# ENTROPY-BASED THEORY OF THERMOMAGNETIC PHENOMENA:
Poynting Vector, Vorticity, and Advanced Sensing

Andrei Sergeev<EMAIL_ADDRESS>*U.S. Army Research Laboratory, Adelphi,
Maryland 20783, USA  Michael Reizer  Baruch Hashem Beseder Lab
<EMAIL_ADDRESS>
###### Abstract

We show that in the linear response approximation only entropy provides
coupling between thermal and electric phenomena. The dissipationless quantum
currents, - magnetization, superconducting, persistent and topological edge
currents, - do not produce and transfer entropy and may be excluded from final
formulas for thermomagnetic coefficients. The magnetization energy flux,
$\displaystyle c\bf{M}\times\bf{E}$, in crossed electric and magnetic fields
strongly modifies the Poynting vector in magnetic materials and metamaterials,
but do not contribute to the heat current. Calculating entropy fluxes of
fluctuating Cooper pairs, we find the fluctuation Nernst coefficient in pure
superconductors. To account electron scattering, we generalize the gauge-
invariant Kubo formalism developed for the Hall effect to thermomagnetic
entropy transfer. We also introduce the thermomagnetic entropy per unit charge
and derive the Nernst coefficient proportional to the difference of the
thermoelectric and thermomagnetic entropies. This explains the Sondheimer
cancellation and high sensitivity of thermomagnetic phenomena to correlations.
In 2D superconductors, the transport entropy transferred by a vortex moving
through the background formed by vortex-antivortex pairs is the configuration
entropy of $\displaystyle k_{B}\cdot\ln 2$, which strongly exceeds the
intrinsic entropy of vortex core. Beyond the linear response, the non-entropic
forces can lead to phenomena unexpected from thermodynamics, such as vortex
attraction to the moving hot spot. Quantum currents do not transfer entropy
and may be used as ideal connectors to quantum nanodetectors.

##  I Introduction

In 1886, Albert von Ettingshausen and Walther Nernst while studying the Hall
effect in bismuth also investigated the “thermal Hall effect.” Instead of
creating electric current in the bismuth ribbon placed in the magnetic field,
they warmed one end of the ribbon and generated the heat flow from the hot to
the cold end of the ribbon.1  The discovered phenomenon, - induction of
electric current or voltage in the direction perpendicular to both magnetic
field and the heat current, - is currently known as the Nernst effect. These
experimental investigations were carried out under the supervision of
Boltzmann, who realized the entropic nature of thermal forces and foreseen the
related phenomenon, - generation of the transverse entropy flow (heat current)
or temperature gradient due to magnetic field and the electric current. In
1931, considering entropy production, Onsager derived reciprocal relations for
certain coefficients between flows and forces, including entropy flow and
entropic forces generated by the temperature gradient.2  In 1948 Callen
applied the Onsager’s relations to thermomagnetic phenomena and derived exact
formal formula for thermomagnetic coefficients.3

Thermomagnetic phenomena observed in wide variety of conducting materials,
including metals, semiconductors, organic materials, and superconductors.4 ; 5
; 6 ; 7  In most metals and semiconductors, thermomagnetic effects are well
described by the model of noninteracting electrons.4 ; 8  In this model,
formulas the thermomagnetic transport coefficient in metals may be obtained by
replacing the electron density in the formulas for the Hall conductivity by
the electron entropy. For semiconductors, electron and hole densities should
be replaced by the electron and hole entropies. For a degenerate conductor,
the electron entropy brings the degeneracy factor of $\displaystyle
k_{B}T/\epsilon_{F}$, where $\displaystyle\epsilon_{F}$ is the Fermi energy.
Therefore, the Fermi statistics puts strong limitations on thermomagnetic
coefficient and significant thermomagnetic effects are expected in conductors
with small value of the Fermi energy.9

Novel strongly correlated materials turn out to be a great challenge for the
theory of thermomagnetic phenomena. Giant thermomagnetic effects observed in
CeCoIn5, Bechgaard salts, URu2Si2 and PrFe4P12 are usually linked with exotic
electronic orders and interaction effects are described in terms specific
configuration entropy transferred by electrons.10 ; 11 ; 12  Problems of
microscopic theory of entropy flux for interacting electrons were discussed in
a recent review by Kadanoff.13  In any approximation for electron
correlations, the transport entropy is defined via combination of the
distribution functions (Green’s functions), which are affected by
correlations. While this approach does not work for the entanglement entropy,
due to its strongly nonlocal nature, it is applicable to many popular physical
models.13  Anyway, the correlation energy by itself cannot be equated to the
heat.

Meanwhile numerous recent papers directly associate giant thermomagnetic
effects with the flow of large correlation energy, which has quantum and/or
electromagnetic origin. In particular, many papers14 ; 15 ; 16 ; 17 ; 18 ; 19
; 20 ; 21  associate the flux of magnetization energy, $\displaystyle
c\bf{M}\times\bf{E}$, in crossed magnetic and electric fields with the heat
(entropy) flow. In the electron systems with large magnetization currents,
e.g. near the superconducting transition, the obtained magnetization
contribution to the thermomagnetic coefficient may exceed the thermomagnetic
coefficient in metals by the factor of $\displaystyle\epsilon_{F}/k_{B}T$.14 ;
15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22  The corresponding thermal forces are
associated with the electromagnetic interaction between an electron and the
magnetization currents generated by the temperature gradient,
$\displaystyle-c(d\bf{M}/dT)\times\nabla T$. At the same time, it is well
understood that the coherent electron motion in magnetization,
superconducting, persistent and topological edge currents has quantum origin
and the entropy of any coherent current is zero.23 ; 24 ; 25 ; 26 ; 27 ; 28
The nonzero transport entropy of magnetization and superconducting currents is
a long term enigma.

In particular, above the superconducting transition, the fluctuating
superconducting magnetization currents are large and the contribution of
$\displaystyle c\bf{M}\times\bf{E}$ to the thermomagnetic heat flow is huge.
To satisfy the Onsager relations modified by magnetization contribution, many
theoretical papers obtained the ”giant” Nernst coefficient due to the Gaussian
fluctuations of Cooper pairs.14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 29 ; 30 ;
31 ; 32 ; 33  In a weak magnetic field the fluctuation contribution to the
Nernst coefficient exceeds its value in a normal state by a huge factor of
$\displaystyle(\epsilon_{F}/k_{B}T)^{2}$. At the same time, it is well
understood that superconducting fluctuations give small corrections to the
electron entropy. Therefore, the origin of this giant fluctuation Nernst
effect is still unclear. Some papers34 ; 35 ; 36  associate it with the strong
temperature-dependence of the thermodynamic chemical potential of Cooper
pairs. However, the thermodynamic chemical potential of Cooper pairs is always
zero due to the lack of constraint on the number of Cooper pairs in the
system.23  For the same reason, the chemical potential of the photons in the
equilibrium black-body radiation is zero, while a non-zero chemical potential
is reached in solar cells.37 ; 38  Thermodynamic limitations on quasi-
equilibrium (the linear response) and strongly nonequilibrium thermomagnetic
transport are not well understood yet.

The electromagnetic energy and thermal energy balance equations could shed
some light on the form of the heat current in the magnetic field. However,
this question also turns out to be rather complicated. Recent theories
associate the magnetization currents in electric field with dissipation and
entropy production.15 ; 39 ; 40  Assumption of the dissipative character of
magnetization currents immediately leads to the magnetization contribution to
the Poynting vector of $\displaystyle c\bf{M}\times\bf{E}$ and, therefore, the
Poynting vector in magnetic materials has a form $\displaystyle c[{\bf
E}\times{\bf B}]/4\pi$.39 ; 40  While this modified form of the Poynting
vector is widely accepted in the area of metamaterials, some authors41  find
that this result contradicts optical properties of metamaterials and this
question is still far from resolution.

In this work, we develop consistent entropy-based description of
thermomagnetic phenomena and investigate a role of magnetization and
superconducting currents in thermomagnetic transport and energy transfer
(Poynting vector). We show that entropy and only entropy provides coupling
between thermal and electric phenomena in the linear response approximation.
Non-entropic energy fluxes and non-entropic thermal forces do not contribute
to thermomagnetic phenomena. In Section II, we consider the textbook model of
noninteracting electrons in quantized magnetic field and show that the
magnetization energy flux, $\displaystyle\bf{M}\times\bf{E}$, does not
transfer the entropy. In Section III, we discuss entropic and non-entropic
thermal forces induced by the temperature gradient and show that to satisfy
the Onsager relations the non-entropic forces should be canceled among
themselves. In Section IV we review a divergent-free nature of magnetization
currents, which leads to cancellation of bulk and surface magnetization
currents42  and cancellation of corresponding non-entropic forces in
thermomagnetic phenomena. In Section V we consider transfer of magnetization
energy, $\displaystyle\bf{M}\times\bf{E}$, and derive a new formula for the
Poynting vector in magnetic materials and metamaterials. In Sections VI and
VII we investigate fluctuation thermomagnetic phenomena in superconductors
above the transition. In the collisionless limit, we derive exact formulas for
the thermomagnetic coefficient, which is proportional to electron entropy
corrected due to superconducting fluctuations. To calculate the fluctuation
Nernst coefficient in a disordered superconductor, we employ the gauge -
invariant Kubo formalism developed for the fluctuation Hall effect.43

In Section VIII we generalize the entropic approach to electrons with finite
mobility and derive exact thermodynamic equations, which correct recently
proposed phenomenological formulas for the Nernst coefficient.34 ; 35 ; 36  In
Section IX we consider thermomagnetic transport in the vortex liquid. While,
in accordance with the Onsager approach, non-entropic thermal forces are
canceled among themselves in quasi-equilibrium conditions, this cancellation
is not valid for the local or time modulated temperature gradient. In this
context we review and explain recent experiments that observe motion of single
superconducting vortices, which follow to the hot spot generated by a laser
beam.44  We also discuss thermomagnetic phenomena in the Berezinskii -
Kosterlitz - Thouless (BKT) state of 2D superconductors and show that the
transport entropy transfered by a vortex moving through the background formed
by vortex-antivortex pairs is the configuration entropy of $\displaystyle
k_{B}\cdot\ln 2$. Finally, the unique property of quantum currents that
transfer electric charge without entropy is discussed in Section X for
applications to ultra-sensitive detectors.

##  II Entropy and Magnetization Fluxes in Collisionless Thermomagnetic
Transport

Thermomagnetic coefficient, $\displaystyle\mathcal{L}$, describes the
transport electric current as a response to the temperature gradient in the
magnetic field,

$\displaystyle\displaystyle{\bf j}^{e}_{tr}=\mathcal{L}\cdot{[\nabla
T\times{\bf H}]\over H}.$ (1)

Taking into account the Onsager relation, the corresponding equation for the
heat current in crossed electric and magnetic fields has a form,

$\displaystyle\displaystyle{\bf j}^{h}=T\mathcal{L}\cdot{[{\bf H}\times{\bf
E}]\over H}$ (2)

The Nernst effect may be considered as the transverse thermopower in the
magnetic field. The Nernst coefficient, $\displaystyle\mathcal{N}$, relates
the electric field induced in the open circuit by crossed temperature gradient
and magnetic field,

$\displaystyle\displaystyle\mathcal{N}={\nabla\varphi_{\bot}\over H\cdot\nabla
T}={1\over
H}{(\mathcal{L}+\mathcal{S}\sigma_{H})\sigma\over\sigma^{2}+\sigma_{H}^{2}},$
(3)

where $\displaystyle\varphi_{\bot}$ is the measured voltage,
$\displaystyle\sigma$ is the electrical conductivity,
$\displaystyle\sigma_{H}$ is the Hall conductivity, and
$\displaystyle\mathcal{S}$ is the thermopower or Seebeck coefficient. In
particular, in the Fermi liquid with the energy independent electron
scattering, the transport coefficients are related as
$\displaystyle\mathcal{L}=\mathcal{S}\sigma_{H}$ and, as a result, the Nernst
coefficient is zero. This fact is well known as Sondheimer cancellation.4  In
general, for the Fermi liquid in a weak magnetic field both the thermomagnetic
coefficient $\displaystyle\mathcal{L}$ and the product of the Seebeck and Hall
coefficients are proportional to product of $\displaystyle\omega_{c}\tau$ and
$\displaystyle k_{B}T/\epsilon_{F}$, and, therefore, the Nernst coefficient is
very small.

The Ettingshausen effect may be considered as the thermal Hall effect and the
Ettingshausen coefficient is defined as

$\displaystyle\displaystyle\mathcal{E}={\nabla T_{\bot}\over
H\cdot{j}^{e}_{tr}}.$ (4)

Due to the Onsager relation given by Eqs. 1 and 2, the Nernst and
Ettingshausen coefficients are interconnected by a fundamental relation
obtained by Bridgman,45

$\displaystyle\displaystyle\mathcal{E}={T\over\kappa}\mathcal{N},$ (5)

where $\displaystyle\kappa$ is the thermal conductivity. Thus, all measurable
transverse thermomagnetic effects may be presented via the thermomagnetic
coefficient, $\displaystyle\mathcal{L}$, and other well-known coefficients
that describe the electric and thermal transport in the absence of magnetic
field.

