# On the Behavior of Superconductors of High Critical Temperatures Outside
Schwarzchild Black Holes in AdS Space

D. R. Musk  Stanford University Online High School

###### Abstract

The physical analysis of condensed matter systems can be difficult due to
strong coupling, but the mathematical context of the AdS/CFT correspondence
enables non-perturbative descriptions in terms of dual weakly coupled systems.
This brief review explores the holographic condensed matter applications of
AdS/CFT, particularly through the lens of a high-$T_{c}$ superconductor
outside a Schwarzchild black hole in Anti-de Sitter space. A simple two-
dimensional electron condensate Lagrangian is examined first, as employed by
G. T. Horowitz, later used to calculate a frequency-dependent conductivity and
a free energy analysis; the asymptotics of both in this procedure, as examined
by P. Säterskog, are also reviewed. An extended Lagrangian with a higher order
Maxwell term is assessed thereafter, with a conductivity peak obtained at low
frequencies described well by the Drude model in certain limits. The behavior
of Drude model parameters in these limits is also investigated.

AdS/CFT correspondence, Condensed matter, Superconductor, Critical temperature

###### pacs:

74.25.-q, 04.62.+v, 04.70.-s, 11.25.Hf

††preprint: AIP/123-QED

##  I Introduction to the Correspondence in Classical Bulk Theory

The AdS/CFT correspondence was first conjectured in 1997 by Juan Maldacena,
relating the physics of a string theory in Anti-de Sitter space (AdS) to a
conformal field theory (CFT) on the boundary of a AdS space, a maximally
symmetric space of constant negative curvature. There is currently no proof of
the correspondence, although it has been extensively tested. The strength of
the duality is predominantly due to the bulk theory being weakly coupled when
the boundary theory is strongly coupled and vice versa. This lets us solve
otherwise computationally intractable problems on the strongly coupled side by
solving them on the weakly coupled side.

Examples of strongly coupled systems exhibiting critical behavior (that are
thereby also applications of the AdS/CFT correspondence) are quark-gluon
plasmaJanik and Peschanski (2006), high-$T_{c}$, high-$T_{c}$
superconductorsHartnoll (2009a), and possibly grapheneHartnoll (2009a). The
focus of the present article, however, is on that middle item: superconductors
with high critical temperatures. These types of superconductors are generally
layered and electrons within them move in two dimensions.

There is currently no widely accepted theory describing high-$T_{c}$
superconductors possibly due to the strong coupling that makes such a
theoretical understanding rather difficultHartnoll (2009a). These devices may
be in the vicinity of a quantum critical pointBroun (2008) and therefore
exhibit scale invariance, further motivating the use of a conformal field
theory.

Mathematically, the correspondence can be formulated through Hartnoll (2009b)

$\displaystyle
Z_{\text{bulk}}(\delta\psi_{(0)})=\braket{\exp(i\int{\text{d}^{d}x\sqrt{g_{0}}\delta\psi_{(0)}\mathcal{O})}}_{\text{CFT}},$
(1)

for dual function $\mathcal{O}$ and partition function for the bulk theory
with boundary condition $\psi_{(0)}$ at the conformal boundary
$Z_{bulk}(\psi_{(0)})$, with boundary background field $\psi_{(0)}$ as source
of the operator $\mathcal{O}$ such that

$\displaystyle\mathcal{O}=\frac{\delta S_{\text{CFT}}}{\delta\psi_{(0)}},$ (2)

with CFT action $S_{\text{CFT}}$. Overall, the expectation value of Equation 1
is then of a field theory at a temperature given by the Euclidean time-
periodicity of the path integral for the partition function.

The bulk theory becomes classical for a boundary gauge theory with a large
number of colors, a large $N$. This analysis will not involve a large-$N$
theory, but a similar effect is expected for certain strongly coupled boundary
theoriesMcGreevy (2010a), thus one can be assumed for this paper.

Given some source $J^{\alpha}$ for a vector field $A_{a}$ such that

$\displaystyle J^{\alpha}=\frac{\delta S_{\text{CFT}}}{\delta A_{(0)a}},$ (3)

we may determine CFT expectation values from the classical bulk

$\displaystyle\braket{J(x)}_{\text{CFT}}=\frac{\delta S_{c}(\psi_{0})}{\delta
A_{(0)a}(x)}|_{A_{(0)a=0}}.$ (4)

The functional derivative needed to calculate these expectation values is the
change in total on-shell action when the boundary value of the source field is
changed. This can in the case of the operator $\mathcal{O}$ and the source
$\psi$ be calculated through the following procedure:

  1. 1.

Denote all the fields in the bulk theory by $\psi_{i}$.

  2. 2.

The bulk action $S_{c}$ has two components, (1) a bulk Lagrangian density
$\mathcal{L}$ and (2) possibly a boundary term with a boundary density
$\mathcal{L}_{\text{bdy}}$.

$\displaystyle S_{c}=S_{\text{bdy}}+\int{\text{d}^{d+1}y\sqrt{g}\mathcal{L}}.$
(5)

  3. 3.

The functional derivative therefore becomes

$\displaystyle\frac{\delta
S_{c}(\psi_{(0)})}{\delta\psi_{(0)}(x)}|_{\psi_{(}0)=0}=$
$\displaystyle\int{\text{d}^{d+1}y\sqrt{g}(\frac{\partial\mathcal{L}(y)}{\partial\psi_{i}(y)}\frac{\partial\psi_{i}(y)}{\partial\psi_{(0)}(x)}}$
$\displaystyle+\frac{\partial\mathcal{L}(y)}{\partial(\nabla_{a}\psi_{i}(y))}\frac{\partial(\nabla_{a}\psi_{i}(y))}{\partial\psi_{(0)}(x)})$
$\displaystyle+\frac{\delta
S_{\text{bdy}}}{\delta\psi_{0}(x)}|_{\psi_{(0)}=0},$ (6)

where $i$ goes over all fields and summation is implied as in the Einstein
summation convention, with a bulk Lagrangian assumed to only depend on fields
and their first derivatives.

  4. 4.

Now integrate by parts such that

$\displaystyle\frac{\delta
S_{c}(\psi_{(0)})}{\delta\psi_{(0)}(x)}|_{\psi_{(0)}=0}$
$\displaystyle=\int\text{d}^{d+1}y\sqrt{g}(\frac{\partial\mathcal{L}(y)}{\partial\psi_{i}(y)}$
$\displaystyle-\nabla_{a}\frac{\partial\mathcal{L}(y)}{\partial(\nabla_{a}\psi_{i}(y))})\frac{\partial\psi_{i}(y)}{\partial\psi_{(0)}(x)}$
$\displaystyle+\int_{\partial\text{AdS}}\text{d}^{d}y\sqrt{g_{(0)}}n_{a}\frac{\partial\mathcal{L}(y)}{\partial(\nabla_{a}\psi_{i}(y))}\frac{\partial\psi_{i}(y)}{\partial\psi_{(0)}(x)}$
$\displaystyle+\frac{\delta
S_{\text{bdy}}}{\delta\psi_{(0)}(x)}|_{\psi_{(0)}=0},$ (7)

where $n_{a}$ is an outward normal to the boundary of AdS. The first integral
vanishes because the fields obey the Euler-Lagrange equation, and the CFT
expectation value can therefore be read off from the boundary behavior of the
bulk fields through this expanded relation, given Equation 2.

A CFT expectation value can then, via this process, be obtained from each of
the bulk fields once the boundary behavior of the on-shell bulk fields are
known. The relation

$\displaystyle\braket{\mathcal{O}}_{\text{CFT}}$
$\displaystyle=\int_{\partial\text{AdS}}\text{d}^{d}y\sqrt{g_{(0)}}n_{a}\frac{\partial\mathcal{L}(y)}{\partial(\nabla_{a}\psi_{i}(y))}\frac{\partial\psi_{i}(y)}{\partial\psi_{(0)}}$
$\displaystyle+\frac{\delta
S_{\text{bdy}}}{\delta\psi_{(0)}(x)}|_{\psi_{(0)}=0}$ (8)

will be the basis of much of the upcoming mathematical framework, as later
demonstrated.

