


\documentclass[
    ,final            % use final for the camera ready runs
%%  ,draft            % use draft while you are working on the paper
%%  ,numberedheadings % uncomment this option for numbered sections
%%  ,                 % add further options here if necessary
  ]
  {aipproc}
\usepackage{amsmath,amssymb}

\layoutstyle{8x11double}





%\documentclass[aps,prl,twocolumn,superscriptaddress,showpacs]{revtex4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage[varg]{txfonts}

\def\Im{\mathop{\rm Im}}
\def\Re{\mathop{\rm Re}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\Landau{\alpha}
\def\fl{{\mathrm{fl}}}


\begin{document}

\title{Giant Nernst Effect due to Fluctuating Cooper Pairs in Superconductors}
\classification{74.40.+k, 74.25.Fy, 72.15.Jf}
\keywords{Nerst effect, supercondictors}
\author{M. N. Serbyn}{address=
{Landau Institute for Theoretical Physics, Chernogolovka, Moscow Region,
142432, Russia\\
Moscow Institute of Physics and Technology, Moscow 141700, Russia}}
\author{M. A. Skvortsov}{address=
{Landau Institute for Theoretical Physics, Chernogolovka, Moscow Region,
142432, Russia\\
Moscow Institute of Physics and Technology, Moscow 141700, Russia}}
\author{A. A. Varlamov}{address=
{COHERENTIA-INFM, CNR, Viale del Politecnico 1, I-00133 Rome, Italy}}
\author{V. Galitski}{address=
{Joint Quantum Institute and CNAM, Department of Physics, University of
Maryland, College Park, MD 20742-4111, USA}}

\begin{abstract}
A theory of the fluctuation-induced Nernst effect is developed for a
two-dimensional superconductor in a perpendicular magnetic field. First, we
derive a simple phenomenological formula for the Nernst coefficient, which
naturally explains the giant Nernst signal due to fluctuating Cooper pairs.
The latter signal is shown to be large even far from the transition and may
exceed by orders of magnitude the Fermi liquid terms. We also present a
complete microscopic calculation of the Nernst coefficient for arbitrary
magnetic fields and temperatures, which is based on the standard definition
of heat current vertices. It is shown that the magnitude and the behavior of
the Nernst signal observed experimentally in disordered superconducting
films can be well understood on the basis of superconducting fluctuation
theory.
\end{abstract}


\maketitle


%\date{June 27, 2008}


\section{Introduction}

A series of recent experimental studies have revealed an anomalously strong
thermomagnetic signal in the normal state of the high-temperature
superconductors~\cite{Ong1,Ong2,Ong3,hTc1,hTc3,Ong4,Ong5,Rick} and
disordered superconducting films~\cite{Aubin1,Aubin2}. In the pioneering
experiment \cite{Ong1}, Xu \emph{et al}.\ observed a sizeable Nernst effect
in the La$_{2-x}$Sr$_{x}$CuO$_{4}$ compounds up to $130$~K, well above the
transition temperature, $T_{c}$. This and further similar experiments on the
cuprates have sparked theoretical interest in the thermomagnetic phenomena.
Theoretical approaches to the anomalously large Nernst-Ettingshausen effect
currently include models based on the proximity to a quantum critical point~%
\cite{Sachdev}, vortex motion in the pseudogap phase~\cite%
{Ong2,Ashvin,Ashvin_Huse}, as well as a superconducting fluctuation scenario~%
\cite{Uss1,Uss2,Uss3}. While the two former theories are specific to the
cuprate superconductors, the latter scenario should apply to other more
conventional superconducting systems as well. Very recently, a large Nernst
coefficient was observed in the normal state of disordered superconducting
films~\cite{Aubin1,Aubin2}. These superconducting films are likely to be
well-described by the usual BCS model and, hence, the new experimental
measurements provide an indication that the superconducting fluctuations are
likely to be the key to understanding the underlying physics of the giant
thermomagnetic response.

Various groups have previously calculated the fluctuation-induced Nernst
coefficient in the vicinity of the classical transition~\cite%
{Dorsey,ReizerSergeev1994,Uss1,Uss2,Uss3}. However, these analyses were
limited to the case of very weak magnetic fields and temperatures close to
the zero-field transition, when Landau quantization of the fluctuating
Cooper pair motion can be neglected. In experiment, however, other parts of
the phase diagram (in particular strong fields) are obviously important and
how the quantized motion of fluctuating pairs would figure into the
thermomagnetic response has remained unclear. Here we clarify this
physics, explaining the origin of the giant fluctuation Nernst-Ettingshausen
effect, and develop a complete microscopic theory of Gaussian
superconducting fluctuations at arbitrary magnetic fields and temperatures
above the mean-field transition line \cite{SSVG}.