Numerous recent theoretical papers state that magnetization current directly
contribute to the transverse heat current induced by electric field and in
some materials the magnetization heat current, can provide significant
contribution to the thermomagnetic effects. According to these works14 ; 15 ;
16 ; 17 ; 18 ; 19 ; 20 ; 21  the heat current consists of the transport and
magnetization components and the magnetization heat current is given by

$\displaystyle\displaystyle{\bf J}^{h}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{\bf J}^{h}_{tr}+{\bf J}^{h}_{mag}$ (6)
$\displaystyle\displaystyle{\bf J}^{h}_{mag}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-c[{\bf E}\times{\bf M}(T)],$ (7)

where $\displaystyle{\bf M}$ is the magnetization and $\displaystyle{\bf E}$
is the electric field. Eqs. 6 an 7 were empirically proposed in Ref. 14 to
satisfy the Onsager relation for heat and electric bulk thermomagnetic
currents above the superconducting transition. The same equations were derived
in the book15  on a base of assumption that magnetization currents in the
electric field dissipate the electromagnetic energy in the same way as the
transport currents, i.e. the dissipated power is given by $\displaystyle{\bf
j}_{tr}\cdot{\bf E}+{\bf j}_{mag}\cdot{\bf E}$.

Here we will show that Eqs. 6 and 7 are wrong. The energy flux

${\bf P}_{mag}=-c[{\bf E}\times{\bf M}(T)],$ (8)

has nothing to do with the heat current. In Section V we will show that
$\displaystyle{\bf P}_{mag}$ is the magnetization part of the Poynting vector,
which describes the transfer of electromagnetic energy by the bulk
magnetization currents.

First of all, magnetization currents have zero intrinsic entropy. The
magnetization currents are generated by magnetic vector potential without the
electrical electromotive force. Therefore, the magnetization currents have a
quantum nature. In other words, in accordance with the Bohr - van Leeuwen
theorem,46  the magnetization currents would vanish, if the motion of
electrons is considered within the classical mechanics. Magnetization current
are not much sensitive to scattering of electrons, because the scattering only
slightly shifts the electron orbits without substantial changes of the phase
of the electron wave function. In terms of thermodynamics, the entropy related
to the electromagnetic magnetization part of the free energy, $\displaystyle
F_{em}$ is given by24

$S_{mag}=\left({\partial F_{em}(M(T))\over\partial T}\right)_{M}=0,$ (9)

because the temperature derivative is calculated at constant magnetization and
the temperature dependence of $\displaystyle{\bf M}$ does not contribute to
the entropy.26

The direct relation between the entropy and thermomagnetic coefficients may be
obtained in the collisionless limit. If the electron mean free path,
$\displaystyle\ell$, exceeds the Larmor radius, $\displaystyle r_{L}$, one can
neglect electron scattering. In this case the whole electron system moves with
the drift velocity,

${\bf v}_{dr}=c\cdot{[{\bf F}\times{\bf H}]\over eH^{2}},$ (10)

where $\displaystyle{\bf F}$ is the force, which is perpendicular to the
magnetic field, $\displaystyle{\bf H}$, $\displaystyle c$ is the light
velocity, and $\displaystyle e$ is the electron charge. In particular, in
crossed electric and magnetic fields, the electron drift velocity is
$\displaystyle c[{\bf E}\times{\bf H}]/H^{2}$. Taking into account that the
electron thermal energy is $\displaystyle k_{B}TS_{e}$ ($\displaystyle S_{e}$
is the electron entropy), the heat current may be presented as

${\bf J}^{h}_{tr}={\bf v}_{dr}TS_{e}=cTS_{e}\cdot{[{\bf E}\times{\bf H}]\over
H^{2}},$ (11)

and the thermomagnetic coefficient is given by

$\mathcal{L}=-c\cdot{S_{e}\over H}.$ (12)

Thus, in the collisionless limit the thermomagnetic coefficient is directly
expressed via the thermodynamic entropy of electron subsytem. Eq. 12 is well
known in the theory of thermomagnetic phenomena.8

Let us apply this equation to non-interacting electrons in quantized magnetic
field and calculate the oscillating part of the thermomagnetic transport
coefficient. The entropy may be calculated from the thermodynamic potential,
which is given by47

$\displaystyle\displaystyle\Omega(T,H,\zeta)={m^{3/2}(\hbar\omega_{c})^{5/2}\over
4\pi^{4}\hbar^{3}}$ (13)
$\displaystyle\displaystyle\times\sum_{k}(-1)^{k}k^{-5/2}\cdot\Psi(\lambda\cdot
k)\cdot\cos\left({2\pi\zeta\over\hbar\omega_{c}}\cdot k-{\pi\over 4}\right),$

where $\displaystyle\omega_{c}=eH/mc$ is the cyclotron frequency,
$\displaystyle\lambda(T,H)=2\pi^{2}k_{B}T/\hbar\omega_{c}$,
$\displaystyle\Psi(x)=x/\sinh(x)$, and $\displaystyle\zeta$ is the electron
chemical potential. In Eq. 13 we limit our consideration by the oscillating
part of the thermodynamic potential. Calculating the entropy,

$S=\left({\partial\Omega(T,H,\zeta)\over\partial T}\right)_{H,\zeta}\ \ ,$
(14)

and substituting it in Eg. 12, we get the thermomagnetic coefficient of free
electrons in quantized magnetic field,

$\displaystyle\displaystyle\mathcal{L}={ek_{B}m^{1/2}(\hbar\omega_{c})^{1/2}\over
2\pi^{2}\hbar^{2}}$ (15)
$\displaystyle\displaystyle\times\sum_{k}(-1)^{k}k^{-3/2}\cdot\Psi^{\prime}(\lambda\cdot
k)\cdot\cos\left({2\pi\zeta\over\hbar\omega_{c}}\cdot k-{\pi\over 4}\right).$

Eq. 15 coincides with the result obtained by Bar’yachtar and Peletminskii
using the linear response approach.48

Now we compare the thermomagnetic heat current (Eq. 15) with the
electromagnetic flux related the magnetization. In the limit of quantized
field, the magnetization is given by47

$\displaystyle\displaystyle M=\left({\partial\Omega(T,H,\zeta)\over\partial
H}\right)_{T,\zeta}=-{em^{1/2}(\hbar\omega_{c})^{1/2}\zeta\over
2\pi^{3}\hbar^{2}c}$ (16)
$\displaystyle\displaystyle\times\sum_{k}(-1)^{k}k^{-3/2}\cdot\Psi(\lambda\cdot
k)\cdot\sin\left({2\pi\zeta\over\hbar\omega_{c}}\cdot k-{\pi\over 4}\right).$

Comparing the thermomagnetic heat current, $\displaystyle{\bf
j}^{h}=T\mathcal{L}\cdot[{\bf H}\times{\bf E}]/H$, where
$\displaystyle\mathcal{L}$ is given by Eq. 15, and the magnetization part of
the Poynting vector $\displaystyle{\bf j}^{\epsilon}_{mag}=c[{\bf M}\times{\bf
E}]$, where $\displaystyle M$ is given by Eq. 16, we see that the
electromagnetic flux exceeds the heat flux by the factor of
$\displaystyle\zeta/k_{B}T$. If we would follow to Refs. 14 ; 15 ; 16 ; 17 ;
18 ; 19 ; 20 ; 21  and associate the electromagnetic flux with the heat, we
obtain another ”giant” Nernst effect without the degeneracy factor of
$\displaystyle k_{B}T/\epsilon_{F}$ for free electrons in quantized magnetic
field. Obviously, this conclusion is erroneous.

Thus, in the collisionless limit, the thermomagnetic coefficient may be
directly presented via the entropy of electrons (Eq. 12). A textbook example
of electrons in quantized magnetic field shows that the transverse flux of
magnetization energy, $\displaystyle c[{\bf M}\times{\bf E}]$, drastically
exceeds the transverse heat current and, therefore, the magnetization energy
cannot be associated with the heat.

##  III Thermomagnetic Charge Transfer Induced by Entropic Forces

In this section we discus a thermal force induced by the temperature gradient
and the Onsager relation for thermomagnetic coefficients. To calculate the
thermal force, let us divide the electron system into a large number of small
statistical subsytems of electrons. So, the total number of electrons is given
by $\displaystyle N_{e}=\Sigma_{i}\Delta N_{i}$ and the total entropy is
$\displaystyle S_{e}=\Sigma_{i}\Delta S_{i}$. When one of this subsystems
moves from the point $\displaystyle{\bf R}$ with temperature $\displaystyle T$
to the point $\displaystyle{\bf R}+\Delta{\bf R}$ with temperature
$\displaystyle T+\Delta T$, the entropic force produces the work of
$\displaystyle A={\bf F}_{i}\cdot\Delta{\bf R}$ and transfers the thermal
energy of $\displaystyle Q=T\cdot\Delta S_{i}$ . According to the second law
of thermodynamics,

$\displaystyle\displaystyle{A\over Q}={{\bf F}^{th}_{i}\cdot\Delta{\bf R}\over
T\cdot\Delta S_{i}}=-{\Delta T\over T}.$ (17)

From the last equation we obtained that the thermal force acting on the
subsystem is $\displaystyle{\bf F}^{th}_{i}=-\Delta S_{i}\cdot\nabla T$, and
the thermal force per electron in this subsystem is

$\displaystyle\displaystyle{\bf f}^{th}_{i}=-{\Delta S_{i}\over\Delta
N_{i}}\cdot\nabla T.$ (18)

As in the previous Section, we limit our consideration by the colisionless
limit. In this case the drift velocity of electrons is given by

$\displaystyle\displaystyle{\bf v}_{dr}=-{c\over e}{[{\bf
f}^{th}_{i}\times{\bf H}]\over H^{2}}=-{c\over e}{\Delta S_{i}\over\Delta
N_{i}}\cdot{\nabla T\times{\bf H}\over H^{2}}.$ (19)

Thus, the electric current transferred by the subsystem is

$\displaystyle\displaystyle\Delta{\bf j}^{e}_{i}=e{\bf v}_{dr}\cdot\Delta
N_{i}=-c\Delta S_{i}\cdot{\nabla T\times{\bf H}\over H^{2}},$ (20)

and the total thermomagnetic transport electric current is

$\displaystyle\displaystyle{\bf j}^{e}_{tr}=-cS_{e}\cdot{\nabla T\times{\bf
H}\over H^{2}},$ (21)

Eqs. 12 and 21 directly demonstrate validity of the Onsager relations in
thermomagnetic phenomena (Eqs. 1 and 2). If the temperature gradient is taken
into account via thermal entropic force (Eq. 18), the obtained electric
current does not have any magnetization component (Eq. 21), because
magnetization currents have zero intrinsic entropy. In other words, the
electric transport current is directly calculated as a response to thermal
force. This approach was proposed by Stephen49  for description of
thermomagnetic phenomena in the superconducting vortex state and it is widely
used in this area.26 ; 50

To the best of our knowledge, the entropic approach based on thermal forces
was not used in microscopic theories. Therefore, microscopically calculated
electric current induced by the temperature gradient incorporates
contributions from the all temperature dependent material characteristics,
such as magnetization. In particular, in the Boltzmann transport equation the
temperature gradient is usually introduced via the dependence of
$\displaystyle T(x)$ in the argument, $\displaystyle\varsigma=(\epsilon({\bf
p})-\mu)/T(x)$ of the electron distribution function $\displaystyle
f(\varsigma)$. In the collisionless limit, in the first order in
$\displaystyle\nabla T$ the nonequilibrium electron distribution function has
well-known form,8

$\displaystyle\displaystyle f_{1}({\bf p})={\epsilon-\zeta\over
T}\cdot{\partial
f_{0}\over\partial\epsilon}\cdot{v_{y}^{2}\over\omega_{c}}\cdot\nabla_{x}T,$
(22)

where $\displaystyle v_{y}$ is the $\displaystyle y$-component of the electron
velocity, and $\displaystyle\omega_{c}$ is the cyclotron frequency.

