Proposal for D0-brane anti-D0-brane Action

J. Klusoň 111Email addresses<EMAIL_ADDRESS>(J. Klusoň)

Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

We propose an action for the D0-brane anti-D0-brane system that has its exact
solution corresponding to the marginal tachyon profile defined for arbitrary
constant separation of these D-branes. We also find its covariant formulation
and generalization to higher dimensional Dp-branes.

##  1 Introduction and Summary

The dynamics of Dp-branes with opposite charges is one of the most interesting
processes in string theory. In case, when their separation is larger than
string scale, their instability can be interpreted as a consequence of the
gravity attraction together with the attractive force mediated by Ramond-
Ramond $p+1$-forms that arises between two branes with opposite charges. In
case when their separation is smaller than string length scale the instability
of this system can be interpreted as emergence of the tachyon field [1, 2].
However these modes become massive when the separation of Dp-branes is larger
than string scale. As was nicely argued in [11] the effective action should
depend on the brane separation. When this separation is larger than the string
scale the effective action should be in terms of massless closed string states
that propagate between these branes. On the other hand when their separation
is smaller than the string scale the more appropriate description of this
system should be in terms of the tachyon effective action that captures
dynamics when the separation is below string scale. The first proposal for the
action that describes dynamics of Dp-brane anti-Dp-brane system was given in
[3] that was mainly motivated by the effective actions for non-BPS Dp-branes
in string theory [4, 5, 6, 7]. Coincident Dp-brane anti-Dp-brane system was
also analysed with the help of boundary string field theory [8, 9]. On the
other hand when their separation is larger than string scale the Dp-brane and
anti Dp-brane are attracted by attractive force due to the gravity and RR
fields.

Very interesting proposal for Dp-brane anti-Dp-brane effective action was
given in [11] where the action for this system was derived by $(-1)^{F_{L}}$
projection of non-abelian action for two non-BPS Dp-branes [5] that was
generalization of non-abelian DBI action [10]. The resulting action contains
additional terms with respect to the action introduced in [3]. On the other
hand different approach for the study of Dp-brane anti-Dp-brane system at
finite distance was performed in [13]. It was argued in this paper that there
is a marginal tachyon deformation for all values of relative distance below
critical distance. This tachyon deformation corresponds to half-S-brane
solution in terminology of the time dependent tachyon deformation on the
world-volume of unstable Dp-brane [15, 16, 17, 14]. Then it was shown in [13]
that the effective action proposed in [11] cannot have the marginal tachyon
condensation for constant relative distance as its exact solution. On the
other hand leading order form of new tachyon effective action for Dp-brane
anti-Dp-brane system was suggested in [13] that has tachyon marginal profile
as exact solution for constant relative distance.

This suggested Lagrangian will be the starting point for our proposed
effective action for D0-brane anti-D0-brane system that is valid for any value
of the tachyon field. We determine the form of this Lagrangian using following
procedure. As the first step we consider tachyon effective action that has
similar structure as the Lagrangian given in [14] and which has half S-brane
tachyon condensation as its exact solution. Then we show that the marginal
tachyon profile that describes tachyon condensation on D0-brane anti-D0-brane
system at finite subcritical distance is solution of the equations of motion
that arises from this proposed naive form of the tachyon effective action. On
the other hand when we derive equations of motion for relative distance we
find that they cannot be solved by constant relative distance. For that reason
we modify the proposed naive form of the effective action when we add to it
term linear in the time derivative of relative distance. Then the tachyon
marginal profile at constant relative distance is still its exact solution.
However the presence of this new term will modify equations of motion for
relative distance when now the requirement that the marginal tachyon profile
together with the constant relative distance are solutions of equations of
motion determines the form of the new term. In other words we find an exact
form of the effective action for D0-brane anti-D0-brane system that correctly
describes its dynamics when their separation is smaller than the critical
distance. This action also reduces into the tachyon effective action for non-
BPS Dp-brane when we perform $(-1)^{F_{L}}$ projection which is nice
consistency check of our proposal.

Having derived the effective action for real tachyon and for the relative
distance we then determine its covariant form introducing coordinates that
parametrize transverse positions of D0-branes together with world-line gauge
field. As a result we derive full form of D0-brane anti-D0-brane system in the
flat target space background. Then it is easy task to generalize this form to
the case of Dp-brane anti-Dp-brane system.

Let us outline our results and suggest possible extension of this work. We
propose an action for D0-brane anti-D0-brane tachyon effective action that has
tachyon marginal profile at constant relative distance as its exact solution
at any value of the tachyon field and for the relative distance below the
critical distance. This is very important result that shows that tachyon
effective action can describe dynamics of such a complex system as D0-brane
anti-D0-brane is which is in agreement with the general definition of the
tachyon effective action. Then we generalized this action to the case of Dp-
brane anti-Dp-brane system. It is also important to stress that the action is
not unique. In fact, it would be very important to study this action further
and try to see whether it is related to the action proposed in [11]. It is
clear that they cannot be identified but rather they should be related by some
redefinition of the tachyon field in the similar way as the tachyon effective
action given in [14] is related to the DBI like tachyon effective action
suggested in [4, 5, 6, 7]. We hope to return to these problem in future.

This paper is organized as follows. In the next section (2) we review basic
facts about effective action for Dp-brane anti-Dp-brane system. Then in
section (3) we present construction of the new action for this system. Finally
in section (4) we determine covariant form of this action and present its
generalization to Dp-brane anti-Dp-brane system.

##  2 Review of Standard Formulation of $Dp-\overline{D}p$-Brane Action

In this section review basic facts about effective action that describes Dp-
brane anti-Dp-brane system. The first proposal for the action that describes
this system was given in [3] and it has the form

$S=-\int d^{p+1}\xi
V(T,X^{(1)}-X^{(2)})(\sqrt{-\det\mathbf{A}_{(1)}}+\sqrt{-\det\mathbf{A}_{(2)}})\
,$ (1)

where

$\displaystyle|$
$\displaystyle\mathbf{A}_{(i)\alpha\beta}=\eta_{\alpha\beta}+\partial_{\alpha}X^{(i)I}\partial_{\beta}X^{(i)J}\delta_{IJ}+\lambda
F_{\alpha\beta}^{(i)}+\lambda\frac{1}{2}((D_{\alpha}T)^{*}D_{\beta}T+\frac{1}{2}(D_{\beta}T)^{*}D_{\alpha}T)\
,$ $\displaystyle
F_{\alpha\beta}^{(i)}=\partial_{\alpha}A_{\beta}^{(i)}-\partial_{\beta}A_{\alpha}^{(i)}\
,\quad D_{\alpha}T=(\partial_{\alpha}-iA_{\alpha}^{(1)}+iA_{\alpha}^{(2)})T\
,$

and where the tachyon potential has the form

$V(T,X^{(1)}-X^{(2)})=T_{p}[1+\frac{1}{2}(\sum_{I}\frac{(X_{(1)}^{I}-X_{(2)}^{I})^{2}}{\lambda})T^{2}+O(T^{4})]\
,$ (3)

where $T_{p}$ is Dp-brane tension defined as

$T_{p}=\frac{2\pi}{(2\pi
l_{s})^{p+1}}=\frac{1}{2\pi^{\frac{2p+1}{2}}\alpha^{\prime\frac{p+1}{2}}}\
,\quad l_{s}^{2}=2\pi\alpha^{\prime}\equiv\lambda\ .$ (4)

Let us explain our notation and convention.
$\xi^{\alpha},\alpha,\beta=0,1,\dots,p$ label world-volume of Dp-branes where
we presume static gauge

$\xi^{\alpha}=X^{(1)\alpha}=X^{(2)\alpha}\ .$ (5)

Note that Dp-brane anti-Dp-brane pair are extended in flat Minkowski space-
time with the target space metric
$\eta_{\alpha\beta}=\mathrm{diag}(-1,1,\dots,9)$ where the transverse
positions of Dp-brane and anti-Dp-brane are labelled with coordinates
$X^{(i)I}$, where $i=1$ corresponds to Dp-brane and $i=2$ corresponds to anti-
Dp-brane and where $I=p+1,\dots,9$. Finally, $A_{\alpha}^{(1)}$ is the gauge
field living on the world-volume of Dp-brane and $A_{\alpha}^{(2)}$ living on
anti-Dp-brane.

