# Orbital Kerr effect and terahertz detection via the nonlinear Hall effect

Diego García Ovalle<EMAIL_ADDRESS>Armando Pezo
<EMAIL_ADDRESS>Aurélien Manchon
<EMAIL_ADDRESS>Aix-Marseille Université, CNRS, CINaM, Marseille,
France

###### Abstract

We investigate the optical response induced by a d.c. current flowing in a
nonmagnetic material that lacks inversion symmetry. In this class of
materials, the flowing current experiences a nonlinear Hall effect and induces
a nonequilibrium orbital magnetization, even in the absence of spin-orbit
coupling. As a result, an orbital-driven Kerr effect arises that can be used
to probe not only the orbital magnetization, but also the nonlinear Hall
effect. In addition, in the long wavelength limit, the nonlinear Hall effect
leads to a rectification current that can be used to detect terahertz
radiation. We apply the theory to selected model systems, such as WTe2
bilayer, as well as to realistic materials, i.e., bulk Te and metallic
superlattices. The nonequilibrium orbital Kerr efficiencies obtained in these
systems are comparable to the largest values reported experimentally in GaAs
and MoS2, exceeding the values reported in metals and suggesting a large
terahertz current responsivity.

##  I Introduction

Nonreciprocal transport in noncentrosymmetric quantum materials has been
attracting increasing interest over the past decade [1], emphasizing the
connection between nonlinear optical and electrical responses and the geometry
of the Bloch states [2, 3, 4]. Among these phenomena, the nonlinear Hall
effect associated with the Berry curvature dipole [5] (BCD) is particularly
intriguing as it enables the generation of transverse currents even in the
absence of magnetic field [6, 7]. In a recent proposal, Zhang and Fu [8]
suggested that the rectification induced by the nonlinear Hall effect could be
used to detect terahertz radiations, a topic gaining increasing momentum [9].
As discussed in detail below, the nonlinear Hall effect, at the second order
in the electric field, is a companion of the nonequilibrium orbital
magnetization [10]. Indeed, both effects only require inversion symmetry
breaking and can emerge in the absence of spin-orbit coupling. Therefore, the
nonlinear Hall effect is not only a signature of the BCD but also a signature
of the nonequilibrium orbital magnetization. This connection is remarkably
instrumental as it enables the optical probe of the nonequilibrium orbital
moment via the so-called orbital-driven Kerr effect, as illustrated in Fig. 1.

In recent years, it has been proposed that orbital currents and densities can
be electrically induced in noncentrosymmetric metals via the so-called orbital
Hall effect [11, 12, 13] and orbital Rashba-Edelstein effect [14, 15],
respectively, not necessitating spin-orbit coupling. This prediction opens
wide perspectives for materials research and device development as entire
families of metallic compounds made of light elements, often cheap and
abundant, could in principle host interconversion phenomena between charge and
orbital currents [13, 16, 17, 18]. The central role of orbital currents in
spin torque [19, 20, 21, 22, 23], spin pumping [24] and magnetoresistance
experiments [25] has been clearly demonstrated recently.

In all the experiments mentioned above, the clear distinction between spin and
orbital transport phenomena remains challenging as both classes of effects are
intermingled via spin-orbit coupling. Since the orbital accumulation of
conduction electrons produces a nonequilibrium orbital magnetization, it can
be detected by converting this orbital signal into a chemical potential using
a proximate magnetic layer for instance. This technique is typically used to
detect spin accumulation in magnetic bilayers [26] or in nonlocal geometries
[27]. When it comes to detecting the orbital accumulation though, a difficulty
arises. Indeed, assuming that this proximate ferromagnet is made out of
transition metals, its magnetization mostly comes from the spin angular
momentum because of orbital quenching [28, 29]. Therefore, one needs to first
convert the orbital signal into a spin signal, typically via spin-orbit
coupling, as achieved in Ref. [25]. In other words, the electrical detection
of nonequilibrium orbital magnetization using a bilayer configuration requires
the coexistence of spin-orbit coupling and magnetism. As a result, it is
virtually impossible to selectively probe the orbital accumulation using
electrical means only.

Figure 1: (Color online) Schematics of the connection between the
nonequilibrium orbital Kerr effect and the second-order anomalous Hall effect
in a noncentrosymmetric, nonmagnetic metal.

An alternative method consists of using optical methods [10, 30, 31] to
directly probe the orbital accumulation without relying on any magnetic
material or nonlocality. Indeed, the magnetooptical Kerr (and Faraday) effect
is routinely used to probe the magnetization of ferro- and ferrimagnets and
image magnetic domains [32]. In this process, the spin-orbit coupling enables
the generation of the spin accumulation itself as well as its coupling with
the circularly polarized light. This method has been successfully used in the
past to detect the current-driven spin accumulation generated via spin Hall
effect in GaAs [33], Pt and W thin films [34] and Bi2O3 surface [35]. In light
metals though, both the generation of orbital accumulation out of equilibrium
and its coupling with light occur in the absence of spin-orbit coupling, which
results, in principle, in a much larger nonequilibrium Kerr signal [30, 31].

In this Article, we show that the Kerr effect can be used to probe this
nonequilibrium orbital magnetization, via the nonlinear anomalous Hall effect
[5]. In ferromagnets, the magnetooptical Kerr effect does not directly probe
the magnetization itself, but it rather probes the anomalous Hall effect of
the magnetic material, so that the Kerr angle is directly related to the
anomalous Hall conductivity $\theta_{k}+i\phi_{k}\sim\sigma_{xy}/\sigma_{xx}$.
Hence, the mere presence of the nonlinear Hall effect guarantees the current-
induced orbital-driven Kerr effect. In addition, the rectified Hall current
can be used to detect the presence of low-energy photons [8], typically in the
terahertz range (10 THz=12 meV). Remarkably, the ”current responsivity”, i.e.,
the performance criterion for terahertz detection, is directly proportional to
the orbital Kerr efficiency in the low-frequency limit, $R\propto\theta_{k}$.
This relation opens appealing perspectives in the search for optimal materials
for terahertz detection.

Our manuscript is organized as follows: In Section II we present the general
theory of the nonequilibrium orbital Kerr effect in nonmagnetic,
noncentrosymmetric crystals, and establish the connection between the
nonequilibrium orbital magnetization and the nonlinear Hall effect. Then, in
Section III we demonstrate the proof-of-principle on bilayer WTe2 [36] and
Nb2n+1SinTe4n+2 [37]. In Section IV, we present first-principles calculations
on bulk Te and on a multilayer made out of transition metals. Finally, in
Section V we provide a general discussion and draw the main conclusions.

##  II General Theory

###  II.1 Nonlinear magnetooptics

Without loss of generality, let us consider the magnetooptical Kerr effect in
its polar version [32, 38]. Under the application of a perpendicularly
incident light $\textbf{E}_{\rm
ac}=\mathcal{E}_{0}e^{i(\mathbf{k}\cdot\mathbf{z}-\omega t)}$ with frequency
$\omega$ and circular polarization, the Kerr rotation angle $\theta_{k}$ and
the ellipticity $\phi_{k}$ can be approximated in terms of the longitudinal
($\sigma_{xx}$) and transverse ($\sigma_{xy}$) optical conductivity [38],

$\displaystyle\theta_{k}+i\phi_{k}=\frac{\sigma_{xy}}{\sigma_{xx}\left(1+\frac{4\pi
i}{\omega}\sigma_{xx}\right)^{\frac{1}{2}}}.$ (1)

The longitudinal conductivity in Eq. (1) can be accounted for through the
Green function formula [39]

$\displaystyle\sigma_{xx}=\frac{e^{2}}{4\pi\hbar}\int
d\epsilon\partial_{\epsilon}f(\epsilon)\operatorname{Re}\left\\{\operatorname{Tr}\left[v_{x}G^{\rm
R-A}v_{x}G^{\rm R-A}\right]\right\\},$ (2)

where $G^{\rm R(A)}=\left[(\epsilon+\hbar\omega)-\mathcal{H}_{\mathbf{k}}\pm
i\Gamma\right]^{-1}$, $G^{\rm R-A}=G^{\rm R}-G^{\rm A}$ and $\Gamma$ is a
disorder broadening. Besides, the transverse optical conductivity is given by
[40]

