M$ is anti-invariant then $\widetilde{\bar{R}}(X,Y)V=0$ i.e.
$\widetilde{\bar{R}}(X,Y)V$ normal to $M$ for any $X,Y,\in\Gamma(TM)$ and
$V\in\Gamma(T^{\bot}M)$. ∎

######  Lemma 3.5.

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$,
$f_{1}(p)=f_{3}(p)$ and $TM$ is invariant under the action of
$\widetilde{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is either invariant
or anti-invariant.

###### Proof.

For $X,Y\in\Gamma(TM)$, we have from (2.10) that

$\displaystyle\widetilde{\bar{R}}(X,Y)X$ $\displaystyle=$
$\displaystyle(f_{1}-1)\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
X)\phi Y$ $\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi
X\big{\\}}+(f_{3}-1)\big{\\{}\eta(X)\eta(X)Y$ $\displaystyle-$
$\displaystyle\eta(Y)\eta(X)X+g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})\\{g(\phi Y,X)X-g(\phi
X,X)Y+g(Y,X)\phi X$ $\displaystyle-$ $\displaystyle g(X,X)\phi Y\\}.$

Note that $\widetilde{\bar{R}}(X,Y)X$ should be tangent if $[-3f_{2}g(Y,\phi
X)\phi X+(f_{1}-f_{3})\\{g(Y,X)\phi X-g(X,X)\phi Y\\}]$ is tangent. Since
$f_{2}(p)\neq 0$, $f_{1}(p)=f_{3}(p)$ at any point $p$ then by similar way of
proof of Lemma $3.2$ of [3], we can prove that either $M$ is invariant or
anti-invariant. This proves the Lemma. ∎

######  Remark 3.1.

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{1}(p)\neq f_{3}(p)$ and $TM$ is
invariant under the action of $\widetilde{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$,
then $M$ is invariant.

From Lemma $3.3$ and Lemma $3.5$, we have

######  Theorem 3.1.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$,
$f_{1}(p)=f_{3}(p)$ then $M$ is either invariant or anti-invariant if and only
if $TM$ is invariant under the action of $\widetilde{\bar{R}}(X,Y)$ for all
$X,Y\in\Gamma(TM)$.

######  Proposition 3.1.

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to $\widetilde{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant
under the action of $\widetilde{\bar{R}}(U,V)$ for any
$U,V\in\Gamma(T^{\bot}M)$.

###### Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2.10), we get

$\displaystyle\widetilde{\bar{R}}(U,V)X$ $\displaystyle=$
$\displaystyle(f_{1}-1)\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi
X)\phi V$ $\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi
X\big{\\}}+(f_{3}-1)\big{\\{}\eta(U)\eta(X)V$ $\displaystyle-$
$\displaystyle\eta(V)\eta(X)U+g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})\\{g(\phi V,X)U-g(\phi
U,X)V+g(V,X)\phi U$ $\displaystyle-$ $\displaystyle g(U,X)\phi V\\}.$

As $M$ is invariant, $U,V\in\Gamma(T^{\bot}M)$, we have

(3.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$

for any $X\in\Gamma(TM)$. Using (3.5) in (3), we have

(3.6) $\widetilde{\bar{R}}(U,V)X=2f_{2}g(U,\phi V)\phi X,$

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

######  Proposition 3.2.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$,
$f_{1}(p)=f_{3}(p)$ for each $p\in M$ and $T^{\bot}M$ is invariant under the
action of $\widetilde{\bar{R}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then $M$ is
either invariant or anti-invariant.

###### Proof.

The proof is similar as it is an Lemma $3.4$, just assuming that
$\widetilde{\bar{R}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎

##  4\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
${\bar{\nabla}}^{{}^{\prime}}$

######  Lemma 4.1.

If $M$ is either invariant or anti-invarint submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
${\bar{\nabla}}^{{}^{\prime}}$, then ${\bar{R}}^{{}^{\prime}}(X,Y)Z$ is
tangent to $M$ and ${\bar{R}}^{{}^{\prime}}(X,Y)V$ normal to $M$ for any
$X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

###### Proof.

If $M$ is invariant then from (2) we say that ${\bar{R}}^{{}^{\prime}}(X,Y)Z$
is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.
If $M$ is anti-invariant then

(4.1) $g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.$

From (2) and (4.1) we have

$\displaystyle{\bar{R}}^{{}^{\prime}}(X,Y)Z$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-$
$\displaystyle\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle[\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y],$

which is tangent.
If $M$ is invariant then from (2), it follows that
${\bar{R}}^{{}^{\prime}}(X,Y)V$ is normal to $M$, and if $M$ is anti-invariant
then ${\bar{R}}^{{}^{\prime}}(X,Y)V=0$ i.e. ${\bar{R}}^{{}^{\prime}}(X,Y)V$ is
normal to $M$ for any $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This
proves the Lemma. ∎

######  Lemma 4.2.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in
M$ and $TM$ is invariant under the action of $\bar{R}^{{}^{\prime}}(X,Y)$,
$X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

###### Proof.

For $X,Y\in\Gamma(TM)$, we have from (2) that

$\displaystyle{\bar{R}}^{{}^{\prime}}(X,Y)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi X)\phi Y$
$\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi
X\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(X)Y$ $\displaystyle-$
$\displaystyle\eta(Y)\eta(X)X+g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle-$ $\displaystyle(f_{1}-f_{3})g(\phi
X,Y)X+\\{\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y\\}.$

Note that ${\bar{R}}^{{}^{\prime}}(X,Y)X$ should be tangent if
$3f_{2}(p)g(Y,\phi X)\phi X$ is tangent. Since $f_{2}(p)\neq 0$ for each $p\in
M$, as similar as proof of Lemma $3.2$ of [3], we may conclude that either $M$
is invariant or anti-invariant. This proves the Lemma. ∎

From Lemma $4.1$ and Lemma $4.2$, we have

######  Theorem 4.1.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in
M$, then $M$ is either invariant or anti-invariant if and only if $TM$ is
invariant under the action of ${\bar{R}}^{{}^{\prime}}(X,Y)$ for all
$X,Y\in\Gamma(TM)$.

######  Proposition 4.1.

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to ${\bar{\nabla}}^{{}^{\prime}}$. If $M$ is invariant, then $TM$ is invariant
under the action of ${\bar{R}}^{{}^{\prime}}(U,V)$ for any
$U,V\in\Gamma(T^{\bot}M)$.

###### Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2), we get

$\displaystyle{\bar{R}}^{{}^{\prime}}(U,V)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi X)\phi V$
$\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi
X\big{\\}}+f_{3}\big{\\{}\eta(U)\eta(X)V$ $\displaystyle-$
$\displaystyle\eta(V)\eta(X)U+g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})\\{g(U,\phi X)V-g(V,\phi X)U\\}$
$\displaystyle+$ $\displaystyle\\{\eta(V)\eta(X)U-\eta(U)\eta(X)V\\}.$

As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have

(4.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$

for any $X\in\Gamma(TM)$. Using (4.5) in (4), we have

(4.6) ${\bar{R}}^{{}^{\prime}}(U,V)X=2f_{2}g(U,\phi V)\phi X,$

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

######  Proposition 4.2.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in
M$ and $T^{\bot}M$ is invariant under the action of ${\bar{R}}(U,V)$,
$U,V\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

###### Proof.

The proof is similar as the proof of Lemma $4.2$, just imposing that
${\bar{R}}^{{}^{\prime}}(U,V)U$ is normal for any $U,V\in\Gamma(TM)$. ∎

##  5\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
$\hat{\bar{\nabla}}$

######  Lemma 5.1.

If $M$ is either invariant or anti-invarint submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$, then
$\hat{\bar{R}}(X,Y)Z$ is tangent to $M$ and $\hat{\bar{R}}(X,Y)V$ is normal to
$M$ for any $X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

###### Proof.

If $M$ is invariant then from (2.14) we say that $\hat{\bar{R}}(X,Y)Z$ is
tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.

If $M$ is anti-invariant then

(5.1) $g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.$

From (2.14) and (5.1) we have

$\displaystyle\hat{\bar{R}}(X,Y)Z$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$
$\displaystyle+$ $\displaystyle g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}},$

which is tangent.
If $M$ is invariant from (2.14) we have $\hat{\bar{R}}(X,Y)V$ is normal to
$M$, and if $M$ is anti-invariant then $\hat{\bar{R}}(X,Y)V=0$ i.e.
$\hat{\bar{R}}(X,Y)V$ is normal to $M$ for any $X,Y\in\Gamma(TM)$ and
$V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎

######  Lemma 5.2.

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq(f_{1}-f_{3})^{2}$ on $M$ and
$TM$ is invariant under the action of $\hat{\bar{R}}(X,Y)$,
$X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

###### Proof.

For $X,Y\in\Gamma(TM)$, we have from (2.14) that

$\displaystyle\hat{\bar{R}}(X,Y)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi X)\phi Y$
$\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi X\big{\\}}$
$\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(X)Y-\eta(Y)\eta(X)X$
$\displaystyle+$ $\displaystyle g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(X,\phi X)\phi
Y-g(Y,\phi X)\phi X\big{\\}}.$

Now, we see that $\hat{\bar{R}}(X,Y)X$ should be tangent if
$\\{3f_{2}+(f_{1}-f_{3})^{2}\\}g(Y,\phi X)\phi X$ is tangent. Since
$3f_{2}\neq-(f_{1}-f_{3})^{2}$ then in similar way of proof of Lemma $3.2$ of
[3] we may conclude that either $M$ is invariant or anti-invariant. This
proves the Lemma. ∎

From Lemma $5.1$ and Lemma $5.2$, we can state the following:

######  Theorem 5.1.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq-(f_{1}-f_{3})^{2}$, then $M$
is either invariant or anti-invariant if and only if $TM$ is invariant under
the action of $\hat{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.