\section{Qualitative discussion}

We start with a qualitative discussion of the Nernst effect. Consider a
conductor in the presence of a magnetic field, $H_{z}$, and electric field,
$E_{y}$, directed along the $z$- and $y$-axes respectively. The charged
carriers (of charge $q$) subject to these crossed fields acquire a drift
velocity $\overline{v}_{x}=cE_{y}/H_{z}$ in the $x$-direction.
The latter would result in the appearance of a transverse drift
current $j_x^{(d)}=nq\overline{v}_x$.
When the circuit along $x$-axis is broken, no current flows, and the
drift of carriers is prevented by the spacial variation of the electric
potential: $\nabla _{x}\varphi =-E_{x}=(nqc/\sigma )(E_{y}/H_{z})$,
where $\sigma $ is the conductivity.
Due to electroneutrality, this generates the spacial gradient
of the chemical potential: $\nabla_x\mu(n,T)+q\nabla_x\varphi =0$,
which corresponds to the appearance of the transverse
temperature gradient $\nabla_x T=\nabla_x\mu (d\mu /dT)^{-1}$ along the
$x$-direction. Hence, the Nernst coefficient can be expressed
in terms of the temperature derivative of the chemical potential:
\be
  \nu_N
  \equiv
  \frac{E_y}{(-\nabla_xT)H_z}
  \sim
  \frac{\sigma }{nq^{2}c}
    \frac{d\mu}{dT}
  .
\label{nernstdef}
\ee

To verify Eq.~(\ref{nernstdef}),
consider, for example, a degenerate electron gas ($q=-|e|$).
In this case, the conductivity is given by the Drude formula,
$\sigma =ne^{2}\tau /m$, and the chemical potential $\mu
(T)=\mu _{0}-(\pi ^{2}T^{2}/6)(d\ln \nu /d\mu )$, where $\nu (\mu )$ is the
density of states. Therefore one easily reproduces the value of the Nernst
coefficient in a $3D$ normal metal: $\nu _{N}=\pi ^{2}T\tau /(6mcE_{F})$
\cite{Sondheimer1948,footnote}, where $\tau $ is the elastic scattering time
(here and below $\hbar =k_{B}=1$). Thus the Nernst effect in metals is small
due to the large value of the Fermi energy $E_{F}$.

We note that the regime of applicability of the qualitative Eq.~(\ref{nernstdef})
is expected to coincide with that of the quasiclassical Drude formula.
Equation~(\ref{nernstdef}) should be relevant to the
description of classical Aslamazov-Larkin fluctuations and provide a correct
order-of-magnitude estimate of the effect. As shown below, Eq.~(\ref{nernstdef})
does in fact remain quantitatively valid in a very wide
quasiclassical parameter regime. We note however that one should be careful
in applying Eq.~(\ref{nernstdef}) to quantitatively describe other systems,
especially those where quantum transport dominates.

The simple form of Eq.~(\ref{nernstdef}) suggests that in order to get a
large Nernst signal, \emph{a strong temperature dependence of the chemical
potential of carriers is required}. This simple and intuitive result alone
may shed light on the physics behind the strong Nernst signal often seen in
various superconductors. Indeed, a strong temperature-dependence of the
chemical potential can be achieved in the vicinity of the transition where
the fluctuating Cooper pairs appear side by side with the normal electrons.
These excitations are unstable, have the characteristic lifetime of order $%
\tau _{\mathrm{GL}}=\pi /8(T-T_{c})$, and form an interacting Bose gas with
a variable number of particles. In two dimensions, their concentration is $%
n_{\mathrm{c.p.}}^{(2)}(T)=(mT_{c}/\pi )\ln [T_{c}/(T-T_{c})]$ \cite{LV}.

Near transition, the chemical potential of the fluctuating pairs
can be found by identifying its value in the Bose distribution to
give $n_{\mathrm{c.p.}}^{(2)}(T)$ above. This leads to
$\mu_{\mathrm{c.p.}}^{(2)}(T) = T_{c}-T$. Since
$d\mu_{\mathrm{c.p.}}^{(2)}/dT=-1$, the fluctuation contribution
to the Nernst signal exceeds parametrically the
Fermi liquid term. In this sense it is similar to the fluctuation
diamagnetism (which also exceeds the Landau/Pauli terms and is
effectively a correction to the perfect diamagnetism of a
superconductor).

Based on the qualitative Eq.~(\ref{nernstdef}) and using the
known expression for paraconductivity in a magnetic field,
$\sigma_{\fl}=(e^2/2\epsilon)F(\epsilon/2\tilde h)$ \cite{LV}, one
can estimate the value of the Nernst coefficient due to fluctuating
Cooper pairs in the Ginzburg-Landau (GL) region:
\begin{gather}
\nu _{N}^{(2)}(T,H)\sim \frac{1}{mc}\frac{F(x)}{T-T_{c}}\sim
\begin{cases}
\lbrack mc(T-T_{c})]^{-1}, & x\gg 1, \\
(meDH)^{-1}, & x\ll 1,%
\end{cases}
\label{nernstmagn} \\
F(x)=x^{2}\left[ \psi \left( 1/2+x\right) -\psi \left( x\right) -1/(2x)%
\right] ,  \label{F(x)}
\end{gather}%
where $x=\epsilon /2\tilde{h}$, $\epsilon =\ln (T/T_{c})$ and $\tilde{h}=H/%
\widetilde{H}_{c2}(0)$ are the reduced temperature and magnetic field, $%
\widetilde{H}_{c2}(0)=4cT_{c}/\pi eD$ is the linearly extrapolated value of
the upper critical field, and $D$ is the diffusion coefficient.
The estimate (\ref{nernstmagn}) corresponds to the results
in the GL region~\cite{Uss1,Dorsey}.
We will see below that Eq.~(\ref{nernstmagn}) indeed provides
an order-of-magnitude estimate of the Nernst effect close to the
classical transition point, thus giving an additional justification
of qualitative arguments leading to Eq.~(\ref{nernstdef}).