$\displaystyle\sigma_{xy}=\frac{e^{2}}{\hbar}\int_{BZ}\frac{d^{2}k}{(2\pi)^{2}}\sum_{nm}\Omega_{nm\mathbf{k}}^{z}(f_{n\mathbf{k}}^{(0)}-f_{m\mathbf{k}}^{(0)}),$
(3)

where $\mathbf{\Omega}_{n\mathbf{k}}$ is the Berry curvature (BC)

$\displaystyle\mathbf{\Omega}_{n\mathbf{k}}(\omega)$
$\displaystyle=i\sum_{m\not=n}\frac{\bra{u_{n\mathbf{k}}}\hat{v}_{\mathbf{k}}\ket{u_{m\mathbf{k}}}\times\bra{u_{m\mathbf{k}}}\hat{v}_{\mathbf{k}}\ket{u_{n\mathbf{k}}}}{(\epsilon_{n\mathbf{k}}-\epsilon_{m\mathbf{k}})^{2}-(\hbar\omega+i\Gamma)^{2}}$
$\displaystyle=\sum_{m\not=n}\mathbf{\Omega}_{nm\mathbf{k}}$ (4)

in the Bloch eigenbasis, where
$\hat{v}_{\mathbf{k}}=\frac{\partial\mathcal{H}_{\mathbf{k}}}{\partial\mathbf{k}}$.
Notice that in the limit $\hbar\omega\to 0$, Eq. (3) converges to the d.c.
Hall conductivity [41]. In the present work, the Fermi distribution
$f_{n\mathbf{k}}^{(0)}$ is taken at zero temperature such that
$f_{n\mathbf{k}}^{(0)}\to-\Theta(\epsilon_{n\mathbf{k}}-\mu)$, where
$\Theta(x)$ is the Heaviside step function and $\mu$ the chemical potential.
From Eq. (3), it is clear that if the system is time-reversal symmetric then
$\sigma_{xy}\to 0$, and thus the magnetooptical Kerr effect completely
vanishes in the absence of flowing current. Notwithstanding, when time
reversal is preserved and inversion symmetry is absent, a nonequilibrium Kerr
effect is allowed by symmetry, revealing the underlying generation of a
nonequilibrium magnetization in the process. Now, let us suppose that an
additional d.c. electric field $\mathbf{E}_{0}$ is applied in the sample.
Following the rationale developed by Sodemann and Fu [5] for the nonlinear
Hall effect, and Yoda and Murakami [15] for the nonequilibrium orbital
magnetization, the nonequilibrium Fermi distribution function reads
$f_{n\mathbf{k}}=f_{n\mathbf{k}}^{(0)}+f_{n\mathbf{k}}^{(1)}$, and within the
relaxation time approximation,

$\displaystyle
f_{n\mathbf{k}}^{(1)}=\frac{e\tau}{\hbar}\mathbf{E}_{0}\cdot\partial_{\mathbf{k}}f_{n\mathbf{k}}^{(0)},$
(5)

where $\tau$ is a constant scattering time. Replacing Eq. (5) into Eq. (3), we
deduce that [42]

$\displaystyle\sigma_{xy}$ $\displaystyle=$
$\displaystyle\frac{e^{3}\tau\mathbf{E}_{0}}{\hbar^{2}}\int_{BZ}\frac{d^{2}{\bf
k}}{(2\pi)^{2}}\sum_{nm}\Omega_{nm\mathbf{k}}^{\mathrm{z}}(\omega)(\partial_{\mathbf{k}}f_{n\mathbf{k}}^{(0)}-\partial_{\mathbf{k}}f_{m\mathbf{k}}^{(0)}).$
(6)

Therefore, the combination of Eqs. (6) and (1) indicates that the
nonequilibrium Kerr effect is proportional to an a.c. BCD [5] [the integral in
Eq. (6)], which in fact is the leading order contribution to the second-order
optical conductivity at small frequencies [43]. As such, the nonequilibrium
orbital magnetization, nonlinear Hall current, and nonequilibrium Kerr effect
are companion phenomena in noncentrosymmetric nonmagnetic materials and
heterostructures. Because of this intimate relationship between the orbital
moment and the Kerr effect, we tag the latter ”orbital Kerr effect”. As
complementary evidence of this connection, in the next section we demonstrate
a general decomposition of the nonequilibrium orbital magnetization in terms
of the BCD in the d.c. limit $\hbar\omega\to 0$.

Before closing this section, let us comment on the rectification effect
associated with the nonlinear Hall response. As pointed out in Ref. [8], a
high-frequency electric field ${\bf E}_{\rm ac}={\bf E}_{0}\cos(\omega t)$
induces an oscillatory nonlinear Hall current [5]

$\displaystyle
j_{i}=\epsilon_{ibk}\frac{e^{3}\tau}{\hbar^{2}}D_{jk}E_{b}E_{j}\cos^{2}(\omega
t),$ (7)

where $D_{jk}$ is the BCD coefficient [see Eq. (12)]. The d.c. rectified Hall
current $j^{r}_{i}=\langle j_{i}(t)\rangle$ averaged over a cycle period can
be used to detect the incoming radiation. Following Ref. [8], the performance
indicator of this detector is called the current responsivity $R$ and is given
by the ratio between the Hall current $I_{H}$ and the absorbed power $P$,

$\displaystyle
Rw=\frac{I_{H}}{P}=\frac{j^{r}}{\sigma_{xx}E_{0}^{2}}=\frac{\theta_{k}}{2E_{0}},$
(8)

where $w$ is the width of the rectifier. Remarkably, the current
responsitivity is simply equal to half of the orbital Kerr electrical
efficiency.

###  II.2 Decomposition of the nonequilibrium orbital magnetization

Let us start with the definition of the nonequilibrium orbital magnetization
in the limit $\hbar\omega\to 0$, which is based on the orbital moment within
the unit cell [44],

$\displaystyle\mathbf{m}_{n\mathbf{k}}=i\sum_{m\not=n}\frac{\braket{u_{n\mathbf{k}}}{\hat{v}_{\mathbf{k}}}{u_{m\mathbf{k}}}\times\braket{u_{m\mathbf{k}}}{\hat{v}_{\mathbf{k}}}{u_{n\mathbf{k}}}}{(\epsilon_{n\mathbf{k}}-\epsilon_{m\mathbf{k}})}.$
(9)

Since it shares the same symmetries as the Berry curvature, the equilibrium
orbital magnetization is zero in nonmagnetic materials. Nevertheless, a
nonequilibrium orbital magnetization $\mathbf{M}_{\rm
orb}=\left(\frac{e^{2}\tau}{2\hbar^{2}}\right)\alpha\mathbf{E}$ is allowed in
systems without inversion symmetry. The orbital Edelstein coefficients are
defined [45, 15]

$\displaystyle\alpha_{ij}(\mu)$
$\displaystyle=\int_{BZ}\frac{d^{2}k}{(2\pi)^{2}}\sum_{n}m_{n\mathbf{k}}^{{i}}v_{n\mathbf{k}}^{{j}}\partial_{\epsilon_{n\mathbf{k}}}f_{n\mathbf{k}}^{(0)}.$
(10)

Using the definition of the Berry curvature in Eq. (4), Eq. (10) becomes

$\displaystyle\alpha_{ij}(\mu)=-i\int_{BZ}\frac{d^{2}k}{(2\pi)^{2}}\sum_{m\not=n,n}(\epsilon_{n\textbf{k}}-\epsilon_{m\textbf{k}})v_{n\mathbf{k}}^{j}$
$\displaystyle\left(\frac{\bra{u_{n\mathbf{k}}}\hat{v}_{\mathbf{k}}\ket{u_{m\mathbf{k}}}\times\bra{u_{m\mathbf{k}}}\hat{v}_{\mathbf{k}}\ket{u_{n\mathbf{k}}}}{(\epsilon_{n\textbf{k}}-\epsilon_{m\textbf{k}})^{2}}\right)_{i}\delta(\epsilon_{n\mathbf{k}}-\mu)$
$\displaystyle=-\int_{BZ}\frac{d^{2}k}{(2\pi)^{2}}\sum_{n}\left(\epsilon_{n\textbf{k}}{\Omega}_{{n}\textbf{k}}^{i}-b_{{n}\textbf{k}}^{i}\right)v_{n\mathbf{k}}^{j}\delta(\epsilon_{n\mathbf{k}}-\mu)$
$\displaystyle=\mu D_{ji}(\mu)+\mathcal{B}_{ji}(\mu).$ (11)