In said framework, however, a strongly coupled boundary theory will be
considered. Conformal field theories are characterized by not having any
particular length scale and the physics of critical points, which include
thermodynamic and quantum phase transitions, often has this property. The
physics near a critical point can be expected as similar to the critical
system and finding the critical behavior is then of interest.

##  II Mathematical Approach Overview

The AdS/CFT correspondence will be used to model a high-$T_{c}$
superconductor, both below and above $T_{c}$. This will first be demonstrated
through the simplest possible Lagrangian, with a determined frequency-
dependent conductivity. The Lagrangian is then extended with a higher-order
term and outlying behavior will be identified. Finally, the Drude model will
be used to further describe and explain outlying trends in conductivity.

##  III Two-Dimensional Condensed Matter Systems

We are to model a superconductor with a high $T_{c}$. While conventional
superconductors are well-described by the BCS (Bardeen–Cooper–Schrieffer)
theory, in which electrons, photons, and phonons are the degrees of freedom of
interest. The importance of phonon interactions was understood from the
isotope effect, the mass of the atoms in the lattice changing the
superconductivity behavior. The isotope effect is much weaker, however in
high-temperature superconductors and the phonons are thus not believed to be
important for high temperature superconductivityLeggett (2006). The important
degrees of freedoms are the electrons and photons. These electrons are, as in
BCS theory, expected to form Cooper pairsLeggett (2006), pairs of electrons of
opposite spin but otherwise effectively in the same state becoming spin 0
particles.The methodology presented here is adapted from the holographic
methods review by P. Säterskog, who computed the numerics shown throughout
this section using a Prince-Dormand methodSäterskog (2013), to which end our
superconductor model will thus contain two fields: a spin 1 field $A_{a}$ for
the photons and a spin 0 field $\psi$ for the Cooper-pairs.

###  III.1 Symmetry assumptions

The bulk theory should present the same symmetries as the boundary theory. A
U(1) gauge symmetry of the complex $\psi$ field will therefore be imposed.
Lorentz invariance will also be employed for both theories, even though
relativistic phenomena are largely negligible for superconductivity.

###  III.2 A bulk Lagrangian

There are many different possible bulk Lagrangians for the fields $A_{a}$ and
$\psi$ and the metric $g_{ab}$. A Lagrangian previously successfully used to
model two-dimensional electron condensates,

$\displaystyle\mathcal{L}=\frac{1}{2\kappa}(R-2\Lambda)-\frac{1}{4}F_{ab}F^{ab}-m^{2}|\psi|^{2}-|D_{a}\psi|^{2},$
(9)

having been obtained through Wilsonian naturalnessHartnoll, Herzog, and
Horowitz (2008); Horowitz and Santos (2013). Of Equation 9, the first term is
an Einstein-Hilbert term with cosmological constant $\Lambda$, with a negative
constant giving the required asymptotically AdS space, the Ricci scalar
curvature $R$, and coupling constant between metric and other fields $\kappa$;
the second term is an ordinary Maxwell term where electromagnetic tensor
$F_{ab}$ is the exterior derivative of the electromagnetic field tensor
$F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}$; and the third and fourth terms
are the kinetic and mass terms for the scalar field, respectively, with
covariant derivative $\nabla_{a}$ and gauge covariant derivative
$D_{a}=\nabla_{a}-iqA_{a}$. The action of this Lagrangian is

$\displaystyle
S=\int{\text{d}^{d+1}x\sqrt{g}\mathcal{L}+S_{\text{boundary}}},$ (10)

with $g$ as the absolute value of the determinant of the metric tensor
$g=|\text{det}(g_{ab})|$. $S_{\text{boundary}}$ is a boundary term needed to
cancel divergences when integrating the action towards the boundary. It does
not significantly affect equations of motion but is needed to get
normalizeable modes.

In accordance with the previous symmetry assumptions, the described gauge
coupling makes the Lagrangian invariant under the U(1) gauge transformation

$\displaystyle\psi$ $\displaystyle\rightarrow e^{i\theta(x)}\psi$ (11)
$\displaystyle A_{a}$ $\displaystyle\rightarrow
A_{a}+\frac{1}{q}\nabla_{a}\theta(x).$ (12)

This Lagrangian is also manifestly Lorentz invariant, imposing the Lorentz
invariance of the boundary theory.

###  III.3 Equations of motion

The bulk equations of motion are obtained by varying the bulk Lagrangian with
respect to all fields, a procedure possible through Noether’s theorem. This
can be done with the Euler-Lagrange equation since the action does not contain
any higher derivatives. The Euler-Lagrange equation for a scalar field $\chi$
states that

$\displaystyle\nabla_{a}(\frac{\partial\mathcal{L}}{\partial(\nabla_{a}\chi)})-\frac{\partial\mathcal{L}}{\partial\chi}=0.$
(13)

We first vary $\psi$, giving

$\displaystyle(m^{2}-\nabla^{2}+q^{2}A^{2}+iq(\nabla_{a}A^{a}))\psi=0.$ (14)

Varying $A_{a}$ gives the equations of motion

$\displaystyle-\nabla_{a}F^{ab}+2q^{2}|\psi|^{2}A^{b}+iq(\overline{\psi}\nabla^{b}\psi-\psi\nabla^{b}\overline{\psi})=0.$
(15)

A real $\psi$ simplifies calculations and can be obtained since the gauge
invariance lets us relate any configuration to a real one through a gauge
transformation. The Lorentz gauge

$\displaystyle\nabla_{a}A^{a}=0,$ (16)

removes the last term in the parentheses of the equation of motion for $\psi$.
The equation of motion for $\psi$ does not mix the real and imaginary parts
after this choice and $\psi$ can then be taken to be real since a global shift
of phase does not affect $A_{a}$. The gauge is still not completely fixed, a
gauge transformation $\theta(x)$ such that $\nabla_{a}\nabla^{a}\theta(x)=0$
can still be done without violating the gauge condition of Equation 16. After
choosing the Lorentz gauge and a real $\psi$, the equations of motion are

$\displaystyle\begin{split}(m^{2}-\nabla^{2}+q^{2}A^{2})\psi=0,\\\
-\nabla_{a}F^{ab}+2q^{2}\psi^{2}A^{b}=0.\end{split}$ (17)

###  III.4 Lagrangian Parameters

As seen, there are four free parameters in the bulk Lagrangian: $\kappa$,
$\Lambda$, $m^{2}$, and $q$. These must be investigated to find values that
give us the boundary theory we are interested in. Some of these parameters may
be redundant since we can make different symmetry transformations of fields
and coordinates. The physics of the bulk are treated in the classical limit
and the Lagrangian can thus be changed as long as the equations of motion for
$\psi$ and $A_{a}$ are left unchanged.

####  III.4.1  On the $\kappa$ Parameter

This Einstein-Hilbert term makes the theory employed here gravitational.
$\kappa$ is proportional to Newton’s gravitational constant, with a small
$\kappa$ giving the probe limit where the metric equations of motion can be
solved independently of the other fields, although it is not guaranteed here
that boundary theories of interest are dual to bulk theories. This can be
understood by varying the Lagrangian with respect to the metric; the Einstein-
Hilbert part gives a term inversely proportional to $\kappa$ and the rest of
the Lagrangian gives the stress-energy tensor independently of $\kappa$.

This greatly simplifies calculations and will therefore be used throughout
this investigation.

####  III.4.2  On the $\Lambda$ Parameter

The scale invariance of the system allows an arbitrary $\Lambda$, as two
systems with different cosmological constants can be equated through a
rescaling. It will later be shown that the cosmological constant and length
scale $L$ can be related via

$\displaystyle L=\sqrt{-\frac{3}{\Lambda}},$ (18)

which also allows for other parameters to be related and scale invariance can
thereby not be used to choose these parameters freely.

####  III.4.3  On the $q$ Parameter

The gauge coupling strength $q$ is also the charge of the scalar field, hence,
the letter used. Taking conjugates $\tilde{\psi}=q\psi$ and
$\tilde{A}_{a}=qA_{a}$ as fields will give a Lagrangian identical to our
previous Equation 9, but divided by $q^{2}$ except for the term originally
containing $q^{2}$, which is divided by $q^{4}$. Scaling the Lagrangian by a
constant does not affect the equations of motion, so the system can be solved
for any value of $q$. Other solutions can be obtained by rescaling the fields.