\section{Microscopic calculation}

\subsection{General framework}

We now proceed with the microscopic calculation of the Nernst coefficient,
\be
  \nu_N(T,H) = R_\Box\beta^{xy}/H ,
\label{nu-via-beta}
\ee
where $R_\Box=1/\sigma_{xx}$,
and $\beta^{\alpha\beta}$ is the thermoelectric tensor
relating the transport heat current $\mathbf{j}^Q_\text{tr}$
to the applied electric field $\mathbf{E}$ in the absence
of a temperature gradient:
\be
  j^{Q\alpha}_\text{tr} = T\beta^{\alpha\beta}E^\beta .
\label{jQ}
\ee
In Eq.~(\ref{nu-via-beta}) we neglect the contribution of the
diagonal component, $\beta^{xx}$,
since it is small in the parameter $T/E_F$ even in the presence
of fluctuations \cite{LV}. Thus our aim will be to find the fluctuation
contribution to the off-diagonal component, $\beta^{xy}$.

First we recall a deep relation between
thermomagnetic effects and magnetization as emphasized by Obraztsov
\cite{Obraztsov} already in 1965 (later on his arguments have been widely
used in Refs.~\cite{diaNernst,CHR97,Uss1}):
In the presence of a magnetic field, the measurable
transport heat current $\mathbf{j}^Q_\text{tr}$ differs from the microscopic
heat current $\mathbf{j}^Q$ by the circular magnetization current
$\mathbf{j}^Q_M=c\mathbf{M}\times\mathbf{E}$, where $\mathbf{M}$ is the induced
magnetization. As a result, the thermoelectric tensor $\beta^{\alpha\beta}$
in Eq.~(\ref{jQ})
%relating $j^{Q\alpha}_\text{tr}=T\beta^{\alpha\beta}E^\beta$ with the
%applied electric field $\mathbf{E}$
can be found as a sum of the kinetic, $\tilde\beta^{\alpha\beta}$,
and thermodynamic, $\beta_M^{\alpha\beta}$, contributions:
\be
  \beta^{\alpha\beta}
  =
  \tilde\beta^{\alpha\beta} + \beta^{\alpha\beta}_M,
\qquad
  \beta^{\alpha\beta}_M = \epsilon^{\alpha\beta\gamma} c M^\gamma / T .
\label{betagen}
\ee

To calculate the kinetic term, $\tilde\beta^{\alpha\beta}$,
we employ the Matsubara-Kubo approach and express it via the thermal and
quantum mechanical averaging of the electric-heat currents correlator
$Q^{\alpha\beta}(\omega_\nu)
= \bigl\langle j^{e\alpha}(-\omega_\nu) j^{Q\beta}(\omega_\nu) \bigr\rangle$.
Then $\tilde\beta^{\alpha\beta}$ can be found by analytic
continuation from bosonic Matsubara frequencies, $\omega_\nu=2\pi T\nu$, to
real frequencies:
\be
  \tilde{\beta}^{\alpha\beta}
  =
  \frac{1}{T} \lim_{\omega \rightarrow 0}
  \mathop{\rm Im} \frac{Q^{\alpha\beta} (-i\omega+0)}{\omega}
  .
\ee
The thermodynamic term $\beta_M^{\alpha\beta}$ accounts
for the magnetization heat current $\mathbf{j}^Q_M$ \cite{Obraztsov}.
It is expressed in terms of the fluctuation magnetization, $M(T,H)$, which
has been calculated previously in the GL region~\cite{Kurkijarvi,Klemm,LV}
and at low temperatures, close to $H_{c2}(0)$~\cite{VGL}.