As we can see from Eq. (11), the nonequilibrium orbital magnetization can be
divided into two contributions: A term linear in the chemical potential and
proportional to the BCD [5]

$\displaystyle
D_{ji}=\int_{BZ}\frac{d^{2}k}{(2\pi)^{2}}\left[\sum_{n}v_{n\mathbf{k}}^{j}\Omega_{n\mathbf{k}}^{i}\partial_{\epsilon_{n\mathbf{k}}}f_{n\mathbf{k}}^{(0)}\right],$
(12)

and a term that is nonlinear in $\mu$ arising from an effective magnetic field
with the same symmetries as the Berry curvature and the orbital magnetic
moment,

$\displaystyle\textbf{b}_{n\mathbf{k}}(\omega)$
$\displaystyle=i\sum_{m\not=n}\epsilon_{m\mathbf{k}}\frac{\bra{u_{n\mathbf{k}}}\hat{v}_{\mathbf{k}}\ket{u_{m\mathbf{k}}}\times\bra{u_{m\mathbf{k}}}\hat{v}_{\mathbf{k}}\ket{u_{n\mathbf{k}}}}{(\epsilon_{n\mathbf{k}}-\epsilon_{m\mathbf{k}})^{2}-(\hbar\omega+i\Gamma)^{2}}.$

Remarkably, expression (11) goes one step forward from previous studies
relating Berry curvature and orbital magnetic moment in the vicinity of the
gap [36] and for two-band systems with particle-hole symmetry [46].

##  III Two dimensional nonmagnetic systems

In this section, we compute the Kerr angle and analyze the previous
decomposition for the current-induced orbital magnetization in the cases of
WTe2 bilayer and Nb2n+1SinTe4n+2 monolayer.

###  III.1 WTe2 bilayer

In order to illustrate our theory, we start by computing $\theta_{k}$ and
$\phi_{k}$ in the exemplary case of the WTe2 bilayer [36], in which strong
signatures of second-order Hall effect has been reported experimentally [6,
7]. The minimal model is composed of four tilted Dirac Hamiltonians in the
form [36]

$\displaystyle\mathcal{H}_{\mathbf{k}}$
$\displaystyle=\begin{pmatrix}\mathcal{H}_{\mathbf{k}}^{d1}&\mathcal{P}_{\mathbf{k}}&0&\gamma\\\
\mathcal{P}_{\mathbf{k}}^{{\dagger}}&\mathcal{H}_{\mathbf{k}}^{d1}&\gamma&0\\\
0&\gamma&\mathcal{H}_{\mathbf{k}}^{d2}&\mathcal{P}_{\mathbf{k}}\\\
\gamma&0&\mathcal{P}_{\mathbf{k}}^{{\dagger}}&\mathcal{H}_{\mathbf{k}}^{d2}\end{pmatrix},$
(14)

where $\mathcal{H}_{\mathbf{k}}^{di}$ describes the tilted Dirac cone at
$\mathbf{K_{i}}$,

$\displaystyle\mathcal{H}_{\mathbf{k}}^{di}$
$\displaystyle=[E_{i}+t_{i}(k_{x}+K_{i})]\hat{\sigma}_{0}$
$\displaystyle+v_{i}[k_{y}\hat{\sigma}_{1}+\eta_{i}(k_{x}+K_{i})\hat{\sigma}_{2}]+\frac{m_{i}\hat{\sigma}_{3}}{2},$
(15)

where $\hat{\sigma}_{i}$, $i=1..3$ are the Pauli spin matrices,
$\hat{\sigma}_{0}=\mathrm{1}_{2\times 2}$, and $\gamma$ is an interlayer
coupling between the layers. In addition,

$\displaystyle\mathcal{P}_{\mathbf{k}}$
$\displaystyle=\begin{pmatrix}\nu_{x}k_{x}-i\nu_{y}k_{y}&0\\\
0&-\nu_{x}k_{x}-i\nu_{y}k_{y}\end{pmatrix}$ (16)

is a matrix that contains the spin-orbit coupling strength $\nu_{x}$
($\nu_{y}$) along $\mathbf{x}$ ($\mathbf{y}$). Since the Berry curvature of
the eigenstates of Eq. (14) is highly concentrated around the Dirac cones
[36], we perform our calculations in the domain
$\left[-\pi,\pi\right]\times\left[-\pi,\pi\right]$ for simplicity. The Berry
curvature and the nonequilibrium orbital moment are perpendicular to the
$(k_{x},k_{y})$ plane, and since the system also has a mirror symmetry along
$k_{y}$, only $D_{xz}$ and $\alpha_{zx}$ are non zero. Therefore, when the
electric field is applied along $\hat{x}$ and the Berry curvature is along
$\hat{z}$, the resulting orbital magnetization is along $\hat{z}$ and the Hall
current flows along $\hat{y}$. From the symmetry perspective, we can interpret
the second-order Hall current along $\hat{y}$ as the interplay between the
nonequilibrium orbital magnetization along $\hat{z}$ and the external electric
field applied along $\hat{x}$.

Figure 2: (Color online) Real (a) and imaginary (b) part of the a.c. BCD as a
function of the energy of the incident light for different values of spin-
orbit coupling $\nu_{x}$ and taking $\mu=0$. (c) Corresponding longitudinal
conductivity, (d) Kerr rotation, and (e) ellipticity. We consider the values
adopted in Ref. [36]: $\nu_{y}=0$, $\gamma=0.05$ eV, $K_{1}=0.1\pi$ Å-1,
$K_{2}=0.15\pi$ Å-1, $v_{1}=v_{2}=2$ eVÅ, $t_{1}=t_{2}=1.5$ eV Å,
$m_{1}=m_{2}=0.1$ eV, $\eta_{1}=-\eta_{2}=-1$, $E_{1}=0.02$ eV, $E_{2}=-0.08$
eV and $\Gamma=0.01$ eV.

The real and imaginary parts of the a.c. BCD and the longitudinal conductivity
as a function of the incoming light energy are reported in Figs. 2(a,b) and
(c), respectively. In the calculation discussed below, the electric field is
applied at $45^{\circ}$ with respect to the $\hat{x}$ axis in order to retain
the polar configuration. While the real part of the BCD converges to a finite
but non zero value at $\hbar\omega\to 0$, the imaginary part goes to zero at
zero frequency. In addition, $\operatorname{Re}(D)$ and $\operatorname{Im}(D)$
are concentrated at energies $\hbar\omega\leq 250$ meV, corresponding to the
model system’s bandwidth. The longitudinal conductivity [Fig. 2(c)] steadily
increases with $\hbar\omega$ and remains mostly independent of the spin-orbit
coupling parameter $\nu_{x}$. In contrast, both real and imaginary parts of
the BCD are highly sensitive to spin-orbit coupling and present a sharply
peaked structure. Nonetheless, there is no simple relationship between the BCD
peaks and peculiarities in the band structure (avoided band crossing and Berry
curvature maxima). From the BCD, one can compute the Kerr angle and
ellipticity, displayed in Figs. 2(d,e), that show a similar energy dependence
as the BCD: at $\nu_{x}=0$, the peak around $125$ meV corresponds to the
maximum of $\operatorname{Re}(D)$ and $\operatorname{Im}(D)$ for the same
energy. For $\nu_{x}\not=0$, the average value of $\theta_{k}$ and $\phi_{k}$
becomes large by increasing the spin-orbit coupling. The high sensitivity of
the nonlinear Hall effect and orbital Kerr response on spin-orbit coupling
suggests that these properties can be controlled externally with a gate
voltage [36].