The nice generalization of this action was presented in [11](see also [12])
where this action was derived from an effective action for two non-BPS Dp-
branes by projecting it with $(-1)^{F_{L}}$ where $F_{L}$ is space-time left
handed fermion number. In the flat background the resulting action has the
form

$S=-\int
d^{p+1}\xi\mathcal{V}(T,r)\left(\sqrt{-\det\mathbf{A}_{(1)}}+\sqrt{-\det\mathbf{A}_{(2)}}\right)\
,$ (6)

where

$\mathcal{V}(T,r)=V(|T|)\sqrt{1+\frac{T^{2}}{\lambda}r^{I}r_{I}}\ ,\quad
r^{I}=X^{(1)(I)}-X^{(2)I}\ ,$ (7)

and where $\mathbf{A}_{(i)}$ has more general form

$\displaystyle\mathbf{A}^{(i)}_{\alpha\beta}=\eta_{\alpha\beta}+\delta_{IJ}\partial_{\alpha}X^{(i)I}\partial_{\beta}X^{(i)J}-\frac{TT^{*}}{\lambda(1+\frac{TT^{*}}{\lambda}r^{I}r_{I})}\partial_{\alpha}X^{(i)I}\delta_{IJ}r^{J}r^{K}\delta_{KL}\partial_{\beta}X^{(i)L}+\lambda
F_{\alpha\beta}^{(i)}+$
$\displaystyle+\frac{1}{1+\frac{TT^{*}}{\lambda}r^{I}r_{I}}\frac{\lambda}{2}(D_{\alpha}T(D_{\beta}T)^{*}+D_{\beta}T(D_{\alpha}T)^{*})+$
$\displaystyle+\frac{i}{2}\partial_{\alpha}X^{(i)I}\delta_{IJ}r^{J}(T(D_{\beta}T)^{*}-T^{*}(D_{\beta}T))-\frac{i}{2}(T(D_{\alpha}T)^{*}-T^{*}D_{\alpha}T)r^{I}\delta_{IJ}\partial_{\beta}X^{(n)J}\
.$

Following [11] we can gain more physical insight into this action when
consider pure time dependent situation and presume that Dp-branes are
separated in the one direction only. We further write the tachyon field as
$T=\tau e^{i\rho}$ and also write $X^{(1)}=R+\frac{r}{2}\
,X^{(2)}=R-\frac{r}{2}$. Then it is easy to see that the effective action does
not depend on $R$ so that without lost of generality we take
$R=\mathrm{const}$. Then we have

$\displaystyle\mathbf{A}^{(1)}_{00}=\mathbf{A}_{00}^{(2)}=-1+\frac{1}{4}\dot{r}^{2}-\frac{\tau^{2}}{4\lambda(1+\frac{\tau^{2}}{\lambda}r^{2})}\dot{r}^{2}r^{2}+\frac{\lambda}{(1+\frac{\tau^{2}}{\lambda}r^{2})}(\dot{\tau}^{2}+\tau\dot{\rho}^{2})\
,$

so that the Lagrangian density takes the form

$\mathcal{L}=-2V(\tau^{2})\sqrt{1+\frac{1}{\lambda}\tau^{2}r^{2}-\frac{1}{4}\dot{r}^{2}-\lambda\tau^{2}-\lambda\tau^{2}\dot{\rho}^{2}}\
.$ (10)

The properties of this Lagrangian were very extensively studied in [11]. On
the other hand it was argued in [13] that this Lagrangian does not capture an
important exact solution of the tachyon condensation corresponding to the
profile

$\tau_{marg}=\tau_{0}\exp\frac{\omega}{\sqrt{\lambda}}\xi^{0}\ ,$ (11)

where $\omega$ is given by condition

$\omega^{2}=\frac{1}{2}-\frac{r^{2}}{\lambda}\ .$ (12)

In more details, let us consider the leading order form of Dp-brane anti-Dp-
brane Lagrangian that was proposed in [13]

$\mathcal{L}=-2+\sqrt{1-2\lambda^{-1}r^{2}}\left(\frac{\tau^{2}}{2}+\lambda\frac{\dot{\tau}^{2}}{1-2\lambda^{-1}r^{2}}\right)\
.$ (13)

The equations of motion for $r$ and $\tau$ that follow from this Lagrangian,
have the form

$\displaystyle-\frac{2r\lambda^{-1}}{\sqrt{1-2\lambda^{-1}r^{2}}}\left(\frac{\tau^{2}}{2}+\lambda\frac{\dot{\tau}^{2}}{1-2\lambda^{-1}r^{2}}\right)+\sqrt{1-2\lambda^{-1}r^{2}}\frac{4\lambda^{-1}r}{(1-2\lambda^{-1}r^{2})^{2}}\lambda\dot{\tau}^{2}=0\
,$
$\displaystyle\tau-2\lambda\frac{d}{d\xi^{0}}\left(\frac{\dot{\tau}}{1-2\lambda^{-1}r^{2}}\right)=0\
.$

Then using the fact that for $\tau_{marg}$ we have
$\dot{\tau}_{marg}=\frac{\omega}{\sqrt{\lambda}}\tau$ we find that two
equations given above are obeyed for $\tau_{marg}$ and for $r=\mathrm{const}$.

The goal of this paper is to find possible generalization of the Lagrangian
density (13) for any value of the tachyon field that has marginal solution as
its exact solution in the similar way how the tachyon effective action for
unstable Dp-brane has half S-brane solution as its exact solution. This was
nicely shown in [14] 222For related work, see for example [18, 19]. where it
was also shown that full S-brane tachyon profile is exact solution of
corresponding equations of motion. The construction of such an action for D0
anti-D0-brane system will be presented in the next section.

##  3 Proposal for D0-brane anti-D0-brane Action

In this section we propose form of D0-brane anti-D0-brane action.

We start our construction with the observation that the Lagrangian density
(13) can be written in the leading order as

$\displaystyle\mathcal{L}=-2T_{0}\left(1-\sqrt{1-2\lambda^{-1}r^{2}}\frac{\tau^{2}}{4}-\lambda\frac{\dot{\tau}^{2}}{2\sqrt{1-2\lambda^{-1}r^{2}}}\right)=$
$\displaystyle=-2T_{0}(1-\sqrt{1-2\lambda^{-1}r^{2}}\frac{\tau^{2}}{2})(1+\frac{\tau^{2}}{4}\sqrt{1-2\lambda^{-1}r^{2}}-\lambda\frac{\dot{\tau}^{2}}{2\sqrt{1-2\lambda^{-1}r^{2}}})\approx$
$\displaystyle\approx-2T_{0}\frac{1}{1+\sqrt{1-2\lambda^{-1}r^{2}}\frac{\tau^{2}}{2}}\sqrt{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}-\lambda\frac{\dot{\tau}^{2}}{\sqrt{1-2r^{2}}}}\
,$

where overall pre factor $2$ comes from the fact that this action describes
two D0-branes and in the last step we extrapolated the expression in the
second bracket to the square root structure. Then the last form of the action
for $r=0$ reduces to the tachyon effective action for D0-brane in Type IIB
theory [14]

$\mathcal{L}_{non}=-\frac{T_{0}^{non}}{1+\frac{\tau^{2}}{2}}\sqrt{1+\frac{\tau^{2}}{2}-\lambda\dot{\tau}^{2}}\
.$ (16)

where $T_{0}^{non}=\sqrt{2}T_{0}$. This is the first important sign how the
effective action for D0-brane anti-D0-brane system should have the form since
as was argued in [3] this action should reduce for $r=0$ to the action for
unstable D0-brane. In fact, the original unstable D0-brane was defined as
moding out D0-brane anti-D0-brane by $(-1)^{F_{L}}$ operation [2].

To proceed further we introduce $\mathbf{B}$ defined as

$\mathbf{B}=1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}-\lambda\frac{\dot{\tau}^{2}}{\sqrt{1-2\lambda^{-1}r^{2}}}$
(17)

that has the property that for $\tau=\exp\omega\frac{\xi^{0}}{\sqrt{\lambda}}$
it is equal to

$\mathbf{B}=1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}-\frac{\omega^{2}\tau^{2}}{\sqrt{1-2\lambda^{-1}r^{2}}}$
(18)

and this is equal to $1$ when $\omega$ is equal to

$\omega^{2}=\frac{1}{2}-\frac{r^{2}}{\lambda}\ .$ (19)

This is condition of marginality of the tachyon profile for constant $r$ [13].

Then we should ask the question whether the form of the Lagrangian density
given on the last line on (3) could be the right one. Explicitly, let us
denote this Lagrangian density as

$\mathcal{L}_{naiv}=-\frac{2T_{0}}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\sqrt{\mathbf{B}}\
.$ (20)

Now we should explicitly show that the equations of motion for $r$ and $\tau$
that follow from (20) have tachyon marginal profile $\tau_{marg}$ and
$r=\mathrm{const}$ as exact solutions. First of all the equation of motion for
$\tau$ that follow from (20) has the form

$\displaystyle-\frac{\tau\sqrt{1-2\lambda^{-1}r^{2}}}{(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}})^{2}}\sqrt{\mathbf{B}}+\frac{\sqrt{1-2\lambda^{-1}r^{2}}}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\frac{\tau}{2\sqrt{\mathbf{B}}}$
$\displaystyle+\frac{d}{d\xi^{0}}\left(\frac{1}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\frac{\lambda\dot{\tau}}{\sqrt{\mathbf{B}}\sqrt{1-2\lambda^{-1}r^{2}}}\right)=0\
.$

Then inserting $\tau_{marg}$ given in (11) into (3) we get

$\displaystyle-\frac{\tau}{(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}})^{2}}(\sqrt{1-2\lambda^{-1}r^{2}}+\omega^{2}\tau^{2})+\frac{\tau}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\sqrt{1-2\lambda^{-1}r^{2}}=0\
$

that shows that $\tau_{marg}$ solves the equation of motion which suggests
that (20) could be step in the right direction. To proceed further let us
study equation of motion for $r$ that follows from (20)