In the quantum transport equation, besides the nonequilibrium distribution
function proportional to the temperature gradient, one should include quantum
corrections in the form of the Poisson brackets calculated with any two
electron Green functions (polarization operators, fluctuations propagators
etc), $\displaystyle A(\omega,{\bf q},T)$ and $\displaystyle B(\omega,{\bf
q},T)$, that correspond to different time intervals in the diagrammatic
technique,27 ; 51 ; 52 ; 53

$\displaystyle\displaystyle\\{A(\omega,{\bf q},T),B(\omega,{\bf
q},T)\\}={i\over 2}\nabla T\left({\partial A\over\partial T}{\partial
B\over\partial{\bf p}}-{\partial B\over\partial T}{\partial A\over\partial{\bf
p}}\right).$ (23)

The $\displaystyle\nabla T$-induced electric current calculated in the
Boltzmann formalism or in the quantum transport approach will include
contributions of all temperature dependent characteristics, such as
magnetization, Cooper pair density etc. In the case of the temperature
dependent magnetization, $\displaystyle{\bf M}(T)$, the electric bulk current
is given by

$\displaystyle\displaystyle{\bf j}^{e}_{bulk}={\bf j}^{e}_{tr}+{\bf
j}^{e}_{mag},$ (24)

where the magnetization current density in the homogeneous sample is

$\displaystyle\displaystyle{\bf j}^{e}_{mag}=c\nabla\times{\bf M}(T)=c\nabla
T\times{\partial{\bf M}(T)\over\partial T}.$ (25)

In other words, the thermomagnetic transport current is given by

$\displaystyle\displaystyle{\bf j}^{e}_{tr}={\bf j}^{e}_{bulk}+c{\partial{\bf
M}(T)\over\partial T}\times\nabla T.$ (26)

Eq. 26 was derived by Obraztsov,54  who has pointed out that to satisfy the
Onsager relation presented by Eqs. 1 and 2 the bulk electric current generated
by the temperature gradient should be corrected by magnetization currents.
Employing the quantum transport equation, in our work27  we have confirmed
that bulk thermomagnetic electric current should be corrected by magnetization
currents, while the heat bulk current does not have such corrections.

As we have shown in the previous section, the magnetization currents do not
contribute the thermomagnetic heat transfer. In the first part of this
section, we have demonstrated that the magnetization currents do not appear in
the thermomagnetic charge transfer, which is produced by the entropy-related
thermal force generated by $\displaystyle\nabla T$ (Eq. 18). Therefore,
magnetization currents do not arise in the thermomagnetic theory formulated in
terms of entropy and thermal forces. If $\displaystyle\nabla T$ is taken into
account in the electron distribution function (Green function Poisson brackets
in quantum formalism), the temperature-dependent magnetization currents,
$\displaystyle{\bf j}^{e}_{mag}(T)$, give contribution to the bulk electric
current. To obtained electric transport current, $\displaystyle{\bf
j}^{e}_{tr}$, the magnetization contribution should be removed from the bulk
current (Eq. 26).

The textbook example of electrons in quantized magnetic field considered in
the previous section shows that the magnetization correction to the electric
current, $\displaystyle c\cdot({\partial M/\partial T)}\times\nabla T$, where
$\displaystyle M$ is given by Eq. 16, drastically exceeds the thermomagnetic
transport current. Therefore, in the Boltzmann type formalism it is important
accurately calculate the magnetization component in the bulk current and then
remove it, in accordance with the Obraztsov’s formula (Eq. 26), to find the
transport current. If one would neglect the thermomagnetic component in the
bulk current of free electrons in quantized magnetic field, the obtained
thermomagnetic coefficient $\displaystyle\mathcal{L}=c\cdot(\partial
M/\partial T)$ exceeds the correct coefficient given by Eq. 15 by a huge
factor of $\displaystyle\epsilon_{F}/T$. Such mistake occurs in some papers,
in particular in the recent work,55  where the authors calculated the Righi -
Leduc coefficient and accepted that $\displaystyle\mathcal{L}=c\cdot(\partial
M/\partial T)$ (Eq. 4 in Ref. 55 ). As a result, they obtained a huge
transverse heat current in crossed magnetic field and $\displaystyle\nabla T$,

$\displaystyle\displaystyle{\bf j}^{h}_{y}=-{cT\over e}\left({dM\over
dT}\right)\left({d\zeta\over dT}\right)\nabla_{x}T,$

where $\displaystyle\zeta$ is the chemical potential. The above equation was
applied to the temperature range near the superconducting transition, where
the magnetization has strong temperature dependence and, therefore, the
magnetization current drastically exceeds the thermomagnetic transport
current.

The examples above directly show that all non-entropic forces should be
canceled among themselves. Only in this case, the entropy flux generated by
electric field corresponds to the electric current generated by the
temperature gradient in accordance with Onsager relations.

##  IV Gedankenexperiment in Corbino Disk Geometry

According to Maxwell’s equations, the magnetization currents are divergence-
free, i.e. $\displaystyle\nabla\cdot{\bf
j}_{mag}=c\nabla\cdot[\nabla\times{\bf M}]=0$. It is well understood that as a
consequence the net magnetization current that is measured in any transport
experiment is always zero.42 ; 56 ; 57  In other words, the bulk magnetization
current is always compensated by the surface currents,

$\displaystyle\displaystyle{\bf J}_{mag.bulk}+{\bf J}_{mag.sf}=0.$ (27)

For example, if the two-dimensional Corbino ring is placed in the
perpendicular magnetic field, the radial temperature gradient generates the
circular bulk magnetization current and edge magnetization currents
circulating along the inner and outer edges of the ring (see Fig. 1.A). The
magnetization currents at inner and outer edges, $\displaystyle{\bf
J}_{mag.sf.1}$ and $\displaystyle{\bf J}_{mag.sf.2}$, may be substantially
different, but the net electric magnetization current, $\displaystyle{\bf
J}_{mag.bulk}+{\bf J}_{mag.sf.1}+{\bf J}_{mag.sf.2}$, in any Corbino ring
cross-section is always zero.42

In the recent paper22 , the authors considered the thermomagnetic effects in
the Corbino disk geometry and concluded that the surface magnetization
currents provide the leading contribution to the Nernst effect. In this work,
the authors assume that the inner and outer edges of the disk are kept at
temperature $\displaystyle T_{1}$ and $\displaystyle T_{2}$, but ignore the
temperature gradient inside the ring. As electrons in the bulk are not
affected by the temperature gradient and the thermal entropic force given by
Eq. 18 is absent, the bulk thermomagnetic current is also absent. However,
this model does not corresponds to the experimental setup for measurements of
thermomagnetic coefficients. As it is highlighted in Section IV of Ref. 42 ,
in the thermomagnetic measurements the homogeneous temperature gradient is
created in a substrate and the thermal coupling to the substrate should be
sufficiently strong to establish the local electron temperature in the sample
to be equal to the local temperature of the substrate. In this setup the bulk
and surface magnetization currents cancel each other and the thermomagnetic
transport coefficient is solely determined by the electric transport current
in the bulk.42  In the entropic approach, only the bulk transport current is
generated by the entropic force created by temperature gradient (Eqs. 18 and
21).

Figure 1: A. Cancellation of bulk and surface electric magnetization currents
induced by the radial temperature gradient in any cross-section S of the
Corbino ring.42 B. Gedankenexperiment setup with azimuth inductive electric
field and radial entropy/heat flux. There is no any surface/edge magnetization
contributions to the radial current, which gives the thermomagnetic
coefficient.

While the Corbino disk geometry may be convenient for some calculations, the
obtained thermomagnetic coefficient should not depend on geometry (if
geometrical lengths are much larger than characteristic quantum wavelengths).
Obviously, in the case of the radial temperature gradient (Fig. 1.A), the
usual experimental strip geometry is obtained if the inner radius increases
and becomes much larger than the width of the Corbino ring.

Gedankenexperiment is a thermomagnetic setup with the circular inductive
electric field directed along the azimuth of the Corbino geometry (see Fig.
1.B). In this case, the magnetic field perpendicular to the disc plane and the
azimuth electric field generate the radial entropy/heat flux.27  As the
entropy and heat are transfered in the radial direction, any surface (edge)
magnetization contribution to the entropy/heat transfer is excluded in this
setup. This gedankenexperiment should give the same thermomagnetic coefficient
as the usual strip setup and, therefore, we deduce that edge magnetization
currents do not contribute to the entropy/heat transfer. Taking into account
the divergence-free nature of magnetization currents, we conclude that bulk
magnetization currents also do not transfer the entropy and heat. This
consideration confirms conclusions of Sections II and III. According to
magnetization thermomagnetic theory,14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22
; 29 ; 30 ; 31 ; 32 ; 33  the large heat current is transfered by the
magnetization currents and, therefore, the gedankenexperiment will give the
thermomagnetic coefficient strongly different from that in the strip geometry.
Let us also note, that the same type of the thermomagnetic gedankenexperiment
with radial electric current is impossible, because the circular temperature
gradient does not exists in nature. This is a reason, why there is no analog
of the Obraztsov formula (Eq. 26) for the entropy/heat transfer.

Finally, in previous sections we discussed the Onsager relation between the
bulk heat current induced by electric field and the electric transport current
generated by $\displaystyle\nabla T$ (see Eqs. 1 and 2). As the net electric
magnetization is zero and magnetization does not contribute to the heat
transfer, we see that in thermomagnetic phenomena the net electric and heat
currents naturally satisfy the Onsager relation.

##  V Contribution of Magnetization Energy to the Poynting Vector

The study and design of metamaterials with unique electromagnetic properties
has generated a lot of attention to electrodynamics of magnetic materials, in
particular to the energy flow and corresponding Poynting vector. Negative
phase velocity propagation and negative refraction and their interrelation
have remained to be one of the most difficult aspect in the theory of
metamaterials.

In this section we consider electrodynamics of magnetic materials and revisit
the Poynting vector, $\displaystyle{\bf P}$, taking into account transfer of
the bulk and surface electromagnetic energies related to magnetization. To
simplify our consideration we assume zero dielectric polarization (the
permittivity equals one) and concentrate our attention at the magnetic
contribution. According to the Maxwell equations, in a steady state the
Poynting satisfy the continuity equation which expresses the energy
conservation law in the following form,

$\displaystyle\displaystyle\nabla\cdot{\bf P}={\bf j}\cdot{\bf E},$ (28)

where the current $\displaystyle{\bf j}$ includes both transport component,
$\displaystyle{\bf j}_{tr}$, and magnetization component, $\displaystyle{\bf
j}_{mag}$. It is well understood that the term $\displaystyle{\bf
j}_{tr}\cdot{\bf E}$ describes dissipation of electromagnetic energy into the
heat.

According to classical textbooks the magnetization does not affect the
Poynting vector and in the magnetic materials it has the same form as in
vacuum,56 ; 57

$\displaystyle\displaystyle{\bf P}_{0}={c\over 4\pi}[{\bf E}\times{\bf H}].$
(29)

In particular, according to Ref. 56  the magnetization currents do not
dissipate electromagnetic energy, do not change energy balance of
electromagnetic energy, and, therefore, do not change the form of the Poynting
vector. Another augment of Ref. 55  in favor of Eq. 29 for magnetic materials
is that the Eq. 29 ”follows independently from the obvious condition that the
normal component of the Poynting vector must be continuous at the surface of a
conductor, if we use the continuity of $\displaystyle{\bf H}_{t}$ and the
validity of Eq. 29 in the vacuum outside the body.”

Two recent papers39 ; 40  independently pointed out that in magnetic materials
the tangential component of $\displaystyle{\bf H}$ was not continuous at the
surface due to surface (edge) magnetization currents. To take into account
magnetization currents in the energy balance, the authors of Refs. 39  and 40
accepted that the magnetization currents, $\displaystyle{\bf j}_{mag}$,
contributed to the dissipation of electromagnetic energy and produced entropy
in same way as electric transport current. The dissipation increases the
internal energy,

$\displaystyle\displaystyle{\partial U\over\partial t}={\bf j}_{tr}\cdot{\bf
E}+{\bf j}_{mag}\cdot{\bf E}$ $\displaystyle\displaystyle=-{c\over
4\pi}\nabla\cdot[{\bf E}\times{\bf H}]-c\nabla\cdot[{\bf E}\times{\bf M}].$
(30)

Then, the Poynting vector is given by39 ; 40

$\displaystyle\displaystyle{\bf P}={c\over 4\pi}[{\bf E}\times({\bf
H}+4\pi{\bf M})]={c\over 4\pi}[{\bf E}\times{\bf B}].$ (31)

This form of the Poynting vector is widely accepted,58 ; 59 ; 60 ; 61 ; 62 ;
63  while some authors arguer in favor of the classical formula (Eq. 29).41 ;
64  However, to the best of our knowledge, the transfer of electromagnetic
energy by surface magnetization currents are still overlooked in this
discussion.