####  III.4.4  On the $m^{2}$ Parameter

The values of $m$, the mass of the scalar field in the bulk, that are suitable
will later be investigated when solving equations of motion in the bulk.

##  IV Solutions of the Classical Bulk Theory

The Lagrangian describes a general system, so there exist many solutions to
the described equations of motion. We wish to investigate two properties of a
superconductor, the development of a condensate at low temperatures and the
conductivity at different frequencies. We are interested in a superconductor
subject to spatially uniform conditions, the applied electric field uniform
and the chemical potential uniform. The atomic lattice and its imperfections
are thus not accounted for, but interesting superconductivity behavior can be
obtained anywayHartnoll, Herzog, and Horowitz (2008). It is thus enough to
examine a system with translational symmetry in the $x$ and $y$ directions. A
rotationally invariant superconductor will further be studied. The system is
subject to conditions constant in time, such as there being no time-dependent
chemical potential. This allows for the assumption of time independence when
solving the non-linear field equations.

The conductivity is the linear electrical current response to an applied
transverse electrical field. We apply this in the $x$ direction due to the
rotational symmetry and let the applied field have a harmonic time dependence
$\exp(it\omega)$ so we can get the response function in the frequency domain.
The linear response is sought so the applied field should be infinitesimal.
The applied field breaks the rotational and time symmetries but since it is
infinitesimal and we are not interested in the effect it has on another
fields, it can be neglected in calculations. The applied field is later added
with the other fields as a background solution.

The electrical field in the $x$-direction is
$E_{x}=F_{xt}=\partial_{x}A_{t}-\partial_{t}A_{x}$. Translational symmetry
gives $E_{x}=\partial_{t}A_{x}$. These limitations allow for the following
definitions:

$\displaystyle\text{d}s^{2}$
$\displaystyle=g_{tt}(z)\text{d}t^{2}+g_{xx}(z)(\text{d}x^{2}+\text{d}y^{2})+g_{zz}(z)\text{d}z^{2},$
(19) $\displaystyle\psi$ $\displaystyle=\psi(z),$ (20)

and

$\displaystyle A_{a}=(\phi(z),A_{x}(z)\exp(it\omega),0,A_{z}(z)),$ (21)

where $\phi(z)$ is infinitesimal. The gauge condition requires

$\displaystyle\nabla_{a}A^{a}=\partial_{a}A^{a}+\Gamma^{a}_{ba}A^{b}=0,$ (22)

which gives a homogeneous first-order linear ordinary differential equation
for $A_{z}$ because the contracted Christoffel symbol only has a $z$
component. The remaining gauge symmetry lets us add a function to $A_{z}$ and
can be used to set $A_{z}(z)=0$ for a specific $z$. The above differential
equation then requires $A_{z}(z)$ to be identically 0 for all $z$. We will
hereafter work with $A_{z}(z)=0$.

The explicit $z$\- and $t$-dependence of these functions will hereafter be
omitted.

###  IV.1 The Schwarzchild metric

The path integral for the bulk partition function is approximated in a semi-
classical approximation where we need the saddle point of the action. We first
wish to find the metric saddle point of the periodic imaginary time path
integral. The bulk equation of motion for the metric $g_{ab}$ is the Einstein
equation

$\displaystyle R_{ab}-\frac{1}{2}g_{ab}R+g_{ab}\Lambda=\kappa T_{ab},$ (23)

where $R_{ab}$ is the Ricci curvature tensor and $T_{ab}$ is the stress-energy
tensor. The probe limit was assumed, so we may assume a small $\kappa$ and
neglect the right-hand side of Equation 23. We desire a translation-invariant
solution in the $t$, $x$, and $y$ directions that is asymptotically AdS. The
solution is known to be a black holeMcGreevy (2010b), the Schwarzchild metric
in AdS space. The metric has the following form in a particular choice of
coordinates where the radial coordinate $z$ is 0 at the boundary and $z_{h}$
at the horizon,

$\displaystyle
g_{ab}\text{d}x^{a}\text{d}x^{b}=\frac{L^{2}}{z^{2}}(\frac{\text{d}z^{2}}{f(z)}-f(z)\text{d}t^{2}+\text{d}x^{2}+\text{d}y^{2}),$
(24)

where $f(z)=1-z^{3}z_{h}^{-3}$. This function of $z$ approaches $1$ at the
boundary and the space is asymptotically AdS. There is a horizon at $z=z_{h}$
where $f(z_{h})=0$. The space behind the horizon can not affect the physics of
the boundary and can therefore be neglected in these calculations. This
solution is periodic in imaginary time. Consider the near-horizon metric where

$\displaystyle
f(z)=f(z_{h})-(z_{h}-z)f^{\prime}(z_{h})+\mathcal{O}((z_{h}-z)^{2})\approx
3(1-zz_{h}^{-1})$ (25)

. With a change of variables of $\rho^{2}=(4L^{2})/3(1-zz_{h}^{-1})$, we have
$f(z)\approx 9\rho^{2}/4L^{2}$ and
$\rho^{2}\text{d}\rho^{2}=\text{d}z^{2}z_{h}^{-2}\cdot 4L^{4}/9$. The near-
horizon metric is then

$\displaystyle
g_{ab}\text{d}x^{a}\text{d}x^{b}=\frac{L^{2}}{z_{h}^{2}}(\frac{\rho^{2}\text{d}\rho^{2}}{z_{h}^{-2}\cdot\frac{4L^{4}}{9}\rho^{2}\frac{9}{4L^{2}}}-\rho^{2}\frac{9}{4L^{2}}\text{d}t^{2}+\text{d}x^{2}+\text{d}y^{2}).$
(26)

We now extend to the imaginary time $\tau=it$, where

$\displaystyle
g_{ab}\text{d}x^{a}\text{d}x^{b}=\text{d}\rho^{2}+\rho^{2}(\frac{3}{2z_{h}}\text{d}\tau)^{2}+\frac{L^{2}}{z_{h}^{2}}(\text{d}x^{2}+\text{d}y^{2}).$
(27)

The near-horizon metric is then that of a Euclidean plane in polar
coordinates, and there is thus a deficit angle unless $(3/2z_{h})\tau$ has a
periodicity of $2\pi$. The imaginary time has periodicity $\beta$, so we must
have

$\displaystyle\frac{3}{2z_{h}}=\frac{2\pi}{\beta}.$ (28)

This gives the relationship between $z_{h}$ and the temperature

$\displaystyle T=\frac{3}{4\pi z_{h}}.$ (29)

This expression for the temperature agrees with the Beckenstein-Hawking
temperature of a black hole.

We have assumed $\kappa T_{ab}=0$ in finding the metric. The backreaction,
$\delta g_{ab}$, from the non-zero fields will be of order $\kappa T_{ab}$, as
according to Equation 23. The Einstein equation is obtained by varying the
Lagrangian with respect to $g_{ab}$ and finding the saddle point so $\delta
S\propto\kappa^{-1}\delta g_{ab}^{2}$ for the variation around the saddle
point. We thus have that $\delta S\propto\kappa T^{2}_{ab}$ and the
backreaction can be safely neglected when calculating the action from
different configurations.

The background metric can now be used instead of solving the equations of
motion for the metric together with the fields. The gravitational part of the
Lagrangian must be kept when calculating the value of the total action, which
is dominated by the gravitational part.

The horizon $z_{h}$ and the curvature length $L$ set length scales in the
metric. Length units in the numerical solution can be chosen such that
$z_{h}=1$. This means that for different temperatures, we have different
units, since $z_{h}$ is related to the temperature. We will have to convert
between these units when comparing results from different temperatures.

