An excursion into modern
superconductivity: from
nanoscience
to cold atoms and holography
Yuzbashyan
Rutgers
Altshuler
Columbia
Urbina
Regensburg
Richter
Regensburg
Sangita
Bose, Tata,
Max Planck Stuttgart
Kern
Stuttgart
Diego Rodriguez
Queen Mary
Sebastian Franco
Santa Barbara
Masaki
Tezuka
Kyoto
Jiao Wang
NUS
Antonio M.
Garc
í
a

Garc
í
a
Superconductivity in
nanograins
New forms of
superconductivity
New tools
String Theory
Increasing the
superconductor
T
c
Superconductivity
Practical
Technical
Theoretical
Enhancement and control of
superconductivity in nanograins
Phys. Rev. Lett. 100,
187001 (2008)
Yuzbashyan
Rutgers
Altshuler
Columbia
Urbina
Regensburg
Richter
Regensburg
Sangita Bose, Tata,
Max Planck Stuttgart
Kern
Ugeda, Brihuega
arXiv:0911.1559
Nature Materials
L
1. Analytical description
of
a
clean, finite

size
BCS superconductor?
2.
Are
these results
applicable to realistic
grains?
Main goals
3
. Is it possible to
increase the critical
temperature?
The
problem
Semiclassical
1/k
F
L <<1
Berry, Gutzwiller, Balian
Can I combine
this?
Is it already
done?
BCS gap equation
?
V
finite
Δ
=
?
V bulk
Δ
~
D
e

1/
Relevant Scales
Mean level spacing
Δ
0
Superconducting gap
F
Fermi Energy
L typical length
l coherence length
ξ
Superconducting
coherence length
Conditions
BCS
/
Δ
0
<< 1
Semiclassical
1/k
F
L << 1
Quantum coherence
l >> L
ξ
>> L
For Al
the optimal region is L ~ 10nm
Go ahead!
This has not been
done before
Maybe it is possible
It is possible but,
is it relevant?
If so, in what range of
parameters?
Corrections
to BCS
smaller or
larger?
Let’s think about this
A little history
Parmenter
,
Blatt
, Thompson (60’s) : BCS in a
rectangular
grain
Heiselberg (2002): BCS in harmonic potentials, cold atom appl.
Shanenko, Croitoru (2006): BCS in a wire
Devreese (2006): Richardson equations in a box
Kresin, Boyaci, Ovchinnikov (2007) Spherical grain, high T
c
Olofsson (2008): Estimation of fluctuations in BCS, no correlations
Superconductivity in
particular geometries
Nature of superconductivity (?)
in ultrasmall systems
Breaking of superconductivity for
⼠
Δ
0
> 1?
Anderson (1959)
Experiments
Tinkham
et al. (1995)
.
Guo
et al., Science
306, 1915, Superconductivity Modulated by
quantum Size
Effects.
Even for
/
Δ
0
~ 1 there is
“
supercondutivity
T
= 0 and
⼠
Δ
0
>
1
(1995

)
Richardson, von Delft, Braun, Larkin,
Sierra,
Dukelsky
,
Yuzbashyan
Thermodynamic properties
Muhlschlegel
,
Scalapino
(1972
)
Description beyond
BCS
Estimation.
No rigorous!
1.Richardson’s
equations:
Good but
Coulomb,
phonon
spectrum
?
2.BCS fine until
/
Δ
0
~ 2
/
Δ
0
>> 1
We are in business!
No
systematic
BCS
treatment of the dependence
of size and shape
Hitting a bump
Fine, but the
matrix
elements?
I ~1/V?
I
n,n
should admit a
semiclassical expansion
but how to proceed?
For the cube
yes but for a
chaotic grain
I am not sure
λ
/V ?
Yes, with help,
we
can
From desperation to hope
)
,
,
'
(
)
'
,
(
2
2
L
f
L
k
B
L
k
A
I
V
F
F
F
?
Regensburg, we have got a problem!!!
Do not worry. It is not
an easy job but you are
in good hands
Nice closed
results that do
not depend on
the chaotic
cavity
f(L
,