Figure 2: Transfer of electromagnetic energy by bulk and surface magnetization
currents

As we highlighted in previous sections, the quasi-stationary magnetization
currents in natural magnetic materials have a quantum nature and do not
dissipate energy. The same is valid for superconducting metamaterials. Let us
first neglect the imaginary part of the magnetic permeability. Without
dissipation, the electromagnetic energy may be only redistributed in the
sample volume. Due to circulating character of magnetization currents, the net
electromagnetic energy transferred by magnetization through any cross-section
of the sample is always zero. Thus, the electromagnetic energy transferred by
the bulk magnetization currents is always compensated by the surface currents.
This fundamental compensation is described by the divergent free form of the
net energy current related to magnetization,42

$\displaystyle\displaystyle{\bf j}^{\epsilon}_{mag}=\nabla\times(c\phi{\bf
M})=\varphi{\bf j}^{e}_{mag}+c{\bf M}\times{\bf E},$ (32)

where $\displaystyle\varphi$ is the electric potential. In a homogeneous
sample the bulk magnetization current $\displaystyle{\bf
j}^{e}_{mag}=c\nabla\times M$ is zero. (we neglect the temperature gradient in
this section). Therefore, the first term in the rhs of Eq. 32 describes the
surface magnetization currents and the second term is the bulk energy flux
related to magnetization. Finally, the Poynting vector, i.e. the flux of
electromagnetic energy, is given by

$\displaystyle\displaystyle{\bf P}={\bf P}_{0}+{\bf P}_{mag}={c\over
4\pi}[{\bf E}\times{\bf H}]+c{\bf M}\times{\bf E}$
$\displaystyle\displaystyle={c\over 4\pi}[{\bf E}\times({\bf H}-4\pi{\bf
M})].$ (33)

Let us illustrate the above statements by the energy transfer in the simple
two-dimensional rectangular geometry of the sample presented in Fig. 2 The
surface electric magnetization current is $\displaystyle{\bf
J}^{e}_{mag.sf}=c{\bf M}\times{\bf n}$, where $\displaystyle{\bf n}$ is the
unit vector normal to the surface and directed outward from the sample. Then,
the energy current transferred by the right and left edge currents is given
$\displaystyle\varphi_{l}\cdot{\bf J}^{e}_{mag.l}+\varphi_{r}\cdot{\bf
J}^{e}_{mag.r}$. Its absolute value in the direction $\displaystyle[{\bf
M}\times{\bf E}]$ may be presented as $\displaystyle
j^{e}_{mag.sf}(\varphi_{r}-\varphi_{l})=-cME\cdot w$, where $\displaystyle w$
is the width of the sample. Certainly, the surface energy current is
compensated by the bulk current $\displaystyle w\cdot c[{\bf M}\times{\bf
E}]$.

Figure 3: Conservation of the electromagnetic energy in a inhomogeneous
sample. The magnetization currents across the sample (red arrows) transfer the
energy $\displaystyle{\bf E}\cdot\Delta{\bf J}^{e}_{mag.s}$ from one surface
to another without dissipation.

To clarify meaning of the term $\displaystyle{\bf J}_{mag}\cdot{\bf E}$ that
was previously associated with dissipation,15 ; 39 ; 40  we consider a sample,
where the permeability and corresponding magnetization $\displaystyle{\bf M}$
changes in the direction of $\displaystyle{\bf E}\times{\bf M}$ as it is shown
in Fig. 3. We will analyze the energy balance in a small volume, which is
formed by two close cross-sections, $\displaystyle(A,B)$ and
$\displaystyle(A^{\prime},B^{\prime})$, shifted in the direction
$\displaystyle{\bf E}\times{\bf M}$. The net power carried to the volume by
the bulk energy currents related to magnetization is

$\displaystyle\displaystyle cw[({\bf M}+\Delta{\bf M})\times{\bf E}]-cw[{\bf
M}\times{\bf E}]$ (34) $\displaystyle\displaystyle=cw[\Delta{\bf M}\times{\bf
E}].$

Let us show that the incoming bulk electromagnetic energy increases the energy
of the surface currents in the same volume. First, let us note that the change
of the magnetization $\displaystyle\Delta{\bf M}$ between $\displaystyle(A,B)$
and $\displaystyle(A^{\prime},B^{\prime})$ is created by the magnetization
current $\displaystyle\Delta J^{e}_{mag.s}=c\Delta M$, which flows from the
left side of the sample $\displaystyle(A,A^{\prime})$ to the right side
$\displaystyle(B,B^{\prime})$ between the two cross-sections (A,B) and (A’,B’)
as it is shown in Fig. 3 by the red arrows. Because of the charge
conservation, the current $\displaystyle\Delta J^{e}_{mag.s}$ across the
sample is exactly equal to the changes in the surface magnetization currents
between $\displaystyle A$ and $\displaystyle A^{\prime}$, and $\displaystyle
B$ and $\displaystyle B^{\prime}$:
$\displaystyle\Delta{J}^{e}_{mag.s}={J}^{e}_{mag.B^{\prime}}-{J}^{e}_{mag.B}=-({J}^{e}_{mag.A^{\prime}}-{J}^{e}_{mag.A})$.
The current $\displaystyle\Delta{J}^{e}_{mag.s}=c\Delta M$ increases the
surface current at right side with high electric potential and decreases the
surface current at the left side with small electric potential. Thus, the
energy of surface currents increases by

$\displaystyle\displaystyle(\phi_{B}-\phi_{A})\Delta J^{e}_{mag.s}=-{\bf
E}\cdot\Delta{\bf J}^{e}_{mag.s}=cw|E|\Delta M,$ (35)

where $\displaystyle\Delta{\bf J}^{e}_{mag.s}$ is shown in Fig. 3 by red
arrows. Compare Eqs. 34 and 35, we see that in any volume of a non-homogeneous
sample, the energy conservation is reached by the redistribution between the
bulk and surface energy currents.

We limit our consideration above by the dissipationless magnetization. Phase
delay between magnetization and external magnetic field in natural magnetic
materials or resistivity of metallic coils in metamaterials lead to magnetic
dissipation, which is described by the imaginary part of the permeability,
$\displaystyle\mu^{\prime\prime}$. In this case, the attenuation of the
Poynting vector is given by a factor of
$\displaystyle\exp[-(\mu^{\prime\prime}/\mu^{\prime}){\bf k}\cdot{\bf r}]$,
where $\displaystyle{\bf k}$ is a wavevector and
$\displaystyle\mu^{\prime\prime}\ll\mu^{\prime}$.

Thus, the magnetization redistributes the electromagnetic energy between the
bulk and surface currents. Without magnetic dissipation, the divergent free
character of magnetization energy current (Eq. 32) lead to the Poyinting
vector in the form of Eq. 33. For example, if $\displaystyle 4\pi{\bf M}={\bf
H}$, i.e. the permeability $\displaystyle\mu=2$, the whole electromagnetic
energy is transferred by the surface currents and the Poynting flux is zero.
If $\displaystyle\mu>2$, the energy flux transferred by surface energy current
exceeds the energy current incoming from the vacuum. In this case, the
Poynting flux in the bulk moves in the direction opposite to the wavevector of
the incoming flux to compensate the surface energy current.

##  VI Nernst Effect of Fluctuating Cooper Pairs in Collisionless Limit

In accordance with expectations based on the Fermi liquid picture,
experimental studies did not demonstrate large thermomagnetic effects in
ordinary superconductors above the superconducting transition. Therefore,
experimental observations of huge Nernst effect in various high-$\displaystyle
T_{c}$ superconductors is usually considered as an evidence of non-BCS nature
of superconducting transition and the non-Fermi liquid character of
thermomagnetic phenomena in these materials. According to the concept proposed
by Emery and Kivelson65  the low electron density and quasi-two dimensional
electron spectra in superconducting cuprates lead to strong phase fluctuations
and the finite temperature Berezinskii - Kosterlitz - Thouless (BKT) type
transitions. Above the transition temperature the long-range superconducting
coherence is destroyed by strong phase fluctuations which lead to the
generation of free vortices. The amplitude of the local superconducting order
parameter remains non-vanishing in significant temperature range above
$\displaystyle T_{c}$. Elementary excitations in a form of vortices determine
all transport phenomena above the BKT-type transition. As it is highlighted in
Ref. 66  “it is vital to recognize the elephant in the room, namely the vortex
liquid above and below $\displaystyle T_{c}$.” Also, large Nernst signals are
observed in the pseudogap state of cuprates, which is believed unrelated to
vorticity.67 ; 68  Strongly correlated pseudogap state is often associated
with the holon type excitations, which originates due to mutual inelastic
decay channel for electrons and holes into the charge reservoir.69
Modification of the of charge-conjugation symmetry can provides large
thermomagnetic effects.69  Huge thermomagnetic effects based on vortices or
holons do not require the Fermi liquid degeneracy factor. Surprisingly, in
recent papers14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 29  the giant Nernst
effect without the $\displaystyle k_{B}T/\epsilon_{F}$ factor have been
calculated in the frame of traditional Gaussian fluctuation model within the
Fermi liquid concept. In the next three sections we will show that giant
thermomagnetic effects in the Fermi liquid without the $\displaystyle
k_{B}T/\epsilon_{F}$ factor are impossible.

Here we will employ the entropic formalism of Sections II and III to calculate
the fluctuation Nernst effect in the collisionless limit that provides the
upper boundary for the Nernst effect in materials with a finite electron
mobility. Aronov et all.43  calculated the Hall coefficient in the
collisionless limit, $\displaystyle\omega_{c}\tau\gg 1$, and demonstrated that
at $\displaystyle\tau\to\infty$ the fluctuation correction to the Hall
coefficient vanishes, because without scattering processes it is impossible to
differ the motion of electrons from the motion of Cooper pairs. In other
words, in crossed electric and magnetic fields the whole electron system moves
with the drift velocity,

${\vec{v}}_{dr}=c\cdot{{\vec{E}}\times{\vec{H}}\over H^{2}}.$ (36)

Corrections to the entropy may be calculated from the corresponding
corrections to electron free energy in the fluctuation region, $\displaystyle
S=-\partial F/\partial T$. For two dimensional superconductor ($\displaystyle
d\ll\xi$, $\displaystyle d$ is the conductor thickness, and $\displaystyle\xi$
is the coherence length in pure superconductor) the singular part of the
fluctuation free energy is15

$F_{fl}(\epsilon,h)=-{T\ h\over 2\pi\xi^{2}}\cdot\ln{\Gamma\left({1\over
2}+{\epsilon\over 2h}\right)\over\sqrt{2\pi}},$ (37)

where dimensionless parameters $\displaystyle\epsilon=1-T/T_{c}$ and
$\displaystyle h=H/H_{c2}(T)$ are described temperature and magnetic field
dependency of the free energy. Thus, the singular part of the entropy is

$S_{fl}=-\begin{cases}{1\over 4\pi\xi^{2}}\cdot\ln\epsilon,&h\ll\epsilon\ll
1;\\\ {\ln 2\over 4\pi^{2}r_{H}^{2}},&\epsilon\ll h\ll 1;\\\ {1\over
2\pi\xi^{2}}\cdot{h\over\epsilon+h},&\epsilon+h\ll h\ll 1.\end{cases}$ (38)

Finally, we can calculate the heat current,
$\displaystyle{\vec{J}}^{h}=TS_{fl}{\vec{v}}_{dr}$, and determine the
thermomagnetic coefficient,

$\mathcal{L}=-{1\over
2}\beta_{0}\begin{cases}\left({r_{H}\over\xi}\right)^{2}\ln{T\over
T-T_{c}},&h\ll\epsilon\ll 1;\\\ \ln 2,&\epsilon\ll h\ll 1;\\\
\left({r_{H}\over\xi}\right)^{2}{H_{c2}(T)\over H-H_{c2}(T)},&\epsilon+h\ll
h\ll 1,\end{cases}$ (39)

where $\displaystyle\beta_{0}=ke/\pi\hbar$=6.68 nA/K is the quantum of
thermoelectric conductance.

Taking into account that $\displaystyle\xi\sim\hbar v_{F}/k_{B}T_{c}$, we see
that in small magnetic field the fluctuation correction to the electronic
entropy (Eq. 38) is of the order of $\displaystyle kT_{B}/\epsilon_{F}$ with
respect to the normal state entropy, which is
$\displaystyle\sim(p_{F}/\hbar)^{2}k_{B}T/\epsilon_{F}$. Therefore, the
entropy related to superconducting fluctuations cannot provide the giant
Nernst effect.

##  VII Nernst Effect of Fluctuating Cooper Pairs: Microscopic Calculations in
Weak Magnetic Field

Theoretical and experimental investigations of the Nernst effect due to
fluctuating Cooper pairs above the superconducting transition temperature,
$\displaystyle T_{c}$, has a long history. As we discussed in previous
sections, in normal metals in a weak magnetic field the Nernst effect is
proportional to $\displaystyle(\omega_{c}\tau)\cdot(k_{B}T/\epsilon_{F})$ and,
therefore, it is very small. Considering the Nernst effect, as the Seebeck
effect in magnetic field or as a thermal Hall effect, one can say that the
Nernst coefficient in normal metals accumulates both the entropy related
degeneracy factor of the Seebeck coefficient and the particle - hole asymmetry
(PHA) factor of the Hall effect related to time-reversal asymmetry in magnetic
field. As was shown by Fukuyama et all.70 , for the fluctuating Cooper pairs
both the Hall and Seebeck coefficients are proportional to small PHA term in
the fluctuation propagator. These results were confirmed by dozens of
experimental investigations in traditional and high-Tc superconductors.15  The
Nernst coefficient due to fluctuations was calculated by Maki29  employing the
Kubo method. Surprisingly, Maki obtained the giant Nernst effect (GNE),29
which does not require any PHA and, therefore, it
$\displaystyle(\epsilon_{F}/k_{B}T)^{2}\sim 10^{8}-10^{10}$ times larger then
the Nernst effect in normal metals. Ten years later, Ullah and Dorsey obtained
the same results using phenomenological time dependent Ginzburg Landau
equation.30  After discovery of high-Tc superconductors, several theoretical
groups14 ; 15 ; 18 ; 19 ; 31 ; 32 ; 33 ; 34 ; 35 ; 36  reproduced the giant
Nernst effect due to fluctuating Cooper pairs to explain the experimental data
in high-Tc materials without appealing to vorticity.