Our goal now is to evaluate the linear response operator $Q^{xy}(\omega_\nu)$
and analytically continue it to real frequencies.
%We do this diagrammatically in the imaginary-time (Matsubara)
%representation using the electric and heat vertices shown
%in Fig.~\ref{F:vertices}. For the heat current, which is not
%a well-defined quantity in the Hamiltonian formalism, we use
%the vertex $i(\varepsilon_l+\varepsilon_{l+\nu})\mathbf{v}/2$
%as suggested in Ref.~\cite{LV}.
%
We follow Ref.~\cite{VGL}
and perform calculations in the Landau basis, without expanding the Green
functions, propagators, current and heat vertices in the magnetic field.
This guarantees that gauge invariance is preserved and allows to access the
high-field regime. The fluctuation part of the correlator $%
Q^{xy}(\omega_\nu) $ is generally represented by ten diagrams\ \cite{VGL,LV}%
. However, in the case of the Nernst effect, the Maki-Thompson contribution
can be shown to be exactly zero and some of the DOS diagrams turn out to be
less singular: The graphs containing three cooperons (see Fig.~\ref{figa1})
are dominant. The positive Aslamazov-Larkin (AL) term dominates in the
classical Ginzburg-Landau (GL) region and competes with the negative
density-of-states (DOS) contribution everywhere else.
\begin{figure}[tbp]
\includegraphics[width=.9\columnwidth]{diagrams3.eps}
%\includegraphics[width=\columnwidth]{diagrams.eps}
%\vspace*{-2mm}
\caption{The Aslamazov-Larkin (AL) and density-of-states (DOS) diagrams for
the thermoelectric response $\protect{\tilde\beta^{xy}}$. The DOS diagram has
a symmetric counterpart. The white and black circles correspond to the
different heat and electric vertices, the shadowed blocks represent
cooperons, and the wavy lines denote the fluctuation propagator (see text).
All objects on these graphs are generally matrices in the Landau basis.}
\label{figa1}
\end{figure}
These AL and DOS contributions, and the fluctuation magnetization are given
by \cite{B-sym}
\begin{gather}
Q_{\text{AL}}^{xy}(\omega_\nu) = -4 \nu_H T \sum_{\Omega _{k}}\sum_{n,m}
\hat{q}_{mn}^{x} B_{nm}^{(e)} L_n(\Omega_k) \hat{q}_{nm}^{y} B_{nm}^{(Q)}
L_m(\Omega_{k+\nu}),  \label{AL} \\
2 Q_\text{DOS}^{xy}(\omega_\nu) = 4 \nu_H T \sum_{\Omega_k}\sum_{n,m}\hat{q}%
_{mn}^{x}\Sigma_{nm}^{(e,Q)} \hat{q}_{nm}^{y}L_{n}(\Omega_k),  \label{DOS} \\
M^z = - \frac{\partial}{\partial H} \nu_H T \sum_{\Omega_k}\sum_n \ln
L_n^{-1}(\Omega_k).  \label{Mag}
\end{gather}%
Here $L_n(\Omega_k)= -\nu^{-1} \left[ \ln(T/T_{c})+\psi _{n}(|\Omega
_{k}|)-\psi(1/2) \right]^{-1}$ is the fluctuation propagator, $%
\psi_n(\Omega) $ is a short-hand notation for $\psi[1/2+(\Omega
+\alpha_n)/4\pi T]$, with $\alpha_n=(4eDH/c)(n+1/2)$ being the Landau
spectrum, $\nu_H=eH/\pi c$, and the matrix elements of the momentum operator
in the Landau basis are given by $\hat{q}^{x,y}_{mn} =\sqrt{eH/c} \, {\binom{%
i}{1}} \bigl( \sqrt{m}\,\delta _{m,n+1}\mp \sqrt{n}\,\delta_{n,m+1}\bigr)$.
%(in the gauge $\mathbf{A}=(0,Hx,0)$)

\begin{figure}
\includegraphics[width=.9\columnwidth]{vertices.eps}
\caption{Electric-current (left) and heat-current (right) vertices
($\varepsilon_{l}$ and $\varepsilon_{l+\nu}$ are the fermionic
Matsubara energies).}
\label{F:vertices}
\end{figure}


\subsection{Electric and heat current blocks}

We now calculate exactly the three-Green-function blocks,
$B_{nm}^{(e)}(\Omega_k,\omega_\nu)$ and $B_{nm}^{(Q)}(\Omega_k,\omega_\nu)$,
with two cooperons and electric or heat vertices shown in Fig.~\ref{F:vertices}.
We note here that there exists a
long-standing controversy of the proper definition of a heat current and,
more generally, the applicability of Kubo-type linear response theory for
thermal transport. In our microscopic calculations, we do assume that the
latter holds and use the standard definition of the heat current vertex,
$i(\varepsilon_l+\varepsilon_{l+\nu})\mathbf{v}/2$ \cite{LV},
which is known to reproduce all known results,
e.g., in a Fermi gas and interacting
Fermi liquids. We also evaluate the six-Green-function block with three
cooperons and electric and heat vertices, $\Sigma_{nm}^{(e,Q)}(\Omega_k,%
\omega_\nu)$ and find ($\omega_\nu\geq0$):
\begin{multline}
B_{nm}^{(e)}(\Omega_k,\omega_\nu) = e \nu D \left[ \frac{\psi_m(\omega
_{\nu}+|\Omega _{k}|) - \psi_n(|\Omega_{k}|)} {\omega_\nu+\alpha_m-\alpha_n}
\right. \\
\left. {} + \frac{\psi_n(\omega _{\nu}+|\Omega _{k+\nu}|) -
\psi_m(|\Omega_{k+\nu}|)} {\omega_\nu-\alpha_m+\alpha_n} \right] ,
\label{Be}
\end{multline}
\vspace{-7mm}
\begin{multline}
B_{nm}^{(Q)}(\Omega_k,\omega_\nu) = \frac{-i\nu D}{2} \\
{} \times \left[ \frac{(\Omega_{k}-\alpha_m)\psi_m(|\Omega_k|+\omega_\nu) -
(\Omega_{k+\nu}-\alpha_n)\psi_n(|\Omega_k|)} {\omega_\nu+\alpha_m-\alpha_n}
\right. \\
\left. {} + \frac{(\Omega_{k+\nu}+\alpha_n)
\psi_n(|\Omega_{k+\nu}|+\omega_\nu) -
(\Omega_k+\alpha_m)\psi_m(|\Omega_{k+\nu}|) } {\omega_\nu+\alpha_n-\alpha_m} %
\right] ,  \label{BQ}
\end{multline}
\begin{multline}
\Sigma_{nm}^{\left(e,Q\right) }(\Omega_k,\omega_\nu) = - i e \nu {D}^2 \left[
\frac{\Omega_{k+\nu}-\alpha_n}{\omega_\nu+\alpha_m-\alpha_n}\psi_n^{\prime
}(|\Omega_k|) \right. \\
- \frac{\Omega_{k+\nu}+\alpha_n}{\omega_\nu-\alpha_m+\alpha_n}\psi_n^{\prime
}(|\Omega_{k+\nu}|+\omega_\nu) \\
-\frac{\Omega_k-\alpha_m}{(\omega_\nu+\alpha_m-\alpha_n)^2}%
\left(\psi_m(|\Omega_k|+\omega_\nu)-\psi_n(|\Omega_k|)\right) \\
\left.{} +\frac{\Omega_k+\alpha_m}{(\omega_\nu-\alpha_m+\alpha_n)^2}%
\left(\psi_n(|\Omega_{k+\nu}|+\omega_\nu)-\psi_m(|\Omega_{k+\nu}|)\right)%
\right] .  \label{Sigma}
\end{multline}