$\displaystyle-\frac{1}{(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}})^{2}}\frac{\tau^{2}r}{\sqrt{1-2\lambda^{-1}r^{2}}}\sqrt{\mathbf{B}}$
$\displaystyle+\frac{1}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\frac{1}{\sqrt{\mathbf{B}}}\frac{r}{\sqrt{1-2\lambda^{-1}r^{2}}}\left(\frac{\tau^{2}}{2}+\frac{\lambda\dot{\tau}^{2}}{(1-2\lambda^{-1}r^{2})}\right)=0\
,$

where we used

$\frac{\delta\sqrt{\mathbf{B}}}{\delta
r}=-\frac{1}{\sqrt{\mathbf{B}}}\frac{\lambda^{-1}r}{\sqrt{1-2\lambda^{-1}r^{2}}}\left(\frac{\tau^{2}}{2}+\frac{\lambda\dot{\tau}^{2}}{1-2\lambda^{-1}r^{2}}\right)\
.$ (24)

Inserting (11) into (3) we find that this equation is equal to

$\displaystyle\frac{r\tau^{4}}{2(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}})^{2}}$

that shows that the solution $r=\mathrm{const}$ and $\tau_{marg}$ are not
solutions of the equation of motion (3). In other words we should modify (20)
in such way that $\tau_{marg}$ and $r=\mathrm{const}$ will be solutions of the
equations of motion for $\tau$ and $r$. For that reason we add new term to
Lagrangian density and we demand that the presence of this new term does not
change equation of motion for $\tau$ in case when $r=\mathrm{const}$.
Explicitly, we add term linear in $\dot{r}$ to $\mathbf{B}$ so that

$\mathbf{B}=1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}-2f(\tau,r)\dot{r}r\dot{\tau}-\lambda\frac{1}{\sqrt{1-2\lambda^{-1}r^{2}}}\dot{\tau}^{2}\
$ (26)

that is again equal to $1$ for $\tau=\tau_{marg}$ and for $r=\mathrm{const}$.
Note that $f(\tau,r)$ is unknown function that will be determined by
requirement that $\tau_{marg}$ and $r=\mathrm{const}$ solve the equations of
motion for $r$. First of all we see that the new term in (26) does not
contribute to the equation of motion for $\tau$ since it is equal to zero for
$\dot{r}$. On the other hand it gives following contribution to the equation
of motion for $r$

$\displaystyle\frac{1}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\frac{1}{\sqrt{\mathbf{B}}}(\partial_{r}f(r,\tau)r+f(\tau,r))\dot{r}\dot{\tau}-\frac{d}{d\xi^{0}}\left(\frac{1}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}f(\tau,r)\frac{r\dot{\tau}}{\sqrt{\mathbf{B}}}\right)\
.$

Let us now evaluate this expression when $\tau=\tau_{marg}$ and when
$r=\mathrm{const}$. Then the first term in (3) is zero while the second one is
equal to

$\displaystyle\frac{d}{d\xi^{0}}\left(\frac{r\dot{\tau}}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}f(\tau,r)\right)=$
$\displaystyle=\frac{r\lambda^{-1}\omega^{2}\left[f(r,\tau)(\tau-\frac{\tau^{3}}{2}\sqrt{1-2\lambda^{-1}r^{2}})+\partial_{\tau}f(\tau^{2}+\frac{\tau^{4}}{2}\sqrt{1-2\lambda^{-1}r^{2}})\right]}{(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}})^{2}}\
.$

Then in order to obey equation of motion for $r$ we should now demand that (3)
is equal to

$\frac{r\tau^{4}}{2(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}})^{2}}$
(29)

that implies differential equation for $f$ in the form

$f(\tau-\frac{\tau^{3}}{2}\sqrt{1-2\lambda^{-1}r^{2}})+\partial_{\tau}f(\tau^{2}+\frac{\tau^{4}}{2}\sqrt{1-2\lambda^{-1}r^{2}})=\frac{\lambda}{2\omega^{2}}\tau^{4}\
.$ (30)

This equation can be easily solved with result

$\displaystyle
f=\frac{2\lambda}{(1-2\lambda^{-1}r^{2})^{2}}(1+\ln(1+\frac{1}{2}\tau^{2}\sqrt{1-2\lambda^{-1}r^{2}})(1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}))\
.$

This is central result of our analysis since we have shown that it is possible
to have exact form of the action for D0-brane anti-D0-brane system where the
marginal tachyon profile at constant relative distance is its exact solution.
The crucial point is the presence of the mixed term under square root that is
proportional to the function $f(\tau,r)$ given above. Of course, this form of
this function is not fixed uniquely since we can still add to it term which
depends on $r$ only. Further, this action is valid for the separation of
D0-brane anti-D0-brane below critical distance
$r_{c}=\frac{1}{\sqrt{2\lambda}}$. On the other hand there is nothing wrong
with this restriction since the tachyon profile $\tau_{marg}$ is marginal for
$r<r_{c}$ and hence it is natural to define the tachyon effective action for
$r<r_{c}$ only.

##  4 Covariant Form of the Action and Dp-Brane Generalization

In this section we generalize the form of the proposed action into its
covariant form. Recall that the action for D0-brane anti-D0-brane system when
restricted to the dynamics of relative separation and real tachyon has the
form

$\displaystyle S=\int d\xi^{0}\mathcal{L}\ ,$
$\displaystyle\mathcal{L}=-\frac{2T_{0}}{\sqrt{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}}\sqrt{1-2F(\tau,r)r\dot{\tau}\dot{r}-\frac{1}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\frac{\lambda\dot{\tau}^{2}}{\sqrt{1-2\lambda^{-1}r^{2}}}}\
,$

where we introduced function $F(\tau,r)$ defined as

$F(\tau,r)=\frac{f(\tau,r)}{1+\frac{\tau^{2}}{2}\sqrt{1-2\lambda^{-1}r^{2}}}\
.$ (33)

Let us introduce coordinates $X^{(i)I},I=1,\dots,9,i=1,2$ that label
transverse positions of D0-brane anti-D0-brane respectively. Further, we
generalize $r$ to $9-$dimensional vector $r^{I}$ defined

$r^{I}=X^{(1)I}-X^{(2)I}\ .$ (34)

It is important to stress that $\tau$ is real part of the complex tachyon $T$
that is defined as

$T=\tau e^{i\rho}\ ,$ (35)

Then it is natural to replace $\tau^{2}$ as

$\tau^{2}\rightarrow TT^{*}\ .$ (36)

We further generalize expression $2F(\tau)r\dot{\tau}\dot{r}$ into the form

$F(T)\dot{T}(X^{(1)I}-X^{(2)I})(\dot{X}^{(1)}_{I}-\dot{X}^{(2)}_{I})+F(T)^{*}\dot{T}^{*}(X^{(1)I}-X^{(2)I})(\dot{X}^{(1)}_{I}-\dot{X}^{(2)}_{I})\
,$ (37)

where $F(T)\equiv\tilde{F}(T^{2})\frac{1}{\tau}e^{-i\rho}\
,F^{*}(T)=\tilde{F}(T^{2})\frac{1}{\tau}e^{i\rho}$ so that

$\displaystyle
F(T)\dot{T}+F(T)^{*}\dot{T}^{*}=2\tilde{F}(T^{2})\frac{1}{\tau}\dot{\tau}\ .$

finally we replace $\dot{T}$ with covariant derivative

$\displaystyle\dot{T}\Rightarrow
D_{0}T\equiv\partial_{t}T-iA^{(1)}_{0}T+iA_{0}^{(2)}T\ ,$
$\displaystyle\dot{T}^{*}\Rightarrow(D_{0}T)^{*}=\partial_{t}T^{*}+iA^{(1)}_{0}T^{*}-iA_{0}^{(2)}T^{*}\
,$

where $A^{(1)}_{0}$ is the gauge field living on the world-line of D0-brane
while $A_{0}^{(2)}$ is the gauge field defined on the world-line of
anti-D0-brane.