######  Proposition 5.1.

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to $\hat{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant under the
action of $\hat{\bar{R}}(U,V)$ for any $U,V\in\Gamma(T^{\bot}M)$.

###### Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2.14), we get

$\displaystyle\hat{\bar{R}}(U,V)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi X)\phi V$
$\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi X\big{\\}}$
$\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(U)\eta(X)V-\eta(V)\eta(X)U$
$\displaystyle+$ $\displaystyle g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(U,\phi X)\phi
V-g(V,\phi X)\phi U\big{\\}}.$

As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have

(5.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$

for any $X\in\Gamma(TM)$. Using (5.5) in (5), we have

(5.6) $\hat{\bar{R}}(U,V)X=2f_{2}g(U,\phi V)\phi X,$

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

######  Proposition 5.2.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq-(f_{1}-f_{3})^{2}$ on $M$ and
$T^{\bot}M$ is invariant under the action of $\hat{\bar{R}}(U,V)$,
$U,V\in\Gamma(T^{\bot}M)$, then $M$ is either invariant or anti-invariant.

###### Proof.

The proof is similar as the proof of Lemma $5.2$, just imposing that
$\hat{\bar{R}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎

##  6\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
$\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$

######  Lemma 6.1.

If $M$ is either invariant or anti-invarint submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$, then
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ is tangent to $M$ and
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ is normal to $M$ for any
$X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

###### Proof.

If $M$ is invariant then from (2.16) we say that
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ is tangent to $M$ because $\phi
X$ and $\phi Y$ are tangent to $M$.

If $M$ is anti-invariant then

(6.1) $g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.$

From (2.16) and (6.1) we have

$\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ $\displaystyle=$
$\displaystyle f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$
$\displaystyle+$ $\displaystyle g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$

which is tangent.
If $M$ is invariant from (2.16) we have
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ normal to $M$ and if $M$ is
anti-invariant then $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V=0$ i.e.
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ normal to $M$ for any
$X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎

######  Lemma 6.2.

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If
$\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$ for each $p\in M$ and
$TM$ is invariant under the action of
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is
either invariant or anti-invariant.

###### Proof.

For $X,Y\in\Gamma(TM)$, we have from (2.16) that

$\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)X$ $\displaystyle=$
$\displaystyle f_{1}\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
X)\phi Y$ $\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi
X\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(X)Y-\eta(Y)\eta(X)X$
$\displaystyle+$ $\displaystyle g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(X,\phi X)\phi
Y-g(Y,\phi X)\phi X\big{\\}}$ $\displaystyle+$ $\displaystyle
2(f_{1}-f_{3})g(X,\phi Y)\phi X.$

Now we see that $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)X$ should be
tangent if $\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)g(Y,\phi X)\phi X$
is tangent. Since $\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$
then by similar way of proof of Lemma $3.2$ of [3] we can proved that either
$M$ is invariant or anti-invariant. This proves the Lemma. ∎

From Lemma $6.1$ and Lemma $6.2$, we have

######  Theorem 6.1.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If
$\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$, then $M$ is either
invariant or anti-invariant if and only if $TM$ is invariant under the action
of $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.

######  Proposition 6.1.

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If $M$ is invariant, then
$TM$ is invariant under the action of
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)$ for any
$U,V\in\Gamma(T^{\bot}M)$.

###### Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2.16), we get

$\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)X$ $\displaystyle=$
$\displaystyle f_{1}\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi
X)\phi V$ $\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi
X\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(U)\eta(X)V-\eta(V)\eta(X)U$
$\displaystyle+$ $\displaystyle g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(U,\phi X)\phi
V-g(V,\phi X)\phi U\big{\\}}$ $\displaystyle+$ $\displaystyle
2(f_{1}-f_{3})g(U,\phi V)\phi X.$

As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have

(6.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$

for any $X\in\Gamma(TM)$. Using (6.5) in (6), we have

(6.6)
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)X=\\{2f_{2}+2(f_{1}-f_{3})\\}g(U,\phi
V)\phi X,$

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

######  Proposition 6.2.

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If
$\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$ for each $p\in M$ and
$T^{\bot}M$ is invariant under the action of
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then
$M$ is either invariant or anti-invariant.

###### Proof.

The proof is similar as the proof of Lemma $6.2$, just considering that
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)U$ is normal for any
$U,V\in\Gamma(T^{\bot}M)$. ∎

Acknowledgement: The first author (P. Mandal) gratefully acknowledges to the
CSIR(File No.:09/025(0221)/2017-EMR-I), Govt. of India for financial
assistance. The Third author (S. K. Hui) are thankful to University of Burdwan
for providing administrative and technical support.

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Pradip Mandal and Shyamal Kumar Hui
Department of Mathematics, The University of Burdwan, Burdwan – 713104, West
Bengal, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Shyam Kishor
Department of Mathematics and Astronomy, University of Lucknow, Lucknow –
226007, India
E-mail<EMAIL_ADDRESS>


# Wiedemann-Franz law violation domain for graphene and nonrelativistic
systems

Thandar Zaw Win, Cho Win Aung, Gaurav Khandal
Sabyasachi Ghosh  Department of Physics, Indian Institute of Technology
Bhilai, Kutelabhata, Durg 491002, India

###### Abstract

Systematic and comparative research on Lorenz ratios for graphene and
nonrelativistic systems has been studied to identify their Wiedemann-Franz law
violation domain. Fermi energy and temperature are the main governing
parameters for deciding the values of the Lorenz ratio, which is basically
thermal conductivity divided by electrical conductivity times temperature
times Lorenz number. Metals as three-dimensional nonrelativistic electron gas
locate at higher Fermi-energy by temperature domain, where Lorenz ratio
remains one. Hence, they obey the Wiedemann-Franz law. By creating higher
doping in a two-dimensional graphene system, one can again reach a higher
Fermi-energy by temperature domain and get a constant Lorenz ratio. For both
graphene and nonrelativistic systems, the Lorenz ratio goes below one if we go
lower Fermi-energy by temperature domain, which is possible for the graphene
system by decreasing the doping concentration. Experimentally observed greater
than one Lorenz ratio in this lower Fermi-energy by temperature domain or
Dirac Fluid domain indicates that non-fluid expressions of Lorenz ratio should
be replaced by fluid-type expressions. We have noticed a divergent trend of
Lorenz ratio in the Dirac Fluid domain using its fluid-type expression, and it
matches with the trend of experimental data.

##  I Introduction

In 1853, Gustav Wiedemann and Rudolph Franz experimentally discovered a
universal or constant value of the ratio between thermal ($\kappa$) and
electrical ($\sigma$) conductivity, which are approximately followed by all
metals. Later, in 1872, Ludvig Lorenz theoretically realized that the ratio in
terms of the universal constants $k_{B}$ (Boltzmann constant) and $e$
(electric charge) is:

$\displaystyle\frac{\kappa}{\sigma T}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}$ (1)
$\displaystyle=$ $\displaystyle L_{0}=2.445\times
10^{-8}watt\frac{\Omega}{K^{2}}~{},$

where $L_{0}$ is known as the Lorenz number. The fact is very well known from
our text book Refs. [1, 2, 3, 4, 5]. In natural unit, $\hbar=c=k_{B}=1$ and
$e^{2}=\frac{4\pi}{137}$, we can express temperature $T$ in eV, $\sigma$ in eV
and $\kappa$ in eV2. So Lorenz number will be a dimensionless ratio
$L_{0}=\frac{137\pi}{12}\approx 35.84$. In the present paper, we will mostly
use the natural unit methodology for convenience, but we may go to the actual
unit in some cases (whenever necessary).

The Wiedemann-Franz (WF) law has proven remarkably robust for many metallic
systems, where electrons are the main electric and thermal charge
transportation carriers. Due to the similar mechanism of transportation in
free electron theory [1, 2, 3, 4, 5], the dimensionless (in natural unit)
ratio of two transport coefficients - thermal and electrical conductivity
becomes constant. However, deviations have been observed in many systems like
$Li_{0.9}Mo_{6}O_{17}$ [6], $CeCoIn_{5}$ [7], $ZrZn_{2}$ [8], $YbRh_{2}Si_{2}$
[9], $(Pr,Ce)_{2}CuO_{4}$ [10] and $VO_{2}$ [11]. For our convenience, if we
define the ratio of thermal and electrical conductivity as
$L=\frac{\kappa}{\sigma T}$, and if we call $L/L_{0}$ as Lorenz ratio, then
the validity of the WF law will be established from the relations $L=L_{0}$ or
$\frac{L}{L_{0}}=1$. On the other hand, relations $\frac{L}{L_{0}}>1$ or
$\frac{L}{L_{0}}<1$ reflect violation of the WF law. A large diverging
outcomes $\frac{L}{L_{0}}>10^{4}$ is expected in one-dimensional exotic
Tomonaga-Luttinger liquids [6]. On the other hand, a strong downward violation
of the WF Law $L<L_{0}$ was observed in $WP_{2}$ semimetal [12, 13] and MoP
binary compound [14]. This downward violation is also observed in heavy
fermion metals [7], marginal ferromagnetic metal [8], anti-ferromagnetic metal
[9] and copper-oxide superconducting material [10] at low-temperature regions
as well as in metallic vanadium dioxide [11] at high-temperature range
($240$-$340$K). Understanding the WF law violation mechanism of that wide
range of systems is a non-trivial task and challenge for the theoretical
community to explain.