GNE due to fluctuating Cooper pairs also looks rather controversial, if one
tries to find excitations which can transfer giant entropy that is required by
GNE. GNE has been challenged in our work.27  Employing the gauge-invariant
form of the electron heat current in magnetic field, we show that the Nernst
effect due to fluctuating Cooper pairs in weak magnetic field is proportional
to the square of PHA. Controversy related to the fluctuation Nernst effect was
discussed in the comments,28 ; 71  but no agreement has been reached.

There are two main problems in the theory of the Nernst effect of fluctuating
Cooper pairs. The first one is the general problem of the gauge invariance of
the Kubo method for superconductors in magnetic field. The second problem is
the microscopic form of the heat current operator of interacting electrons. In
general, the heat current operator is obtained from the Luttinger effective
gravitational field72  or from the energy-momentum tensor, which is calculated
from the Lagrangian of the system (for details see Ref. 52 ). Keeping time
derivative of the electron field operators in the energy-momentum tensor, one
obtains the heat current operator in the
“$\displaystyle\epsilon$-representation”, i.e. in the frequency domain.
Employing the equation of motion, the time derivative of field operators may
be expressed via the electron field operators to get the “$\displaystyle\xi$
\- representation” of the heat current, i.e. the heat current in the energy
domain. The $\displaystyle\xi$\- representation is physically transparent, as
its every term may be directly associated with the kinetic energy or the
interaction energy, but it leads to a large number of diagrams to be
considered.52  For the fluctuation thermomagnetic effects, both representation
of the electron heat current provide consistent results.15 ; 51  For the
fluctuation Nernst effect, In Refs. 73 ; 74  and 75  we employed the the gauge
invariant $\displaystyle\xi$-representation, while in other works14 ; 15 ; 18
; 19 ; 32 ; 33  the heat current was used in the
$\displaystyle\epsilon$-representation.

To finally resolve the long-standing controversy related to the Nernst effect
of fluctuating pairs, in this section we will employ the well-established
gauge-invariant linear response formalism developed by Aronov, Hikami, and
Larkin for the fluctuation Hall coefficient.43  Implementing the heat current
operator in the same form as it was used in Refs. 14 ; 15 ; 18 ; 19 ; 32  and
33  that obtained GNE, we will show that in the gauge invariant formalism the
giant Nernst effect of fluctuating Cooper pairs, i.e. the Nernst effect
without PHA, does not exist.

It is well established that in the Kubo method the magnetization currents do
not contribute to the Hall conductivity, $\displaystyle\sigma_{xy}$, which is
given by

$\displaystyle\displaystyle\sigma_{xy}\equiv\lim_{\Omega\rightarrow 0}\
\langle[(j^{e}_{tr})_{x}(j^{e}_{tr})_{y}]\rangle=\lim_{\Omega\rightarrow 0}\
\langle[(j^{e}_{bulk})_{x}(j^{e}_{bulk})_{y}]\rangle,$ (40)

where $\displaystyle\langle[j^{e}_{x}j^{e}_{y}]$ is the Fourier representation
of the retarded correlation functions of electric currents $\displaystyle
J^{e}_{x}$ and and $\displaystyle J^{e}_{y}$.

Figure 4: Kubo diagrams for the Hall coefficient and thermomagnetic
coefficient of fluctuating Cooper pairs. The wavy line is for the fluctuation
propagator $\displaystyle L_{n}$ (Eq. 53), the $\displaystyle B^{e}$-triangle
is the electric current operator, and the $\displaystyle B^{h}$-triangle is
the heat current operator for Cooper pairs.

The thermomagnetic coefficient, $\displaystyle\mathcal{L}_{xy}$, may be
presented in analogous way,

$\displaystyle\displaystyle\mathcal{L}_{xy}\equiv\lim_{\Omega\rightarrow 0}\
{1\over T}\langle[j^{h}_{x}(j^{e}_{tr})_{y}]\rangle=\lim_{\Omega\rightarrow
0}\ {1\over T}\langle[j^{h}_{x}(j^{e}_{bulk})_{y}]\rangle,$ (41)

where $\displaystyle\langle[j^{h}_{x}\ j^{e}_{y}]\rangle$ is the Fourier
representation of the retarded correlation functions of the heat current
$\displaystyle J^{h}_{x}$ and the electric current $\displaystyle J^{e}_{y}$.
Magnetization currents do not contribute to the heat current, $\displaystyle
j^{h}$, and in the linear response the magnetization currents do not correlate
with the heat transfer. Therefore, in the correlation function
$\displaystyle\langle[j^{h}_{x}(j^{e}_{tr})_{y}]$ the electric transport
current $\displaystyle j^{e}_{tr}$ may be replaced by the bulk electric
current $\displaystyle j^{e}_{bulk}$. Thus, the thermomagnetic coefficient
given by the Kubo formula does not require any correction from magnetization
currents.

To calculate the thermomagnetic coefficient, $\displaystyle\mathcal{L}_{xy}$,
in the Kubo method one should replace the electric current operator,
$\displaystyle B^{e}_{x}$, in the equation for the Hall coefficient,
$\displaystyle\sigma_{xy}$, by the heat current operator $\displaystyle
B^{h}_{x}$ (see Fig. 4):

$\displaystyle\displaystyle\sigma_{xy}=\lim_{\Omega\rightarrow 0}\
\int{d\omega\over\Omega}\ B^{e}_{x}B^{e}_{y}F(\omega,\Omega),$ (42)
$\displaystyle\displaystyle\mathcal{L}_{xy}=\lim_{\Omega\rightarrow 0}\
{1\over T}\int{d\omega\over\Omega}\ B^{h}_{x}B^{e}_{y}F(\omega,\Omega),$ (43)

The heat current vertex for fluctuating Cooper pairs is usually taken in the
form,14 ; 15 ; 18 ; 19 ; 32 ; 33

$B^{h}(\omega)={\omega\over 2e}B^{e}.$ (44)

Let us also note, that some authors33  together with $\displaystyle\omega$
also include the Kubo frequency, $\displaystyle\Omega$, to the heat current
vertex. First of all, appearance of the Kubo frequency in the thermal energy
is erroneous, because the thermal energy is defined via the entropy, which is
the function of the probability distribution in the statistical ensemble and
it does not fluctuate since it is not a function of the state in which the
system happens to be. Nevertheless, even if $\displaystyle\Omega$ is
erroneously included in the thermal energy, this term does not provide any
contribution to the Nernst coefficient (Eq. 43), because $\displaystyle
F(\omega,\Omega=0)=0$ in accordance with the causality principle for the Hall
effect (Eq. 42). Thus, to calculate the thermomagnetic coefficient (Eq. 43)
one should just add $\displaystyle\omega/2eT$ to the expression for the Hall
coefficient (Eq. 42),

$\mathcal{L}_{xy}=\lim_{\Omega\rightarrow 0}\ {1\over
2eT}\int{d\omega\over\Omega}\ \omega B^{e}_{x}B^{e}_{y}F(\omega,\Omega).$ (45)

The most accurate way to obtain the gauge invariant expression for the Hall
coefficient is to apply the Kubo method to electron states in Landau
representation, which directly includes the magnetic field. For fluctuating
Cooper pairs this approach has been developed in Ref. 43 . According to Eqs. 6
and 13 of Ref. 43 , the Hall coefficient in a two dimensional superconductor
in Landau representation is given by

$\displaystyle\displaystyle\sigma_{xy}=-\lim_{\Omega\rightarrow
0}{(4e\xi_{\ell}^{2})^{2}\over 8\pi^{2}r_{H}^{4}}\sum_{n=0}^{\infty}(n+1)$
$\displaystyle\displaystyle\times\int\limits_{-\infty}^{\infty}{d\omega\over\Omega}\coth{\omega\over
2T}\Phi(n,\omega,\Omega),$ (46)

where $\displaystyle\Phi(n,\omega,\Omega)$ is given by

$\displaystyle\displaystyle\Phi(n,\omega,\Omega)=\Im
L_{n}^{R}(\omega)\cdot\left[L_{n+1}^{R}(\omega+\Omega)-L^{A}_{n+1}(\omega-\Omega)\right]$
$\displaystyle\displaystyle-\ \Im
L_{n+1}^{R}(\omega)\cdot\left[L_{n}^{R}(\omega+\Omega)-L_{n}^{A}(\omega-\Omega)\right],$
(47)

where $\displaystyle\xi_{\ell}$ is the coherence length in 2D disordered
superconductor, $\displaystyle r_{H}^{2}=(2eH)^{-1}$, and $\displaystyle
L_{n}^{R}$ is the fluctuation propagator in the Landau representation. In the
zero order in PHA, the fluctuation propagator is

$L_{n}^{R(A)}=\left({T-T_{c}\over T_{c}}\mp{i\pi\omega\over
8T_{c}}+{\xi_{\ell}^{2}\over r_{H}^{2}}(n+1)\right)^{-1}.$ (48)

Calculating the linear response to applied electric field, one should keep in
Eq. 46 only the linear terms in $\displaystyle\Omega$. Also, in a weak
magnetic field ($\displaystyle r_{H}\rightarrow\infty$), we may expand the
integrand $\displaystyle\Phi(n,\omega,\Omega)$ in power of $\displaystyle 1/n$
and take into account only the first term in this expansion. In the first
order in $\displaystyle\Omega$ and $\displaystyle 1/n$, we get

$\displaystyle\displaystyle\Phi(\omega,\Omega)=\Omega\cdot{8\over
3}{\xi_{\ell}^{2}\over r_{H}^{2}}\cdot{\partial\left(\Im
L^{R}(\omega)\right)^{3}\over\partial\omega},$ (49)

where the fluctuation propagator in a weak field is

$L^{R(A)}(\omega,q)=\left({T-T_{c}\over T_{c}}\mp{i\pi\omega\over
8T_{c}}+\xi_{\ell}^{2}q^{2}\right)^{-1}.$ (50)

Substituting the summation over $\displaystyle n$ by the integration over
$\displaystyle q$ in Eq. 46 we obtained the final equation the Hall
conductivity of fluctuating Cooper pairs in a weak magnetic field,

$\displaystyle\displaystyle\sigma_{xy}=-{2(4e\xi_{\ell}^{2})^{2}\xi_{\ell}^{2}\over
3\pi r_{H}^{2}}\int_{0}^{\infty}q^{2}{2\pi qdq\over(2\pi)^{2}}$
$\displaystyle\displaystyle\times\int_{-\infty}^{\infty}d\omega\
\coth{\omega\over 2T}\cdot{\partial\left(\Im
L^{R}(\omega,q)\right)^{3}\over\partial\omega},$ (51)

Let us note that integrating Eq. 51 over $\displaystyle\omega$ by parts we
will get the Eq. 14 of Ref. 43 . As $\displaystyle\Im L^{R}(-\omega)=-\Im
L^{R}(\omega)$, the Hall conductivity is zero in zero order in PHA. To get the
Hall conductivity in the first order in PHA, one should add the PHA term,
$\displaystyle\gamma\omega/2\epsilon_{F}$, to the fluctuation propagator,43 ;
70

$L_{n}^{R(A)}=\left({T-T_{c}\over T_{c}}\mp{i\pi\omega\over
8T_{c}}+{\gamma\omega\over 2\epsilon_{F}}+{\xi_{\ell}^{2}\over
r_{H}^{2}}(n+1)\right)^{-1}.$ (52)

where $\displaystyle\gamma=\partial\ln T_{c}/\partial\ln\mu$, and
$\displaystyle\mu$ is the electron chemical potential.43  Finally, the
fluctuation Hall coefficient is given by43

$\displaystyle\displaystyle\sigma^{fl}_{H}={e\gamma\over
48}\cdot\omega_{c}\tau\cdot\left({T_{c}\over T-T_{c}}\right)^{2}.$ (53)

Replacing the electric current operator by the heat current operator in Eq.
46, we obtained the thermomagnetic coefficient,

$\displaystyle\displaystyle\mathcal{L}^{fl}=-\lim_{\Omega\rightarrow
0}{(4e\xi_{\ell}^{2})^{2}\over 8\pi^{2}r_{H}^{4}}\sum_{n=0}^{\infty}(n+1)$
$\displaystyle\displaystyle\times\int\limits_{-\infty}^{\infty}{d\omega\over\Omega}\
\omega\coth{\omega\over 2T}\Phi(n,\omega,\Omega),$ (54)

where $\displaystyle\Phi(n,\omega,\Omega)$ is given by Eq. 47, which is the
same as for the Hall coefficient. Therefore, in a weak magnetic field in zero
order in PHA (Eqs. 48 -50) we obtained