As the final step we introduce kinetic term for $X^{(1)I}$ and $X^{(2)I}$
when, in order to have contact with the previous forms of the D0-brane
anti-D0-brane actions, we introduce these kinetic terms in each own square
root function. Then the final form of the action for D0-brane anti-D0-brane
system in the form

$\displaystyle\mathcal{L}_{D0+\overline{D0}}=-\frac{T_{0}}{\sqrt{1+\frac{TT^{*}}{2}\sqrt{1-2\lambda^{-1}(X^{(1)I}-X^{(2)I})(X^{(1)}_{I}-X^{(2)}_{I})}}}(\sqrt{\mathbf{A}_{(1)}}+\sqrt{\mathbf{A}_{(2)}})\
,$
$\displaystyle\mathbf{A}_{(i)}=1-\dot{X}^{(i)I}\dot{X}^{(i)}_{I}-F(T)D_{0}T(X^{(1)I}-X^{(2)I})(\dot{X}^{(1)}_{I}-\dot{X}^{(2)}_{I})-$
$\displaystyle-F^{*}(T)D^{*}_{0}T(X^{(1)I}-X^{(2)I})(\dot{X}^{(1)}_{I}-\dot{X}^{(2)}_{I})-$
$\displaystyle-\frac{\lambda}{1+\frac{TT^{*}}{2}\sqrt{1-2\lambda^{-1}(X^{(1)I}-X^{(2)I})(X^{(1)}_{I}-X^{(2)}_{I})}}D_{0}T(D_{0}T)^{*}\
.$

Note that this action is defined in the static gauge that by definition

$X^{0}_{(1)}=X^{0}_{(2)}\equiv t=\xi^{0}\ .$ (41)

Leaving the static gauge we replace $\sqrt{\mathbf{A}_{(i)}}$ with
$\sqrt{-\mathbf{A}_{00}^{(i)}}$ where $\mathbf{A}_{00}^{(i)}$ is defined as

$\displaystyle\mathbf{A}_{00}^{(i)}=\eta_{MN}\partial_{0}X^{(i)M}\partial_{0}X^{(i)N}+F(T)D_{0}T(X^{(1)M}-X^{(2)M})\eta_{MN}(\partial_{0}X^{(1)N}-\partial_{0}X^{(2)N})+$
$\displaystyle+F^{*}(T)D^{*}_{0}T(X^{(1)M}-X^{(2)M})\eta_{MN}(\partial_{0}X^{(1)N}-\partial_{0}X^{(2)N})-$
$\displaystyle-\frac{\lambda}{1+\frac{TT^{*}}{2}\sqrt{1-2\lambda^{-1}(X^{(1)M}-X^{(2)M})\eta_{MN}(X^{(1)N}-X^{(2)N})}}D_{0}T(D_{0}T)^{*}\
.$

Note that this form of the action can be straightforwardly extended to the
case of $p+1$ dimensional objects which are Dp-branes when we replace
$\mathbf{A}_{00}^{(i)}$ with $(p+1)\times(p+1)$ matrix
$\mathbf{A}_{\alpha\beta}^{(i)}$ so that the action has the form

$\displaystyle S=\int d^{p+1}\xi\mathcal{L}\ ,$
$\displaystyle\mathcal{L}=-\frac{T_{p}}{\sqrt{1+\frac{TT^{*}}{2}\sqrt{1-2\lambda^{-1}(X^{(1)M}-X^{(2)M})\eta_{MN}(X^{(1)N}-X^{(2)N})}}}\times$
$\displaystyle\times\left(\sqrt{-\det\mathbf{A}_{\alpha\beta}^{(1)}}+\sqrt{-\det\mathbf{A}_{\alpha\beta}^{(i)}}\right)\
,$
$\displaystyle\mathbf{A}_{\alpha\beta}^{(i)}=\eta_{MN}\partial_{\alpha}X^{(i)M}\partial_{\beta}X^{(i)N}+F(T)D_{\alpha}T(X^{(1)M}-X^{(2)M})\eta_{MN}(\partial_{\beta}X^{(1)N}-\partial_{\beta}X^{(2)N})+$
$\displaystyle+F^{*}(T)(D_{\alpha}T)^{*}(X^{(1)M}-X^{(2)M})\eta_{MN}(\partial_{\beta}X^{(1)N}-\partial_{\beta}X^{(2)N})+\lambda
F_{\alpha\beta}^{(i)}-$
$\displaystyle-\frac{\lambda}{1+\frac{TT^{*}}{2}\sqrt{1-2(X^{(1)M}-X^{(2)M})\eta_{MN}(X^{(1)N}-X^{(2)N})}}\frac{1}{2}(D_{\alpha}T(D_{\beta}T)^{*}+D_{\beta}T(D_{\alpha}T)^{*})\
.$

This is the final form of the action for Dp-brane anti-Dp-brane system. Note
that the matrix $\mathbf{A}_{\alpha\beta}$ contain terms with derivatives of
$X$ and $T$ that have similar structure as in the action derived in [11]. Then
one can ask the question whether there exists field redefinition that maps
these two tachyon effective actions each other. This problem is currently
under study.

Acknowledgement:
This work is supported by the grant “Integrable Deformations” (GA20-04800S)
from the Czech Science Foundation (GACR).

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##  Appendix A Telescope polarization correction

The aluminum coating of the M3 mirror in the VLT introduces instrumental
polarization, which is partly but not completely compensated for by the
following optical components. The details are described in Schmid et al.
(2018) with correction formulas and parameters for different filters. The
measured polarization for uncorrected cycles are expected to lie on a circle
in the $Q/I$-$U/I$ plane because the M3-mirror orientation of a Nasmyth
telescope rotates with respect to the sky during the observations. The radius
of the circle is defined by the telescope polarization $p_{\rm tel}$, and the
position angle $\theta_{\rm tel}$ depends on the parallactic angle
$\theta_{\rm para}(t)$ and a SPHERE-ZIMPOL specific position angle offset
$\delta_{\rm tel}$, which depends on the used passband filter. The midpoint
($q_{\rm m},u_{\rm m}$) of the circle can be offset from (0,0) because of an
intrinsic polarization of the star or interstellar polarization introduced by
dust between us and the target.

Figure 12: Measured $Q/I$ and $U/I$ parameters and correction for the temporal
evolution of the telescope polarization position angle due to the changing
parallactic angle during one polarimetric cycle. The blue circle sections show
the effect of the changing telescope polarization angle $\theta_{\rm tel}$
during the $Q^{+}$ (lightest blue), $Q^{-}$, $U^{+}$, and $U^{-}$ (darkest
blue) integrations. The required corrections for Stokes $Q/I$ and $U/I$ are
indicated by the orange and green arrows, respectively.  Figure 13: Measured
I′-band (blue dots) and R′-band (red dots) polarization for HD 169142 for the
individual, non-corrected, cycle. The best solutions for the corrected
polarization are shown with diamonds for the two bands. The thin circles
illustrate the derived telescope polarization $p_{\rm tel}$, and the colored
lines the derived position angle offset $\theta_{\rm tel}$ for the measured
cycles. Every fourth cycle number, as defined in Table 1, is annotated.

Therefore, the telescope polarization $p_{\rm tel}$ can be derived from the
systematic change of the polarization from cycle to cycle along this circle in
the $Q/I$-$U/I$ plane. Actually, a more accurate correction should even
consider the evolution of $\theta_{\rm para}$ during each polarimetric cycle,
which can be very fast for targets near zenith. In this case $\theta_{\rm
para}(t)$ differs strongly between the Stokes $Q$ and $U$ measurements
introducing a systematic bias effect in the polarization measurement and one
should use the following correction formulas for the four measurements
$Q^{+}$, $Q^{-}$, $U^{+}$, and $U^{-}$ of a polarimetric cycle:

$Q_{\rm cor}^{\pm}=\pm\left(Q-p_{tel}\frac{\int cos(2(\theta_{\rm
para}(t)+\delta_{\rm tel}))dt}{nDIT\cdot DIT}\ I\right),$ $U_{\rm
cor}^{\pm}=\pm\left(U-p_{tel}\frac{\int sin(2(\theta_{\rm para}(t)+\delta_{\rm
tel}))dt}{nDIT\cdot DIT}\ I\right).$

Figure 12 shows an extreme numerical example for a strong evolution of the
parallactic angle by $50^{\circ}$ during a long polarimetric cycle, resulting
in ($q_{\rm c},u_{\rm c}$) position of the cycle in the $Q/I$-$U/I$ plane. The
necessary polarization corrections shown in orange (Stokes $Q$) and green
(Stokes $U$), calculated from the formulas above, represent a time average
$Q/I$ value for the instrument polarization circle sections shown in light
blue for the $Q^{+}$ and $Q^{-}$ measurements and for the average $U/I$ value
for the dark blue $U^{+}$ and $U^{-}$ measurements. For this example using the
mean parallactic angle of the complete cycle for the U correction would lead
to a small correction ($+0.08\%$) instead of the value ($+0.38\%$) indicated
by the green arrow. Neglecting the temporal evolution effect of $\theta_{\rm
para}(t)$ can produce wrong results for the telescope polarization $p_{\rm
tel},\theta_{\rm tel}$ and therefore also the “corrected” target polarization
$(Q_{\rm cor}/I,U_{\rm cor}/I)=(q_{\rm m},u_{\rm m})$.

We analyzed our fast polarimetry cycles of HD 169142 (all cycles without
saturation) using the detailed formula above and searched for the telescope
polarization parameters $p_{\rm tel}$ and $\delta_{\rm tel}$, which minimize
the scatter for the resulting circle center $(Q_{\rm cor}/I,U_{\rm cor}/I)$.
Figure 13 shows the measured polarization $(Q/I,U/I)$ of each cycle for the I′
band and the R′ band and the median values for the corrected polarization
$(Q_{\rm cor}/I,U_{\rm cor}/I)$ = $(-(0.08\pm 0.03)\%,-(0.06\pm 0.03)\%$) for
the I′ band and $(-(0.05\pm 0.05)\%,-(0.05\pm 0.04)\%$) for the R′ band. Also
shown are the circles, which represent the telescope polarization parameters
$p_{\rm tel}=0.34\%$ and $\delta_{\rm tel}=44.4^{\circ}$ for the I′ band and
$p_{\rm tel}=0.59\%$ and $\delta_{\rm tel}=13.8^{\circ}$ for the R′ band. For
our data of HD 169142 the parallactic angle changes during one cycle are less
than 6 degrees and the detailed formulae given above yields almost the same
result like the approximate correction described in Schmid et al. (2018) using
just polarization values for the mean parallactic angle $\theta_{\rm
para}({\rm mean})$ for each cycle $Q_{\rm cor}\approx Q-p_{tel}\
\cos(2(\theta_{\rm para}({\rm mean})+\delta_{\rm tel}))\ I$ and $U_{\rm
cor}\approx U-p_{tel}\ \sin(2(\theta_{\rm para}({\rm mean})+\delta_{\rm
tel}))\ I$.