Recently, Crossno et al. [15] have found a similar kind of the WF law
violation in graphene systems by tuning the doping concentration and
temperature. Unlike standard metal, the Fermi level of graphene can be moved
upwards or downwards from the relative Dirac point depending upon the n- or
p-type doping of graphene. Fermi energy $(\epsilon_{F})$ or a chemical
potential $(\mu)$ at Dirac point is considered zero for undoped graphene and,
via n- or p-type doping, will shift it towards the positive or negative
direction. The net charge carrier concentration will be tuned from zero to
non-zero when one experimentally goes from undoped to doped graphene cases,
and theoretically, one goes from $\epsilon_{F}=0$ to $\epsilon_{F}\neq 0$
cases. This kind of tuning possibility of $\epsilon_{F}$ can not be expected
for metal systems as it almost remains constant. Their typical values remain
within the range $\epsilon_{F}=2-10$ eV, for which one can expect the limit
$\epsilon_{F}/T>>1$ at room temperature $T\approx 0.025$ eV. In this limit,
one can consider electrons in metal as Fermi gas and use Sommerfeld’s
expansion, which provides a linear T dependent on electron-specific heat.
Fermi gas is completely based on the crude non-interacting assumption, but
there is a theory of interacting Fermion system, which is popularly known as
the Fermi Liquid (FL) Theory. Originally, Landau [16, 17, 18] proposed this
phenomenological theory for studying 3He. In FL prescription, interaction is
taken care of via the effective mass of electrons by assuming a quasi-particle
picture. Hence, some mass-independent quantities like $L$ remain the same as
$L_{0}$ for both FL and Fermi gas prescriptions. So, if we define
$\epsilon_{F}/T>1$ as the FL domain and $\epsilon_{F}/T>>1$ as the Fermi gas
domain, then electron-doped graphene systems almost follow WF law. When we go
towards un-dopped or clean graphene systems with $\epsilon_{F}/T<1$, Fermi
liquid theory becomes invalid (as it is commonly accepted). It is concluded
from the experimental observation of WF law violation in this
$\epsilon_{F}/T<1$ or $\epsilon_{F}/T<<1$ domains, popularly called Dirac
Fluid (DF) or Dirac Liquid (DL) domain. Theoretical works [19, 20, 21, 22, 23]
are attempted to explain this WF law violation in the DF domain. In this
regards, electron hydrodynamics (eHD) or fluid dynamics in graphene system,
recently observed by Refs [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37], is thought to be linked with this WF law violation. In this context, the
present work is planned to explore a transition from non-fluid to fluid type
expressions of Lorenz ratio for a 2-dimensional (2D) graphene (G) system with
linear energy-momentum relation ($E\propto p$) as well as a 3-dimensional (3D)
metal system with nonrelativistic (NR) energy-momentum relation ($E\propto
p^{2}$). For mathematical completeness, we have demonstrated all possible
cases like 2D-NR, 3D-NR, 2D-G, and 3D-G but our final destination is kept to
show how fluid expressions of 2D-G case can be associated with WF law
violation and how non-fluid expressions of 3D-NR case, followed by metal
systems, obey WF law. In the direction of eHD, our earlier comparative
research on ideal [38] and viscous dissipating [39] parts of energy-momentum
tensor calculation clearly demonstrate that the graphene hydrodynamics is
neither relativistic hydrodynamics nor nonrelativistic hydrodynamics. The
present work has gone through similar comparative research on electrical
conductivity, thermal conductivity, and Lorenz ratio for a systematic search
of the WF law violation domain.

The article is organized as follows, in Sec. II, the formalism part of WF law
calculations for different cases. The results and discussions are in Sec. III
and summarised by Sec. IV.

##  II Formalism

Graphene is a 2-dimensional single atomic layer of carbon atoms tightly bound
into a honeycomb lattice [40]. Near the Dirac point, electrons in graphene
follow the linear dispersion relation

$\epsilon\left(k\right)=pv_{F},$ (2)

where $v_{F}$ is the Fermi velocity of electrons in graphene, whose values
remain within the range $0.003c$-$0.01c$ [41]. In one direction, electrons in
graphene do not follow a quadratic relation between energy and momentum
($\epsilon=p^{2}/(2m)$) like in the nonrelativistic case. On the other hand,
its velocity ($v_{F}$) is not close to the speed of light ($c$), so some
relativistic corrections will appear. In that sense, electrons in graphene
follow neither relativistic nor nonrelativistic. So, its different properties
may not be the same as expected for traditional nonrelativistic or
relativistic matters as shown in Refs. [38, 39], for thermodynamics [38] and
dissipative viscous [39] properties. In this regard, thermal and electrical
conductivity for the graphene (G) case may also be interesting to compare with
nonrelativistic (NR) and ultrarelativistic (UR) cases, which are studied in
the present work. In Sub-sec. II.1, we have gone through the first non-fluid
type framework description, which is traditionally adopted in standard books
of solid state physics [1, 2, 3, 4, 5]. Next, in Sub-sec. II.2, we have
extended our framework towards fluid type description.

###  II.1 Non fluid description

For systematic study and understanding, we will calculate the Lorenz ratios
for the 3D system and then the 2D system. The NR, UR, and G case expressions
of the Lorenz ratio will be addressed step by step for each system. Then, we
will also examine the limiting conditions using Sommerfeld’s expansion.

3D-NR: Let us first discuss the 3D-NR case. The well-known Drude’s formula for
the electrical conductivity of nonrelativistic electrons in 3D solids (metals)
is

$\sigma_{NR}^{3D}=\frac{ne^{2}\tau_{c}}{m}=\frac{ne^{2}\lambda}{mv_{F}},$ (3)

and the thermal conductivity is

$\kappa_{NR}^{3D}=\frac{1}{3}nv_{F}\lambda\,\prescript{3D}{NR}{[C_{V}]_{e}},$
(4)

where, $n$ is the number (electron) density, $\tau_{c}$ is the relaxation
time, $m$ is electron mass, $\lambda=v_{F}\tau_{c}$ is the mean free path, and
$[C_{V}]_{e}$ is the electronic specific heat per particle. After taking the
ratio of thermal and electrical conductivity, we get

$\frac{\kappa_{NR}^{3D}}{\sigma_{NR}^{3D}}=\frac{2}{3}\frac{\epsilon_{F}}{e^{2}}\,\prescript{3D}{NR}{[C_{V}]_{e}},$
(5)

where $\epsilon_{F}=\mu=\frac{1}{2}mv_{F}^{2}$ is the Fermi energy.

Here, in Eq. (5), $[C_{V}]_{e}$ is the main important thing to calculate the
conductivity ratio. We will use the two definitions of $[C_{V}]_{e}$:

  1. 1.

$[C_{V}]_{e1}=\frac{\partial}{\partial
T}\Big{(}\frac{U}{N}\Big{)}\Bigg{|}_{V,\ \epsilon_{F}}$,

  2. 2.

$[C_{V}]_{e2}=\frac{1}{N}\frac{\partial U}{\partial T}\Bigg{|}_{V,\
\epsilon_{F}}$.

Here, for subsequent simplicity, we will use the notations like NR1 and G1
concerning $[C_{V}]_{e1}$ and NR2 and G2 dealing with $[C_{V}]_{e2}$. The
former definition prescribes taking $T$ derivative of internal energy per
particle $u=U/N$, while the latter definition says to take $T$ derivative of
internal energy $U$ and then normalize by $N$. The detailed calculation can be
seen in Appendix C. Let us first put the former specific heat,

$\prescript{3D}{NR1}{[C_{V}]_{e1}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]},$
(6)

in Eq. (5), the Lorenz ratio for 3D-NR will be

$\frac{L_{NR1}^{3D}}{L_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]}.$
(7)

Here,

$\displaystyle
f_{\nu}(A)=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}\frac{x^{\nu-1}}{A^{-1}e^{x}+1}dx,$
(8)

is the standard Fermi integral function (see details in Appendix B) with
$A=e^{\epsilon_{F}/k_{B}T}$. Using Sommerfeld’s lemma, in the limit of
$\epsilon_{F}/k_{B}T=lnA>>1$, the electronic-specific heat becomes

$\prescript{3D}{NR1}{[C_{V}]_{e1}}=\frac{\pi^{2}}{2}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(9)

and then Eq. (7) becomes

$L_{NR1}^{3D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (10)

which is the so-called Wiedemann-Franz law for metals.