The terahertz current responsivity, defined in Eq. (7), is readily obtained
from Fig. 2(d). Contrary to what was claimed in Ref. [8], the responsivity is
frequency-dependent, which means that it can be optimized through band
structure engineering. The current responsivity obtained at 50 meV ($\equiv$12
THz$\equiv$3.3 mm) is about $10^{-2}$ (A$\cdot\mu$m)/W, midway between GeTe
and NbP [8].

As an additional probe of the connection between the nonequilibrium orbital
magnetization and the second-order Hall effect, in the d.c. limit
$\hbar\omega\to 0$, we evaluate the decomposition of the orbital magnetization
given by Eq. (11). In fact, such a relationship has been proposed by Son et
al. [10] in the vicinity of the gap of strained MoS2, and we intend to assess
the validity of this connection in the more complex case of WTe2 bilayer. As
explained above, the low-energy Hamiltonian of WTe2 couples four tilted Dirac
cones defined as $h_{0}=E_{i}+t_{i}(k_{x}+K_{i})$. The term $E_{i}+t_{i}K_{i}$
results in a rigid energy shift and $t_{i}k_{x}$ is an odd function in
$k_{x}$. Consequently, we expect that the first term in Eq. (11) must dominate
over $\mathcal{B}$ close to the neutrality point. In other words, the orbital
Edelstein effect in WTe2 bilayer must follow $\alpha_{zx}\simeq\mu D_{zx}$, as
long as the chemical potential $\mu$ remains in the vicinity of the Dirac
nodes. These two terms, $\alpha_{zx}$ and $\mu D_{zx}$, are reported in Fig. 3
as a function of $\mu$ for different values of the spin-orbit coupling. When
$\nu_{x}=0$ [Fig. 3(a)], the system possesses two tilted Dirac cones and
$\alpha_{zx}\simeq\mu D_{xz}$. Upon increasing the spin-orbit coupling [Figs.
3(b,c)], the system exhibits band inversion and anticrossing (see Ref. [36])
such that the linear relationship between the nonlinear Hall effect and
nonequilibrium orbital moment fails due to discontinuity in the Berry
curvature related to topological transitions. At $\nu_{x}=0.4$ eV Å[Fig.
3(d)], a gap opens and the linear relation between $\alpha_{zx}$ and $D_{xz}$
is restored. In other words, the validity of the phenomenological relationship
between the BCD and the orbital Edelstein effect claimed in Ref. [10] is not
universal and only holds as long as the band structure can be approximated by
a gapped Dirac cone (as in MoS2 and numerous transition metal
dichalcogenides).

Figure 3: (Color online) Comparison between the orbital Edelstein coefficients
and the BCD for the minimal model of WTe2 bilayer, for different values of
$\nu_{x}$ and as a function of $\mu$. We fix the same microscopic parameters
that in Fig. 2, exemplifying the cases (a) $\nu_{x}=0$, (b) $\nu_{x}=0.1$ eV
Å, (c) $\nu_{x}=0.15$ eV Åand (d) $\nu_{x}=0.4$ eV Å.

###  III.2 Nb2n+1SinTe4n+2 monolayer

Large nonlinear Hall effect and orbital Kerr effect are expected to be found
in materials displaying diverging Berry curvature close to Fermi level, i.e.,
typically Dirac or Weyl semimetals [47]. Recently, it has been proposed that
the family of van der Waals compounds Nb2n+1SinTe4n+2 can also display Dirac
nodes [48] and nodal lines [49]. A model Hamiltonian for such a system reads
$\mathcal{H}_{\mathbf{k}}=\mathcal{H}_{\mathbf{k}}^{(0)}+\mathcal{H}_{\mathbf{k}}^{(soc)}$,
where [37]

$\displaystyle\mathcal{H}_{\mathbf{k}}^{(0)}$
$\displaystyle=t\begin{pmatrix}0&1+e^{-ik_{x}}\\\
1+e^{ik_{x}}&0\end{pmatrix}\otimes\hat{\sigma}_{0}$ $\displaystyle+\delta
t\begin{pmatrix}0&e^{-ik_{y}}(1+e^{-ik_{x}})\\\
e^{ik_{y}}(1+e^{ik_{x}})&0\end{pmatrix}\otimes\hat{\sigma}_{0},$ (17)
$\displaystyle\mathcal{H}_{\mathbf{k}}^{(soc)}$
$\displaystyle=t\begin{pmatrix}\lambda_{1}\sin k_{x}&0\\\ 0&-2\lambda_{1}\sin
k_{x}\end{pmatrix}\otimes\hat{\sigma}_{3}+$ $\displaystyle+\delta
t\begin{pmatrix}2\lambda_{3}\sin
k_{y}&i\lambda_{2}e^{ik_{y}}(1+e^{-ik_{x}})\\\
-i\lambda_{2}e^{-ik_{y}}(1+e^{ik_{x}})&2\lambda_{3}\sin
k_{y}\end{pmatrix}\otimes\hat{\sigma}_{3}.$ (18)

In Eqs. (17)-(18), $t$ and $\delta t$ are the intra and inter-chain hopping
parameters, respectively, and $\lambda_{i}$ ($i=1...3$) are the Rashba
parameters. Considering that the Berry curvature and the orbital moment are
parallel to $\hat{z}$, the nonequilibrium orbital magnetization is along
$\hat{z}$ and a current flowing along $\hat{y}$ generates a second-order Hall
current along $\hat{x}$ with the BCD coefficient $D_{yz}$. Let us first study
the nonequilibrium orbital Kerr effect in this system, showing the behavior of
the BCD, the longitudinal conductivity, and the Kerr angle as a function of
the energy of the incident light. Our results are reported in Fig. 4 for the
same incidence angle as for WTe2 bilayer.

Figure 4: (Color online) (a) Real (blue line) and imaginary (red line) part of
the a.c. BCD as a function of the energy of the incident light for the
Nb2n+1SinTe4n+2 monolayer. (b) Longitudinal conductivity of the system, (c)
Kerr angle (blue line), and ellipticity (red line) as a function of the energy
of the incident light. (d,e,f) Same quantities as a function of inter-chain
parameter $\delta$. We set the parameters to $t=-0.2$ eV, $t^{\prime}=\delta
t=-0.16$ eV, $\lambda_{1}=1$, $\lambda_{2}=\lambda_{3}=0.1$, $\mu=-0.028$ eV,
$\hbar\omega=100$ meV and $\Gamma=0.01$ eV.

In Fig. 4(a), we verify that at zero frequency the real part of the BCD is
finite while its imaginary part vanishes, as expected. The coefficients also
tend to zero upon increasing the frequency, with peaks close to
$\hbar\omega\to 100$ meV. Besides, the longitudinal conductivity is almost a
constant for the region $\hbar\omega\leq 300$ meV [Fig. 4(b)] and then
decreases slowly with the frequency. This signal has the same order of
magnitude as the response reported in previous experiments on Nb3SiTe6 [50,
51], whereas the nonequilibrium orbital Kerr effect is comparable to the one
computed in WTe2 bilayer in the previous section. We also find that restoring
inversion symmetry by tuning the inter-chain hopping to zero progressively
quenches the BCD [Fig. 4(d,e)] and nonequilibrium orbital Kerr effect [Fig.
4(f)]. Finally, Fig. 5 shows the orbital Edelstein coefficient and BCD,
$\alpha_{zy}$ and $\mu D_{yz}$, as a function of the chemical potential $\mu$
in the d.c. limit. Whereas both functions have a similar behavior, their
magnitude is markedly different. This again suggests that although both the
nonequilibrium orbital moment and nonlinear anomalous Hall effect are
companion phenomena that exist under the same symmetry-breaking conditions, an
explicit relationship between these two mechanisms can only be established in
the limiting case of gapped Dirac systems.