$\displaystyle\displaystyle\mathcal{L}^{fl}=-{e(4\xi_{\ell}^{2})^{2}\xi_{\ell}^{2}\over
3\pi r_{H}^{2}}\int_{0}^{\infty}{q^{3}dq\over(2\pi)}$
$\displaystyle\displaystyle\times\int_{-\infty}^{\infty}d\omega\
\omega\coth{\omega\over 2T}\cdot{\partial\left(\Im
L^{R}(\omega,q)\right)^{3}\over\partial\omega},$ (55)

The heat current vertex brings additional $\displaystyle\omega$, which changes
the parity of the integrant for the thermomagnetic coefficient (Eq. 55) with
respect to the integrant for the Hall coefficient (Eq. 51). According to Refs.
18,19, this leads to giant thermomagnetic coefficient in zero order in PHA.
However, one needs to be careful with this conclusion. Note that the function
$\displaystyle\Phi(\omega)$ is the full derivative (see Eq. 49) and,
therefore, integrating Eq. 55 over $\displaystyle\omega$ by parts, we get

$\displaystyle\displaystyle\mathcal{L}^{fl}={e(4\xi_{\ell}^{2})^{2}\xi_{\ell}^{2}\over
3\pi r_{H}^{2}}\int_{0}^{\infty}{q^{3}dq\over(2\pi)}$
$\displaystyle\displaystyle\times\int_{\infty}^{\infty}d\omega\ \left(\Im
L^{R}(\omega,q)\right)^{3}\cdot{\partial\over\partial\omega}\left(\omega\coth{\omega\over
2T}\right).$ (56)

The only essential difference of Eq. 56 from the corresponding equation for
the Hall coefficient given by Eq. 51 (Eg. 14 in AHL paper) is the presence of
an additional $\displaystyle\omega$, which appears in Eq. 56 in combination
with the distribution function, $\displaystyle\coth(\omega/2T)$. As the
characteristic range of $\displaystyle\omega$ is $\displaystyle\sim T-T_{c}\ll
T_{c}$, the derivative of $\displaystyle\omega\coth(\omega/2T)$ over
$\displaystyle\omega$ is $\displaystyle\omega/3T\ll 1$. For this reason,
fluctuations of Cooper pairs do not provide any singular contribution to the
thermomagnetic coefficient.

To obtain singular contribution to the Nernst coefficient, one should take
into account PHA term in the fluctuation propagator (Eq. 52). Calculating the
Nernst coefficient (Eq. 54) we should expand the fluctuation propagator up to
the second order in PHA (the first order contribution is zero due to the
parity). In this way we reproduce our result in Ref. 27  obtained with the
$\displaystyle\xi$ \- presentation of the heat current operator,

$\displaystyle\displaystyle\mathcal{L}^{fl}=-{5\gamma^{2}\over
4}\beta_{0}\left({\xi_{\ell}\over
r_{H}}\right)^{2}\left({4T_{c}\over\pi\epsilon_{F}}\right)^{2}{T_{c}\over
T-T_{c}}$
$\displaystyle\displaystyle=-{5\gamma^{2}\over\pi}\beta_{0}\cdot\omega_{c}\tau\cdot{T_{c}\over\epsilon_{F}}\cdot{T_{c}\over
T-T_{c}}$ (57)

where $\displaystyle\beta_{0}=k_{B}e/\pi\hbar=6.68$nA/K is the quantum of
thermoelectric conductance. In Ref. 27  the fluctuation thermomagnetic
coefficient was calculated by two different methods. In the Kubo method, we
employ the gauge - invariant form of the heat current operator in the
$\displaystyle\xi$-representation, which was derived from the energy-momentum
tensor (Refs. 73  and 74 ) and the quantum energy balance equation (Ref. 75 ).
Using the quantum transport equation in the Keldysh formalism, we calculated
both the heat current generated by the electric field and the electric current
induced by temperature gradient.

Summarizing our calculations of the Nernst coefficient of fluctuating Cooper
pairs, we would like to highlight mathematical and physical rationales for the
absence of giant Nernst effect, which was predicted in a number of theoretical
papers in frames of BSC model,14 ; 15 ; 18 ; 19 ; 31 ; 32 ; 33  but has never
been observed in traditional superconductors. Mathematically, this is a result
of the form of the integrands in Eqs. 51 and 54. The only dereference between
the integrands is the appearance of $\displaystyle\omega$ in the
thermomagnetic coefficient. Let us note that the thermal energy of Cooper
pairs, $\displaystyle\omega$, always appears in combination with the
distribution function $\displaystyle\coth(\omega/2T)$ (Eq. 56) and never
appears in combination with its derivative. Close to $\displaystyle T_{c}$,
the function $\displaystyle(\omega/2T)\coth(\omega/2T)$ is $\displaystyle\sim
1$ and, therefore, it does not provide any singular contribution to the
thermomagnetic coefficient without taking into account PHA in the fluctuation
propagators.

According to Eq. 57, the fluctuation thermomagnetic coefficient scales as the
product of $\displaystyle\omega_{c}\tau$ and $\displaystyle T/\epsilon_{F}$,
i.e. exactly in the same way as the the normal state thermomagnetic
coefficient. Let us note, that the fluctuation Hall coefficient (Eq. 53)
scales as $\displaystyle\omega_{c}\tau$, i.e. as the Hall coefficient in the
normal state. Fluctuation thermoelectric coefficient scales as $\displaystyle
T_{c}/\epsilon_{F}$, i.e. as the thermoelectric coefficient in the normal
state.15 ; 51  Therefore, the scaling $\displaystyle\omega_{c}\tau\cdot
T/\epsilon_{F}$ is highly expected for the fluctuation thermomagnetic
coefficient. The characteristic thermal energy transfered by fluctuation
Cooper pairs, $\displaystyle\omega$, is of the order of $\displaystyle
T-T_{c}$. Therefore, the fluctuation thermoelectric power is given by the
fluctuation conductivity multiplied by the product of $\displaystyle
T_{c}/\epsilon_{F}$ and $\displaystyle(T-T_{c})$. In the same way, the
thermomagnetic coefficient (Eq. 57) is given by the fluctuation Hall
coefficient (Eq. 53) multiplied by the product of $\displaystyle
T_{c}/\epsilon_{F}$ and $\displaystyle(T-T_{c})$. The giant thermomagnetic
coefficient obtained in Refs. 14 ; 15 ; 18 ; 19 ; 31 ; 32  and 33  is by a
factor of $\displaystyle(\epsilon_{F}/T_{c})^{2}$ larger than that given by
Eq. 57.

Thus, the fluctuating Cooper pairs cannot give the giant Nernst effect because
the Cooper pairs do not transfer the entropy. The only possible effect of
superconducting pairing is the reduction of the entropy of normal electrons.
Therefore, the expected thermomagnetic coefficient in the fluctuation region
cannot exceed the thermomagnetic effect of normal electrons, which is
proportional to the square of PHA and small. Giant thermomagnetic effects may
be associated with vortices and will be discussed in Section IX.

##  VIII Thermomagnetic Entropy per Charge

In Sections II and III we consider thermomagnetic effects in the collisionless
limit in terms of entropy fluxes and entropy - related thermal forces. This
approach provides clear physical picture, which is not obscured by various
temperature - dependent properties of materials and structures that are not
related to the entropy. In this Section we generalize the entropic approach to
the disordered conductors with interacting (correlated) electrons.

Again, we divide the electron system into a large number of small statistical
subsytems of charge carriers and calculate the linear response of these
subsytems to the crossed temperature gradient and magnetic field. The thermal
force per electron in the $\displaystyle i$-subsytem, $\displaystyle{\bf
f}^{th}_{i}$ (Eq. 18), leads to the drift velocity of a carrier,
$\displaystyle{\bf v}^{\parallel}_{i}=\mu^{\parallel}_{i}{\bf f}^{th}_{i}$,
where $\displaystyle\mu^{\parallel}_{i}$ is the electron mobility in the
direction of the temperature gradient. In the direction perpendicular to the
temperature gradient, the Lorentz force, $\displaystyle{\bf f}^{L}_{i}$
provides the electron drift, $\displaystyle{\bf
v}^{\perp}_{i}=\mu^{\perp}_{i}{\bf f}^{L}_{i}$. Calculating corresponding
thermomagnetic current, we obtained the thermomagnetic coefficient,

$\displaystyle\displaystyle\mathcal{L}=-{H\over
c}\sum_{i}\mu^{\parallel}_{i}\mu^{\perp}_{i}\Delta S_{i}=-{H\over
c}\sum_{i}\mu^{\parallel}_{i}\mu^{\perp}_{i}\tilde{s}_{i}\Delta N_{i},$ (58)

where $\displaystyle{\tilde{s}}_{i}=\Delta S_{i}/\Delta N_{i}$ is the entropy
per carrier in the $\displaystyle i$-group of charge carriers. Eq. 58
reproduces the thermomagnetic coefficient for noninteracting electrons.

In particular, in an isotropic conductor with the energy - independent
electron momentum relaxation time, $\displaystyle\tau$, we obtain
$\displaystyle\mathcal{L}=-H\mu^{2}S_{e}/c$, where
$\displaystyle\mu=|e|\tau/m$. Taking into account the thermodynamic relation
between the entropy per electron and the temperature derivative of the
electron chemical potential, $\displaystyle
S_{e}/n_{e}=(\partial\zeta/\partial T)_{n_{e}}$, the thermomagnetic
coefficient in this case is given by

$\displaystyle\displaystyle\mathcal{L}=-{H\over c}\mu^{2}S_{e}=-{H\over
c}\mu^{2}n_{e}\left({\partial\zeta\over\partial T}\right)_{n_{e}}.$ (59)

Let us note, that the same-type formula, but with full temperature derivative
was proposed by Varlamov et all.34 ; 35 ; 36  directly for the Nernst
coefficient $\displaystyle\mathcal{N}$, given by Eq. 3. Our consistent
derivation shows that applicability of Eq. 59 is rather limited by the
isotropic and energy-independent electron mobilities. These limitations do not
allow for account the delicate balance of all transport coefficients in the
equation for the Nernst effect (Eq. 3). In particular, according to
Sondheimer4 , if the momentum relaxation time does not depend on electron
energy, the Nernst coefficient is zero.

Let us also note, that Eq. 59 presents thermomagnetic coefficient via the
thermodynamic entropy of electrons or, alternately, via the temperature
derivative of the thermodynamic chemical potential. In some papers34 ; 35 ; 36
the giant fluctuation Nernst coefficient is explained by the strong
temperature-dependence of the chemical potential of Cooper pairs. Like the
chemical potential of the photons in equilibrium black-body radiation, the
thermodynamic chemical potential of Cooper pairs is always zero due to the
lack of constraint on the number of Cooper pairs in the system.23

As the Nernst effect is described in terms of thermoelectric and
thermomagnetic transport coefficients, let us present these coefficients in
terms of the entropy. It is well understood that the thermoelectric Seebeck
coefficient is given by the entropy per unit charge,

$\displaystyle\displaystyle\mathcal{S}={\bar{s}_{th-e}\over|e|},$ (60)

where the entropy per carrier charge, $\displaystyle\bar{s}_{th-e}/|e|$, is
defined as a ratio of the entropy current to the electric current induced by
the same electric field. In terms of electron mobility, the the entropy per
charge is given by

$\displaystyle\displaystyle{\bar{s}_{th-e}\over|e|}={\sum_{i}q_{i}\tilde{s}_{i}\mu^{\parallel}_{i}\Delta
N_{i}\over e^{2}\sum_{i}\mu^{\parallel}_{i}\Delta N_{i}},$ (61)

where $\displaystyle q_{i}=\mp|e|$ for electrons and holes, correspondingly.

In analogy with the thermoelectric phenomena, it will be convenient to
introduce the average thermomagnetic entropy per unit charge,
$\displaystyle\bar{s}_{th-m}$, that is defined as a ratio of transverse
entropy current to the transverse electric current induced by crossed electric
and magnetic fields. Then transverse transport coefficients are interrelated
as

$\displaystyle\displaystyle{\mathcal{L}\over\sigma_{H}}=-{\bar{s}_{th-m}\over|e|}.$
(62)

In accordance with Eq. 58, in the disordered limit,
$\displaystyle\omega_{c}\tau\ll 1$, the thermomagnetic entropy per unit charge
is

$\displaystyle\displaystyle{\bar{s}_{th-m}\over|e|}={\sum_{i}\tilde{s}_{i}\cdot\mu^{\parallel}_{i}\mu^{\perp}_{i}\Delta
N_{i}\over\sum_{i}q_{i}\mu^{\parallel}_{i}\mu^{\perp}_{i}\Delta N_{i}}.$ (63)

In the opposite, collisionless limit, $\displaystyle\omega_{c}\tau\gg 1$, the
whole electron system drifts with the same velocity and the entropy per charge
is

$\displaystyle\displaystyle{\bar{s}_{th-m}\over|e|}={S_{e}\over|e|N_{e}},$
(64)

where $\displaystyle S_{e}$ is the total electron entropy and $\displaystyle
N_{e}$ is the number of electrons.