Next, using another definition of $[C_{V}]_{e2}$ for the 3D-NR case, whose
general expression will be (see Appendix C)

$\prescript{3D}{NR2}{[C_{V}]_{e2}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}~{},$
(11)

we will get the Lorenz ratio as

$\frac{L_{NR2}^{3D}}{L_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(12)

In the limit of $\epsilon_{F}/T=lnA>>1$, Sommerfeld’s expansion of Eq. (11)
becomes

$\prescript{3D}{NR2}{[C_{V}]_{e2}}=\frac{3\pi^{2}}{4}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(13)

and then the Eq. (12) becomes

$L_{NR2}^{3D}=\frac{\pi^{2}}{2}\left(\frac{k_{B}}{e}\right)^{2}=1.5L_{0}.$
(14)

3D-G: Now, we consider a hypothetical 3D system, which follow graphene-type
dispersion relation $\epsilon=v_{F}p$. In this case, the electrical
conductivity

$\sigma_{G}^{3D}=\frac{ne^{2}\tau_{c}}{\epsilon_{F}}v_{F}^{2}=\frac{ne^{2}\lambda}{\epsilon_{F}}v_{F},$
(15)

and the thermal conductivity,

$\kappa_{G}^{3D}=\frac{1}{3}nv_{F}\lambda\prescript{3D}{G}{[C_{V}]_{e}},$ (16)

will form the ratio

$\frac{\kappa_{G}^{3D}}{\sigma_{G}^{3D}}=\frac{1}{3}\frac{\epsilon_{F}}{e^{2}}\prescript{3D}{G}{[C_{V}]_{e}},$
(17)

where the relation between the Fermi energy and the Fermi momentum will be
$\epsilon_{F}=p_{F}v_{F}$ for graphene.

Using two possible expressions (see Appendix C) of specific heat for 3D-G
system,

$\prescript{3D}{G1}{[C_{V}]_{e1}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}$
(18)

and

$\prescript{3D}{G2}{[C_{V}]_{e2}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(19)

in Eq. (17), we get the Lorenz ratios

$\frac{L_{G1}^{3D}}{L_{0}}=\frac{\kappa_{G1}^{3D}}{\sigma_{G1}^{3D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}$
(20)

and

$\frac{L_{G2}^{3D}}{L_{0}}=\frac{\kappa_{G2}^{3D}}{\sigma_{G2}^{3D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
(21)

respectively. In the Sommerfeld’s limit, Eqs. (18) and (19) will be converted
to

$\prescript{3D}{G1}{[C_{V}]_{e1}}=\pi^{2}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(22)

and

$\prescript{3D}{G2}{[C_{V}]_{e2}}=3\pi^{2}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(23)

and so, Eqs. (20) and (21) become

$L_{G1}^{3D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (24)

and

$L_{G2}^{3D}=\pi^{2}\left(\frac{k_{B}}{e}\right)^{2}=3L_{0}.$ (25)

respectively.

2D-NR: Now, let us go for 2D cases. In the case of 2D nonrelativistic system,
the ratio of thermal and electrical conductivity can be written as

$\frac{\kappa_{NR}^{2D}}{\sigma_{NR}^{2D}}=\frac{\epsilon_{F}}{e^{2}}\prescript{2D}{NR}{[C_{V}]_{e}}.$
(26)

Using two possible expressions (see Appendix C) of specific heat for 2D-NR
system,

$\prescript{2D}{NR1}{[C_{V}]_{e1}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]},$
(27)

and

$\prescript{2D}{NR2}{[C_{V}]_{e2}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(28)

in Eq. (26), we get the Lorenz ratios

$\frac{L_{NR1}^{2D}}{L_{0}}=\frac{\kappa_{NR1}^{2D}}{\sigma_{NR1}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]}$
(29)

and

$\frac{L_{NR2}^{2D}}{L_{0}}=\frac{\kappa_{NR2}^{2D}}{\sigma_{NR2}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
(30)

respectively. In the Sommerfeld’s limit (SL), Eqs. (29) and (30) will be
converted to

$L_{NR1}^{2D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (31)
$L_{NR2}^{2D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0}.$ (32)

2D-G: Now, we are taking an actual 2D graphene case. The ratio of
$\kappa_{G}^{2D}$ and $\sigma_{G}^{2D}$ is given by

$\frac{\kappa_{G}^{2D}}{\sigma_{G}^{2D}}=\frac{1}{2}\frac{\epsilon_{F}}{e^{2}}\prescript{2D}{G}{[C_{V}]_{e}}.$
(33)

Using two possible expressions (see Appendix C) of specific heat for 2D-G
system,

$\prescript{2D}{G1}{[C_{V}]_{e1}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]},$
(34)

and

$\prescript{2D}{G2}{[C_{V}]_{e2}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(35)

in Eq. (33), we get the Lorenz ratios

$\frac{L_{G1}^{2D}}{L_{0}}=\frac{\kappa_{G1}^{2D}}{\sigma_{G1}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}$
(36)

and

$\frac{L_{G2}^{2D}}{L_{0}}=\frac{\kappa_{G2}^{2D}}{\sigma_{G2}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
(37)

respectively. The SL of Eqs. (36) and (37) will be

$L_{G1}^{2D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (38)

and

$L_{G2}^{2D}=\frac{2\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=2L_{0}$ (39)

respectively.

3D-UR and 2D-UR: The expressions of electrical and thermal conductivity for
3D-UR and 2D-UR will be same as those expressions for 3D-G and 2D-G if $v_{F}$
is replaced by the speed of light $c$. Readers can notice that specific heat
and Lorenz ratios for 3D-G and 2D-G are independent of $v_{F}$, so those
expressions can also be used for 3D-UR and 2D-UR systems. Since our focal
quantity is the Lorenz ratio in the results, we will not discuss it further.
The expressions of 3D-UR and 2D-UR systems coincide with those of 3D-G and
2D-G systems.

###  II.2 An approach towards the fluid description

The previous subsection provides the expression of $\kappa$ and $\sigma$ using
a standard solid-state physics framework, where no fluid concept has been
entered. So, we can call that description a non-fluid description. In the
present section, we will try to build an approach towards a fluid description
by using relaxation time approximation-based Boltzmann transport equations.
Let us first address the brief calculations of the Lorenz ratio for 2D-G
systems; then, we will write down its final expressions for other possible
systems like 2D-NR, 3D-NR, and 3D-G instead of repeating similar calculations.

2D-G: Let us assume a local thermalization picture of electron fluid in a 2D-G
system, where the equilibrium distribution function (see Appendix A)

$f_{0}\left(T(x^{\mu})\right)=\frac{1}{e^{\left(\epsilon-\epsilon_{F}\right)/k_{B}T(x^{\mu})}+1},$
(40)

will be $x^{\mu}$-dependent due to temperature profile $T(x^{\mu})$. Here,
$x^{\mu}=(ct,x^{i})$, a four-dimensional coordinate is considered for general
notation, but we have to take care of $i=1,2$ for the 2D system and $i=1,2,3$
for the 3D system.

Now, let us first write down the macroscopic definitions for electrical and
thermal conductivity

$\displaystyle{\vec{J}}$ $\displaystyle=$ $\displaystyle\sigma{\vec{E}},$
$\displaystyle{\vec{Q}}$ $\displaystyle=$
$\displaystyle\kappa{\vec{\nabla}}T~{},$ (41)

where electrical current vector ${\vec{J}}$ and heat flow vector ${\vec{Q}}$
can have microscopic expressions:

$\displaystyle{J_{i}}$ $\displaystyle=$ $\displaystyle
ge\int\frac{d^{2}p}{h^{2}}v_{i}\delta f_{\sigma},$ $\displaystyle{Q_{i}}$
$\displaystyle=$ $\displaystyle g\int\frac{d^{2}p}{h^{2}}\epsilon v_{i}\delta
f_{\kappa}.$ (42)

Here, we are assuming that the external electric field $E_{i}$ and the
temperature gradient ${\vec{\nabla}}T$ will create deviations $\delta
f_{\sigma}$ and $\delta f_{\kappa}$ respectively from the equilibrium
distribution $f_{0}$.