\subsection{Results in various asymptotic regions}

The general calculation of Eqs.~(\ref{AL})--(\ref{Mag}) is straightforward
but cumbersome. However, one can identify nine qualitatively different
regions of the phase diagram (Fig.~\ref{F:H-T-plane}), where the asymptotic
behavior has a simple analytical form. Before proceeding to the
corresponding details, we emphasize that the result for the Nernst
coefficient is universal in the sense that the function $\beta^{xy}(T,H)$
depends only on $T/T_c$ and $H/H_{c2}(0)$, but not the elastic scattering
time $\tau$ (unlike conductivity). This universality is due to the
magnetization contribution, $\beta_M^{xy}$, which regularizes the otherwise
divergent (and thus $\tau$-dependent) terms in $\tilde\beta^{xy}$. These
seemingly accidental cancellations between the two physically distinct terms
appear in a wide parameter range and are unlikely to be a coincidence (e.g.,
they may not occur if a different definition of heat vertices is used).
Therefore, these remarkable cancellations provide a strong evidence that the
standard definition of the heat vertices is indeed appropriate to describe
the effect.

\begin{figure}[b]
%\vspace{55mm}
%\includegraphics[width=.9\columnwidth]{figa2.eps}
\includegraphics[width=.9\columnwidth]{regimes2.eps}
\caption{Different asymptotic regions for the fluctuation
Nernst effect on the $(H,T)$ phase diagram. Landau quantization of the Cooper
pair motion is important in the regions II--VII and IX.}
\label{F:H-T-plane}
\end{figure}

\subsubsection{Ginzburg-Landau region}

We start by discussing \emph{the classical regime close to the critical
temperature $T_{c}$}: The regions I, II, III in Fig.~\ref{F:H-T-plane} are
characterized by $\epsilon =\ln (T/T_{c})\ll 1$ and $\tilde{h}=H/\widetilde{H%
}_{c2}(0)\ll 1$. In these domains, only the classical AL contribution is
important and is given by [cf.~Eq.~(\ref{nernstmagn})]: $\tilde{\beta}^{xy}
= 2\beta_0 F(x)/x$, where $x=\epsilon /2\tilde{h}$, $\beta_0=k_Be/\pi%
\hbar=6.68$ nA/K is the quantum of thermoelectric conductance, and the
function $F(x)$ is given by Eq.~(\ref{F(x)}). The magnetization
contribution to the observable $\beta^{xy}$ [see Eq.~(\ref{betagen})] is
given by
\begin{equation}
\beta_\text{M}^{xy} = \beta_0 \Big[\ln \frac{\Gamma (1/2+x)}{\sqrt{2\pi}}%
-x\,\psi (1/2+x)+x\Big] .  \label{beta-M-123}
\end{equation}

In the limit of vanishingly small magnetic fields $\tilde{h}\ll \epsilon$
(region I), we find $\tilde\beta^{xy}=\beta_0(\tilde{h}/2\epsilon)$, which
is two times larger than the result of Refs.~\cite{Uss1,Uss2,LV}
where the same microscopic Matsubara-Kubo approach has been used.
The additional factor is due to the complicated analytic structure of the
heat-current block (\ref{BQ}) overlooked in the previous diagrammatic
calculations \cite{Uss1,Uss2,LV}, but properly accounted for in
Ref.~\cite{ReizerSergeev1994}.
Note that our diagrammatic calculation also differs from by a
factor-of-two from the result of the phenomenological
Ginzburg-Landau approach~\cite{LV}.
This difference may
be related to a more fundamental issue (as compared to a calculational
mistake) and may signal, e.g. a problem with the definition of the heat
currents within time-dependent Ginzburg-Landau (TDGL) theory or/and diagrammatics.
The exact origin of the factor-of-two discrepancy remains unclear at this stage.