###  IV.2 Field equations of motion

The equations of motion for $\psi(z)$, $\phi(z)$, and $A_{x}(z)$ can now be
obtained. With the past definitions for $\text{d}s^{2}$, $\psi$, and $A_{a}$,
the equations of motion of Equation 30 through 32, and the metric of Equation
24, we have

$\displaystyle\Big{(}q^{2}z^{2}\phi^{2}-L^{2}m^{2}f+zf(zf^{\prime}-2f)\partial_{z}+z^{2}f^{2}\partial_{z}\partial_{z}\Big{)}\psi=0$
(30)
$\displaystyle\Big{(}-2q^{2}\psi^{2}L^{2}+z^{2}f\partial_{z}\partial_{z}\Big{)}\phi=0$
(31)
$\displaystyle\Big{(}-2q^{2}\psi^{2}L^{2}f+z^{2}\omega^{2}+z^{2}ff^{\prime}\partial_{z}+z^{2}f^{2}\partial_{z}\partial_{z}\Big{)}A_{x}=0.$
(32)

A Frobenius expansion Teschl (2012) of these equations can be done at the
boundary, $z=0$. The leading behavior of the functions is

$\displaystyle\psi=\psi_{(0)}\left(\frac{z}{L}\right)^{\Delta_{\psi}}$ (33)
$\displaystyle\phi=\phi_{(0)}\left(\frac{z}{L}\right)^{\Delta_{\phi}}$ (34)
$\displaystyle A_{x}=A_{x(0)}\left(\frac{z}{L}\right)^{\Delta_{A_{x}}}$ (35)

where $\Delta_{\psi}$, $\Delta_{\phi}$ and $\Delta_{A_{x}}$ are constants that
are to be determined. This is a slight assumption since not all functions have
this type of leading behavior111The function $\log$ for example does not allow
an expansion like this, we have assumed the function does not have an
essential singularity at $z=0$.. Entering this in the equations of motion
yields

$\displaystyle q^{2}z^{2}\phi_{(0)}^{2}s^{2\Delta_{\phi}}$ $\displaystyle-$
$\displaystyle L^{2}m^{2}f$ $\displaystyle+$ $\displaystyle
f(zf^{\prime}-2f)\Delta_{\psi}$ $\displaystyle+$ $\displaystyle
f^{2}\Delta_{\psi}(\Delta_{\psi}-1)$ $\displaystyle=0,$ (36)
$\displaystyle-2q^{2}\psi_{(0)}^{2}s^{2\Delta_{\psi}}L^{2}+f\Delta_{\phi}(\Delta_{\phi}-1)=0,$
(37)

and

$\displaystyle-2q^{2}\psi_{(0)}^{2}s^{2\Delta_{\psi}}L^{2}f$ $\displaystyle+$
$\displaystyle z^{2}\omega^{2}$ $\displaystyle+$ $\displaystyle
zff^{\prime}\Delta_{A_{x}}$ $\displaystyle+$ $\displaystyle
f^{2}\Delta_{A_{x}}(\Delta_{A_{x}}-1)$ $\displaystyle=0.$ (38)

where $s=zL^{-1}$. This immediately gives $\Delta_{\psi}\geq 0$ and
$1+\Delta_{\phi}\geq 0$ since the first terms otherwise diverge at the horizon
where the other terms are finite. First consider the case of strict
inequalities. The leading order behavior is then

$\displaystyle-L^{2}m^{2}-2\Delta_{\psi}+\Delta_{\psi}(\Delta_{\psi}-1)=0$
(39) $\displaystyle\Delta_{\phi}(\Delta_{\phi}-1)=0$ (40)
$\displaystyle\Delta_{A_{x}}(\Delta_{A_{x}}-1)=0.$ (41)

with solutions

$\displaystyle\Delta_{\psi}=\frac{3}{2}\pm\sqrt{\frac{9}{4}+L^{2}m^{2}}$ (42)
$\displaystyle\Delta_{\phi}=0,1$ (43) $\displaystyle\Delta_{A_{x}}=0,1.$ (44)

Observe that each of these three exponents have two solutions, each
_independent_ of one another. Now assume $\Delta_{\psi}=0$. Equation 37 gives

$-2q^{2}\psi_{(0)}^{2}L^{2}+\Delta_{\phi}(\Delta_{\phi}-1)=0$ (45)

while Equation 36 gives $\Delta_{\phi}=-1$. We then have

$\displaystyle q^{2}\phi_{(0)}^{2}z_{h}^{2}=L^{2}m^{2}$ (46) $\displaystyle
q^{2}\psi_{(0)}^{2}L^{2}=1$ (47)
$\displaystyle\Delta_{A_{x}}(\Delta_{A_{x}}-1)=2.$ (48)

with solutions $\Delta_{A_{x}}=-1,2$. First assuming $\Delta_{\phi}=-1$ yields
the same result. There are however no solutions to Equation 46 for the
negative $m^{2}$ we later will consider and infinities are encountered when
calculating the action for these solutions so they will not be considered. All
useful solutions are thus given by Equation 42 to Equation 44.

###  IV.3 Field behavior at horizon

The same kind of Frobenious expansion can be made at the horizon but there are
some simplifying conditions. The time component of the metric vanishes at the
horizon, $f(z_{h})=0$. This means that $A_{t}(z_{h})$ must be zero because a
finite $A_{t}(z_{h})$ would give a finite Wilson loop around the periodic
imaginary time circle whose length in time is 0. A Wilson loop is contrary to
$A_{t}$ a physical quantity ($A_{t}$ is gauge-dependent). This gives a
singular gauge connection which is unphysical Hartnoll (2009b). We thus have
$A_{t}(z_{h})=0$. Expand the fields as

$\displaystyle\psi=\psi_{(h)}s^{\Delta^{(h)}_{\psi}}$ (49)
$\displaystyle\phi=\phi_{(h)}s^{\Delta^{(h)}_{\phi}}$ (50) $\displaystyle
A_{x}=A_{x(h)}s^{\Delta^{(h)}_{A_{x}}}$ (51)

where $s$ now is $(1-z/z_{h})$ and $\Delta^{(h)}_{\phi}>0$222The notation
$\Delta^{(h)}$ is used to signify that these exponents describe the _horizon_
behavior, $\Delta$ was earlier used for the conformal boundary behavior.. The
function $f$ can be expanded as $f=3s-3s^{2}+s^{3}$. Inserting these leading
terms in the equations of motion gives

$\displaystyle q^{2}z^{2}\phi_{(h)}^{2}s^{2\Delta^{(h)}_{\phi}}$
$\displaystyle+$ $\displaystyle 9z_{h}^{2}\Delta^{(h)}_{\psi}$
$\displaystyle+$ $\displaystyle
z_{h}^{2}9\Delta^{(h)}_{\psi}(\Delta^{(h)}_{\psi}-1)$ $\displaystyle=0,$ (52)

$\displaystyle-2q^{2}\psi_{(h)}^{2}s^{2\Delta^{(h)}_{\psi}}L^{2}$
$\displaystyle+$ $\displaystyle
z_{h}^{2}3s^{-1}\Delta^{(h)}_{\phi}(\Delta^{(h)}_{\phi}-1)$ $\displaystyle=0,$
(53)

and

$\displaystyle-6q^{2}\psi_{(h)}^{2}s^{2\Delta^{(h)}_{\psi}+1}L^{2}$
$\displaystyle+$ $\displaystyle z_{h}^{2}\omega^{2}$ $\displaystyle+$
$\displaystyle 9\Delta^{(h)}_{A_{x}}$ $\displaystyle+$ $\displaystyle
9\Delta^{(h)}_{A_{x}}(\Delta^{(h)}_{A_{x}}-1)$ $\displaystyle=0.$ (54)

Solving for the leading terms of these equations and using
$\Delta^{(h)}_{\phi}>0$ gives

$\displaystyle\Delta^{(h)}_{\psi}=0$ (55) $\displaystyle\Delta^{(h)}_{\phi}=1$
(56) $\displaystyle\Delta^{(h)}_{A_{x}}=\pm\frac{\i\omega z_{h}}{3}.$ (57)

The two possible $\Delta^{(h)}_{A_{x}}$ represent solutions going into or
coming out of the horizon. Close to the horizon is $A_{x}(z,t)$ given by

$A_{x}(z,t)=s^{\pm\frac{\i\omega z_{h}}{3}}\exp(\i\omega
t)=\exp\big{(}\i\omega(t\pm\frac{z_{h}\log s}{3})\big{)}.$ (58)

The phase is constant for $s=\exp(\mp 3t/z_{h})$ so the plus sign in Equation
57 gives the ingoing solution.