Finally, in accordance with Eq. 3, the measured Nernst coefficient is given by

$\displaystyle\displaystyle\mathcal{N}=\left({\bar{s}_{th-e}\over|e|}-{\bar{s}_{th-m}\over|e|}\right)\cdot{1\over
H}\cdot{\sigma\cdot\sigma_{H}\over\sigma^{2}+\sigma_{H}^{2}}.$ (65)

Thus, the Nernst coefficient is proportional to the difference of
thermoelectric and thermomagnetic entropies per unit charge. If the electron
momentum relaxation time is independent on energy and momentum direction, both
thermoelectric and thermomagnetic entropies per unit charge equal
$\displaystyle S_{e}/|e|N_{e}$ (Eqs. 61 and 63), and the Nernst coefficient is
zero. This explains the Sondheimer cancellation in terms of the entropy. Also,
in weak disorder (strong magnetic field) the thermoelectric entropy per charge
is also given by $\displaystyle S_{e}/|e|N_{e}$ (see Ref. 42 ) and, according
to Eqs. 64 and 65, the Nernst coefficient is zero. Large Nernst effect
requires large entropy of some group of electrons,
$\displaystyle\tilde{s}_{i}$, as well as substantially different averaging
procedures for thermoelectric (Eq. 61) and thermomagnetic (Eqs. 63) entropies.
Significant difference between thermoelectric and thermomagnetic entropies per
carrier charge is also expected for conductors with strongly anisotropic
mobility. The consideration above shows that in the Fermi liquid the
thermomagnetic coefficient is proportional to the entropy per carrier charge,
which is always proportional to $\displaystyle k_{B}T/\epsilon_{F}$. The
Nernst coefficient may be further reduced due to small difference between
thermoelectric and thermomagnetic entropies per unit charge.

##  IX  $\displaystyle{\bf\nabla T}$ \- induced Transverse Vortex Transport

###  IX.1 Nonentropic Forces Generated by Moving Temperature Gradient

Thermomagnetic effects have been intensively studied in vortex state of
traditional and high-$\displaystyle T_{c}$ superconductors.7 ; 49 ; 50 ; 76 ;
77  Superconducting vortex consists of a normal core with the size of the
coherent length, $\displaystyle\xi$, and superconducting currents circulating
around the core in the area of the order of the magnetic penetration length,
$\displaystyle\lambda$. Both the normal core and superconducting currents
contribute to the free energy of the vortex. These contributions with respect
to the superconducting background per a unit of vortex line are given by7

$\displaystyle\displaystyle F_{\phi}^{n}(T)={H_{c}^{2}\over
8\pi}\pi\xi^{2}=\left({\Phi_{0}\over 4\pi\lambda}\right)^{2},$ (66)
$\displaystyle\displaystyle F_{\phi}^{s}(T)={H_{c}^{2}\over
8\pi}\pi\xi^{2}\ln(\lambda/\xi)=\left({\Phi_{0}\over
4\pi\lambda}\right)^{2}\ln(\lambda/\xi),$ (67)

where $\displaystyle H_{c}^{2}(T)$ is the critical magnetic field, and
$\displaystyle\Phi_{0}$ is the quantum of the magnetic flux. The free energy
of superconducting currents is logarithmically exceeds the free energy of the
core and, therefore, the core contribution is usially neglected.7 ; 49 ; 50 ;
76

In the Nernst effect, vortices under the thermal force move in the direction
of $\displaystyle\nabla T$ and transfer the magnetic flux, which generates the
transverse Nernst voltage. In the Ettingshausen effect, vortices under the
Lorentz force move perpendicular to the electric field ( electric current) and
transfer the thermal energy in this direction.7 ; 49 ; 50 ; 76 ; 77  In the
original work by Stephen49  and other papers the leading term in the transport
entropy was associated with the superconducting currents (Eq. 67). The vortex
entropy was revisited in our recent paper,26  where we have highlighted that
the thermodynamic entropy should be calculated at constant magnetization (Eq.
9) and, therefore, the entropy of superconducting currents is zero. This is
also highly expected from the quantum nature of superconducting currents.
Besides the configuration entropy of vortices, which is usually small (see
next subsection), the only entropy that appears in vortex thermomagnetic
phenomena is the entropy of a vortex core. Both the thermal force and the heat
current are described in terms of the transport entropy of a vortex,
$\displaystyle S_{d}$,26

$\displaystyle\displaystyle S_{d}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-{\partial F_{\phi}^{n}(T)\over\partial T},$ (68)
$\displaystyle\displaystyle{\bf f}_{th}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-S_{d}\nabla T=-\left({\Phi_{0}\over
4\pi}\right)^{2}{1\over(\lambda(T))^{3}}{\partial\lambda(T)\over\partial
T}\nabla T,$ (69) $\displaystyle\displaystyle{\bf j}^{h}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle TS_{d}{\bf v}_{dr}.$
(70)

Figure 5: Single vortex moves from the cold to the hot area to minimize the
kinetic and electromagnetic energy of superconducting magnetization currents
circulating around the vortex core. This effect does not contradict to
thermodynamics, because the magnetization currents do not transfer entropy.

Supercurrents around cores neither produce the thermal force, nor participate
in the heat transport. For this reason, the thermomagnetic effects are
negligible in a system of Josephson vortices with a insulator core in a
superconductor-insulator-superconductor junctions.78  At the same time, the
temperature dependence of the free energy of supercurrents, $\displaystyle
F_{\phi}^{s}(T)$, leads to the non-entropic force,

$\displaystyle\displaystyle{\bf f}_{s}=-{\partial
F_{\phi}^{s}(T)\over\partial{\bf r}}=\left({\Phi_{0}\over
4\pi}\right)^{2}{1\over(\lambda(T))^{3}}{\partial\lambda(T)\over\partial
T}\ln{\lambda\over\xi}\cdot\nabla T.$ (71)

As the energy of supercurrents decreases with temperature, this non-entropic
force drives a vortex from the cold to the hot area. Moreover, the non-
entropic force (Eq. 67) exceed the thermal force related to the core entropy
by a large logarithmic factor. Of course, in quasi-equilibrium conditions,
i.e. in the homogeneous temperature gradient, such motion of vortices
contradicts the second thermodynamic law. In Ref. 26 , we have shown that the
non-entropic force given by Eq. 71 is balanced by another non-entropic force,
which is the Lorentz force between the vortex and superconducting
magnetization currents created by $\displaystyle\nabla T$.26  Let us note that
the Lorentz force may be also described as vortex repulsion from the hot area
with high vortex concentration to the cold area with low vortex concentration.
Due to fundamental relation between the electromagnetic energy and
magnetization of interacting vortices,78  this cancellation of
$\displaystyle{\bf f}_{s}$ and the Lorentz force is also valid for strongly
interacting and even overlapping vortices in the homogeneous temperature
gradient. However, the local or transient temperature gradient does not
provide vortex redistribution, does not create corresponding magnetization
currents, and does not generate the Lorentz force. In this case, the non-
entropic force related to the temperature dependent superconducting currents
drives a single vortex in the direction of $\displaystyle\nabla T$.

Such vortex motion has been demonstrated in recent unique experiments with
single superconducting vortices that follow a laser-generated hot spot.44  The
vortex motion was investigated in superconducting niobium films with a
thickness of 90 nm grown by magnetron sputtering on a silicon substrate. For
imaging of individual vortices the authors develop a magneto-optical system
based on the Faraday rotation of light polarization in a magneto-optical
indicator placed onto the superconductor, in a crossed-polarizer beam path
configuration. Ref 44  has demonstrated precise manipulation of individual
vortices by the laser-induced non-entropic force.

Obviously, the static laser-generated hot spot creates classical
thermomagnetic phenomena, where vortices move from the hot to cold area. If
the laser hot spot moves with the velocity below the speed of vortices, the
vortices will be accumulated in the hot sport. Sooner or later the vortex
concentration in the hot sport will reach the quasi-equilibrium value that
corresponds to the hot spot temperature and the many-body repulsive force will
equilibrate the force $\displaystyle{\bf f}_{s}$.

Let us evaluate the minimal velocity of the hot sport to observe the vortex
movement from the cold area to the hot area. This velocity equals to the
vortex driving speed, which is given by the Stephen-Bardeen vortex viscosity,
$\displaystyle\eta_{S-B}$,

$\displaystyle\displaystyle{\bf v}_{\phi}={{\bf f}_{s}\over\eta_{S-B}},\ \ \ \
\eta_{S-B}={\pi\hbar^{2}\over 4e^{2}\xi^{2}\rho}$ (72)

where $\displaystyle\rho$ is the film resistivity. In particular, for
experimental conditions in Ref. 44 , - $\displaystyle T$ = 4.6 K and
$\displaystyle\nabla T\sim$ 1K/$\displaystyle\mu$m, - we evaluate the vortex
driving speed as 10 $\displaystyle\mu$m/s, which is in a good agreement with
the speed experimentally observed in Ref. 44 . Driving speeds of vortices in
electric fields are significantly higher.80  The minimal scanning speed, i.e.
velocity of the hot spot, depends on the laser power. High power increases the
minimal scanning speed due to the increase in $\displaystyle\nabla T$ and the
non-entropic force $\displaystyle{\bf f}_{s}$ (Eq. 69). It would be
interesting to experimentally investigate a change of the direction of vortex
motion as a function of the hot spot velocity and laser power (hot spot
temperature).

We would like to highlight that the non-entropic forces related to the
temperature-dependent magnetization currents (in general, quantum currents)
are completely different from the thermal forces, related to the entropy. In
accordance with the Onsager approach, the all non-entropic forces created by
$\displaystyle\nabla T$ are always equilibrated in quasi-equilibrium
conditions and do not manifest themselves in the thermomagnetic phenomena.
However, some of none-entropic forces and corresponding phenomena can be
observed in strongly nonequilibrium conditions, such as local or transient
temperature gradient. In particular, vortex motion from the cold to hot area
due to non-entropic force generated by mobile hot spot has been observed in
Ref. 44 .

###  IX.2 Thermomagnetic Effects in BKT State of 2D Superconductors

Thermomagnetic phenomena in various 2D superconductors have attracted
significant attention during last years. The Nernst coefficient has been
measured in strongly disordered thin films of Nb0.15Si0.85,81  and amorphous
films $\displaystyle\alpha$-MoSe,82  in flakes of crystalline NbSe2,83  and in
superconducting heterostructures, such as 2D Nb-doped strontium titanate (STO)
placed between undoped STO cap and buffer layers82  and in single-layer FeSe
film on STO.84  It was found that above in 2D conductors the Nernst signal is
observed in a wide temperature, which strongly exceeds the fluctuation region
of traditional superconductors. In particular, the Nernst coefficient 2D NbSe2
flakes was found to be at least two orders in magnitude larger than that in
the bulk material.83  All 2D superconductors show non-monotonic temperature
dependence with the peak of the Nernst signal slightly below the transition
temperature, where thermomagnetic effects are associated with vortices. There
is no peculiarity of the Nernst coefficient at the transition temperature
determined from the resistivity measurements. Recent work82  has analyzed
thermomagnetic characteristics related to 2D superconductors belonging to
different families and found that the maximum Nernst signal corresponds to the
entropy per vortex of

${S}^{*}_{\phi}=k_{B}\cdot\ln 2.$ (73)

As it is highlighted in Ref. 82 , the entropy $\displaystyle{S}^{*}_{\phi}$ is
approximately fifty times smaller than the intrinsic entropy of the vortex
core with respect to superconducting background given by Eqs. 66 and 68.

All results above show strong evidence in favor of the Berezinskii -
Kosterlitz - Thouless (BKT) type superconductivity,85 ; 86  which is well
established in 2D superconducting heterostructures87 ; 88  and thin disordered
films, especially in materials with low superfluid density near the
superconductor-insulator transition.89 ; 90 ; 91  In the BKT superconductor, a
vortex created by magnetic field moves in the dense media of vortex -
antivortex pairs and, therefore, the intrinsic entropy of the vortex with
respect to the BKT background is negligible. This fact is well understood in
the BKT theory, which takes into account only configuration entropy of
dissociated vortex-antivortex pairs. Therefore, the transport vortex entropy,
$\displaystyle{S}^{*}_{\phi}$, may be associated with the configuration
entropy transfered by the vortex in the media of vortex-antivortex pairs. The
factor of $\displaystyle\ln 2$ in Eq. 73, corresponds to the two positions
available to the vortex in the real space, when the vortex and the pair
exchange their positions. It is also intriguing, that according to Ref. 82 ,
for multi-layer high-Tc cuprates, such as La1.92Sr0.08CuO4 studied in Ref. 92
, Eq. 73 provides the thermomagnetic transport entropy per vortex per layer.
This observation is well aligned with the pancake vortex structure in these
materials. Our consideration predicts that the same thermomagnetic phenomena
should be observed in the BKT state of Josephson junction arrays and studies
of the transport entropy in corresponding experiments are very desirable.