It is relaxation time approximation (RTA) based on Boltzmann’s transport
equation (BTE) [42]

$\frac{\partial f}{\partial t}+\frac{\partial x^{i}}{\partial t}\frac{\partial
f}{\partial x^{i}}+\frac{\partial p^{i}}{\partial t}\frac{\partial f}{\partial
p^{i}}=-\frac{\delta f}{\tau_{c}}~{},$ (43)

which will guide us to guess appropriate form of $\delta f_{\sigma}$ and
$\delta f_{\kappa}$. Considering $f=f_{0}+\delta f\approx f_{0}$ in left hand
side of Eq. (43) and local thermalization assumption of $f_{0}$, given in Eq.
(40), we can simplify Eq. (43) as

$\displaystyle v^{i}\frac{\partial f_{0}}{\partial x^{i}}+eE^{i}\frac{\partial
f_{0}}{\partial p^{i}}$ $\displaystyle=$ $\displaystyle-\frac{{\delta
f}_{\sigma}}{\tau_{c}}-\frac{{\delta f}_{\kappa}}{\tau_{c}}$ $\displaystyle
v^{i}\frac{\partial T}{\partial x^{i}}\frac{\partial\epsilon}{\partial
T}\frac{\partial
f_{0}}{\partial\epsilon}+eE^{i}\frac{\partial\epsilon}{\partial
p^{i}}\frac{\partial f_{0}}{\partial\epsilon}$ $\displaystyle=$
$\displaystyle-\frac{{\delta f}_{\sigma}}{\tau_{c}}-\frac{{\delta
f}_{\kappa}}{\tau_{c}}$ $\displaystyle v^{i}\frac{\partial T}{\partial
x^{i}}[C_{V}]_{e}\frac{\partial
f_{0}}{\partial\epsilon}+eE^{i}v_{i}\frac{\partial f_{0}}{\partial\epsilon}$
$\displaystyle=$ $\displaystyle-\frac{{\delta
f}_{\sigma}}{\tau_{c}}-\frac{{\delta f}_{\kappa}}{\tau_{c}}.$ (44)

Here, we consider the approximation
$[C_{V}]_{e}\approx\frac{\partial\epsilon}{\partial T}$, and one can expect
again two possible definitions of specific heat as discussed in the earlier
section. From Eq. (44), we can get the form of $\delta f_{\sigma}$ and $\delta
f_{\kappa}$ as

$\displaystyle\delta f_{\sigma}$ $\displaystyle=$ $\displaystyle
eE^{i}v_{i}\left(-\frac{\partial f_{0}}{\partial\epsilon}\right)\tau_{c},$
$\displaystyle\delta f_{\kappa}$ $\displaystyle=$ $\displaystyle
v^{i}\left(-\frac{\partial
f_{0}}{\partial\epsilon}\right)[C_{V}]_{e}\left(\frac{\partial T}{\partial
x^{i}}\right)\tau_{c},$ (45)

with

$\left(-\frac{\partial f_{0}}{\partial\epsilon}\right)=\beta f_{0}(1-f_{0}).$
(46)

Using Eq. (45) in Eq. (42) and then comparing with Eq. (41), we will get the
final expressions of electrical and thermal conductivity:

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle
ge^{2}\frac{v_{F}^{2}}{2}\tau_{c}\int\frac{d^{2}p}{h^{2}}\beta f_{0}(1-f_{0})$
$\displaystyle\implies\sigma_{G}^{2D}$ $\displaystyle=$ $\displaystyle 2\pi
k_{B}\tau_{c}\left(\frac{e}{h}\right)^{2}f_{1}\left(A\right)T,$ (47)

and

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
g\epsilon\frac{v_{F}^{2}}{2}\tau_{c}[C_{V}]_{e}\int\frac{d^{2}p}{h^{2}}\beta
f_{0}(1-f_{0})$ $\displaystyle\implies\kappa_{G}^{2D}$ $\displaystyle=$
$\displaystyle\frac{4\pi
k_{B}^{2}\tau_{c}}{h^{2}}[C_{V}]_{e}\,f_{2}\left(A\right)T^{2}.$ (48)

Now, using two different forms of specific heat, given in Eqs. (34) and (35)
in Eq. (48), we get

$\kappa_{G1}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}f_{2}\left(A\right)T^{2},$
(49)

and

$\kappa_{G2}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}f_{2}\left(A\right)T^{2},$
(50)

and hence, corresponding Lorenz ratios will be

$\frac{L_{G1}^{2D}}{L_{0}}=\frac{12}{\pi^{2}}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)},$
(51)

and

$\frac{L_{G2}^{2D}}{L_{0}}=\frac{12}{\pi^{2}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}$
(52)

respectively.

2D-NR After doing the same calculation as the previous one for a 2-dimensional
nonrelativistic system, the final general expressions of the electrical and
two possible thermal conductivities and their corresponding Lorenz ratios
$L_{NR}^{2D}/L_{0}$ are given by as below:

$\displaystyle\sigma_{NR}^{2D}=4\pi
k_{B}\tau_{c}\left(\frac{e}{h}\right)^{2}f_{1}\left(A\right)T,$ (53)
$\displaystyle\kappa_{NR1}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]}f_{2}\left(A\right)T^{2},$
(54) $\displaystyle\kappa_{NR2}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}f_{2}\left(A\right)T^{2}~{},$
(55)
$\displaystyle\frac{L_{NR1}^{2D}}{L_{0}}=\frac{6}{\pi^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]}\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)},$
(56)
$\displaystyle\frac{L_{NR2}^{2D}}{L_{0}}=\frac{6}{\pi^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}.$
(57)

3D-G: For the 3D-G case, the expression of the electrical conductivity and two
possible thermal conductivities and their corresponding Lorenz ratios
$L_{G}^{3D}/L_{0}$ are given as below:

$\sigma_{G}^{3D}=16\pi
k_{B}^{2}\tau_{c}\left(\frac{e^{2}}{3h^{3}v_{F}}\right)f_{2}\left(A\right)T^{2},$
(58) $\kappa_{G1}^{3D}=\frac{48\pi
k_{B}^{4}\tau_{c}}{h^{3}v_{F}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}f_{3}\left(A\right)T^{3},$
(59) $\kappa_{G2}^{3D}=\frac{48\pi
k_{B}^{4}\tau_{c}}{h^{3}v_{F}}\left[4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\right]f_{3}\left(A\right)T^{3},$
(60)
$\frac{L_{G1}^{3D}}{L_{0}}=\frac{27}{\pi^{2}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)},$
(61)
$\frac{L_{G2}^{3D}}{L_{0}}=\frac{27}{\pi^{2}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}.$
(62)

3D-NR: Now again, the same type of calculation is done for a 3-dimensional
nonrelativistic system; we get all expressions of the electrical, thermal
conductivity, and the ratio $L_{NR}^{3D}/L_{0}$ in a more general form, which
can be expressed as

$\displaystyle\sigma_{NR}^{3D}=\frac{2e^{2}k_{B}^{\frac{3}{2}}\tau_{c}}{m}\left(\frac{2\pi
m}{h^{2}}\right)^{\frac{3}{2}}f_{\frac{3}{2}}\left(A\right)T^{\frac{3}{2}}~{},$
(63)
$\displaystyle\kappa_{NR1}^{3D}=\frac{15k_{B}^{\frac{7}{2}}\tau_{c}}{2m}\left(\frac{2\pi
m}{h^{2}}\right)^{\frac{3}{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]}$
$\displaystyle f_{\frac{5}{2}}\left(A\right)T^{\frac{5}{2}},$ (64)
$\displaystyle\kappa_{NR2}^{3D}=\frac{15k_{B}^{\frac{7}{2}}\tau_{c}}{2m}\left(\frac{2\pi
m}{h^{2}}\right)^{\frac{3}{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
$\displaystyle f_{\frac{5}{2}}\left(A\right)T^{\frac{5}{2}},$ (65)
$\displaystyle\frac{L_{NR1}^{3D}}{L_{0}}=\frac{45}{4\pi^{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}~{},$
(66)
$\displaystyle\frac{L_{NR2}^{3D}}{L_{0}}=\frac{45}{4\pi^{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}.$
(67)

##  III Results

Figure 1: The Lorenz ratio regarding the chemical potential for non-fluid
descriptions of graphene case (a) for 3D and (b) for 2D expressions.

Figure 2: The Lorenz ratio regarding the chemical potential for non-fluid
descriptions of nonrelativistic case (a) for 3D and (b) for 2D expressions.

In the above section, we have calculated all the general expressions of
electrical and thermal conductivity and their ratios to check the validity of
the Wiedemann-Franz law for all possible systems like 2D and 3D graphene (G)
and nonrelativistic (NR) cases. Using those final expressions, the present
section is intended to go with their numerical estimations. During the result
generation, we will use natural unit $k_{B}=\hbar=c=1$ for our convenience.
Dimensionless quantities Lorenz ratio ($L/L_{0}$) and $\epsilon_{F}/T$ will be
taken as the y and x axes of all graphs.

Figure 3: Left: Lorenz ratio of 2D-G systems by using fluid-type expression.
Right: Transition from fluid to non-fluid domain in 2D-G system.

In Fig. 1 and Fig. 2, we explored the Lorenz ratio ($\frac{L}{L_{0}}$) by
using non-fluid (NF) type expressions and then the same ratios for both non-
fluid (NF) and fluid (F) type expressions in Fig. 3. Fig. 1(a) and (b) have
shown the Lorenz ratio versus $\frac{\epsilon_{F}}{T}$ for 3D-G and 2D-G
systems respectively. By using the non-fluid (NF) type of Eq. (20) and its
Sommerfeld limit Eq. (24), we plotted the black solid and dotted lines in Fig.
1(a). We used the specific heat ${[C_{V}]_{e1}}=\frac{\partial}{\partial
T}\Big{(}\frac{U}{N}\Big{)}$, which is traditionally used in solid-state or
non-fluid systems. We are getting $L/L_{0}=1$ in the limit of
$\epsilon_{F}/T\gg 1$ or Fermi Liquid (FL) regime of graphene case, which is
as expected for most of the metal cases. The deviation of the Lorenz ratio
from 1 has been started below $\frac{\epsilon_{F}}{T}\approx 10$ as we notice
from the black solid line. We have also plotted the red solid line by using
Eq. (21), where another definition of specific heat ${[C_{V}]_{e2}}$ is used
in the thermal conductivity expression. In the case of SL of 3D-G (the red
dotted line), the Lorenz ratio is three instead of one. For the general case
(the red solid line), the Lorenz ratio deviates from this constant line $3$
below $\frac{\epsilon_{F}}{T}\approx 10$. For the 2D-G system, similar kinds
of black and red solid lines are plotted in Fig. 1(b) by using Eq. (36) and
Eq. (37). Their respective SLs by using Eq. (38) and Eq. (39), are plotted as
black and red horizontal dotted lines. We considered the surface area $(S)$ in
place of volume $(V)$ in the 2D system. The Lorenz ratio when we take
${[C_{V}]_{e2}}$, the ratio becomes 2 (the red-dotted line).