##  Appendix B PSF variations

Figure 14: I′-band PSF variations in the HD 169142 observations on July 9,
2015, and impact on the $Q_{\phi}$ simulation. The top row shows the best
cycle PSF (a) and the worst cycle PSF (b). Both are divided by the maximum of
the best cycle PSF. The bottom row illustrates the $Q_{\phi}$ model
(normalized to 1) given in Fig. 6 (c) convolved with the best PSF (d) and the
worst PSF (j).

The shape of the PSF depends strongly on the atmospheric turbulence and the AO
performance. As illustrated in Sect. 4.2.2 this has a strong impact on the
observed $Q_{\phi}$ signal, due to smearing and polarimetric cancelation. In
Fig. 14 we show the I′-band intensity images of HD 169142 with the highest and
lowest peak intensity, selected from $I_{\rm peak}^{n}$ given in Fig. 4. We
use these images as proxy for the best and the worst I′-band PSF and to assess
the effects of the convolution on $Q_{\phi}$. Figure 15 shows the
corresponding best and worst radial intensity profiles for HD 169142, and the
mean profile for all unsaturated (fast polarimetry mode) I′-band observations
from July 7, 2015. Additionally, we show as comparison the profiles of
standard stars for the $\rm N\\_I$ filter from Schmid et al. (2018) taken
under different observing conditions. The profiles are flux normalized and the
peak flux can be used as proxy for the Strehl ratio. According to Schmid et
al. (2018) we derive for the best profiles of HD 169142 an approximate Strehl
ratios of about 15 $\%$ for the I′ band and 10 $\%$ for the R′ band. It should
be noted that this kind of Strehl ratio is underestimating the AO-performance,
because it includes also stray light effects, detector smearing (e.g., from
the frame transfer), and other instrumental effects not related to the AO
system. The lower panel in Fig. 15 shows the encircled flux as cumulative
radial counts normalized to 1 inside an aperture with r=1.5′′ aperture as
illustration of PSF smearing. However, the logarithmic presentations can be
misleading and corresponding profiles on linear scales can be found in Schmid
et al. (2018).

(a) Flux normalized radial intensity profiles.

(b) Normalized encircled flux.

Figure 15: I′-band PSF variations in the HD 169142 observations from July 9,
2015, in comparison with other PSF profiles obtained with SPHERE/ZIMPOL ($\rm
N\\_I$ filter).


# Thermal Transport and Non-Mechanical Forces in Metals

J. Amarel1, D. Belitz1,2, and T.R. Kirkpatrick3 1 Department of Physics and
Institute for Fundamental Science, University of Oregon, Eugene, OR 97403, USA
2 Materials Science Institute, University of Oregon, Eugene, OR 97403, USA
3 Institute for Physical Science and Technology, University of Maryland,
College Park, MD 20742, USA

###### Abstract

We discuss contributions to the thermopower in an electron fluid. A simple
argument based on Newton’s second law with the pressure gradient as the force
suggests that the thermopower is given by a thermodynamic derivative, viz.,
the entropy per particle, rather than being an independent transport
coefficient. The resolution is the existence of an entropic force that results
from a coupling between the mass current and the heat current in the fluid. We
also discuss and clarify some aspects of a recent paper (Phys. Rev. B 102,
214306 (2020)) that provided a method for exactly solving electronic transport
equations in the low-temperature limit.

##  I Introduction

###  I.1 Transport coefficients

Consider the number current density ${\bm{j}}_{n}$ and the heat or entropy
current density ${\bm{j}}_{s}$ in a Fermi liquid without momentum
conservation. These currents are driven by gradients of the chemical (or
electrochemical) potential $\mu$ and the temperature $T$, and the relevant
transport coefficients $L_{ij}$ are defined by the linear-response
relationsCallen (1985); Mahan (1981)

$\displaystyle{\bm{j}}_{n}$ $\displaystyle=$
$\displaystyle\frac{-L_{11}}{T}\,{\bm{\nabla}}\mu-\frac{L_{12}}{T^{2}}\,{\bm{\nabla}}T\
,$ (1a) $\displaystyle{\bm{j}}_{s}$ $\displaystyle=$
$\displaystyle\frac{-L_{21}}{T}\,{\bm{\nabla}}\mu-\frac{L_{22}}{T^{2}}\,{\bm{\nabla}}T\
.$ (1b)

Of the four transport coefficients, only three are independent, since an
Onsager reciprocal relation requires $L_{21}=L_{12}$. The independent
coefficients are given by Kubo formulasKubo (1957); Mahan (1981) that describe
density-current–density-current correlations (for $L_{11}$), density-
current–heat-current correlations (for $L_{12}$), and heat-current–heat-
current correlations (for $L_{22}$), respectively. In a kinetic-theory
framework they take the form of different matrix elements of the inverse
collision operator.Dorfman et al. (2021); Kir  They determine the electrical
conductivity $\sigma$, the thermopower or Seebeck coefficient
${\mathfrak{S}}$, and the heat conductivity $\kappa$ via

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle e^{2}L_{11}/T\ ,$ (2a)
$\displaystyle-e{\mathfrak{S}}$ $\displaystyle=$ $\displaystyle
L_{12}/T\,L_{11}\ ,$ (2b) $\displaystyle\kappa$ $\displaystyle=$
$\displaystyle L_{22}/T^{2}\ ,$ (2c)

with $e$ the electron charge.

###  I.2 A simple argument for the thermopower

An elementary argument for the thermopower can be constructed as follows. The
electron mass current density is ${\bm{j}}_{m}=m{\bm{j}}_{n}$, with $m$ the
electron mass. Let $\tau$ be the momentum relaxation time, which can be due to
electron-impurity scattering, or electron-phonon scattering, or any
combination of scattering processes that do not conserve the electronic
momentum. Then the equation of motion for ${\bm{j}}_{m}$ is, by Newton’s
second law,

$\partial_{t}\,{\bm{j}}_{m}=\frac{-1}{\tau}\,{\bm{j}}_{m}+\frac{1}{V}\,{\bm{F}}\
,$ (3)

with $V$ the system volume and ${\bm{F}}$ the total force on the electron
system.cur  One contribution to the force density is the negative pressure
gradient

${\bm{F}}_{p}/V=-{\bm{\nabla}}p\ ,$ (4)

where $p$ is the electronic contribution to the pressure. Let us assume for
now that this is the only contribution to the force, as would be the case in a
purely mechanical description of the fluid. In steady state,
$\partial_{t}{\bm{j}}_{m}=0$, Eq. (3) then yields

$\displaystyle{\bm{j}}_{n}$ $\displaystyle=$
$\displaystyle-\frac{\tau}{m}\,{\bm{\nabla}}p$ (5) $\displaystyle=$
$\displaystyle-\frac{\tau}{m}\left(\frac{\partial
p}{\partial\mu}\right)_{T,V}{\bm{\nabla}}\mu-\frac{\tau}{m}\left(\frac{\partial
p}{\partial T}\right)_{\mu,V}{\bm{\nabla}}T$ $\displaystyle=$
$\displaystyle-\frac{n\tau}{m}\,{\bm{\nabla}}\mu-\frac{s\tau}{m}\,{\bm{\nabla}}T\
.$

Here $n=N/V$ and $s=S/V$ are the number density and entropy density,
respectively, and in going from the second line to the third one we have used
the fact that the pressure derivatives at constant volume are just derivatives
of the grand canonical potential $\Omega=-pV$. We note that the thermodynamics
of a Fermi liquid with a short-ranged interaction apply since the effective
force on the electrons is the unscreened external force.Pines and Nozières
(1989) Comparing with Eq. (1a) we find for the electrical conductivity the
Drude formula

$\sigma=ne^{2}\tau/m\ ,$ (6)

and for the thermopower

$-e{\mathfrak{S}}=s/n=S/N\ .$ (7)

That is, according to this simple argument the thermopower is given by a
thermodynamic derivative, namely, the electronic entropy per particle. This is
because $L_{11}$ and $L_{12}$ are given by thermodynamic derivatives
multiplying the same relaxation time $\tau$, and therefore their ratio is
simply a thermodynamic derivative. This is at odds with the fact $L_{11}$ and
$L_{12}$ are independent transport coefficients.

Since $L_{11}$ and $L_{12}$ both describe the response of the same current to
a driving force, there is only one relaxation time that can appear in Eq. (5).
The conclusion is therefore that there must be another contribution to the
force in Eq. (3) that has nothing to do with the pressure gradient. In the
next section we will use kinetic theory to elucidate the nature of this non-
mechanical or entropic force.