####  IV.3.1 Boundary conditions

The equations of motion, Equations 30, 31, and 32, can be integrated
numerically. Just one leading horizon behavior is allowed for $\psi$ and
$\phi$, so only two horizon conditions are needed for them, $\psi(z_{h})$ and
$\phi^{\prime}(z_{h})$. The derivative $\psi^{\prime}(h)$ needed for starting
a numerical integration from the horizon can be obtained directly from the
equations of motion as $z\rightarrow 0$:

$\displaystyle\psi^{\prime}(z_{h})=-\frac{L^{2}m^{2}}{3z_{h}}.$ (59)

A two-parameter family of solutions to the equations of motion can then be
obtained for $\psi$ and $\phi$. These solutions give the boundary values of
the fields which describe the background fields of the field theory. The two
horizon parameters must be chosen to obtain the desired background fields.

The field theory operator $\mathcal{O}$ corresponding to the background field
$\psi$ is expected to spontaneously attain a non-zero expectation value
breaking the U(1) symmetry, so we therefore require the source $\psi_{(0)}=0$.
The time component of the electromagnetic potential $A_{a}$ corresponds to the
electric potential in the Lorentz gauge. The electrical potential gives the
energy per charge needed to add a charge to the system and the chemical
potential for the electrons $\mu$ can thus be expressed as $\mu=q\psi_{(0)}$.

Just one of the two horizon behaviors of the Maxwell perturbation $A_{x}$ is
wanted. We want a casual response from the perturbation of the background
field. This corresponds to the solution going into the horizon as time
passesHartnoll (2009b). We thus choose the ingoing horizon behavior. The
equation for $A_{x}$ is linear and we are only interested in the linear
response at the conformal boundary, so the horizon amplitude of the ingoing
solution can be chosen arbitrarily.

The horizon parameters $\psi(z_{h})$ and $\phi^{\prime}(z_{h})$ can now be
varied to find solutions to the two boundary conditions $\psi_{(0)}=0$ and
$\mu=q\psi_{(0)}$.

There is a trivial analytical solution of the equations of motion with the
above boundary conditions.

$\begin{cases}&\psi(z)=0\\\ &\phi(z)=\mu(1-z/z_{h})\\\
&A_{x}(z)=\left[\exp\left(-\sqrt{3}\tan^{-1}\frac{z_{h}+2z}{z_{h}\sqrt{3}}\right)\frac{z_{h}-z}{\sqrt{z^{2}+zz_{h}+z_{h}^{2}}}\right]^{\frac{\i\omega
z_{h}}{3}}\end{cases}$ (60)

The field $\psi$ is here identically zero and there is no spontaneous symmetry
breaking. This solution thus corresponds to the physics above the critical
temperature, $T_{c}$, of the superconductor. We will now make a choice of $m$
to be able to numerically investigate solutions with $\psi\neq 0$.

####  IV.3.2 Choice of scalar mass $m$

The mass squared of a scalar field in flat space must be non-negative for
stability. This is not the case in a space with negative curvature, however.
The Breitenlohner-Freedman (BF) bound is a lower stability bound on $m^{2}$ of
a massive scalar field in AdS space with metric given by Equation 24. It
requires thatKleban, Porrati, and Rabadan (2005)

$\displaystyle L^{2}m^{2}\geq-\frac{d^{2}}{4}.$ (61)

The scalar field $\psi$ should obey this bound far away from the black hole
for normalizable modes. We would however like a spontaneous symmetry breaking
of $\psi$ near the black hole corresponding to the electron condensateGubser
(2008). This can happen because the coupling of $\psi$ to the perturbation
$A_{a}$ gives $\psi$ an effective mass that might break the BF bound near the
black hole. The effective mass is given by

$\displaystyle
m_{\text{eff}}^{2}=m^{2}+A_{a}A^{a}=m^{2}-\frac{z^{2}}{L^{2}(1-z^{d}z_{h}^{-d})}\phi^{2}.$
(62)

This can for large enough values of $\phi$ break the BF bound. Let us consider
the trivial, uncondensed solution of Equation 60. When does this give an
effective mass breaking the BF bound and possibly enabling an additional
condensed solution? The location of the effective mass minimum, $z_{0}$, can
be found by differentiating the above Equation 62 by $z$ and using the
uncondensed solution of Equation 60,

$\displaystyle\frac{z_{0}}{z_{h}}=\frac{1}{3}(\sqrt[3]{37+9\sqrt{17}}-\frac{2}{\sqrt[3]{37+9\sqrt{17}}}-2).$
(63)

The effective mass breaks the BF bound at $z_{0}$ when

$\displaystyle\frac{\mu}{T}>\frac{2\pi}{\sqrt{3}}\sqrt{\frac{4L^{2}m^{2}+9}{8-\sqrt[3]{142-34\sqrt{17}}-\sqrt[3]{142+34\sqrt{17}}}},$
(64)

where Equation 29 has been used. We will choose $m^{2}L^{2}=-2$, which does
not break the BF bound, but is relatively close. It gives integer scalings for
the scalar field at the conformal boundary, which is convenient.

###  IV.4 Expectation values of field theory operators

Expectation values of field theory operators can now be calculated using the
solutions of the equations of motion of Equation 30, 31, and 32, and the
previously discussed CFT expectation value of Equation 8. We will later see,
however, that not only is the leading behavior of the fields needed to
calculate the expectation values, but also the first subleading behavior. We
therefore expand the fields as

$\displaystyle\psi=\psi_{(0)}\frac{z}{L}+\psi_{(1)}(\frac{z}{L})^{2}$ (65)
$\displaystyle\phi=\phi_{(0)}+\phi_{(1)}\frac{z}{L}$ (66) $\displaystyle
A_{x}=A_{x(0)}+A_{x(1)}\frac{z}{L}$ (67)

and obtain $\psi_{(i)}$, $\phi_{(i)}$, and $A_{x(i)}$ from the numerical
solution. For this the boundary terms of the action are required. The boundary
term needed for the scalar field is calculated as

$\displaystyle
S_{\text{bdy}}=-\int_{z=\epsilon}{\text{d}^{d}xL^{-1}\psi^{2}\sqrt{g_{(0)}}}.$
(68)

We may insert this in Equation 8, so

$\displaystyle\braket{\mathcal{O}}_{\text{CFT}}$
$\displaystyle=-\int_{\partial\text{AdS}}{\text{d}^{d}y\sqrt{g_{(0)}}(n_{a}\frac{\partial\mathcal{L}(y)}{\partial(\nabla_{a}\psi(y))}-2L^{-1}\psi)\frac{\delta\psi(y)}{\delta\psi_{(0)}(x)}}$
$\displaystyle=\frac{L^{3}}{z^{3}}(\frac{z}{L}2\nabla_{z}\psi(y)-2L^{-1}\psi)\frac{z}{L}$
$\displaystyle=\frac{L^{2}}{z^{2}}(\frac{z}{L}2(\psi_{(0)}\frac{1}{L}+\psi_{(1)}\frac{2z}{L^{2}})-2L^{-1}(\psi_{(0)}\frac{z}{L}+\psi_{(1)}(\frac{z}{L})^{2})$
$\displaystyle=\frac{2\psi_{(1)}}{L}.$ (69)

This simple relationship gives us the expectation value of the scalar
operator.

We are now in a position to numerically solve the bulk theory and obtain the
expectation values. We do this by sweeping over different horizon values and
for each value finding all solutions to the boundary condition $\psi_{(0)}=0$.
This yields many different solutions $\rho/T$ at the boundary. Scale
invariance lets us interpret this as systems of constant $\rho$ but at
different temperatures $T$. We then get a variation in the chemical potential
$\mu$. The chemical potential of the trivial solution shown there is
calculated through

$\displaystyle\rho=\frac{\mu}{z_{h}}=\mu T\frac{4\pi}{3}.$ (70)

Alternatively, one can let $\mu$ be constant while varying the temperature and
get a variation in $\rho$.

