An excursion into modern superconductivity: from nanoscience to cold atoms and holography Yuzbashyan Rutgers Altshuler Columbia Urbina Regensburg Richter.

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. García-García

Superconductivity in nanograins

New forms of superconductivity

New tools String Theory

Increasing the superconductor

Tc

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/kF L <<1 Berry, Gutzwiller, Balian

Can I combine this?

Is it already done?

BCS gap equation

?V finite

Δ=?

V bulk Δ~

De-1/

Relevant Scales

Mean level spacing

Δ0 Superconducting gap

F Fermi Energy

L typical length

l coherence length

ξ Superconducting coherence length

Conditions

BCS / Δ0 << 1

Semiclassical1/kFL << 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 Tc

Olofsson (2008): Estimation of fluctuations in BCS, no correlations

Superconductivity in particular geometries

Nature of superconductivity (?) in ultrasmall systems

Breaking of superconductivity for / Δ0 > 1? Anderson (1959)

Experiments Tinkham et al. (1995) . Guo et al., Science 306, 1915, Superconductivity Modulated by quantum Size Effects.

Even for / Δ0 ~ 1 there is “supercondutivity

T = 0 and / Δ0 > 1 (1995-)

Richardson, von Delft, Braun, Larkin, Sierra, Dukelsky, Yuzbashyan

Thermodynamic propertiesMuhlschlegel, Scalapino (1972)

Description beyond BCS

Estimation. No rigorous!

1.Richardson’s equations: Good but Coulomb, phonon spectrum?

2.BCS fine until / Δ0 ~ 2

/ Δ0 >> 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?

In,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

),,'()',(22 LfLk

B

Lk

AIV F

FF

?

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/kFL >> 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

kF = 17.5 nm-1

= 7279/N mV

0 = 0.24mV

L = 6nm, Dirichlet, /Δ0=0.67

L= 6nm, Neumann, /Δ0,=0.67

L = 8nm, Dirichlet, /Δ0=0.32

L = 10nm, Dirichlet, /Δ0,= 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 (0) are larger

The correction to (0) proportional to has different sign

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)

VI

BN

STM tip

Pb/Sn nano-particle

Rh(111)

VI

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 THow?

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 gasMasaki 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 D

Our model!!

quasiperiodic

localization transition at finite = D 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

A solution looking for a problem

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 MillsCFT

Anti de Sitter spaceAdS

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 AdS4 geometry

2. For c3 = c4 = 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 transitionsPhys. Rev. D 81, 041901 (2010)

For c4 > 1 or c3 > 0 the transition becomes first order

A jump in the condensate at the critical temperature is clearly observed for c4 > 1

The discontinuity for c4 > 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 c3 < -1

2/112 cTTO

2. For 2/112

Condensate for c = -1 and c4 = 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 c4

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 = Tc

2. It is possible to study different phase transitions

3. A string theory embedding is known

Holographic approach to phase transitionsPhys. Rev. D 81, 041901 (2010)

A U(1) field , p scalars F Maxwell tensor

E. Yuzbashyan, Rutgers

B. AltshulerColumbia

JD Urbina Regensburg

S. Bose Stuttgart

M. Tezuka Kyoto

S. Franco, Santa Barbara

K. Kern, StuttgartJ. Wang

Singapore

D. RodriguezQueen Mary

K. Richter Regensburg

Let’s do it!!

P. NaidonTokyo