##  II Mass transport from kinetic theory

###  II.1 Linearized kinetic eqation

In order to derive a kinetic theory for the mass or number current density, we
recall the formalism developed in Refs. Bel, ; Kir, . Let

$f_{\text{eq}}({\bm{p}})=\frac{1}{e^{\xi_{p}/T}+1}$ (8)

be the equilibrium Fermi-Dirac distribution. Here $\xi_{p}=\epsilon_{p}-\mu$,
with $\mu$ the chemical potential and $\epsilon_{p}$ the equilibrium single-
particle energy. Here, and in what follows, ‘particle’ means quasiparticle in
the sense of Landau Fermi-liquid (LFL) theory.Lifshitz and Pitaevskii (1981);
Baym and Pethick (1991) For simplicity we consider spinless fermions, and we
keep only the first two LFL parameters, $F_{0}$ and $F_{1}$. Let
$f({\bm{p}},{\bm{x}},t)$ be the $\mu$-space or single-particle phase space
distribution function, consider small deviations from equilibrium,

$f({\bm{p}},{\bm{x}},t)=f_{\text{eq}}({\bm{p}})+\delta f({\bm{p}},{\bm{x}},t)\
,$ (9a) and write $\delta f$ as $\delta
f({\bm{p}},{\bm{x}},t)=w({\bm{p}})\,\phi({\bm{p}},{\bm{x}},t)\ ,$ (9b) with
$\displaystyle w({\bm{p}})$ $\displaystyle=$ $\displaystyle-\partial
f_{\text{eq}}({\bm{p}})/\partial{\epsilon_{p}}=\frac{1}{T}\,f_{\text{eq}}({\bm{p}})\left[1-f_{\text{eq}}({\bm{p}})\right]$
(9c) $\displaystyle=$ $\displaystyle\frac{1}{4T\cosh^{2}(\xi_{p}/2T)}\ .$

It is useful to define a scalar product in the space of ${\bm{p}}$-dependent
functions that employs $w$ as a weight:

$\langle
g({\bm{p}})|h({\bm{p}})\rangle=\frac{1}{V}\sum_{\bm{p}}w({\bm{p}})\,g({\bm{p}})\,h({\bm{p}})\
.$ (10)

In terms of this scalar product we can write density fluctuations as

$\delta n({\bm{x}},t)=\langle 1|\phi({\bm{x}},t)\rangle\ .$ (11)

Similarly, we can write velocity fluctuations as

$\delta{\bm{u}}({\bm{x}},t)=\frac{1}{nm}\langle{\bm{p}}|\phi({\bm{x}},t)\rangle$
(12a) or, equivalently, the mass current density as
${\bm{j}}_{m}({\bm{x}},t)=\langle{\bm{p}}|\phi({\bm{x}},t)\rangle\ .$ (12b)

Finally, temperature fluctuations, fluctuations of the entropy density
$s=S/V$, and pressure fluctuations can be written as (see Ref. Bel,  for
derivations)

$\displaystyle\delta T({\bm{x}},t)$ $\displaystyle=$
$\displaystyle\frac{1}{c_{V}}\langle
a_{5}({\bm{p}})|\phi({\bm{p}},{\bm{x}},t)\rangle\ ,$ (13a)
$\displaystyle\delta s({\bm{x}},t)$ $\displaystyle=$
$\displaystyle\langle\epsilon_{p}|\phi({\bm{p}},{\bm{x}},t)\rangle-\mu\,\delta
n({\bm{x}},t)\ ,$ (13b) $\displaystyle\delta p({\bm{x}},t)$ $\displaystyle=$
$\displaystyle\left(\frac{\partial p}{\partial T}\right)_{N,V}\delta
T({\bm{x}},t)+\left(\frac{\partial p}{\partial n}\right)_{T,V}\delta
n({\bm{x}},t)\ .$ (13c)

Here $c_{V}$ is the specific heat at constant volume, and

$a_{5}({\bm{p}})=\epsilon_{p}-\langle\epsilon_{p}|1\rangle/\langle 1|1\rangle\
.$ (14a) The functions $\displaystyle a_{1}({\bm{p}})$ $\displaystyle\equiv$
$\displaystyle 1\ ,$ (14b) $\displaystyle a_{2,3,4}({\bm{p}})$
$\displaystyle=$ $\displaystyle p_{x},\,p_{y},\,p_{z}$ (14c)

together with $a_{5}({\bm{p}})$ constitute the five hydrodynamic modes. They
are mutually orthogonal with respect to the scalar product defined in Eq.
(10), and their normalizations are given byBel

$\displaystyle\langle 1|1\rangle$ $\displaystyle=$ $\displaystyle(\partial
n/\partial\mu)_{T,V}(1+F_{0})\ ,$ (15a)
$\displaystyle\langle{\bm{p}}|{\bm{p}}\rangle$ $\displaystyle=$ $\displaystyle
3nm^{*}\ ,$ (15b) $\displaystyle\langle
a_{5}({\bm{p}})|a_{5}({\bm{p}})\rangle$ $\displaystyle=$ $\displaystyle
c_{V}T\ ,$ (15c)

with $m^{*}=m(1+F_{1}/3)$ the quasiparticle effective mass.

Performing a Fourier transform in space and time, with ${\bm{k}}$ the wave
number and $\omega$ the frequency, we can write the linearized kinetic
equation for the $\mu$-space distribution function in the formBel ; Kir

$\left[-i\omega-\Lambda({\bm{p}})+L_{\bm{k}}^{(1)}({\bm{p}})\right]|\phi({\bm{p}},{\bm{k}},\omega)\rangle=0\
.$ (16)

Here $\Lambda({\bm{p}})$ is the collision operator, and

$\displaystyle L_{\bm{k}}^{(1)}({\bm{p}})=i{\bm{k}}\cdot{\bm{v}}_{p}$
$\displaystyle+$ $\displaystyle\frac{F_{0}}{\langle
1|1\rangle}\,|i{\bm{k}}\cdot{\bm{v}}_{p}\rangle\langle 1|$ (17)
$\displaystyle+$
$\displaystyle\frac{F_{1}}{\langle{\bm{p}}|{\bm{p}}\rangle}\,|(i{\bm{k}}\cdot{\bm{v}}_{p}){\bm{p}}\rangle\cdot\langle{\bm{p}}|\qquad$

is a kinetic operator linear in ${\bm{k}}$ that comprises the streaming term
and the Fermi-liquid interaction. ${\bm{v}}_{p}={\bm{p}}/m^{*}$ is the
quasiparticle velocity. We are interested in a physical situation where
particle number and energy are conserved,

$\Lambda({\bm{p}})|1\rangle=\Lambda({\bm{p}})|a_{5}({\bm{p}})\rangle=0\ ,$
(18a) but momentum is not, $\Lambda({\bm{p}})|{\bm{p}}\rangle\neq 0\ .$ (18b)

###  II.2 Kinetic equation for the mass current

We are interested in a mass current driven by gradients of the chemical
potential, or the density, and the temperature. Accordingly, we want to derive
an effective theory that explicitly keeps the five hydrodynamic modes. To this
end, we define a projection operator

${\cal P}=\sum_{\alpha=1}^{5}|a_{\alpha}\rangle\frac{1}{\langle
a_{\alpha}|a_{\alpha}\rangle}\langle a_{\alpha}|$ (19a) that projects on the
hydrodynamic space ${\cal L}_{h}$ spanned by the hydrodynamic modes, and
another projection operator ${\cal P}_{\perp}=\mathbbm{1}-{\cal P}$ (19b)

that projects onto the space ${\cal L}_{\perp}$ that is orthogonal to ${\cal
L}_{h}$. Operating from the left with $\langle{\bm{p}}|{\cal P}$ on Eq. (16),
and using ${\cal P}+{\cal P}_{\perp}=\mathbbm{1}$ yields

$\displaystyle\left(-i\omega+1/\tau_{0}\right){\bm{j}}_{m}$ $\displaystyle=$
$\displaystyle-\langle{\bm{p}}|L_{\bm{k}}^{(1)}{\cal
P}|\phi\rangle+\langle{\bm{p}}|\Lambda{\cal P}_{\perp}|\phi\rangle$ (20a)
$\displaystyle-\langle{\bm{p}}|L_{\bm{k}}^{(1)}{\cal P}_{\perp}|\phi\rangle$
where
$\frac{1}{\tau_{0}}=\frac{-1}{\langle{\bm{p}}|{\bm{p}}\rangle}\,\langle{\bm{p}}|\Lambda({\bm{p}})|{\bm{p}}\rangle$
(20b)

is a bare relaxation rate for the mass current. Operating on Eq. (16) from the
left with ${\cal P}_{\perp}$, and again using ${\cal P}+{\cal
P}_{\perp}=\mathbbm{1}$, allows us to express ${\cal P}_{\perp}|\phi\rangle$
in terms of ${\bm{j}}_{m}$ and ${\cal P}|\phi\rangle$:

${\cal P}_{\perp}|\phi\rangle=G\,{\cal
P}_{\perp}\Lambda|{\bm{p}}\rangle\cdot\frac{1}{\langle{\bm{p}}|{\bm{p}}\rangle}\,{\bm{j}}_{m}-G\,{\cal
P}_{\perp}L_{\bm{k}}^{(1)}{\cal P}|\phi\rangle\ .$ (21)

Here we have defined a propagator

$G({\bm{p}},{\bm{k}},\omega)=\left(-i\omega-{\cal
P}_{\perp}\Lambda({\bm{p}}){\cal P}_{\perp}+{\cal
P}_{\perp}L_{\bm{k}}^{(1)}({\bm{p}}){\cal P}_{\perp}\right)^{-1}$ (22a) For
later reference we expand $G$ to linear order in the wave number ${\bm{k}}$:
$G=G_{0}-G_{0}{\cal P}_{\perp}L_{\bm{k}}^{(1)}{\cal
P}_{\perp}G_{0}+O({\bm{k}}^{2})\ ,$ (22b) where
$G_{0}({\bm{p}},\omega)=\left(-i\omega-\Lambda_{\perp}({\bm{p}})\right)^{-1}$
(22c)

and $\Lambda_{\perp}={\cal P}_{\perp}\Lambda{\cal P}_{\perp}$.