Figure 1: Chemical potential needed for constant $\rho$ at different $T$. The
multiple curves correspond to multiple solutions at the same temperature. The
dashed lines have different signs of the expectation value of $\mathcal{O}$
and the horizon boundary condition $\psi(z_{h})$. Further solutions (here
omitted) are obtained for lower temperature following the trend shown here.
Figure 2: Expectation value of CFT operator $\mathcal{O}$ at different $T$ and
constant $\mu$. The multiple curves correspond to multiple solutions at the
same temperature. The dashed lines have different signs of the expectation
value of $\mathcal{O}$ and the horizon boundary condition $\psi(z_{h})$.
Further solutions (here omitted) are obtained for lower temperature following
the trend shown here.

###  IV.5 Free energy and electrical conductivity

The free energy, $A=-T\log{Z}$, is the same for the bulk and the boundary
theory since their partition functions are the same. This can be calculated in
the classical limit in the bulk through

$\displaystyle A=-T\log{Z}\overset{\text{classical}}{=}-iTS_{c},$ (71)

where $S_{c}$ is the on-shell periodic Euclidean time action. The on-shell
field solutions only depend on the $z$ coordinate and are thus proportional to
$V=i\beta V_{2}$, where $V_{2}$ is the area considered in coordinates $x_{1}$,
$x_{2}$. This gives the free energy per surface area

$\displaystyle\frac{A}{V_{2}}=\int_{0}^{z_{h}}{\text{d}z\sqrt{-g}\mathcal{L}+V^{-1}S_{\text{bdy}}}.$
(72)

Only the free energy difference of the different solutions is needed. We
therefore calculate the free energy contribution $A_{\text{fields}}$ from the
scalar and electromagnetic fields and neglect the contribution from the
Einstein-Hilbert term of the action. We do not need to account for the
contribution from any backreaction on the metric following the previous
argument. We first consider the trivial solution of

$\displaystyle\frac{A_{\text{fields}}}{V_{2}}$
$\displaystyle=\int_{0}^{z_{h}}{\text{d}z\sqrt{-g}\mathcal{L}+V_{2}^{-1}S_{\text{bdy}}}$
$\displaystyle=-\int_{0}^{z_{h}}{\text{d}z(\frac{z}{L})^{-4}\frac{1}{4}F_{ab}F^{ab}}$
$\displaystyle=-\int_{0}^{z_{h}}{\text{d}z(\frac{z}{L})^{-4}\frac{1}{2}F^{2}_{zt}g^{zz}g^{tt}}$
$\displaystyle=\int_{0}^{z_{h}}\text{d}z\frac{\mu^{2}}{2z_{h}^{2}}$
$\displaystyle=z_{h}^{-1}\frac{\mu^{2}}{2}=\frac{4\pi T}{3}\frac{\mu^{2}}{2}.$
(73)

This agrees with the result from thermodynamics, with $\mu$ is the change in
free energy for increasing the expectation value of the number of particles by
one while keeping the temperature constant, as in

$\displaystyle u=(\frac{\partial A}{\partial\braket{N}})_{T}.$ (74)

This is easily shown by using $\rho=\braket{N}/V_{2}$ and we take Equation 70
to get the $N$ dependence of $\mu$ for constant $T$. The gravitational part of
the free energy can be neglected since the derivative is set at constant
temperature.

The free energy for the numerical solutions has been calculated and the result
together with the analytical result is shown in Figure 3.

Figure 3: The free energies of the different solutions are shown here for
constant $\rho$ and varying $T$. The trivial solution is shown as the thick
solid line. The other curves correspond to numerical solutions. The lowest one
corresponds to the solution appearing at $T=T_{c}$ and the other roots follow
in order.

It can there be seen that the trivial solution is the physical solution for
temperatures above $T_{c}$ and that the solution appearing at the temperature
$T_{c}$ is the physical solution for all lower temperatures. We will hereafter
only work with these two solutions. The graph is in line with the phase
transition being a second-order phase transition, though this has not been
investigated mathematically.

####  IV.5.1 Preliminary discussion of electrical conductivity

The conductivity of a superconductor can easily be measured experimentally for
a wide range of frequencies and it is therefore an interesting property to
calculate from our model of a superconductor. The agreement in different
frequency ranges tells us about similarities and differences between our model
and the experimental superconductors.

Let the conductivity $\sigma$ be the linear response function for the current
density $J_{x}$ with applied electrical field $E_{x}$ as source

$\displaystyle\sigma(\omega)=\frac{J_{x}(\omega)}{E_{x}(\omega)}.$ (75)

From the Drude model, conductivity is also a complex quantity, with this
current density-electric field relation suggesting that the imaginary part
then represents the resistance to change of the electromagnetic field’s charge
carriers. The direction $x$ has been chosen for concreteness, but since we
consider two-dimensional systems with rotational symmetry, we need only
consider one direction. These functions of $\omega$ are the Fourier transforms
of the time-dependent quantities. The current in the time domain can be
obtained from the conductivity and the applied field through an inverse
Fourier transform

$\displaystyle
J_{x}(t)=\int_{-\infty}^{\infty}{E_{x}(t-\tau)\sigma(\tau)\text{d}\tau}.$ (76)

Causality implies that $\sigma(t)=0$ for $\tau<0$, since the current would
otherwise depend on future values of the electrical field. Using this, the
conductivity can be written

$\displaystyle\sigma(\omega)=\int_{0}^{\infty}{\sigma(\tau)\exp(i\tau\omega)\text{d}\tau}$
(77)

and thus has an analytic extension to the upper half of the complex plane.
Both the current, $J_{x}(t)$, and applied field, $E_{x}(t)$, are real
quantities which make $\sigma(\tau)$ also real and thus
$\text{Re}(\sigma(\omega))$ an even function and $\text{Im}(\sigma(\omega))$
odd. These properties of $\sigma(\omega)$ give the Kramers-Kronig relations

$\displaystyle\text{Re}(\sigma(\omega))$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}{\frac{\omega^{\prime}\text{Im}(\sigma(\omega^{\prime}))}{\omega{{}^{\prime}}{{}^{2}}-\omega^{2}}\
\text{d}\omega^{\prime}}$ (78)

and

$\displaystyle\text{Im}(\sigma(\omega))$
$\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}{\frac{\omega\text{Re}(\sigma(\omega^{\prime}))}{\omega{{}^{\prime}}{{}^{2}}-\omega^{2}}\
\text{d}\omega^{\prime}},$ (79)

which state that the real part of the conductivity uniquely determine the
imaginary part and vice versa.

##  V Determination and Assessment of the Holographic Conductivity

We have discussed how the bulk equations of motion can be solved via an
infinitesimal applied electrical field in the $x$ direction. The field was
oscillating with a frequency $\omega$. The conductivity can now be calculated
from the obtained current in the $x$ direction.

$\displaystyle\sigma=\frac{J_{x}}{-\partial_{t}A_{x}}=-\frac{A_{x(1)}}{i\omega
A_{x(0)}L}.$ (80)

The electrical conductivity for the normal phase can now be found using the
trivial solutions of Equation 60. The boundary behavior is

$\displaystyle A_{x(0)}=\lim_{z\rightarrow 0}A_{x}(z)=\exp(-\frac{i\pi
z_{h}\omega}{6\sqrt{3}}).$ (81)

The conductivity is therefore 1 for all $\omega$ above $T_{c}$.

Figure 4: Real and imaginary part of the conductivity for different
temperatures. $\rho$ is here constant.

The conductivity below $T_{c}$ can be calculated the same way using the
numerical solution. The result is seen in Figure 4. The conductivity is
lowered for low $\omega$ when the condensate forms. An energy gap
$\Delta_{\text{gap}}$ forms and the conductivity for
$\omega<\Delta_{\text{gap}}$ goes to 0 as the temperature is lowered. The
superconductivity is not immediately evident from the obtained conductivity
curves. There is, however, a delta function at $\omega=0$ since translational
invariance of the boundary theory has been assumed and the charged field
$\psi$ has obtained a non-zero expectation value. The delta function can be
seen through the Kramers-Kronig relations. A delta-function in the real part
is equivalent to a pole in the imaginary part:

$\displaystyle\text{Im}(\sigma(\omega))$
$\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}{\frac{\omega\delta(\omega^{\prime})}{\omega{{}^{\prime}}{{}^{2}}-\omega^{2}}\
\text{d}\omega^{\prime}}$ $\displaystyle=\frac{1}{\pi\omega}.$ (82)

This pole is also visible in the figure.