The Fig. 2(a) and (b), similar to Fig. 1, display the Lorenz ratio in terms of
$\frac{\epsilon_{F}}{T}$ for non-fluid expressions of NR case. Though the real
metals have a constant chemical potential within the range,
$\epsilon_{F}=2-10$ eV, we imagine that it can be tunable from $0$ to
$\infty$. In Fig. 2(a), we notice that the WF law is valid in the SL of the 3D
metal case, expressed by Eq. (10) (green dot-dashed line). We have taken
$T=60$ K ($\approx 0.005$ eV), so metal range $\epsilon_{F}=2$-$10$ eV will
give the ratio $\frac{\epsilon_{F}}{T}=400$-$2000$ ($\gg 1$), which is quite
good domain for SL. Therefore, being situated in the SL domain, metal always
follows the WF law. A slight larger value $L=1.5L_{0}$ comes from Eq. (14)
(blue dotted line), when specific heat ${[C_{V}]_{e2}}$ is taken. A similar
kind of behavior can be seen in Fig. 2(b) for 2D-NR systems, but
interestingly, SLs in both cases, given by Eqs. (31) and (32) are coincided at
$L=L_{0}$ (blue-dotted and green dot-dashed lines). It probably indicates that
if one builds 2D metal systems, then it will also follow the WF law as their
Fermi energy range is again expected to be situated in the SL domain.

Before going to the next graphical results, let us briefly analyze what we
have learned from Fig. 1(a), (b) and Fig. 2(a), (b). Grossly, the non-fluid
type expressions of Lorenz ratios for all cases - 3D-NR, 2D-NR, 3D-G, and 2D-G
are saturated near 1 in the SL range, which means that the WF law is obeyed in
the SL range. Among all four cases, 3D-NR and 2D-G are realistic cases. The
former is applicable to metal systems, whose Fermi energies remain constant
and within the SL range, so they always follow the WF law. Later case - the
non-fluid expressions of Lorenz ratio for 2D-G can be applicable for realistic
graphene system with high doping concentration ($\epsilon_{F}$ is quite large
or $\epsilon_{F}/T\gg 1$), where the WF law is obeyed well again [15].
However, by decreasing the doping concentration, when the graphene system
approaches the charge-neutral points ($\epsilon_{F}$ is quite small or
$\epsilon_{F}/T\ll 1$), the WF law violation domain is observed [15].
According to our non-fluid expressions, the WF law violation is also expected
in the $\epsilon_{F}/T\ll 1$ domain but the outcome $L/L_{0}<1$ is not
matching with experimental outcome $L/L_{0}>1$ by Crossno et al. [15]. As a
result, this $\epsilon_{F}/T\ll 1$ domain may need a fluid-type expression
(something different from traditional non-fluid expressions) of $L/L_{0}$.
This unsettled fact is investigated by the RTA-based Boltzmann transport
formalism, which may be considered an approach towards fluid descriptions. It
was already addressed in the formalism section, and its numerical results will
be described in the next paragraph.

For numerical curves of fluid-type expressions, let us focus only on the 2D-G
case. Although formalism of other cases with an Appendix is provided in the
present article, one may go for result generation for other cases if they have
any real system application. Using Eqs. (51) and (52), black and red solid
lines are plotted in the left panel of Fig. 3. They are saturated at the
values of 2 and 4, respectively, in the high doping or $\epsilon_{F}/T\gg 1$
domain. It means that if the fluid type expression is valid in this high
doping domain 2D-G system, then the WF law may not be obeyed. However, the
experimental data [15] claims that the WF law is well obeyed in the high
doping or $\epsilon_{F}/T\gg 1$ domain. This means that the non-fluid
expression of the Lorenz ratio should be applicable to this domain instead of
fluid-type expression. The SL of Eqs. (51) and (52) will also be (see Appendix
E)

$\displaystyle\frac{L_{G1}^{2D}}{L_{0}}=2+\frac{2\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(68)
$\displaystyle\frac{L_{G2}^{2D}}{L_{0}}=4+\frac{4\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(69)

which are plotted by black and red dotted lines, respectively, in the left
panel of Fig. 3. They blow up from their saturated values (2 and 4) when
$\epsilon_{F}/T$ or doping concentration decreases. So, collecting the results
of all SL expressions -Eqs. (68) and (69) for fluid-type pictures and the
values 1 and 2 from Eqs. (38) and (39) for non-fluid-type pictures, we have
plotted black, red dotted, and dash lines in the right panel of Fig. 3. The
experimental data of Crossno et al. [15] is pasted as an inset figure in the
right panel of Fig. 3, where S1, S2, and S3 represent the samples of the 2D-G
system in terms of lower to higher values of minimum charge carrier density.
Theoretically, zero charge carrier density can be reached at
$\epsilon_{F}/T\rightarrow 0$ but experimentally, a minimum charge carrier
density [15] will be achieved. So, connecting the experimental and theoretical
outcomes, we may have to focus on the qualitative trend of curves. From the
experimental side, we may take the gross level phenomenon that the transition
from higher to lower values of $\epsilon_{F}/T$ pushes the Lorenz ratio from
$L/L_{0}=1$ to $L/L_{0}>1$. On the other hand, from the theoretical side, when
we compile our SL results, the fluid-type expression of Lorenz ratio at lower
values of $\epsilon_{F}/T$ provides $L/L_{0}>1$, while its non-fluid-type
expression at higher values of $\epsilon_{F}/T$ provides $L/L_{0}=1$. So, one
may expect a transition from non-fluid to fluid behavior of electrons in the
2D-G system during the transition from higher to lower $\epsilon_{F}/T$ or
charge carrier concentration. Blue arrows in the right panel of Fig. 3 roughly
denote this transition. This indirectly supports the expectation of the Dirac-
fluid nature of electrons in the domain of lower $\epsilon_{F}/T$ of the 2D-G
system, where the WF law is violated ($L/L_{0}>1$). Eventually, one may
conclude that fluid nature is the reason for the violation.

##  IV Summary

In summary, we have gone through comparative research on Lorenz ratios for 3D,
2D NR and G systems for a systematic search of the WF law violation domain.
During our analysis, we found the WF law violation domain of Fermi-energy
(normalized by a fixed temperature), where the Lorenz ratio deviates from 1.
First, we have used standard solid state book-based non-fluid expressions of
Lorenz ratio for the 3D-NR case, which are also applied for 2D-NR, 3D-G, and
2D-G cases. For all cases, the Lorenz ratio remains constant in the higher
values of normalized Fermi energy. This fact is well in agreement with the
universal validity of the WF law in metal systems, which can be described by
the 3D-NR case of the electron gas having larger Fermi-energy ($2-10$ eV) or
normalized Fermi-energy 400-2000 at $60$ K. Depending on the definition of
specific heat at constant volume and Fermi-energy, we will get two possible
saturated values of Lorenz ratios - 1 and 2, where former is experimentally
observed (average) value but one may mark the saturating domain as the WF law
obeying domain. If one identifies the less and greater than one value of
normalized Fermi-energy, then former domain of all cases shows the deviation
of the WF law. This tuning of normalized Fermi-energy is conveniently possible
for 2D-G system only via tuning the electron doping. Interestingly, non-fluid
expressions of the 2D-G case and recently measured Lorenz ratio by Crossno et
al. [15] both indicate the violation of the WF law in the lower normalized
Fermi-energy domain, which is popularly called as Dirac Fluid (DF) domain.
However, the theory of non-fluid expressions provides Lorenz ratio as lower
than one, but the experiment pointed out that its values are larger than one
in the DF domain. It demands that we may have to transit towards a fluid-type
description from the standard non-fluid expressions of the Lorenz ratio. So
far, to the best of our knowledge, this straightforward thermodynamical phase
space analysis of Lorenz ratio is missing in the literature, and it may be an
important first step before going to a fluid-type description.

Next, we have built a framework towards the fluid-type description of the
Lorenz ratio by using the Boltzmann transport equation, whose Sommerfeld
expansion limit becomes divergent instead of constant for non-fluid
description. In the lower normalized Fermi-energy domain or DF domain, this
divergence trend of the Lorenz ratio may be associated with the experimentally
observed enhancement of the Lorenz ratio. Interestingly, by increasing the
cleanness of the sample, the Lorenz ratio is also enhanced. Therefore, this
connection between our present calculation and experimental data indicates a
necessity for fluid prescription. The present work aims for a systematic
comparative study to identify the the WF law violation domain from a non-fluid
to fluid type framework. After going through this systematic analysis, the
present work is aimed for the future to explain or understand the Crossno et
al. [15] experimental data by doing more phenomenological analysis.