’,
F
) is a
simple function
For l>>L ergodic
theorems assures
universality
Semiclassical (1/k
F
L >> 1) expression
of the matrix elements valid for l >> L!!
ω
=

’
A few
months
later
Relevant in
any mean field approach with
chaotic one body dynamics
Now it is easy
3d chaotic
Sum is
cut

off
ξ
Universal function
Boundary
conditions
Enhancement of SC!
3d chaotic
Al grain
k
F
= 17.5 nm

1
㴠㜲㜹⽎/浖
0
= 0.24mV
L = 6nm, Dirichlet,
/
Δ
0
=0.67
L= 6nm, Neumann,
/
Δ
0,
=0.67
L = 8nm, Dirichlet,
/
Δ
0
=0.32
L = 10nm, Dirichlet,
/
Δ
0
,= 0.08
For L< 9nm leading
correction
comes
from
I(
,
’)
3d integrable
Numerical & analytical
Cube & rectangle
From theory to
experiments
Real (small) Grains
Coulomb interactions
Surface Phonons
Deviations from mean field
Decoherence
Fluctuations
No, but no strong
effect expected
No, but screening
should be effective
Yes
Yes
No
Is it taken into
account?
L ~ 10 nm Sn, Al…
Mesoscopic corrections versus
corrections to mean field
Finite size
corrections to BCS
Matveev

Larkin
Pair breaking
Janko,1994
The leading mesoscopic corrections
contained in
⠰(牥慲来爠
周攠
捯牲散瑩潮c
瑯t
⠰(灲潰p牴楯湡氠瑯t
桡h楦晥牥湴f獩杮
Experimentalists are coming
arXiv:0904.0354v1
Sorry but in
Pb only
small
fluctuations
Are you
300% sure?
Pb and Sn are very different because their
coherence lengths are very different
.
!!!!!!!!!!!!!
!!!!!!!!!!!!!
!!!
However
in Sn is
very
different
BN
STM
tip
Pb/Sn
nano

particle
Rh(111)
V
I
BN
STM
tip
Pb/Sn
nano

particle
Rh(111)
V
I
5
.
33
Å
0
.
00
Å
5
.
33
Å
0
.
00
Å
0 nm
7 nm
dI
/
dV
)
(
T
+
Theory
Direct observation of thermal fluctuations
and the gradual breaking of
superconductivity in single, isolated
Pb
nanoparticles
?
Pb
Theoretical
description of
dI
/
dV
Thermal fluctuations + BCS Finite size
effects + Deviations from mean field
dI
/
dV
)
(
T
?
Solution
Dynes
formula
Dynes fitting
Problem:
>
Thermal fluctuations
Static Path approach
BCS finite size effects
Part I
Deviations from BCS
Richardson formalism
No quantum fluctuations!
Finite T
How?
T=0
BCS finite size effects
Part I
Deviations from BCS
Richardson formalism
No quantum fluctuations!
Not important h ~ 6nm
Altshuler
,
Yuzbashyan
, 2004
Cold atom physics and novel
forms of superconductivity
Cold atoms
settings
Temperatures can be lowered
up to the nano Kelvin scale
Interactions can be controlled
by
Feshbach
resonances
Ideal
laboratory to
test quantum
phenomena
Until
2005
2005

now
1. Disorder &
magnetic fields
2. Non

equilibrium
effects
3. Efimov physics
Test
ergodicity
hypothesis
Bound states of three
quantum particles do exist
even if interactions are
repulsive
Test of Anderson
localization, Hall Effect
Stability of the
superfluid
state in a disordered 1D
ultracold
fermionic
gas
Masaki
Tezuka
(U. Tokyo),
Antonio M. Garcia

Garcia
What is the effect of disorder
in 1d Fermi gases?
arXiv:0912.2263
Why?
DMRG analysis of
Speckel potential
pure random with correlations
localization for any
Our model!!
quasiperiodic
localization transition at finite
2
speckle
incommensurate lattice
Modugno
Only two types of disorder can be
implemented experimentally
Results I
Attractive interactions
enhance localization
U = 1
c
= 1<2
Results II
Weak disorder
enhances
superfluidity
Results III
A pseudo gap
phase
exists.
Metallic
fluctuations
break long
range order
Results IV
Spectroscopic
observables are
not
related
to long
range order
Strongly coupled
field theory
Applications in high
Tc
superconductivity
Why?
Powerful tool to deal with
strong interactions
What
is
next?
Transition from qualitative
to
quantitative
Why
now?
New field.
Potential for high impact
N=4 Super