The Kramers-Kronig relations and the independence of low energy properties for
the high-frequency conductivity can be used to prove a sum rule for the
conductivityFerrell and Glover (1958). The rule states that

$\displaystyle\int_{0}^{\infty}{\text{Re}(\sigma(\omega))\ \text{d}\omega}=C,$
(83)

where $C$ is a constant depending on what system we are considering. This
integral diverges in our case since $\sigma\rightarrow 1$ for high
frequencies, so the rule must be modified. The proof uses that the imaginary
part of $\sigma(\omega)$ becomes independent of low energy properties such as
the temperature at high enough frequencies. That fact that the Kramers-Kronig
relations apply to $\sigma(\omega)$ is also used. These two properties are
also true for $\sigma(\omega)-1$, so the rule can be modified into

$\displaystyle\int_{0}^{\infty}{\text{Re}(\sigma(\omega)-1)\
\text{d}\omega}=C.$ (84)

The analytical solution above $T_{c}$ now gives $C=0$. This rule can now be
used to verify that our numerics are accurate. The integral of
$\text{Re}(\sigma(\omega)-1)$ should vanish for all temperatures. It is then
important to include the delta function at $\omega=0$, which our numerics do
not catch. We can, however, find the amplitude, $\Sigma_{\delta}$, of the
delta function from the amplitude of the pole in the imaginary part of the
conductivity. The sum of the integral of the continuous part of the
conductivity, $\int{(\sigma_{n}-1)\ \text{d}\omega}$, and $\Sigma_{\delta}$
should then equal 0 for all temperatures, as indicated by the red line in
Figure 13, with the results of each expression as a function of $T/T_{c}$
shown as $[T_{c}]$ on the vertical axis.

A suitable cut-off frequency has been used, but since $\sigma(\omega)$
converges fast to 1, this is not a concern. No large discrepancies are
observed. This is a strong check of the numerics, since the real part of the
conductivity at all frequencies and temperatures and the amplitude of the pole
all have to match up. The largest discrepancy is observed for low
temperatures, which we can expect since the numerical integrator there has
been observed to make smaller steps, indicating a numerically more difficult
problem.

Figure 5: The two contributions to the integral in the modified Ferrell-
Glover-Tinkham sum rule for different temperatures. The red line is expected
to be precisely at 0 for perfect numerics.

##  VI Extending the Lagrangian

This analysis makes several simplifications for the system at hand, the most
prominent being the simple Lagrangian of Equation 9 and that translational
symmetry was assumed. This gives a boundary theory with a scalar field that
condenses below a critical temperature, as expected for a superconductor. The
conductivity shows both similarities and differences with that of high-$T_{c}$
superconductors. A delta function develops at $\omega=0$ for $T<T_{c}$, giving
infinite DC conductivity. An evident difference is that lack of a Drude peak
at low frequencies, a Drude peak being an increase in conductivity for low
frequencies in metals due to impurities that can be well modeled by the Drude
model of conductivityDrude (1900), hence the name. This model has been shown
to agree with experiments on cuprates above $T_{c}$Liu  _et al._ (2006). The
linear dependence found by H. L. Liu et al. is attributed to electron-phonon
interactions. Electrons might not be a suitable type of excitation for these
possibly strongly coupled systems and our model might capture interactions
between some other type of excitations. The assumptions of temperature-
independent Drude model parameters, $\rho$, $q^{2}$ and $m$, might then not be
valid. The true temperature dependence of the scattering rate can easily be
obtained directly from the Drude fit but this has not been done. This can then
be compared with the linear experimental observations. The temperature
dependence of the Drude model parameters can then be found. This might give
important insights as to what types of excitations the Drude model in the
cases we have studied describes with such high accuracy.

The Drude model is obtained by treating the charge carriers classically. They
are expected to obey the differential equation

$\displaystyle\frac{\text{d}v}{\text{d}t}=\frac{q}{m}E-\frac{1}{\tau}v,$ (85)

where $qE$ is the electric force, $m$ is the charge carrier mass, $q$ is the
charge, and $\tau$ is the average time between collisions. The last term is a
drag force supposed to model the collisions slowing the charge carriers down.
Solving for this harmonic, $E=E_{0}\exp(-i\omega t)$, gives

$\displaystyle v=\frac{\tau qE_{0}}{m(1-i\omega t)}\exp(-i\omega t).$ (86)

This gives the conductivity

$\displaystyle\sigma(\omega)=\frac{J(\omega)}{E(\omega)}=\frac{v(\omega)q\rho}{E(\omega)}=\frac{\tau\rho
q^{2}}{m(1-i\omega\tau)}=\frac{\sigma_{0}}{1-i\tau\omega},$ (87)

where $\rho$ is the density of charge carriers of some charge $q$. From this
definition, we have

$\displaystyle\frac{\sigma_{0}}{\tau}=\frac{\rho q^{2}}{m}.$ (88)

Different generalizations of the standard Lagrangian of Equation 9 have been
studied. Higher order corrections using $\psi$ will not make a difference
above $T_{c}$, so a Drude peak cannot be obtained using them. T. Wenger
studied these extensionsWenger (2012), and this reassessment will use his
added higher order Maxwell term, resulting in

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2\kappa}(R-2\Lambda)-\frac{1}{4}f_{ab}F^{ab}$
$\displaystyle-m^{2}\psi\overline{\psi}-D_{a}\psi\overline{D^{a}\psi}$
$\displaystyle+\alpha_{2}F^{a}_{b}F^{b}_{c}F^{c}_{d}F^{d}_{a},$ (89)

which gives equations of motion different from our original Equation 30, 31,
and 32. Using SymPy, $z_{h}=1$ and $q=1$ instead yield

$\displaystyle(-4z^{3}+2z^{2}\phi^{2}+4)$ $\displaystyle\psi$
$\displaystyle+(2z^{7}+2z^{4}-4z)$ $\displaystyle\psi^{\prime}$
$\displaystyle+(2z^{8}-4z^{5}+2z^{2})$ $\displaystyle\psi^{\prime\prime}=0$
(90)

for $\psi$,

$\displaystyle(z^{5}-z^{2})$ $\displaystyle\phi^{\prime\prime}$
$\displaystyle+(24\alpha_{2}z^{9}\phi^{\prime\prime}-24\alpha_{2}z^{6}\phi^{\prime\prime})$
$\displaystyle\phi{{}^{\prime}}{{}^{2}}$
$\displaystyle+(32\alpha_{2}z^{8}-32\alpha_{2}z^{6})$
$\displaystyle\phi{{}^{\prime}}{{}^{3}}$ $\displaystyle+2\psi^{2}$
$\displaystyle\phi=0$ (91)

for $\phi$, and

$\displaystyle(8\alpha_{2}z^{12}\phi{{}^{\prime}}{{}^{2}}-16\alpha_{2}z^{9}\phi{{}^{\prime}}{{}^{2}}+8\alpha_{2}z^{6}\phi{{}^{\prime}}{{}^{2}}+z^{8}-2z^{5}+z^{2})$
$\displaystyle A_{x}^{\prime\prime}$
$\displaystyle+(8\alpha_{2}\omega^{2}z^{6}\phi{{}^{\prime}}{{}^{2}}+\omega^{2}z^{2}+2z^{3}\psi^{2}-2\psi^{2})$
$\displaystyle A_{x}$
$\displaystyle+(16\alpha_{2}z^{12}\phi^{\prime}\phi^{\prime\prime}+56\alpha_{2}z^{11}\phi{{}^{\prime}}{{}^{2}}-32\alpha_{2}z^{9}\phi^{\prime}\phi^{\prime\prime}-88\alpha_{2}z^{8}$
$\displaystyle\phi{{}^{\prime}}{{}^{2}}+$ $\displaystyle
16\alpha_{2}z^{6}\phi^{\prime}\phi^{\prime\prime}+32\alpha_{2}z^{6}\phi{{}^{\prime}}{{}^{2}}+3z^{7}-3z^{4})$
$\displaystyle A_{x}^{\prime}=0$ (92)

for $A_{x}$.