###### Acknowledgements.

This work was partly (T.Z.W. and C.W.A.) supported by the Doctoral Fellowship
in India (DIA) program of the Ministry of Education, Government of India. The
authors thank the other members of eHD club Sesha P. Vempati, Ashutosh
Dwibedi, Narayan Prasad, Bharat Kukkar, and Subhalaxmi Nayak.

##  Appendix A FERMI-DIRAC DISTRIBUTION FUNCTION

The Fermi-Dirac distribution function is

$f_{0}\left(\epsilon\right)=\frac{1}{e^{\beta\left(\epsilon-\epsilon_{F}\right)}+1}=\frac{1}{A^{-1}e^{\beta\epsilon}+1},$
(70)

where $\epsilon$ = energy of fermions and $\epsilon_{F}$ = fermi energy that
is nothing but the energy level up to the fermions filled or the maximum
kinetic energy of fermions at absolute temperature (0 K). The thermodynamics
parameter which is $\beta=\frac{1}{k_{B}T}$ and $k_{B}$ is the Boltzmann
constant and $A$ is the fugacity of the system described by

$A=\exp{\left({\frac{\epsilon_{F}}{k_{B}T}}\right)}.$

And the derivative of the distribution function with respect to energy is

$-\frac{\partial f_{0}}{\partial\epsilon}=\frac{\beta
e^{\beta\left(\epsilon-\epsilon_{F}\right)}}{\left(e^{\beta\left(\epsilon-\epsilon_{F}\right)}+1\right)^{2}}=\frac{\partial}{\partial\epsilon_{F}}\left(\frac{1}{e^{\beta\left(\epsilon-\epsilon_{F}\right)}+1}\right).$
(71)

##  Appendix B FERMI-DIRAC FUNCTION

  * •

The Fermi-Dirac Function

$\displaystyle
f_{\nu}(A)=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}\frac{x^{\nu-1}}{A^{-1}e^{x}+1}dx~{},$
(72)

where $f_{\nu}(A)$ is known as the Fermi-Dirac function.

Case.1. When $A$ is small, then the Fermi-Dirac function can be written in a
series form which is

$\displaystyle f_{\nu}(A)=$ $\displaystyle
A-\frac{A^{2}}{2^{\nu}}+\frac{A^{3}}{3^{\nu}}-\frac{A^{4}}{4^{\nu}}+\dots$
(73)
$\displaystyle=\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{A^{n}}{n^{\nu}}.$
(74)

Case.2. When $A$ is too much small $\left(\epsilon_{F}<<k_{B}T\right)$, then
the function becomes simplified as $A$

$f_{\nu}(A)=A,$ (75)

but if we take $\epsilon_{F}=0$, the function becomes unity.

Case.3. When the temperature is very small, and the Fermi energy has some
finite value, then the Fermi-Dirac function can be written according to
Sommerfeld’s lemma, which gives the expression of the function as

$f_{\nu}\left(A\right)=\frac{\alpha^{\nu}}{\Gamma\left(\nu+1\right)}\Bigg{[}1+\nu\left(\nu-1\right)\frac{\pi^{2}}{6}\frac{1}{\alpha^{2}}+\nu\left(\nu-1\right)\left(\nu-2\right)\left(\nu-3\right)\frac{7\pi^{4}}{360}\frac{1}{\alpha^{4}}+\dots\Bigg{]},$
(76)

where $\alpha$ is given by

$\alpha=\ln{A}=\frac{\epsilon_{F}}{k_{B}T}~{}.$

And

$\Gamma\left(\nu+1\right)=\nu!~{},$

so, using this lemma, we can calculate the value of the Fermi-Dirac function
for different values of $\nu$ for this particular type of case.

Case.4. When $\left(\epsilon_{F}>>k_{B}T\right)$, then the function can be
written only using the zeroth order term of Sommerfeld’s lemma expression.

  * •

The derivative of Fermi function with respect to Fermi energy

$\frac{\partial f_{\nu}\left(A\right)}{\partial\epsilon_{F}}=\beta
f_{\nu-1}\left(A\right).$ (77)

  * •

The derivative of Fermi function with respect to temperature

$\frac{\partial f_{\nu}\left(A\right)}{\partial
T}=\frac{1}{A}f_{\nu-1}\left(A\right)\frac{\partial A}{\partial T}.$ (78)

The one identity for the function can be written as

$\frac{1}{A}\frac{\partial A}{\partial
T}=-\epsilon_{F}\beta^{2}k_{B}=-\frac{\epsilon_{F}}{k_{B}T^{2}}.$ (79)

##  Appendix C ELECTRONIC SPECIFIC HEAT

For 2D system graphene, which follows the linear dispersion relation, with the
help of the density of state method, we can calculate the number density,
total internal energy, and the electronic-specific heat.

So, the number of energy states in energy range $\epsilon$ to
$\epsilon+d\epsilon$ and the surface area $S$ is written as

$D\left(\epsilon\right)d\epsilon=g\frac{S}{\left(2\pi\hbar\right)^{2}}\frac{2\pi}{v_{F}^{2}}\epsilon
d\epsilon.$ (80)

Now, the total number of particles at any value of temperature can be
calculated as

$N=\int_{0}^{\infty}D\left(\epsilon\right)d\epsilon
f_{0}\left(\epsilon\right).$ (81)

After plugging the value of $D\left(\epsilon\right)d\epsilon$ in the above
equation, we get

$n=\frac{N}{S}=\frac{g}{2\pi\hbar^{2}v_{F}^{2}}\int_{0}^{\infty}\frac{\epsilon}{A^{-1}e^{\beta\epsilon}+1}d\epsilon.$
(82)

After solving this integration, we get the final, more general expression of
number density, which is given by

$N=g\frac{2\pi
S}{h^{2}v_{F}^{2}}\frac{\Gamma\left(2\right)}{\beta^{2}}f_{2}\left(A\right)$
(83) $\implies
n=\frac{k_{B}^{2}}{\pi\hbar^{2}v_{F}^{2}}f_{2}\left(A\right)T^{2}~{}.$ (84)

The total internal energy of a system can be calculated as

$U=\int_{0}^{\infty}D\left(\epsilon\right)d\epsilon
f_{0}\left(\epsilon\right)\epsilon.$ (85)

By substituting the value of $D\left(\epsilon\right)d\epsilon$ in the above
equation and solving the integration, we get the final, more general
expression of total internal energy, which is given by

$U=g\frac{2\pi
S}{h^{2}v_{F}^{2}}\frac{\Gamma\left(3\right)}{\beta^{3}}f_{3}\left(A\right)$
(86) $\implies
u=\frac{U}{S}=g\frac{2\pi}{h^{2}v_{F}^{2}}\frac{\Gamma\left(3\right)}{\beta^{3}}f_{3}\left(A\right).$
(87)

Now, from the above equations, we get

$\frac{u}{n}=\frac{U}{N}=2k_{B}T\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}.$
(88)

Let us define the specific heat capacity of the electron by taking the
temperature derivative of internal energy per electron ($U/N=u/n$) while
keeping surface area $S$ and $\epsilon_{F}$ as constants:

$\displaystyle\prescript{2D}{G1}{[C_{V}]_{e1}}=\Bigg{[}\frac{\partial}{\partial
T}\Big{(}\frac{U}{N}\Big{)}\Bigg{]}_{S,\epsilon_{F}}.$ (89)

Using the Eq. (88), the specific heat is

$\displaystyle\prescript{2D}{G1}{[C_{V}]_{e1}}=2k_{B}\Bigg{[}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}+T\frac{\partial}{\partial
T}\left(\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\right)\Bigg{]}.$ (90)

Now, the derivative part after using the identity (78) of the Fermi-Dirac
function can be written as

$\displaystyle\frac{\partial}{\partial
T}\left(\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\right)$
$\displaystyle=\frac{1}{f_{2}\left(A\right)}\frac{\partial}{\partial
T}\left(f_{3}\left(A\right)\right)-f_{3}\left(A\right)\frac{\partial}{\partial
T}\left(\frac{1}{f_{2}\left(A\right)}\right)$
$\displaystyle=\frac{1}{A}\frac{f_{2}\left(A\right)}{f_{2}\left(A\right)}\frac{\partial
A}{\partial
T}-\frac{f_{3}\left(A\right)}{\Big{[}f_{2}\left(A\right)\Big{]}^{2}}\frac{\partial}{\partial
T}f_{2}\left(A\right)$ $\displaystyle=\frac{1}{A}\frac{\partial A}{\partial
T}-\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\frac{f_{1}\left(A\right)}{f_{2}\left(A\right)}\frac{1}{A}\frac{\partial
A}{\partial T}.$

The final form, we get

$\frac{\partial}{\partial
T}\left(\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\right)=\frac{2}{T}\Bigg{[}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}.$
(91)

After substituting the value from equation (91) into equation (90), we get a
more general form of electronic specific heat given by