Figure 5: (Color online) Orbital Edelstein coefficient $\alpha_{zy}$ and BCD
coefficient $\mu D_{yz}$ as a function of the chemical potential for the
Nb2n+1SinTe4n+2 monolayer. The microscopic parameters are $t=-0.2$ eV,
$t^{\prime}=\delta t=-0.16$ eV, $\lambda_{1}=1$, $\lambda_{2}=\lambda_{3}=0.1$
and $\mu=-0.028$ eV.

##  IV Orbital Kerr effect from first principles

To illustrate the emergence of the orbital Kerr effect in a realistic system,
we perform first principles simulations on two different structures lacking
inversion symmetry, namely, trigonal Te and a superlattice made of an infinite
repetition of W(2)/Pt(2)/V(1) trilayers. We have used a plane-wave basis set
to obtain the optimized structures employing the Perdew-Burke-Ernzerhof (PBE)
[52, 53] exchange-correlation functional implemented in the VASP package [54,
55]. In these simulations, we used 400 eV for the plane-wave expansion cutoff,
and ionic potentials are described using the projector augmented-wave (PAW)
method [56] performing the geometry optimization with a force criterion of
5$\times 10^{-3}$ eV/Å. To the best of our knowledge, bulk Te is the first
noncentrosymmetric material in which current-driven optical activity has been
reported [57]. In Te, the combination of inversion symmetry breaking and spin-
orbit coupling induces a Rashba splitting of the valence band. Therefore,
injecting a current modifies the balance between carriers of opposite spin
chirality, resulting in nonequilibrium circular dichroism. The W(2)/Pt(2)/V(1)
metallic superlattice on the other hand is a rather conventional system that
uses standard transition metals, familiar to mainstream experimental
spintronics. The band structures of both systems are depicted in Fig. 6(a,b),
where the Fermi level is set to zero. The unit cell structure is shown in the
inset.

Figure 6: (Color online) Electronic band structure of (a) trigonal Te and (b)
W(2)/Pt(2)/V(1) superlattice stacked along (110). The unit cells are depicted
in the inset.

The BCD, the longitudinal conductivity, and the resulting nonequilibrium
orbital Kerr effect are reported in Fig. 7 for both Te (left panels) and the
W(2)/Pt(2)/V(1) superlattice (right panels). We emphasize that, in both
systems, the inversion symmetry is broken along $\hat{z}$, which means that
the orbital Edelstein effect generates a nonequilibrium orbital moment in the
plane perpendicular to $\hat{z}$. Therefore, applying a current in the ($x,y$)
plane generates a second-order Hall current along $\hat{z}$. To measure the
second-order Hall current experimentally in the W(2)/Pt(2)/V(1) superlattice,
one needs to probe the current that flows perpendicular to the plane of the
stack, which is a rather unusual configuration.

Figure 7: (Color online) (a) Real (blue line) and imaginary (red line) part of
the BCD, (b) longitudinal conductivity and (c) nonequilibrium orbital Kerr
effect for bulk Te as a function of the energy of the incident light. (d) Real
(blue line) and imaginary (red line) part of the BCD, (e) longitudinal
conductivity, and (f) orbital Kerr effect for W(2)/Pt(2)/V(1) superlattice.

Let us now consider the nonlinear Hall and orbital Kerr effect responses of
Te, as displayed in Fig. 7(a-c). At low photon energy, the BCD of bulk Te is
about $5\times 10^{-1}$ whereas that of the W(2)/Pt(2)/V(1) superlattice is
about $5\times 10^{-2}$. This needs to be compared with the (3D) BCD of WTe2
bilayer [Fig. 2], $D_{3D}=D_{2D}/t=10^{-1}$, where $t=14$ Å is the thickness
of the WTe2 bilayer. Tellurium being a semiconductor, its a.c. conductivity
increases steadily upon increasing the photon energy below 1 eV. As a result,
the orbital Kerr effect is rather large at low photon energy, displaying a
Kerr angle about three orders of magnitude larger than for the WTe2 bilayer.
On the other hand, in the W(2)/Pt(2)/V(1) superlattice [Fig. 7(d-f)], both the
BCD and the orbital Kerr effect are substantially reduced, which results in a
Kerr angle about two orders of magnitude lower than in WTe2.

Noticeably, the terahertz current responsivity of bulk Tellurium is
particularly large, reaching about $Rw\approx 15$ (A$\cdot\mu$m)/W because of
its low conductivity at low frequency. This value, consistent with earlier
experiments [57], is comparable to NbP [8] and among the best current
responsivities reported so far [58, 59].

To complete this study, the orbital Edelstein coefficients of bulk Te and
W(2)/Pt(2)/V(1) superlattice are displayed in Fig. 8 as a function of the
carrier energy. Again, to compare with the value computed in WTe2 bilayer, one
needs to divide the orbital Rashba coefficient of WTe2 bilayer by the
thickness, giving an order of magnitude of about $10^{-3}$, about one order of
magnitude larger than the orbital Edelstein coefficients computed in bulk Te
and W(2)/Pt(2)/V(1) superlattice. This result suggests that interfacial
engineering could be a practical path to obtain large current-driven orbital
magnetization and nonlinear Hall effect.

Figure 8: (Color online) Orbital Edelstein coefficient as a function of the
carrier energy for (a) bulk Te and (b) W(2)/Pt(2)/V(1) superlattice.

##  V Discussion and Conclusion

In order to assess the ability of orbital Kerr effect to probe the
nonequilibrium orbital accumulation in realistic situations, we now compare
our results with experimental data obtained on different material platforms.
We collected the value of the nonequilibrium orbital Kerr effect efficiencies,
$\theta/j$ and $\theta/E_{0}$, for various systems published in the literature
in Table 1, together with representative values obtained in the present work.
We stress that comparing experiments is not an easy task because the
conductivity of the materials (from $10^{3}$ to $10^{7}$
$\Omega^{-1}\cdot$m-1) and the wavelength of light used for magnetooptical
detection (from 10.6 $\mu$m to 514 nm) can be vastly different.

Material | $\lambda$ | $\sigma$ | $\theta/j$ | $\theta/E_{0}$ | Ref.
---|---|---|---|---|---
| nm |  $\Omega^{-1}\cdot$m-1 | rad/(A/cm2) | rad/(V/m) |
WTe2 | 10.6$\times 10^{3}$ | 4.3$\times 10^{5}$ | -3$\times 10^{-12}$ | -1.3$\times 10^{-10}$ |
NbSiTe | $10.6\times 10^{3}$ | 4.7$\times 10^{5}$ | -1.5$\times 10^{-13}$ | -7.2$\times 10^{-12}$ |
W/Pt/V | $10.6\times 10^{3}$ | 1.6$\times 10^{6}$ | -4.9$\times 10^{-13}$ | -7.9$\times 10^{-11}$ |
Te | $10.6\times 10^{3}$ | 1.7$\times 10^{3}$ | 8.4$\times 10^{-5}$ | 1.5$\times 10^{-5}$ |
GaAs | 825 | $2.8\times 10^{3}$ | $7\times 10^{-10}$ | $2\times 10^{-10}$ | [33]
MoS2 |  $\sim$660 | $10^{3}$ | $8.5\times 10^{-11}$ | $8.5\times 10^{-12}$ | [10]
Pt | 514 | $6\times 10^{6}$ | $2.3\times 10^{-15}$ | $1.4\times 10^{-12}$ | [34]
Pt | 800 | $7\times 10^{6}$ | $3\times 10^{-16}$ | $2\times 10^{-13}$ | [31]
W | 514 | $6\times 10^{5}$ | -$6.6\times 10^{-15}$ | -$4\times 10^{-13}$ | [34]
Ti | 780 | $2.5\times 10^{6}$ | -$5\times 10^{-15}$ | -$1.2\times 10^{-12}$ | [30]
Cr | 800 | $5\times 10^{6}$ | -$1.2\times 10^{-15}$ | -$6\times 10^{-13}$ | [31]
Te | $10.6\times 10^{3}$ |  | $3.5\times 10^{-5}$ | - | [57]
Table 1: Summary of the magnetooptical Kerr efficiencies calculated (top) and
measured (bottom) in different materials. (top) In the theoretical data, we
fixed the wavelength of the incident light to $\hbar\omega=117$ meV. (bottom)
The experimental values have been extracted from the literature when
available.