Thus in the region I, the magnetization contribution
$\beta_\text{M}^{xy} = -\beta_0(\tilde{h}/6\epsilon)$
cancels only 1/3 of $\tilde\beta^{xy}$, and the final result
appears to be four times larger than that of Refs.~\cite{Uss1,Uss2,LV}:
\be
  \beta_\text{I}^{xy}
  =
  \beta_0\frac{\tilde{h}}{3\epsilon }
  =
  \beta_0 \frac{\pi eDH}{12c(T-T_{c})},
\qquad
  \tilde{h}\ll \epsilon \ll 1.
\label{beta-I}
\ee
In the limit $\epsilon \ll \tilde{h}$ (region II), and close to the
transition line, at $\tilde{h}+\epsilon \ll \tilde{h}$ (region III), we find
\begin{gather}
\beta_\text{II}^{xy} = \beta_0 \left[ 1- (\ln 2)/2 \right] , \qquad \epsilon
\ll \tilde{h}\ll 1; \\
\beta_\text{III}^{xy} = \beta_0\frac{\tilde{h}}{\epsilon +\tilde{h}} =
\beta_0\frac{H_{c2}\left( T\right) }{H-H_{c2}\left( T\right) }, \qquad
\epsilon +\tilde{h}\ll \tilde h\ll 1.  \label{beta-above-Hc2(Tc)}
\end{gather}

\subsubsection{Low-temperature region}

Now we turn to the \emph{low-temperature regime close to the upper critical
field} $H_{c2}(0)=\pi cT_{c}/2\gamma eD$ (regions IV, V, VI in Fig.~\ref%
{F:H-T-plane}), where $\gamma=1.78\dots$ is the exponential of the Euler
constant.
Here role of magnetization term becomes crucial: The $1/T$
divergence of $\beta_\text{M}^{xy}=M^{z}/T$ exactly cancels the
divergent contribution originating from $\tilde\beta^{xy}$,
which is necessary to satisfy the third law of thermodynamics.
As a result, the total coefficient $\beta_\text{IV}^{xy}$ remains
finite in the limit or zero temperature.
The exact analytical expression at $t=T/T_{c}\ll 1$ and $\eta
=(H-H_{c2}(t))/H_{c2}(t)\ll 1$ is quite lengthy. We present below only the
asymptotic expressions in the regions IV, V, VI.

In the purely quantum limit of vanishing temperature and away from $%
H_{c2}(0) $ ($t\ll \eta$, region IV), $\beta _{xy}$ is negative:
\begin{equation}
\beta_\text{IV}^{xy} = - \frac{2\beta_0\gamma t}{9\eta} = - \frac{\beta_0
\pi c T}{9eD[H-H_{c2}(0)]}, \qquad t\ll \eta \ll 1.
\end{equation}
This change of sign in thermoelectric response is similar to negative
magnetoresistance in the quantum fluctuation transport for the usual
electrical conductivity found in Ref.~\cite{VGL} in the vicinity of $%
H_{c2}(0)$. %the magnetic-field-tuned quantum phase transition.
The sign change is due to the DOS contribution being numerically larger than
the positive AL term. In the quantum-to-classical crossover region, where $H$
tends to $H_{c2}(t)$ but remains limited as $t^{2}/\ln (1/t)\ll \eta \ll t$
(region V), the coefficient $\beta _{xy}$ is positive:
\begin{equation}
\beta_\text{V}^{xy} = \beta_0 \ln(t/\eta) ,\qquad t^{2}/\ln (1/t)\ll \eta
\ll t\ll 1.  \label{beta-log-Hc2(0)}
\end{equation}
Near $H_{c2}(t)$ ($\eta \ll t^{2}/\ln (1/t),$ region VI), we find:
\begin{equation}
\beta_\text{VI}^{xy} = 8 \beta_0 \gamma^{2}t^{2}/3\eta , \qquad \eta \ll
t^{2}/\ln (1/t) \ll 1.  \label{beta-above-Hc2(0)}
\end{equation}

\subsubsection{Above the transition line}

We also address the full classical region \emph{just above the transition
line}, which covers a wide range of temperatures and magnetic fields ($%
\eta\ll 1$, region VII). In this region, $\tilde{\beta}^{xy}$ is generally
comparable to $\beta_\text{M}^{xy}=-\beta_0/\eta$, and we obtain
\begin{equation}
\beta_\text{VII}^{xy} = \frac{\beta_0}{\eta } \left[ 1+\frac{h}{4\gamma t}%
\frac{\psi^{\prime \prime }(1/2+h/4\gamma t)}{\psi^{\prime }(1/2+h/4\gamma t)%
} \right] , \qquad \eta\to0 ,  \label{beta-above-Hc2(T)}
\end{equation}
with $h=H/H_{c2}(0)$. Close to $T_{c}$, Eq.~(\ref{beta-above-Hc2(T)})
matches Eq.~(\ref{beta-above-Hc2(Tc)}), while in the limit $T\rightarrow 0$
it matches Eq.~(\ref{beta-above-Hc2(0)}) provided that $\eta \ll t^{2}/\ln
(1/t)$. We note that in deriving Eq.~(\ref{beta-above-Hc2(T)}), the Landau
quantization of the Cooper pair motion was crucial.