The exponents of the horizon and boundary are identical because the higher
order term vanishes faster both at the horizon and the boundary. The
analytical solution of the equations of motion found above $T_{c}$ for the
original Lagrangian is no longer valid and a numerical solution must now be
used above $T_{c}$ as well. The result of the conductivity calculated with
this extended model can be seen in Figure 6, where small $\alpha_{2}$ has been
used.

Figure 6: Real part of the conductivity for different temperatures using the
extended Lagrangian with $\alpha_{2}=0.01L^{4}$. $\rho$ is here constant.
Figure 7: Real part of the conductivity for different temperatures using the
extended Lagrangian with $\alpha_{2}=0.1L^{4}$. $\rho$ is here constant.

The higher order term introduces a perturbation in the low frequency
conductivity. The gap still appears at the same position,
$\omega_{\text{gap}}\approx\mu/2$. The behavior around the transition from
$\sigma=0$ to $\sigma=1$ is changed. The earlier increase in conductivity for
$\omega$ slightly larger than $\omega_{\text{gap}}$ has been changed. The
increase is lower and at higher $\omega$. The conductivity for a slightly
larger $\alpha_{2}$ is seen in Figure 7. Here, the change around the
transition is more pronounced. A peak develops for low temperatures and a
second peak is also seen to develop. More peaks develop for higher $\alpha+2$
and lower $T$. These peaks seem to approach delta functions when the
temperature is lowered further. The peaks are not believed to be a numerical
error, since their contribution is needed for the sum rule to be satisfied.

The transition to higher conductivity is now seen to happen in two steps.
First, the conductivity increases to $\sigma\approx 0.7$ at $\omega$ slightly
lower than $\omega_{\text{gap}}$. Now this transition also happens for higher
temperatures, the conductivity approaches 0.7 from above. Secondly, the
conductivity increases to reach 1 at $\omega\approx 2$; this second part of
the transition seems to be mostly independent of temperature.

The conductivity behavior above $T_{c}$ resembles that of the Drude model,
however. A fit of the Drude parameters $\sigma_{0}$ and $\tau$ can be made to
see how well this model agrees with our conductivity. One point of the complex
conductivity is enough to obtain both of these real parameters. The drude
model conductivity approaches 0 at high frequencies whereas the conductivity
of our holographic model approaches 1. We can thus not expect the model to
work well for high frequencies. We make the Drude fit by taking the value of
the conductivity in the limit $\omega\rightarrow 0$. First, obtain
$\sigma_{0}$ through

$\displaystyle\sigma_{0}=\lim_{\omega\rightarrow 0}\sigma(\omega).$ (93)

Use this to obtain $\tau$:

$\displaystyle\lim_{\omega\rightarrow
0}\frac{\text{Im}(\sigma)}{\omega\sigma_{0}}=\lim_{\omega\rightarrow
0}\frac{\tau}{1+\tau^{2}\omega^{2}}=\tau.$ (94)

A fit using these obtained parameters is shown in Figure 8. The fit agrees
well for low frequencies but a difference of 1 in the real part appears as the
Drude model conductivity approaches 0 and the holographic conductivity
approaches 1. This agreement might not be very impressive, but the Drude peak
grows when $\alpha_{2}$ is increased and relative error vanishes in the limit
of large $\alpha_{2}$. See Figure 9 for a Drude fit with a higher
$\alpha_{2}$. The low frequency conductivity is here much larger but the error
is still of order 1 and only appears at higher frequencies where the
holographic conductivity approaches 1.

Figure 8: Conductivity for $\alpha_{2}=0.1L^{4}$ and $T=T_{c}$ together with
Drude model fit. Figure 9: Conductivity for $\alpha_{2}=10L^{4}$ and $T=T_{c}$
together with Drude model fit.

This agreement then motivates an investigation of the Drude parameters
dependence on system parameters. Figure 10 shows the dependence of the
parameters on the strength of the higher order term, $\alpha_{2}$. A power-law
was fit to both $\sigma_{0}$ and $\tau$ at high $\alpha_{2}$. $\sigma_{0}$
clearly approaches a linear dependence on $\alpha_{2}$. $\tau$ did not permit
a power-law description.

Figure 10: Drude parameters as functions of $\alpha_{2}$ at $T=2T_{c}$.

Now we investigate the temperature dependence of $\sigma_{0}$ to try to see
how the coefficient $C$ in the linear dependence on $\alpha_{2}$ depends on
temperature. Figure 11 shows the dependence of $\sigma_{0}$ on both
$\alpha_{2}$ and $T$. The coefficient $C$ also seems to follow a power-law
since the separations between the curves corresponding to different
temperatures are equal. A plot of the temperature dependence of $C$ is seen in
Figure 12. A formula for $\sigma_{0}$ valid for large
$\alpha_{2}T_{c}^{-4/3}/(T^{-1/3}L)^{4}$ can according to these results be
expressed as

$\sigma_{0}=C\frac{\alpha_{2}}{L^{4}}\left(\frac{T}{T_{c}}\right)^{-4/3}.$
(95)

This is valid for a large range of values of $\alpha_{2}$ and $T$.

Figure 11: $\omega\rightarrow 0$ limit of the conductivity as a function of
$\alpha_{2}$ for different temperatures. Figure 12:
$\sigma_{0}L^{4}/\alpha_{2}$ as a function of temperature for large
$\alpha_{2}$.

A test of the numerics through a consistency check has also been performed,
with the results shown in Figure 13. There is a substantial error at low
temperatures, but it is decreased by increasing the number of samples for the
numerical integral, so it is believed to originate from the very sharp peaks,
approaching delta functions, appearing at low temperatures making numerical
integration difficult.

Figure 13: The two contributions to the integral in the modified Ferrell-
Glover-Tinkham sum rule for different temperatures. The red line is expected
to be precisely at 0 for perfect numerics. The small bump at about
$T=0.2T_{c}$ is due to the numerical integration missing the very sharp
conductivity peak developing at low temperature.

##  VII Conclusions

Through the modeling work of researchers like G. T. Horowitz and P. Säterskog,
the AdS/CFT correspondence can be used to model a high-$T_{c}$ superconductor
both below and above $T_{c}$. This was initially done using the simplest
possible Lagrangian and then calculating the frequency-dependent conductivity.
A higher order term was added to the Lagrangian and a conductivity peak was
obtained at low frequencies. The Drude model describes this peak very well
within certain limits. The behavior of the Drude model parameter $\sigma_{0}$
was investigated in certain limits. The extended Lagrangian seems to give an
effective description of the lattice introduced by G. T. Horowitz and J. E.
SantosHorowitz and Santos (2013), but further investigations are of interest.

Overall, the topic is in need of a more phenomenological perspective regarding
real-world experimental data. The ramifications of Equation 70, for example,
which relates the chemical potential and charge density for the non-
superconducting solution, could be further examined. This has been done by S.
A. HartnollHartnoll (2009a), but this could be replicated. A thorough
treatment of the differential equations would also be of interest. All curves
in Figure 2 approach the same value. The differential equation has been
numerically investigated and it has the same boundary behavior, regardless of
initial conditions, so it seems possible to find a mathematical explanation
for the behavior by looking at the boundary behavior of the equations.

As P. SäterskogSäterskog (2013) suggests, the results from the extended
Lagrangian could be investigated further. A physical interpretation of the
conductivity peaks for low temperatures is required for an understanding of
the extension to the Lagrangian. The temperature dependence of the scattering
rate $1/\tau$ can be found and compared to experiment. A linear temperature
dependence would agree with experiments and thus indicate that the extended
Lagrangian truly captures the physics giving rise to the conductivity peak. A
more thorough comparison with the results of G. T. HorowitzHorowitz and Santos
(2013) is of interest, since we could hope to have found an effective
description of their system. The power law behavior of the conductivity at
intermediate frequencies could also further be investigated.

##  VIII Acknowledgements

The author would like to gratefully acknowledge Professor David Kaplan of
Johns Hopkins University for his recommendations and research overview.

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