Let us start by considering the results obtained on GaAs in the seminal work
of Kato et al. [33]. Except for Te, the electrical orbital Kerr efficiency
obtained by Kato et al. is the largest reported to far, $\theta_{\rm
GaAs}/E_{0}\simeq 2\times 10^{-10}$ rad m V-1, which is comparable to our
numerical results for the WTe2 bilayer and the W/Pt/V superlattice.
Interestingly, this observation is interpreted in the framework of the spin
Hall effect, which is rather surprising considering the very small spin-orbit
coupling of GaAs. Although it would require further investigation, we
speculate that this unexpectedly large value could perhaps be related to the
nonequilibrium orbital accumulation rather than to spin accumulation.

When considering the ratio between the Kerr angle and the current, $\theta/j$,
we see a drastic difference between semiconductors (GaAs, MoS2, and Te), and
metallic systems (Cr, Ti, Pt, and W). The formers display an orbital Kerr
efficiency that is about four orders of magnitude larger than their metallic
counterpart. When considering the ratio $\theta/E_{0}$, the orbital Kerr
efficiency of MoS2 remains one order of magnitude larger than in metals. The
theoretical values computed in the present work are all larger than the values
collected in metallic systems. Notice that trigonal Te stands out as the most
efficient material for the orbital Kerr effect, exceeding the Kerr
efficiencies of all the other materials by orders of magnitude. This
intriguing observation calls for further theoretical and experimental study of
this material.

The connection between the orbital Kerr effect and the current responsivity to
terahertz radiation is particularly interesting. We found that the narrow-
gapped semiconductor Te supports a giant current responsivity ($Rw\approx 15$
(A$\cdot\mu$m)/W) whereas (semi)metallic systems like WTe2 and W/Pt/V
superlattices, that possess a relatively large BCD, display a smaller
responsivity due to their high conductivity ($Rw\approx 10^{-4}$
(A$\cdot\mu$m)/W). This finding opens inspiring outlooks for the search for
efficient terahertz rectifiers among Weyl semimetals and other topological
materials with diverging Berry curvature.

In conclusion, we investigate the current-induced orbital Kerr effect and
terahertz current responsivity in nonmagnetic, noncentrosymmetric materials
displaying nonlinear Hall effect. In this case, the Kerr rotation is
proportional to the a.c. BCD, and is therefore directly connected to the
nonlinear Hall effect. The terahertz current responsivity is itself equal to
half the orbital Kerr efficiency. We examined the orbital Kerr effect in
selected noncentrosymmetric materials, using both effective models and
realistic materials based on ab initio simulations, and compared our results
with available experimental data, pointing out that the orbital Kerr
efficiency tends to be larger in semiconductors than in metals, irrespective
of the magnitude of the material’s spin-orbit coupling. This opens
particularly interesting promises for the observation of this effect in light
compounds. In particular, our simulations on a metallic superlattice suggest
that interfacial engineering can be used to optimize the nonequilibrium
orbital Kerr effect, the nonlinear Hall effect, and its ability to detect
terahertz radiations. Along the same line, van der Waals heterostructures [60,
61] that combine materials with very different electronic properties (e.g.,
semiconducting, metallic, semimetallic) could also provide an interesting
platform for the study of these effects.

###### Acknowledgements.

D.G.O. and A. M. acknowledge support from the Excellence Initiative of Aix-
Marseille Université - A*Midex, a French ”Investissements d’Avenir” program.
A.P. was supported by the ANR ORION project, grant ANR-20-CE30-0022-01 of the
French Agence Nationale de la Recherche.

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# Entropy-variation with respect to the resistance in quantized RLC circuit
derived by generalized Hellmann-Feynman theorem

Hong-yi Fan1, Xue-xiang Xu1,2and Li-yun Hu${}^{1,2\text{*}}$ 1Department of
Physics, Shanghai Jiao Tong University, Shanghai 200030, China
2College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

By virtue of the generalized Hellmann-Feynman theorem for ensemble average, we
obtain internal energy and average energy consumed by the resistance $R$ in a
quantized RLC electric circuit. We also calculate entropy-variation with
respect to $R$. The relation between entropy and $R$ is also derived. By
depicting figures we indeed see that the entropy increases with the increment
of $R$.

PACS numbers: 05.30.-d, 42.50.-p, 03.65.-w

##  I Introduction

In the field of mesoscopic physics Louisell was the first who quantized a
mesoscopic L-C (inductance $L$ and capacitance $C$) circuit as a quantum
harmonic oscillator 1 . He made it by quantizing electric charge as the
coordinate operator $q,$ while quantizing electric current $I$ multiplied by
$L$ as the momentum operator $p$. Louisell’s work has become more and more
popular because mesoscopic L-C circuits may have wide applications in quantum
computer. However, Louisell only calculated quantum fluctuation of L-C circuit
at zero-temperature. In Ref.2  Fan and Liang pointed out that since electric
current generates Joule thermal effect, one should take thermo effect into
account, thus every physical observable should be evaluated in the context of
ensemble average. Besides, since entropy increases with the generation of
Joule heat, one should consider how the resistance $R$ in RLC electric circuit
affects the variation of entropy. We shall use the generalized Hellmann-
Feynmam theorem (GHFT) for ensemble average to discuss this topic. The usual
Hellmann-Feynman (H-F) theorem states 3 ; 4

$\frac{\partial E_{n}}{\partial\chi}=\left\langle\psi_{n}\right|\frac{\partial
H}{\partial\chi}\left|\psi_{n}\right\rangle,$ (1)

where $H$ (a Hamiltonian which involves parameter $\chi$) possesses the
eigenvector $\left|\psi_{n}\right\rangle,$
$H\left|\psi_{n}\right\rangle=E_{n}\left|\psi_{n}\right\rangle.$ For many
troublesome problems in searching for energy level in quantum mechanics,
people can resort to the H-F theorem to make the analytical calculation.
However, this formula is only available for the pure state, quantum
statistical mechanics is the study of statistical ensembles of quantum
mechanics. A statistical ensemble is described by a density matrix $\rho$,
which is a non-negative, self-adjoint, trace-class operator of trace 1 on the
Hilbert space describing the quantum system. Extending Eq. (1) to the ensemble
average case is necessary and has been done in Refs 5 ; 6 ; 7 .

Our paper is arranged as follows: In Sec. 2 we briefly introduce the GHFT for
ensemble average $\left\langle H\left(\chi\right)\right\rangle_{e},$ where the
subscript $e\ $denotes ensemble average. In Sec. 3 based on von Neuman’s
quantum entropy definition $S=-k\mathtt{tr}\left(\rho\ln\rho\right)$ and using
the GHFT we derive the entropy-variation formula for $\frac{\partial
S}{\partial\chi}$ and its relation to
$\frac{\partial}{\partial\chi}\left\langle
H\left(\chi\right)\right\rangle_{e}.$ In Sec. 4 we use the GHFT to calculate
internal energy of the quantized RLC circuit and its fluctuation, as well as
the average energy consumed by resistance $R$. In Sec. 5 we employ the GHFT to
find the relation between entropy and $R$. By depicting figures we indeed see
that the entropy increases with the increment of resistance $R.$

##  II Brief review of the generalized Hellmann-Feynman theorem

For the mixed states in thermal equilibrium described by density operators

$\rho=\frac{1}{Z}e^{-\beta H},\text{ \ \ }\beta=\left(kT\right)^{-1},$ (2)

where $Z=$tr$\left(e^{-\beta H}\right)$ is the partition function ($k$ is
Boltzmann constant and $T$ is temperature), we have proposed the GHFT 5 . Thus
the ensemble average of the Hamiltonian $H$ (which is dependent of parameter
$\chi$) is

$\left\langle H\left(\chi\right)\right\rangle_{e}=\text{tr}\left[\rho
H\left(\chi\right)\right]=\frac{1}{Z\left(\chi\right)}\sum_{j}e^{-\beta
E_{j}\left(\chi\right)}E_{j}\left(\chi\right)\equiv\bar{E}\left(\chi\right),$
(3)

and $\left\langle A\right\rangle_{e}\equiv\mathtt{tr}\left(\rho e^{-\beta
H}\right)$ for arbitrary operator $A$ of system. Performing the partial
differentiation with respect to $\chi,$ we have 5