\subsubsection{Far from the transition line}

Finally, we address the \textquotedblleft non-singular\textquotedblright\
\emph{regions VIII and IX far from the transition line}. In this limit, the
Kubo contribution $\tilde{\beta}^{xy}$ diverges as $[\ln \ln
(1/T_{c}\tau)-\ln \ln \max (h,t)]$, with $1/\tau$ playing the role of the
ultra-violet cutoff of the cooperon modes. Remarkably, the same divergence
of the opposite sign occurs in the magnetization contribution $\beta_\text{M}%
^{xy}$. Hence, the total expression for $\beta^{xy}$ remains finite:
\begin{gather}
\beta_\text{VIII}^{xy} = \beta_0 \frac{eDH}{6\pi c T\ln (T/T_{c})}, \qquad
(1,h)\ll t ;
\label{beta-high-T}
\\
\beta _{\text{\textrm{IX}}}^{xy} = \beta_0 \frac{\pi c T}{12eDH\ln
[H/H_{c2}(0)]}, \qquad (1,t)\ll h.
\label{beta-high-H}
\end{gather}%
We see that even far from the transition the fluctuation Nernst coefficient
can be comparable or parametrically larger than the Fermi liquid terms. In
fact, it is conceivable that in some materials the Cooper channel contribution to
thermal transport at low temperatures can be dominant even in the absence
of any superconducting transition at all (e.g., if superconductivity
is \textquotedblleft hidden\textquotedblright\ by another order).

\begin{figure}[tbp]
\includegraphics[width=.9\columnwidth]{NbSi125.eps}
\caption{Comparison with experiment. Circles: experimental
data for $\lim_{H\to0}\protect\beta^{xy}/H$ vs.\ $\protect\epsilon=\ln T/T_c$
obtained for the 12.5-nm-thick Nb$_{0.15}$Si$_{0.85}$ film~\protect\cite%
{Aubin1}. Dashed line: theoretical prediction for the strictly 2D geometry.
Solid line: theoretical prediction for the real sample \protect\cite{Aubin1}
with the 2D-3D crossover taken into account. }
\label{F:teor-and-exp}
\end{figure}


\subsection{Comparison with experiment}

Plotted in Fig.~\ref{F:teor-and-exp} is a comparison between our theory and
the experimentally measured Nernst coefficient~ \cite{Aubin1} for a Nb$%
_{0.15}$Si$_{0.85}$ film of thickness $d=12.5$ nm. The dashed line
corresponds to the coefficient $\lim_{H\to0}\beta^{xy}/H$ in a wide range of
temperatures up to $30\, T_c$. We used the diffusion coefficient $D=0.087$ cm%
$^2$/s which is 60\% of that reported in Ref.~\cite{Aubin1} (with $k_Fl\sim1$%
, the precise determination of $D$ is questionable). Note that far from the
transition point ($\epsilon>2$), the superconducting coherence length $%
\xi(T) $ becomes shorter than $d$ and 3D nature of diffusion manifests
itself. It can be described by substituting $\alpha_n\to\alpha_n+D(\pi p/d)^2
$ and performing an additional summation over $p=0,1,\dots$ in Eqs.~(\ref{AL}%
)--(\ref{Mag}). The resulting curve is shown in Fig.~\ref{F:teor-and-exp} by
the solid line.


\section{Conclusion}

In summary, we have developed a complete microscopic theory of the
fluctuation Nernst effect in a two-dimensional superconductor. Our results
provide a natural explanation for a large Nernst signal observed in
superconducting films~\cite{Aubin1,Aubin2} and probably should be relevant
to the cuprates. %high-temperature
%superconductors~\cite{Ong1,Ong2,Ong3,hTc1,.Tc2,hTc3,Ong4,Ong5,Rick},
%where the energy scale of the chemical potential of preformed Cooper pairs
%is set by the pseudogap temperature $T^*$.
%We identified nine qualitatively different regimes for thermal
%transport on the $H-T$ phase diagram (see Fig.~\ref{F:H-T-plane}).
Another interesting theoretical predictions is a slow decay of the
transverse thermoelectric response away from the transition line, which is
expected to persist well into the metallic phase.

Finally, we remark on a recent alternative approach \cite{Fin} to precisely
the same problem, where the Keldysh nonequilibrium technique has been used.
Technical details of the calculation~\cite{Fin} are not available
but their findings look pretty similar to our results: They
identify exactly the same asymptotic regions on the $(H,T)$ diagram,
with the asymptotic expressions coinciding with our
Eqs.~(\ref{beta-I})--(\ref{beta-high-H}) up to some numerical
factors of order one. In full analogy with our treatment,
in Ref.~\cite{Fin} the magnetization term $\beta_M^{xy}$ was also crucial
to regularize the otherwise divergent term $\tilde\beta^{xy}$
in the regions IV, VII, IX.
Contrary to our approach, Ref.~\cite{Fin} reproduces the TDGL result
in the AL region I, thus making a critical reexamination
of various approaches to the heat transport highly demanding.

\begin{theacknowledgments}
We are grateful to Herve Aubin, Mikhail Feigel'man and Alexei Kavokin for
useful discussions. M.N.S acknowledges partial support from Dynasty
Foundation and hospitality of the University Paris-Sud. V.G. acknowledges BU
visitors program's hospitality. The work of M.N.S. and M.A.S. was partially
supported by RFBR Grant No.\ 07-02-00310.
\end{theacknowledgments}
\bibitem{Ong1}
Z. A. Xu et al., \emph{Nature} \textbf {406}, 486 (2000).