Our vorticity-based interpretation of thermomagnetic phenomena in thin films
of disordered superconductors is well aligned with recent studies of the
Nernst effect near the superconductor - insulator transition (SIT).93 ; 94
Due to vorticity a large Nernst signal is observed on both the superconducting
and the insulating sides with a signal peak close to the critical point.
However, the nature of the transport entropy transfered by vortices near SIT
is still under investigations.94

##  X Electrical Readout with Perfect Thermal Isolation for Quantum Sensors

Quantum currents, - magnetization, superconducting, persistent and topological
edge currents, - transfer the electric charge and spin, but not entropy. This
unique combination of thermal and electric/spin properties is valuable for
many applications related to quantum computing, networking, and sensing.

Resistive and kinetic-inductance micro and nanobolometers are used as direct
detectors, wide band mixers, single photon counters, and quantum colorimeters
(for a review see Ref. 95 ). Andreev mirrors formed at the interface of a
micro/nano-sensor and superconducting interconnectors are widely used to
prevent cooling of ultra-sensitive THz and IR hot-electron nanobolometers.96 ;
97  To achieve high sensitivity, the detector should be very small and
exceptionally well thermally isolated from the environment, including the heat
transfer to connectors related to the electric readout. Fast out-diffusion of
hot electrons from the nanoscale sensor leads to fast sensor cooling, which
reduces the detector sensitivity. Andreev mirrors eliminates cooling of hot
electrons in the sensor via interconnectors. Thermal processes at the
interface between a normal metal and a superconductor have been studied by A.
F. Andreev to describe the anomalous thermal resistance of a superconductor in
the intermediate state.98  Electron scattering from the interface involves
conversions between electron and hole Fermi liquid excitations to a Cooper
pair, which forms a superconducting condensate. As superconducting condensate
does not transfer entropy and. therefore, superconducting nanowires are ideal
interconnectors that provides electric coupling without thermal coupling.

Inter-connectors based on Quantum Hall edge states in topological insulators
may be employed to semiconductor sensors such as the hot electron bolometer
based on radiation-induced intersubband transitions in a high mobility quantum
wells at nitrogen temperatures.99  The sensitivity of this detector is
determined by the cooling time of hot electrons excited by the radiation from
the first subband and redistributed between between subbands due to inter-
electron interaction. Due to high mobility of electrons ($\displaystyle\sim
5\cdot 10^{3}$ cm2/ V s) the out-diffusion cooling becomes important at 10
$\displaystyle\mu$m length of the quantum well detector. Thus, the micron and
sub-micron length detectors require quantum mirrors such as quantum Hall edge
states to prevent the out-diffusion cooling.

Polarization-sensitive detectors have a potential to enhance sensing
technologies. Such detectors can employ quantum wells with spin polarized
states in subbands, which have different electron mobilities. So, the electron
transitions between spin-polarized states due to polarized radiation change
the device resistance, which is measured by a readout. To prevent out-
diffusion cooling of spin polarized detectors one can use quantum spin Hall
helical edge states protected by Kramers’ theorem.100  The quantum phase
coherence of wave functions in spin Hall helical edge states prevents the
heat/entropy transfer by these spin current inter-connectors.

##  XI Discussion

In this work we have investigated interplay of transport thermomagnetic
currents and quantum currents in charge, energy, and heat (entropy) fluxes. In
a small electric field or temperature gradient, quantum currents do not
generate and transfer intrinsic entropy. For example, magnetization energy
flux, $\displaystyle c{\bf M}\times{\bf E}$, does not contribute to the heat
(entropy) current. In Section II we support this rather obvious statement with
a textbook example of the thermomagnetic coefficient of noninteracting
electrons in quantized magnetic field. Even in this rather simple case, the
“magnetization heat current” $\displaystyle c{\bf M}\times{\bf E}$ would lead
to the giant Nernst coefficient. Thus, the transport heat current in crossed
electric and magnetic fields equals the bulk heat current without any
magnetization corrections.

The bulk electric current has the magnetization component,
$\displaystyle-c(d\bf{M}/dT)\times\nabla T$. According to Obraztsov54 , this
component should be removed from the the bulk electric current to obtained the
transport current (Eq. 26). As Onsager relation is valid for electric and heat
transport currents, the Onsager relation for the bulk thermomagnetic currents
is not symmetric with respect to magnetization. Alternatively, the transport
electric current in crossed magnetic field and temperature gradient may be
calculated as a response to entropic thermal forces induced by the temperature
gradient (Eq. 17). As the temperature-dependent magnetization does not
generate entropy and does not produce entropic thermal forces, the
magnetization contribution is absent here. Approach based on entropic forces
is widely used for the vortex thermomagnetic transport.49  In this case,
entropy flux and entropic forces directly provides heat and electric transport
currents.

Nonlocality of magnetization currents is well understood.42  Magnetization
currents are circulating and divergent free and, therefore, the net electric
magnetization current through any conductor cross-section is zero.
Cancellation of bulk and surface magnetization contributions in the net charge
and heat flows is a direct consequence of the Maxwell equations. It is
applicable to any geometry of the conductor including the Corbino disk
geometry. Thus, the net electric and heat thermomagnetic currents equal
transport electric and heat currents and satisfy the same Onsager relations as
transport currents.

Let us highlight, that zero entropy of magnetization currents is a direct
consequence of basic classical and quantum concepts. In thermodynamics, the
entropy is calculated as a temperature derivative of the free energy with
respect to temperature at constant magnetization (Eq. 9) and, therefore, the
entropy does not consists of terms proportional to $\displaystyle d\bf{M}/dT$,
i.e. to the magnetization current, $\displaystyle-c(d\bf{M}/dT)\times\nabla
T$. Also, the surface magnetization currents, $\displaystyle c{\bf
M}\times{\bf n}$ ($\displaystyle{\bf n}$ is the unit vector normal to the
surface), present the macroscopic motion of a group of electrons and,
according to thermodynamics (e.g. see §10 of Ref. 23 ), the kinetic energy and
electromagnetic energy related to this motion do not contribute to the
internal energy, free energy, and entropy. In particular, in the Corbino disk
geometry the magnetic filed perpendicular to the disk plane and the radial
electric field generate the azimuth bulk and edge magnetization electric
currents. In the rotating frame, these currents are absent and corresponding
entropy is absent. As entropy is an invariant with respect to rotation with
constant angular velocity,23  the magnetization currents have zero entropy in
the usual frame.

Quantum thermodynamics also gives zero entropy of magnetization currents
(quantum currents). In the resource theory of coherence101 , zero entropy is a
direct consequence of the rigidness of the wave function and may be derived in
a way similar to London’s approach to the wave function in a superconductor.
In the dynamical view of quantum thermodynamics based on the theory of open
quantum systems,102  the thermal forces affected electrons in thin films may
be introduced via the electron interaction with substrate phonons, which are
placed in the temperature gradient (the short phonon mean free path is
required to define a smooth temperature gradient.42 ; 52 ) As electrons in
quantum currents are not scattered by phonons, any drag effects of quantum
currents by phonons are absent and the entropic thermal forces are also
absent. Thus, both classical and quantum concepts lead to zero magnetization
entropy, zero magnetization heat flux in crossed electric and magnetic fields,
and zero thermal forces related to magnetization currents in crossed
temperature gradient and magnetic field. This description satisfies the
Onsager relation.

While the net magnetization energy current is always zero, its bulk and
surface counterparts in magnetic materials and metamaterials can be very
significant. To calculate the Poynting vector, the bulk magnetization energy
flux, $\displaystyle c{\bf M}\times{\bf E}$, should be added to the
electromagnetic energy flux, $\displaystyle c[{\bf E}\times{\bf H}]/4\pi$.
Thus, the Poynting vector is given by $\displaystyle c[{\bf E}\times({\bf
H}-4\pi{\bf M})]/4\pi$ (Eq. 33). We have shown that the term
$\displaystyle{\bf j}^{e}_{mag}\cdot{\bf E}$ previously associated with the
dissipated power (see Refs. 15 ; 39 ; 40 ) in fact, describes the energy
redistribution between the surface and bulk energy currents (see Fig. 3). The
new expression for the Poynting vector and dissipationless redistribution of
the magnetization energy are critically important for understanding of the
energy transfer in metamaterials.

Besides magnetization currents, we also investigate currents created by
fluctuating Cooper pairs above the transition. In strong magnetic fields,
$\displaystyle\omega_{c}\tau\gg 1$, the thermomagnetic transport is
collisionless and thermomagnetic coefficient is directly presented via the
entropy of electron system (Eq. 12). We obtained thermomagnetic coefficient of
fluctuating Cooper pairs for various relation between temperature and magnetic
field (Eq. 39). As all fluctuation corrections to the electron entropy are
always small (Eq. 38), the corresponding corrections to the thermomagnetic
coefficient are also small.

In general case with respect to the parameter $\displaystyle\omega_{c}\tau$,
many-body corrections to the thermomagnetic coefficient may be calculated by
the Kubo method. As magnetization currents do not contribute to the heat
current, the heat current - electric current correlator does not contain any
magnetization component and, therefore, the Kubo approach does not require any
magnetization correction. In Section VII, using the gauge invariant formalism
developed for the Hall coefficient in the fluctuating region,43  we have
calculated the thermomagnetic coefficient at small magnetic fields and show
that it scales as $\displaystyle\omega_{c}\tau\cdot T/\epsilon_{F}$ (Eq. 57),
i.e. as the normal state thermomagnetic coefficient that combines the Hall
angle $\displaystyle\omega_{c}\tau$ and the Fermi liquid $\displaystyle
T/\epsilon_{F}$ factor in the electron entropy.

Let us note that our microscopic calculations are based on entropic approach
and corresponding “$\displaystyle\epsilon$ -presentation”
(“$\displaystyle\omega$ -presentation”) of the heat/entropy flux, where
$\displaystyle\epsilon$ ($\displaystyle\omega$) is an argument in the fermion
(boson) distribution function. In this approach the thermal energy equals
$\displaystyle\epsilon$ ($\displaystyle\hbar\omega$), because in the linear
approximation the functional derivative of the entropy with respect to the
distribution function is $\displaystyle\epsilon/k_{B}T$
($\displaystyle\hbar\omega/k_{B}T$).13  The entropic approach directly defines
the thermal energy via statistics and does not require any corrections related
to magnetization energy and significantly simplify calculations. As it was
highlighted by Johnson and Girvin,106, by neglecting the magnetization energy
fluxes in the energy balance, one would obtain huge thermoelectric coefficient
in quantized magnetic field with the characteristic “thermal energy” of the
order of the magnetic energy $\displaystyle\hbar\omega_{c}$.

In Section VIII we develop the entropy - based theory of the thermomagnetic
transport in the Fermi liquid at arbitrary values of
$\displaystyle\omega_{c}\tau$. In the approximation of energy-independent
electron mobility we derive exact thermodynamic equation for the
thermomagnetic coefficient (Eq. 59), which corrects empiric formulas for the
Nernst coefficient obtained in recent works 34 ; 35  and 36 . We also
introduce the thermomagnetic entropy per unit charge as the entropy per
carrier averaged over the product of longitudinal and transverse mobilities
(Eq. 63) and show that the Nernst coefficient is proportional to the
difference between thermoelectric and thermomagnetic entropies per charge (Eq.
65). If the electron mobility (momentum relaxation time) is isotropic and
energy independent, both thermoelectric and thermomagnetic entropies per
charge are the same (Eqs. 61 and 63) and Eq. 65 gives the Sondheimer
cancellation in terms of the transport entropy. In the case of
$\displaystyle\omega_{c}\tau\gg 1$ the Nernst coefficient is also negligible,
because the thermoelectric entropy per charge equals the thermomagnetic
entropy. Specific many-body corrections to the entropy and mobility of Fermi
liquid quasiparticles may be incorporated in this approach.

In accordance with thermodynamics of irreversible processes, in the linear in
electric field (temperature gradient) approximation all non-entropic forces
are always canceled among themselves and, as we discussed above, contributions
of quantum currents to the net electric current are absent (see Section III).
However, in strongly nonequilibrium conditions, the non-entropic forces can be
unbalanced. In particular, the non-entropic thermal force that affect
superconducting electrons placed in the transient temperature gradient leads
to the vortex motion from the cold to hot area. In Section IX we explain
results of recent experiments with single vortices, which follow to the hot
spot created by laser beam.44  We also analyze thermoelectric transport in the
BKT state of 2D superconductors, such as thin disordered films and
superconducting heterostructures, and show that the transport entropy
transfered by a single vortex over the background of vortex - antivortex pairs
is given by the configuration entropy, $\displaystyle k_{B}\cdot\ln 2$ (Eq.
73). This result explains experimental data of the recent work82  and other
previous publications.

Quantum currents transfer the electric charge and spin, but not entropy. This
unique combination of thermal and electric properties of superconducting
currents is already used in many applications. Quantum Hall edge currents and
quantum spin Hall edge currents also do not transfer entropy and have a strong
potential to enhance sensing applications by preventing electron cooling in
nanosensors.

## Acknowledgements

We thank I. Aleiner, P. Ong, Y. Kopelevich, K. Behnia, N. Noginova, and B.
Karasik for valuable discussions.

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