###  II.3 Analysis of the equation for the mass current

Consider the right-hand side of Eq. (20a). To zeroth order in a gradient
expansion the only contribution is from the second term with $G_{0}$
substituted for $G$ in Eq. (21). This term is proportional to ${\bm{j}}_{m}$,
and thus a contribution to the relaxation rate. To first order in a gradient
expansion, all three terms formally contribute. However, the part of the third
one that is formally of $O({\bm{k}})$ is proportional to ${\bm{j}}_{m}$, and
in steady state ($\omega=0$) the mass current itself is proportional to
${\bm{k}}$. The third therefore is effectively of $O({\bm{k}}^{2})$. The
equation for the mass current to linear order in the gradients thus reads

$-i\omega\,{\bm{j}}_{m}=\frac{-1}{\tau}\,{\bm{j}}_{m}+{\bm{f}}_{1}+{\bm{f}}_{2}\
,$ (23)

where

$\frac{1}{\tau}=\frac{1}{\tau_{0}}-\frac{1}{\langle{\bm{p}}|{\bm{p}}\rangle}\langle{\bm{p}}|\Lambda
G_{0}{\cal P}_{\perp}\Lambda|{\bm{p}}\rangle\ .$ (24)

Of the two force density terms, the first one is

$\displaystyle{\bm{f}}_{1}$ $\displaystyle=$
$\displaystyle-\langle{\bm{p}}|L_{\bm{k}}^{(1)}{\cal P}|\phi\rangle$ (25)
$\displaystyle=$ $\displaystyle-i{\bm{k}}\left[\frac{n}{(\partial
n/\partial\mu)_{T,V}}\delta n+\left(\frac{\partial p}{\partial
T}\right)_{N,V}\delta T\right]\ ,\qquad$

where we have used the expression (19a) for the projection operator and
various of the thermodynamic identities derived in Appendix A of Ref. Bel, .
Transforming back to real space and using general thermodynamic identities as
well as Eq. (13c) this can be written

${\bm{f}}_{1}({\bm{x}},t)=-{\bm{\nabla}}p({\bm{x}},t)\ .$ (26)

We see that ${\bm{f}}_{1}$ is the density of the mechanical or Newtonian force
${\bm{F}}_{p}$ from Eq. (4).

The second force density term is

$\displaystyle{\bm{f}}_{2}$ $\displaystyle=$
$\displaystyle-\langle{\bm{p}}|\Lambda G_{0}{\cal
P}_{\perp}L_{\bm{k}}^{(1)}{\cal P}|\phi\rangle$ (27a) $\displaystyle=$
$\displaystyle-\Big{\langle}{\bm{p}}|\Lambda\,\Lambda_{\perp}^{-1}|i({\bm{k}}\cdot{\bm{v}}_{p})\psi_{5}^{\text{L}(0)}\Big{\rangle}\frac{1}{T}\,\delta
T\ .$ Here $\displaystyle\psi_{5}^{\text{L}(0)}({\bm{p}})$ $\displaystyle=$
$\displaystyle a_{5}({\bm{p}})-\frac{T}{n}\left(\frac{\partial p}{\partial
T}\right)_{N,V}$ (27b) $\displaystyle=$ $\displaystyle\epsilon_{p}-(Ts/n+\mu)$

is the heat mode from Eq. (3.16) in Ref. Bel, , and the ket vector in Eq.
(27a) is the divergence of the heat current, see Appendix B in Ref. Bel, . The
inverse projected collision operator $\Lambda_{\perp}^{-1}$ acting on the heat
current is to be interpreted as follows. Let
$|x\rangle=\Lambda_{\perp}^{-1}|{\bm{v}}_{p}\psi_{5}^{\text{L}(0)}\rangle$.
Then $|x\rangle$ is the solution of the integral equation

$\Lambda_{\perp}|x\rangle=|{\bm{v}}_{p}\psi_{5}^{\text{L}(0)}\rangle,$ (28)

with the solution made unique by the requirement $|x\rangle\in{\cal
L}_{\perp}$.

Transforming back to real space we have

${\bm{f}}_{2}({\bm{x}},t)=\frac{-1}{3T}\Big{\langle}{\bm{p}}\Big{|}\Lambda({\bm{p}})\Lambda_{\perp}^{-1}({\bm{p}})\Big{|}{\bm{v}}_{p}\psi_{5}^{\text{L}(0)}({\bm{p}})\Big{\rangle}{\bm{\nabla}}T({\bm{x}},t)\
.$ (29)

Note that ${\bm{f}}_{2}$ is a pure temperature gradient and involves no
density gradient. The prefactor is a matrix element that involves a heat
current and a mass current. ${\bm{f}}_{2}$ thus results from the coupling
between the number density and the heat mode and represents a non-mechanical
or entropic force. It is generically nonzero, but vanishes for simple model
collision operators where
$\Lambda\Lambda_{\perp}^{-1}|{\bm{v}}_{p}\psi_{5}^{\text{L}(0)}\rangle\propto|{\bm{v}}_{p}\psi_{5}^{\text{L}(0)}\rangle$,
since the mass and heat currents are mutually orthogonal.

###  II.4 Contributions to the thermopower

By comparing the coefficients in Eqs. (1a) and (23), with ${\bm{f}}_{1}$ and
${\bm{f}}_{2}$ from Eqs. (26) and (29), respectively, we can determine the
Onsager coefficients $L_{11}$ and $L_{12}$ that determine the electrical
conductivity $\sigma$ and the thermopower $\mathfrak{S}$ according to Eqs.
(2). For the former we obtain the Drude formula (6), with $\tau$ given by Eq.
(24). For the latter we find

$-e{\mathfrak{S}}=\frac{S}{N}+\frac{1}{3nT\tau}\langle{\bm{p}}|\Lambda\Lambda_{\perp}^{-1}|{\bm{v}}_{p}\psi_{5}^{\text{L}(0)}\rangle\
,$ (30)

which is Eq. (7) augmented by a contribution from the entropic force.

##  III Examples

An evaluation of the entropic force for a given collision operator involves
solving the integral equation (28). This is equivalent to solving the
Boltzmann equation with the same collision operator. Alternatively, one can
employ the hydrodynamic theories developed in Refs. Bel, ; Kir, . In this
section we use two common scattering processes to illustrate how the entropic
force contributes to the thermopower.

###  III.1 Disordered Fermi liquid

As a simple example, consider the case of a Fermi liquid in the presence of
quenched disorder. In this case, particle number and energy are conserved, but
momentum is not, as we have assumed in Eqs. (18). Hydrodynamic equations for
this problem were derived in Ref. Kir, . Within this hydrodynamic formalism,
the result for the Onsager coefficients $L_{11}$ and $L_{12}$ is (see Eqs.
(3.22, 3.23) in Ref. Kir, )

$\displaystyle L_{11}$ $\displaystyle=$
$\displaystyle\frac{-T}{m^{2}}\,\langle{\hat{\bm{k}}}\cdot{\bm{p}}|\Lambda^{-1}({\bm{p}})|{\hat{\bm{k}}}\cdot{\bm{p}}\rangle\
,$ (31a) $\displaystyle L_{12}$ $\displaystyle=$
$\displaystyle\frac{-T}{m^{2}}\,\langle{\hat{\bm{k}}}\cdot{\bm{p}}|\Lambda^{-1}({\bm{p}})|({\hat{\bm{k}}}\cdot{\bm{p}})(\epsilon_{p}-\mu)\rangle\
.$ (31b)

Here the collision operator describes both electron-impurity scattering and
electron-electron scattering, and its inverse exists in this context since
none of the vectors in the matrix elements are conserved quantities. The two
coefficients are given by different matrix elements of the inverse collision
operator, and thus in general are independent. However, for the simplest
possible model of a constant relaxation rate,

$\Lambda=\frac{-1}{\tau}\,\left[\mathbbm{1}-\frac{|1\rangle\langle 1|}{\langle
1|1\rangle}-\frac{|a_{5}\rangle\langle a_{5}|}{\langle
a_{5}|a_{5}\rangle}\right]$ (32)

the rate drops out of the ratio $L_{12}/L_{11}$ and performing the integrals
yields

$-e{\mathfrak{S}}=\frac{\pi^{2}}{2}\,T/\epsilon_{\text{F}}\ ,$ (33)

which is equal to the entropy per particle of a Fermi liquid in the low-
temperature limit.Baym and Pethick (1991); pre  For this simple model
collision operator we thus recover the result (7) of the naive argument in
Sec. I.2. This is consistent with the analysis in Sec. II.3: With Eq. (32) for
the collision operator, the solution of Eq. (28) is
$|x\rangle=-\tau|{\bm{v}}_{p}\psi_{5}^{\text{L}}(0)\rangle$, and hence the
entropic force, Eq. (29), vanishes.