\bibitem{Ong2}
Y. Wang et al., \emph{Phys. Rev. B} \textbf {64}, 224519 (2001).

\bibitem{Ong3}
Y. Wang et al., \emph{Phys. Rev. Lett.} \textbf {88}, 257003
  (2002).

\bibitem{hTc1}
C. Capan et al., \emph{Phys. Rev. Lett.} \textbf {88}, 056601
  (2002).

\bibitem{hTc3}
H. H. Wen et al., \emph{Europhys. Lett.} \textbf {63}, 583 (2003).

\bibitem{Ong4}
Z. A. Xu et al., \emph{Phys. Rev. B} \textbf {72}, 144527 (2005).

\bibitem{Rick}
P.~Li and
R.~L. Greene,
  \emph{Phys. Rev. B} \textbf P76}, 174512 (2007).

\bibitem{Ong5}
Y.~Wang,
  L.~Li and
  N.~P. Ong,
  \emph{Phys. Rev. B} \textbf{73}, 024510 (2006).

\bibitem{Aubin1}
A. Pourret et al., \emph{Nature Phys.} \textbf {2}, 683 (2006).

\bibitem{Aubin2}
A. Pourret et al., \emph{Phys. Rev. B} \textbf{76}, 214504
  (2007).

\bibitem{Sachdev}
S. A. Hartnoll et al., \emph{Phys. Rev. B} \textbf{76}, 144502
  (2007).

\bibitem{Ashvin_Huse}
S. Raghu et al., arXiv:0801.2925v2 (2008).

\bibitem{Ashvin}
D.}~Podolsky,
  S.~Raghu and
  A.~Vishwanath,
  \emph{Phys. Rev. Lett.} \textbf{99}, 117004 (2007).

\bibitem{Uss2}
I.~Ussishkin,
  \emph{Phys. Rev. B} \textbf{68}, 024517 (2003).

\bibitem{Uss3}
I.~Ussishkin and
  S.~L. Sondhi,
  \emph{Int. J. Mod. Phys. B} \textbf{18}, 3315 (2004).

\bibitem{Uss1}
I.~Ussishkin,
  S.~L. Sondhi and D.~A. Huse,
  \emph{Phys. Rev. Lett.} \textbf{89}, 287001 (2002).

\bibitem{ReizerSergeev1994}
M. Yu. Reizer and A. V. Sergeev,
  \emph{Phys. Rev. B} \textbf{50}, 9344 (1994).

\bibitem{Dorsey}
S.~Ullah and
  A.~T. Dorsey,
  \emph{Phys. Rev. Lett.} \textbf{65}, 2066 (1990); \emph{Phys. Rev. B}
  \textbf{44}, 262 (1991).

\bibitem{SSVG}
M.~N. Serbyn,
  M.~A. Skvortsov,
  A.~A. Varlamov
  and V.~Galitski,
  e-print arXiv:0806.4427.

\bibitem{footnote}
We assume the white-noise disorder, i.e.,
  $\tau(\epsilon)\nu(\epsilon)=\text{const}$.

\bibitem{Sondheimer1948}
E.~H. Sondheimer,
  \emph{Proc. R. Soc. A} \textbf{193}, 484 (1948).

\bibitem{LV}
A.~I. Larkin and
  A.~A. Varlamov,
  in ``The Physics of Superconductors'', Eds.
  K.~Bennemann and J.~Ketterson, Springer, 2002.

\bibitem{Obraztsov}
Y.~N. Obraztsov,
  \emph{Fiz. Tverd. Tela.} \textbf{7}, 573 (1965) [\emph{Sov. Phys.
  Solid State} \textbf{7}, 455-461 (1965)].

\bibitem{diaNernst}
L. Li et al., \emph{Europhys. Lett.} \textbf{72}, 451 (2005).

\bibitem{CHR97}
N.~R. Cooper,
  B.~I. Halperin,
  and I.~M. Ruzin,
  \emph{Phys. Rev. B} \textbf{55}, 2344 (1997).

\bibitem{Klemm}
R.~A. Klemm,
  M.~R. Beasley
  and A.~Luther,
  \emph{Phys. Rev. B} \textbf{8}, 5072 (1973).

\bibitem{Kurkijarvi}
J.~Kurkij\"arvi,
  V.~Ambegaokar
  and
  G.~Eilenberger,
  \emph{Phys. Rev. B} \textbf{5}, 868 (1972).

\bibitem{VGL}
V.~M. Galitski and
  A.~I. Larkin,
  \emph{Phys. Rev. B.} \textbf{63}, 174506 (2001).

\bibitem{B-sym}
The unusual subscript order in Eq.~(\ref{AL}) is related
  to the vertex symmetry property: $B^{(e)}_{mn}(\Omega_k,\omega_\nu) =
  B^{(e)}_{nm}(\Omega_k+\omega_\nu,-\omega_\nu)$.

\bibitem{Fin}
K.~Michaeli and
  A.~M. Finkel'stein,
  e-print arXiv:0812.4268.

\begin{thebibliography{99}

\end{thebibliography}
\end{document}