Yang Mills
CFT
Anti de Sitter space
AdS
String theory meets
condensed matter
Phys. Rev.
D 81
, 041901 (2010)
JHEP
1004:092
(2010)
Collaboration with string theorists
Weakly
coupled
gravity dual
Problems
1.
Estimation of the validity of the
AdS

CFT approach
2. Large
N
limit
For what condensed matter systems
these problems are minimized?
Phase Transitions triggered by thermal
fluctuations
1. Microscopic Hamiltonian is not
important
2. Large N approximation
OK
Why?
1.
d=2
and AdS
4
geometry
2. For c
3
= c
4
= 0
mean field results
3. Gauge field A is U(1) and
is a scalar
4.
A realization in string theory and M theory is
known
for certain choices of ƒ
5.
By tuning ƒ we can reproduce many
types of phase transitions
Holographic approach to phase transitions
Phys. Rev.
D 81
, 041901 (2010)
For c
4
> 1 or c
3
> 0 the transition becomes first order
A
jump in the condensate at the
critical temperature is clearly
observed for c
4
>
1
The
discontinuity for c
4
> 1 is
a signature of a first order
phase transition.
Results I
Second order phase transitions with non mean field
critical exponents different are also accessible
1. For c
3
<

1
2
/
1
1
2
c
T
T
O
2. For
2
/
1
1
2
Condensate for c
=

1
and c
4
= ½.
β
= 1, 0.80,
0.65, 0.5 for
=
3, 3.25,
3.5, 4,
respectively
2
1
Results II
The spectroscopic gap
becomes
larger and
the coherence peak narrower as c
4
increases
.
Results III
Future
1. Extend results to
β
<1/2
2. Adapt holographic techniques to spin discrete
3. Effect of phase fluctuations.
Mermin

Wegner
theorem?
4. Relevance in high temperature superconductors
THANKS!
Unitarity regime
and Efimov states
3 identical bosons with a
large
scattering length
a
1
/
a
Energy
trimer
trimer
trimer
3 particles
Ratio
= 514
Efimov trimers
Naidon,
Tokyo
Bound states exist even
for repulsive interactions!
Predicted by V.
Efimov in 1970
Form an infinite series
(scale invariance)
Bond is purely
quantum

mechanical
What would I bring to
Seoul
National
University
?
Expertise in interesting problems in condensed
matter theory
Cross disciplinary profile and interests with the
common thread of superconductivity
Collaborators
Teaching and leadership experience from a top US
university
Decoherence and
geometrical deformations
Decoherence
effects and small geometrical
deformations
weaken
mesoscopic
effects
How much?
To what extent is
our formalism
applicable?
Both effects can be
accounted analytically by
using an effective cutoff in
the trace formula for the
spectral density
Our approach
provides
an effective
description of
decoherence
Non oscillating
deviations present
even for L ~ l
What
next?
Quantum Fermi gases
From few

body to many

body
Discovery of new forms
of quantum matter
Relation to high Tc
superconductivity
1. A condensate that is non zero at low T and that
vanishes at a certain T =
T
c
2. It is possible to study different
phase transitions
3. A string theory embedding is
known
Holographic approach to phase transitions
Phys. Rev.
D 81
, 041901 (2010)
A U(1) field
, p scalars
F Maxwell tensor
E. Yuzbashyan,
Rutgers
B. Altshuler
Columbia
JD Urbina
Regensburg
S. Bose
Stuttgart
M. Tezuka
Kyoto
S. Franco,
Santa Barbara
K. Kern,
Stuttgart
J. Wang
Singapore
D. Rodriguez
Queen Mary
K. Richter
Regensburg
Let’s do
it!!
P. Naidon
Tokyo
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