$\displaystyle\frac{\partial\left\langle H\right\rangle_{e}}{\partial\chi}$
$\displaystyle=$
$\displaystyle\frac{1}{Z\left(\chi\right)}\left\\{\sum_{j}e^{-\beta
E_{j}\left(\chi\right)}\right.$ (4) $\displaystyle\left.\times\left[-\beta
E_{j}\left(\chi\right)+\beta\left\langle
H\right\rangle_{e}+1\right]\frac{\partial
E_{j}\left(\chi\right)}{\partial\chi}\right\\}.$

Then using Eq.(1) we can further write Eq.(4) as

$\frac{\partial}{\partial\chi}\left\langle
H\right\rangle_{e}=\left\langle\left(1+\beta\left\langle
H\right\rangle_{e}-\beta H\right)\frac{\partial
H}{\partial\chi}\right\rangle_{e}.$ (5)

Noting the relation

$\left\langle H\frac{\partial
H}{\partial\chi}\right\rangle_{e}=-\frac{\partial}{\partial\beta}\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}+\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}\left\langle H\right\rangle_{e},$ (6)

when $H$ is independent of $\beta,$ we can reform Eq. (5) as

$\frac{\partial}{\partial\chi}\left\langle
H\right\rangle_{e}=\frac{\partial}{\partial\beta}\left[\beta\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}\right]=\left(1+\beta\frac{\partial}{\partial\beta}\right)\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}.$ (7)

The integration of Eq.(7) yields two forms. One is

$\beta\left\langle\frac{\partial
H\left(\chi\right)}{\partial\chi}\right\rangle_{e}=\int
d\beta\frac{\partial}{\partial\chi}\left\langle H\right\rangle_{e}+K,$ (8)

which deals with integration over $d\beta$ and $K$ is an integration constant;
and the other is

$\left\langle
H\right\rangle_{e}=\int_{0}^{\chi}\left(1+\beta\frac{\partial}{\partial\beta}\right)\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}d\chi+\left\langle
H\left(0\right)\right\rangle_{e},$ (9)

which tackles integration over $d\chi.$ Note that the fluctuation of $H$ can
be obtain by virtue of

$\left(\Delta H\right)^{2}=\left\langle
H^{2}\right\rangle_{e}-\bar{E}^{2}=-\frac{\partial\left\langle
H\right\rangle_{e}}{\partial\beta}.$ (10)

##  III Deriving entropy-variation $\frac{\partial S}{\partial\chi}$ and its
relation to $\frac{\partial\left\langle H\right\rangle_{e}}{\partial\chi}$
from $S=-k\mathtt{tr}\left(\rho\ln\rho\right)$

Entropy $S$ in classical statistical mechanics is defined as

$F=U-TS,$ (11)

where $U$ is system’s internal energy or the ensemble average of Hamiltonian
$\left\langle H\right\rangle_{e}$, and $F$ is Helmholtz free energy,

$F=-\frac{1}{\beta}\ln\sum_{n}e^{-\beta E_{n}}.$ (12)

According to Eq.(12) the entropy can not be calculated until systems’ energy
level $E_{n}$ is known. In this work we consider how to derive entropy without
knowning $E_{n}$ in advance, i.e., we will not diagonalize the Hamiltonian
before calculating the entropy, instead, our starting point is using entropy’s
quantum-mechanical definition,

$S=-k\mathtt{tr}\left(\rho\ln\rho\right).$ (13)

It is von Neuman who extended the classical concept of entropy (put forth by
Gibbs) into the quantum domain. Note that, because the trace is actually
representation independent, Eq. (13) assigns zero entropy to any pure state.
However, in many cases $\ln\rho$ is unknown until $\rho$ is diagonalized, so
we explore how to use the GHFT to calculate entropy of some complicated
systems, which, to our knowledge, has not been calculated in the literature
before. Rewriting Eq. (13) as

$S=\beta k\mathtt{tr}(\rho H)+k\mathtt{tr}\left(\rho\ln
Z\right)=\frac{1}{T}\left\langle H\right\rangle_{e}+k\ln Z,$ (14)

where $\left\langle H\right\rangle_{e}$ corresponding to $U$ in Eq.(11), it
then follows

$\frac{\partial
S}{\partial\chi}=\frac{1}{T}\left(\frac{\partial}{\partial\chi}\left\langle
H\right\rangle_{e}-\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}\right),$ (15)

which indicates that the entropy-variation is proportional to the difference
between internal energy’s variation and the ensemble average of
$\frac{\partial H}{\partial\chi}.$ In particular, when $\rho$ is a pure state,
then $\frac{\partial}{\partial\chi}\left\langle
H\right\rangle_{e}=\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e},$ $\frac{\partial S}{\partial\chi}=0,$ $S$
is a constant (zero). Supposing the case is

$H=\sum_{i}\chi_{i}H_{i},\text{ }\left\langle
H\right\rangle_{e}=\sum_{i}\chi_{i}\left\langle H_{i}\right\rangle_{e},$ (16)

then due to $\left\langle\frac{\partial
H}{\partial\chi_{i}}\right\rangle_{e}=\left\langle H_{i}\right\rangle_{e},$ we
also have

$\frac{\partial S}{\partial\chi_{i}}=0.$ (17)

Eq. (15) also appears in Ref. 6 , but it does not mention the von Neuman
entropy $S=$ $-k\mathtt{tr}\left(\rho\ln\rho\right)$. Substituting Eq. (15)
into Eq. (7) yields

$T\frac{\partial
S}{\partial\chi}=\beta\frac{\partial}{\partial\beta}\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e},$ (18)

this is another form of the entropy-variation formula. It then follows

$TS=\left\langle H\right\rangle_{e}-\int\left\langle\frac{\partial
H}{\partial\chi}\right\rangle_{e}d\chi+C,$ (19)

where $C$ is an integration constant of parameters involved in $H$ other than
$\chi.$

##  IV Internal energy and average energy consumed by resistance in the RLC
circuit

In terms of $q-p$ quantum variables $\left(\left[q,p\right]=i\hbar\right)$,
Louisell’s Hamiltonian for the quantized RLC circuit is

$H=\frac{1}{2L}p^{2}+\frac{1}{2C}q^{2}+\frac{R}{2L}(pq+qp).$ (20)

We now use GHFT to calculate the internal energy $\left\langle
H\right\rangle_{e}$. Substituting Eq.(20) into Eq.(5) and letting $\chi$ be
$L$, $C$, and $R,$ respectively, we obtain

$\displaystyle-2L^{2}\frac{\partial\left\langle H\right\rangle_{e}}{\partial
L}$ $\displaystyle=\left\langle\left(1+\beta\left\langle
H\right\rangle_{e}-\beta
H\right)\left(p^{2}+R(pq+qp)\right)\right\rangle_{e},$ (21)
$\displaystyle-2C^{2}\frac{\partial\left\langle H\right\rangle_{e}}{\partial
C}$ $\displaystyle=\left\langle\left(1+\beta\left\langle
H\right\rangle_{e}-\beta H\right)\left(q^{2}\right)\right\rangle_{e},$ (22)
$\displaystyle 2L\frac{\partial\left\langle H\right\rangle_{e}}{\partial R}$
$\displaystyle=\left\langle\left(1+\beta\left\langle H\right\rangle_{e}-\beta
H\right)\left(pq+qp)\right)\right\rangle_{e}.$ (23)

Supposing the eigenvector of Hamiltonian is $\left|\Psi_{n}\right\rangle$,
$H\left|\Psi_{n}\right\rangle=E_{n}\left|\Psi_{n}\right\rangle,$ $E_{n}$ is
the energy eigenvalue, due to