$\prescript{2D}{G1}{[C_{V}]_{e1}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}~{}.$
(92)

Let us go for another definition of specific heat per electron

$\prescript{2D}{G2}{[C_{V}]_{e2}}=\frac{1}{N}\frac{\partial U}{\partial
T}\Bigg{|}_{S,\epsilon_{F}}.$ (93)

From Eq. (86), the total internal energy is

$\displaystyle\prescript{2D}{G}{U}=\gamma T^{3}f_{3}\left(A\right),$

where

$\gamma=g\frac{2\pi S}{h^{2}v_{F}^{2}}\Gamma\left(3\right)k_{B}^{3}.$

Then, we will take the derivative of just the above equation to $T$ at
constant Fermi energy $\epsilon_{F}$ and surface area; we get

$\displaystyle\frac{\partial U}{\partial T}\Big{|}_{S,\epsilon_{F}}$
$\displaystyle=\gamma\Bigg{[}3T^{2}f_{3}\left(A\right)+T^{3}f_{2}\left(A\right)\frac{1}{A}\frac{\partial
A}{\partial T}\Bigg{]}$ (94) $\displaystyle=\gamma
T^{2}f_{2}\left(A\right)\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}+T\frac{1}{A}\frac{\partial
A}{\partial T}\Bigg{]}~{}.$ (95)

So,

$\prescript{2D}{G2}{[C_{V}]_{e2}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(96)

By doing a similar type of calculation for 2D-NR case, we will get two
different specific heat expressions as

$\prescript{2D}{NR1}{[C_{V}]_{e1}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]},$
(97)

and

$\prescript{2D}{NR2}{[C_{V}]_{e2}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(98)

Next, for 3D-NR and 3D-G cases also, one can repeat the calculations and find
the expressions of specific heat:

$\prescript{3D}{NR1}{[C_{V}]_{e1}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]},$
(99)
$\prescript{3D}{NR2}{[C_{V}]_{e2}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(100)
$\prescript{3D}{G1}{[C_{V}]_{e1}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]},$
(101)
$\prescript{3D}{G2}{[C_{V}]_{e2}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(102)

##  Appendix D NUMBER DENSITY FOR 3D AND 2D SYSTEMS

  * •

Case.1. Number density in nonrelativistic case at temperature $T=0K$

$n^{3D}_{NR}=g\frac{4\pi\left(2m\epsilon_{F}\right)^{\frac{3}{2}}}{3h^{3}}.$
(103)

  * •

Case.2. Number density in ultrarelativistic case at temperature $T=0K$

$n^{3D}_{UR}=g\frac{4\pi}{3h^{3}}\left(\frac{\epsilon_{F}}{c}\right)^{3}.$
(104)

  * •

Case.3. Number density in graphene case at temperature $T=0K$

$n^{3D}_{G}=g\frac{4\pi}{3h^{3}}\left(\frac{\epsilon_{F}}{v_{F}}\right)^{3}.$
(105)

  * •

Case.1. Number density in nonrelativistic case at temperature $T=0K$

$n^{2D}_{NR}=g\left(\frac{2\pi m}{h^{2}}\right)\epsilon_{F}.$ (106)

  * •

Case.2. Number density in ultrarelativistic case at temperature $T=0K$

$n^{2D}_{UR}=g\frac{\pi}{h^{2}}\left(\frac{\epsilon_{F}}{c}\right)^{2}.$ (107)

  * •

Case.3. Number density in graphene case at temperature $T=0K$

$n^{2D}_{G}=g\frac{\pi}{h^{2}}\left(\frac{\epsilon_{F}}{v_{F}}\right)^{2}.$
(108)

##  Appendix E SOMMERFELD-LEMMA EXPRESSIONS OF ELECTRICAL AND THERMAL
CONDUCTIVITY FOR FLUID DESCRIPTION

Table 1: Lorenz ratio values in Sommerfeld expression for 3D. | Non-Fluid | Towards Fluid
---|---|---
| 3D-NR | 3D-G | 3D-NR | 3D-G
$\frac{L_{1}}{L_{0}}$ | $1$ | $1$ | $1.5$ | $4.5$
$\frac{L_{2}}{L_{0}}$ | $1.5$ | $3$ | $2.25$ | $9$
Table 2: Lorenz ratio values in Sommerfeld expression for 2D. | Non-Fluid | Towards Fluid
---|---|---
| 2D-NR | 2D-G | 2D-NR | 2D-G
$\frac{L_{1}}{L_{0}}$ | $1$ | $1$ | $1$ | $2$
$\frac{L_{2}}{L_{0}}$ | $1$ | $2$ | $1$ | $4$

The Sommerfeld-lemma expressions for electrical and thermal conductivity for
2-dimensional graphene is

$\displaystyle\sigma_{G}^{2D}=2\pi\tau_{c}\left(\frac{e}{h}\right)^{2}\epsilon_{F},$
(109)
$\displaystyle\kappa_{G1}^{2D}=\frac{4\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(110)
$\displaystyle\kappa_{G2}^{2D}=\frac{8\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(111)

So, after using Sommerfeld-lemma, the expressions of the electrical(109),
thermal conductivity(110), (111), and the ratio of $L/L_{0}$ for a
2-dimensional graphene system are respectively as;

$\displaystyle\frac{L_{G1}^{2D}}{L_{0}}=2+\frac{2\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(112)
$\displaystyle\frac{L_{G2}^{2D}}{L_{0}}=4+\frac{4\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(113)

For a 3-dimensional graphene system, after using Sommerfeld’s lemma, the
expressions of the electrical and thermal conductivity;

$\displaystyle\sigma_{G}^{3D}=\frac{8\pi
e^{2}\tau_{c}\epsilon_{F}^{2}}{3h^{3}v_{F}}\Biggl{[}1+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}\Biggl{]},$
(114)
$\displaystyle\kappa_{G1}^{3D}=\frac{4\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}^{2}}{h^{3}v_{F}}\Biggl{[}T+\pi^{2}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(115)
$\displaystyle\kappa_{G2}^{3D}=\frac{8\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}^{2}}{h^{3}v_{F}}\Biggl{[}T+\pi^{2}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(116)

For a 3-dimensional graphene system, the electrical conductivity(114), thermal
conductivity(115), (116), and the ratio of $L/L_{0}$ are

$\displaystyle\frac{L_{G1}^{3D}}{L_{0}}=\frac{9}{2}+3\pi^{2}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(117)
$\displaystyle\frac{L_{G2}^{3D}}{L_{0}}=9+6\pi^{2}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(118)

After applying Sommerfeld-lemma on a 2-dimensional nonrelativistic system
then, we get all the expressions of electrical and thermal conductivity,

$\displaystyle\sigma_{NR}^{2D}=4\pi\tau_{c}\left(\frac{e}{h}\right)^{2}\epsilon_{F},$
(119)
$\displaystyle\kappa_{NR1}^{2D}=\frac{4\pi^{3}k_{B}^{2}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(120)
$\displaystyle\kappa_{NR2}^{2D}=\frac{4\pi^{3}k_{B}^{2}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(121)

After applying Sommerfeld-lemma on a 2-dimensional nonrelativistic system
then, we get all the expressions of electrical(119), thermal
conductivity(120), (121), and the ratio of $L/L_{0}$ which are given by

$\displaystyle\frac{L_{NR1}^{2D}}{L_{0}}=1+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(122)
$\displaystyle\frac{L_{NR2}^{2D}}{L_{0}}=1+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(123)

After applying Sommerfeld-lemma on a 3-dimensional nonrelativistic system
then, we get all the expressions of electrical and thermal conductivity;

$\displaystyle\sigma_{NR}^{3D}=\frac{8\pi
e^{2}\tau_{c}}{3h^{3}m}\left(2m\epsilon_{F}\right)^{\frac{3}{2}}\Biggl{[}1+\frac{\pi^{2}}{8}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}\Biggl{]},$
(124)
$\displaystyle\kappa_{NR1}^{3D}=\frac{4\pi^{3}k_{B}^{2}\tau_{c}}{3mh^{3}}\left(2m\epsilon_{F}\right)^{\frac{3}{2}}\Biggl{[}T+\frac{5\pi^{2}}{8}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(125)
$\displaystyle\kappa_{NR2}^{3D}=\frac{2\pi^{3}k_{B}^{2}\tau_{c}}{mh^{3}}\left(2m\epsilon_{F}\right)^{\frac{3}{2}}\Biggl{[}T+\frac{5\pi^{2}}{8}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(126)

After applying Sommerfeld-lemma on a 3-dimensional nonrelativistic system
then, we get all the expressions of electrical(124), thermal
conductivity(125), (126), and the ratio of $L/L_{0}$ which are given by

$\displaystyle\frac{L_{NR1}^{3D}}{L_{0}}=\frac{3}{2}+\frac{3\pi^{2}}{4}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(127)
$\displaystyle\frac{L_{NR2}^{3D}}{L_{0}}=\frac{9}{4}+\frac{9\pi^{2}}{8}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(128)

After compiling all of Sommerfeld-lemma results, the roughly summarizing
results for different cases in three dimensions (3D) in the table (1) and two
dimensions (2D) in the table (2), particularly regarding the Lorenz ratio
validation and violation are shown.

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