This is no longer true, even within a simple relaxation-time model, if we
allow for an energy dependence of the relaxation time. For instance, if we
replace $1/\tau$ in Eq. (32) byWilson (1954)

$\frac{1}{\tau({\epsilon_{p}})}=\frac{1}{\tau}\,\sqrt{\epsilon_{p}/\epsilon_{\text{F}}}$
(34)

and evaluate the integrals in Eqs. (31) we obtain Wilson’s resultWilson (1954)

$-e{\mathfrak{S}}=\frac{\pi^{2}}{3}\,T/\epsilon_{\text{F}}\ ,$ (35)

consistent with the fact that now the entropic force is no longer zero.

###  III.2 Electron-phonon scattering

As another example we consider the electron-phonon scattering problem with the
commonly used assumption that the phonons remain in equilibrium.Wilson (1954);
Mahan (1981) In Ref. Amarel et al., 2020 we provided an exact solution of
integral equations for transport coefficients based on the Boltzmann equation.
However, the equations solved were not quite equivalent to the Boltzmann
equation since in their derivation various factors of the electron momentum
$p$ were replaced by the Fermi momentum $p_{\text{F}}$. As a result of this
approximation the Onsager relation $L_{21}=L_{12}$ was violated, and the
results for the transport coefficients were different from what is obtained
from the Boltzmann equation proper. However, the solutions of the integral
equations as written were exact. Here we discuss the changes that result from
not making this approximation.

Within the formalism of Ref. Amarel et al., 2020 the Onsager coefficients
$L_{11}$ and $L_{12}$ are are given by

$\displaystyle L_{11}$ $\displaystyle=$
$\displaystyle\frac{nT}{m}\,\langle\nu^{3}|\varphi_{0}\rangle\ ,$ (36a)
$\displaystyle L_{12}$ $\displaystyle=$
$\displaystyle\frac{nT}{m}\,\langle\nu^{3}|\varphi_{1}\rangle\ ,$ (36b)

where

$\nu(\epsilon)=\sqrt{1+\epsilon/\epsilon_{\text{F}}}\ ,$ (37)

and the functions $\varphi_{0}$ and $\varphi_{1}$ are the solutions of
integrals equations

$\displaystyle\Lambda(\epsilon)\,\varphi_{0}(\epsilon)$ $\displaystyle=$
$\displaystyle-\nu(\epsilon)\ ,$ (38a)
$\displaystyle\Lambda(\epsilon)\,\varphi_{1}(\epsilon)$ $\displaystyle=$
$\displaystyle-\epsilon\,\nu(\epsilon)\ ,$ (38b)

with $\Lambda$ a collision operator. The factors $\nu(\epsilon)$ on the right-
hand sides of Eqs. (38) result from a factor $p_{\text{F}}/p$ that appears in
the angular integrations that reduce the linearized Boltzmann equation to a
one-dimensional integral equation, see Eqs. (C2a) and (C3) in Ref. Amarel et
al., 2020.not  The $\nu^{3}$ in Eqs. (36) result from one factor of
$p/p_{\text{F}}$ in the radial $p$-integration measure, and one factor of
$p/p_{\text{F}}$ from each of the two current vertices. All of these factors
were approximated by $\nu(\epsilon)\approx 1$ in Ref. Amarel et al., 2020.

The Onsager coefficients $L_{21}$ and $L_{22}$ can also be obtained from the
solutions of Eqs. (38):

$\displaystyle L_{21}$ $\displaystyle=$
$\displaystyle\frac{nT}{m}\,\langle\epsilon\,\nu^{3}|\varphi_{0}\rangle\ ,$
(39a) $\displaystyle L_{22}$ $\displaystyle=$
$\displaystyle\frac{nT^{2}}{m}\,\langle\epsilon\,\nu^{3}|\varphi_{1}\rangle\
.$ (39b)

The collision operator $\Lambda$ is given by

$\Lambda(\epsilon)=\int du\left[K(\epsilon,u)R_{\epsilon\to
u}-K_{0}(\epsilon,u)\right]$ (40)

with $R_{\epsilon\to u}\,f(\epsilon)=f(u)$ replacement operator. The kernel
$K$ has three contributions,

$K(\epsilon,u)=K_{0}(\epsilon,u)-K_{1}(\epsilon,u)-K_{2}(\epsilon,u)\ .$ (41)

$K_{0}$ and $K_{1}$ are given by Eqs. (2.18a) - (2.18c) in Ref. Amarel et al.,
2020. $K_{2}$ gets modified by a factor of $(p_{\text{F}}/p)^{2}$ in the last
line of Eq. (C3) in that reference that also had been approximated by $1$.
This leads to

$K_{2}(\epsilon,u)=\frac{1}{2}\,\left(1-\frac{T_{1}^{2}}{2\epsilon_{\text{F}}^{2}}\right)\,\left(\frac{u-\epsilon}{T_{1}}\right)^{2}K_{0}(\epsilon,u)\
,$ (42)

which replaces Eq. (2.18d) in Ref. Amarel et al., 2020. Here $T_{1}$ is the
bosonic energy scale that appears in the electron-phonon collision integral
and is on the order of the Debye temperature.Amarel et al. (2020)

The integral equations (38) can be solved exactly in the low-temperature limit
by the same method as in Ref. Amarel et al., 2020. The result is

$\displaystyle\sigma(T\to 0)$ $\displaystyle=$
$\displaystyle\frac{ne^{2}}{m}\,\frac{1}{120\,\zeta(5)g_{0}}\,\frac{1}{1-T_{1}^{2}/4\epsilon_{\text{F}}^{2}}\,\frac{T_{1}^{4}}{T^{5}}$
(43a) $\displaystyle\hskip 70.0pt+O(1/T^{3})\ .$
$\displaystyle-e\,\mathfrak{S}(T\to 0)$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{3}\,\frac{T}{\epsilon_{\text{F}}}+O(T^{3})\ .$
(43b) $\displaystyle\kappa(T\to 0)/T$ $\displaystyle=$
$\displaystyle\frac{n}{m}\frac{1}{g_{0}}\left(\eta+\frac{\pi^{4}/9}{120\zeta(5)}\,\frac{T_{1}^{2}/\epsilon_{\text{F}}^{2}}{1-T_{1}^{2}/4\epsilon_{\text{F}}^{2}}\right)\frac{T_{1}^{2}}{T^{3}}$
(43c) $\displaystyle\hskip 70.0pt+O(1/T)\ .$

Here $g_{0}$ is the electron-phonon coupling constant from Eq. (2.6) in Ref.
Amarel et al., 2020, and $\eta$ is the number from Eq. (3.39b) in that paper.
These results replace Eqs. (3.36a), (3.37), and (3.39a), respectively, in the
same reference. The Onsager relation $L_{21}=L_{12}$ is now satisfied, and the
result for the thermopower agrees with Wilson’s solution of the Boltzmann
equation.Wilson (1954) We emphasize that these results are exact solutions of
the Boltzmann equation in the low-temperature limit. The result for the
thermopower, Eq. (43b), is consistent with the fact that the collision
operator has a complicated energy dependence and hence the entropic force, Eq.
(29), does not vanish.

##  IV Summary and Conclusion

In summary, we have identified two physically different contributions to the
thermopower in a metal. One is due to the mechanical force on the electrons,
i.e., the gradient of the pressure of the Fermi liquid. The other one is an
entropic force that arises from the mass current coupling to the heat current.
This is analogous at some level to the contributions to the sound velocity in
either a classical fluidForster (1975) or a fermionic quantum fluid.Bel  A
purely mechanical theory would conclude that the speed of sound is given by
the isothermal compressibility of the fluid; it is the coupling to the heat
mode that changes this to the adiabatic compressibility. However, an important
difference is that the thermopower is a transport coefficient, whereas the
speed of sound is a thermodynamic derivative; the only question is which
derivative.

We also have clarified some aspects of Ref. Amarel et al., 2020, which gave a
method exactly solving electronic transport problems in the low-temperature
limit. Specifically, the integral equations solved exactly in that reference
were not quite equivalent to the Boltzmann equation due to some approximations
in the procedure that transforms the Boltzmann equation into a one-dimensional
integral equation. These approximations are not necessary, and eliminating
them leads to the exact solution of the Boltzmann equation proper that is
given in Sec. III.2.

## References

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  * (5) T. R. Kirkpatrick and D. Belitz, arXiv:2203.05521.
  * (6) For the scattering processes we will consider explicitly this is the only contribution to the current. For electrons in the presence of quenched disorder this is obviously true. For electron-phonon scattering in the common approximation that treats the phonons as remaining in equilibrium it also is true. More generally, the phonons will contribute to the momentum current, and the hydrodynamic description of the coupled electron-phonon problem will be more complicated.
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