$\left\langle\Psi_{n}\right|\left[q^{2}-p^{2},H\right]\left|\Psi_{n}\right\rangle=0,$
(24)

and

$\left[q^{2}-p^{2},H\right]=\left(\frac{i}{L}+\frac{i}{C}\right)(pq+qp)+2i\frac{R}{L}\left(p^{2}+q^{2}\right),$
(25)

which leads to the following relation

$\left\langle\Psi_{n}\right|\left[\left(\frac{i}{L}+\frac{i}{C}\right)(pq+qp)+2i\frac{R}{L}\left(p^{2}+q^{2}\right)\right]\left|\Psi_{n}\right\rangle=0,$
(26)

and noticing $\left\langle\beta\left\langle H\right\rangle_{e}-\beta
H\right\rangle_{e}=0,$ thus we can have the ensemble average

$\displaystyle\left\langle\left(1+\beta\left\langle H\right\rangle_{e}-\beta
H\right)\right.$
$\displaystyle\left.\times\left[\left(\frac{i}{L}+\frac{i}{C}\right)(pq+qp)+2i\frac{R}{L}\left(p^{2}+q^{2}\right)\right]\right\rangle_{e}\left.=\right.0.$
(27)

Substituting Eqs.(21)-(23) into Eq.(27), we obtain a partial differential
equation

$L^{2}\frac{\partial\left\langle H\right\rangle_{e}}{\partial
L}+C^{2}\frac{\partial\left\langle H\right\rangle_{e}}{\partial
C}+\left(LR-\frac{L^{2}}{2RC}-\frac{L}{2R}\right)\frac{\partial\left\langle
H\right\rangle_{e}}{\partial R}=0,$ (28)

which can be solved by virtue of the method of characteristics 8 ; 9 .
According to it we have the equation

$\frac{dL}{L^{2}}=\frac{dC}{C^{2}}=\frac{dR}{\left(LR-\frac{L^{2}}{2RC}-\frac{L}{2R}\right)},$
(29)

it then follows that

$\frac{1}{L}-\frac{1}{C}=c_{1},\text{
}\frac{R^{2}}{L^{2}}-\frac{1}{LC}=c_{2},$ (30)

where $c_{1}$ and $c_{2}$ are two arbitrary constants. We can now appy the
method above in which the general solution of the partial differenial equation
(28) is found by writing $\left\langle
H\right\rangle_{e}=f\left[c_{1},c_{2}\right],$ i.e.,

$\left\langle
H\right\rangle_{e}=f\left[\frac{1}{L}-\frac{1}{C},\frac{R^{2}}{L^{2}}-\frac{1}{LC}\right],$
(31)

where $f\left[x,y\right]$ is some function of $x,y$. In order to determine the
form of this function, we examine the special case when $R=0$, i.e

$H_{0}=\frac{1}{2L}p^{2}+\frac{1}{2C}q^{2}=\hbar\omega_{0}\left(a^{\dagger}a+\frac{1}{2}\right),$
(32)

where $a=\sqrt{\frac{L\omega_{0}}{2\hbar}}q+i\sqrt{\frac{1}{2\hbar
L\omega_{0}}}p$ with $\omega_{0}=1/\sqrt{LC}$. According to the well-know Bose
statistics formula $\left\langle
H_{0}\right\rangle_{e}=\frac{\hbar\omega_{0}}{2}\coth\frac{\hbar\omega_{0}\beta}{2}$,
then we know

$\left\langle
H\left|{}_{R=0}\right.\right\rangle_{e}=f\left[\frac{1}{L}-\frac{1}{C},-\frac{1}{LC}\right]=\frac{\hbar\omega_{0}}{2}\coth\frac{\hbar\omega_{0}\beta}{2}.$
(33)

To determine the form of function $f\left[x,y\right],$ we let
$x=\frac{1}{L}-\frac{1}{C},$ $y=\frac{-1}{LC},$ then its reverse relations are
$L=\frac{x+\sqrt{x^{2}-4y}}{2y},$ $C=\frac{-x+\sqrt{x^{2}-4y}}{2y},$ and
$\omega_{0}=\sqrt{-y}$. This implies that the form of function
$f\left[x,y\right]$ is

$f\left[x,y\right]=\frac{\hbar\sqrt{-y}}{2}\coth\frac{\hbar\beta\sqrt{-y}}{2},$
(34)

and we obtain the internal energy

$\left\langle
H\right\rangle_{e}=f\left[\frac{1}{L}-\frac{1}{C},\frac{R^{2}}{L^{2}}-\frac{1}{LC}\right]=\frac{\hbar\omega}{2}\coth\frac{\hbar\omega\beta}{2},$
(35)

where $\omega=\sqrt{\frac{1}{LC}-\frac{R^{2}}{L^{2}}}.$

Then according to Eq.(10) the fluctuation of $H$ is

$\left(\Delta
H\right)^{2}=\frac{\hbar^{2}\omega^{2}}{4}\frac{1}{\sinh^{2}\frac{\beta\hbar\omega}{2}}.$
(36)

Using Eq.(8) and the following integration formula,

$\int\frac{1}{e^{ax}-1}dx\allowbreak=\frac{1}{a}\left(\ln\left(e^{ax}-1\right)-ax\right),$
(37)

we have

$\displaystyle\left\langle\frac{\partial H}{\partial R}\right\rangle_{e}$
$\displaystyle=\frac{1}{2L}\left\langle\left(pq+qp\right)\right\rangle_{e}$
$\displaystyle=\frac{1}{\beta}\int\frac{\partial\left\langle
H\right\rangle_{e}}{\partial R}d\beta=-\frac{\hbar\allowbreak R}{2\omega
L^{2}}\coth\frac{\hbar\omega\beta}{2},$ (38)

so the average energy consumed by the resistance is

$\frac{R}{2L}\left\langle\left(pq+qp\right)\right\rangle_{e}=-\frac{\hbar
R^{2}}{2\omega L^{2}}\coth\frac{\hbar\omega\beta}{2},\text{
}\omega=\omega_{0}\sqrt{1-R^{2}C/L},$ (39)

where the minus sign implies that the resistance is a kind of energy consuming
element.

##  V Entropy-variation with respect to the resistance

In this section, based on the above results we investigate the influence of
the resistance on the entropy of RLC electric circuit. By substituting
Eqs.(31) and (38) into Eq.(15), it is easily obtained that

$\frac{\partial S}{\partial R}=\allowbreak\frac{\beta
R\hbar^{2}}{TL^{2}}\frac{\exp\left(\hbar\beta\omega\right)}{\left(\exp\left(\hbar\beta\omega\right)-1\right)^{2}}=\frac{\beta
R\hbar^{2}}{4TL^{2}}\frac{1}{\sinh^{2}\frac{\beta\hbar\omega}{2}}.$ (40)

Further, making use of the integral formula

$\int\frac{\ln y}{\left(y-1\right)^{2}}dy=\ln\left(y-1\right)-\frac{y\ln
y}{y-1},$ (41)

we derive the relation between the entropy and the resistance as follows

$S=-k\ln\left[\exp\left(\hbar\beta\omega\right)-1\right]+\frac{1}{T}\frac{\hbar\omega\exp\left(\hbar\beta\omega\right)}{\exp\left(\hbar\beta\omega\right)-1}.$
(42)

Obviously when $R=0$, the entropy in (42) corresponds to LC electric circuit.
Based on the relation in Eq.(42), we in Figure 1 depict the variation of the
entropy as a function of resistance in the range of
$\left[0,\sqrt{L/C}\right]$. The figure illustrates that the entropy has a
monotonically increasing with the resistance $R.$ When $R$ goes to the
limit$\sqrt{L/C},$ the entropy tends to infinity.

In summary, by virtue of the generalized Hellmann-Feynman theorem for ensemble
average, we have obtained internal energy and average energy consumed by the
resistance, we have also calculated entropy-variation with respect to the
resistance in quantized RLC electric circuit. The relation between entropy and
resistance is also derived. By depicting figure we indeed see that the entropy
increases with the increment of $R.$

ACKNOWLEDGMENTS

The work was supported by the National Natural Science Foundation of China
under Grant Nos.10775097 and 10874174.

## References

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Figure 1: Entropy S as a function of resistance.