Emergent Phenomena in mesoscopic systems

S. Frauendorf

Department of Physics

University of Notre Dame

An emergent behaviour or emergent property can appear when a number of simple entities (agents) operate in an environment, forming more complex behaviours as a collective

Emergent structures are patterns not created by a single event or rule. There is nothing that commands the system to form a pattern, but instead the interactions of each part to its immediate surroundings causes a complex process which leads to order

Emergent structures and properties in nature

The complex behaviour or properties are not a property of any single such entity, nor can they easily be predicted or deduced from behaviour in the lower-level entities: they are irreducible. 1

Living systems-ant colony

  A termite "cathedral" mound produced by a termite colony:

a classic example of emergence in nature.

A more detailed biological example is an ant colony. The queen does not give direct orders and does not tell the ants what to do. Instead, each ant reacts to stimuli in the form of chemical scent from larvae, other ants, intruders, food and build up of waste, and leaves behind a chemical trail, which, in turn, provides a stimulus to other ants. Here each ant is an autonomous unit that reacts depending only on its local environment and the genetically encoded rules for its variety of ant. Despite the lack of centralized decision making, ant colonies exhibit complex behavior and have even been able to demonstrate the ability to solve geometric problems. For example, colonies routinely find the maximum distance from all colony entrances to dispose of dead bodies.

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Emergence means complex organizational structure growing out of simple rule. (p. 200)

Macroscopic emergence, like rigidity, becomes increasingly exact in the limit of large sample size, hence the idea of emerging. There is nothing preventing organizational phenomena from developing at small scale,…. (p. 170)

Protection generates exactness and reliability,… The universal properties of ordering of rigid bodies,the flow of superfluids, and even the emptiness ofspace are among the many concrete,well documented examples of this effect. (p. 144)

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Physics

Emergent phenomena

• Liquid-Gas Phase boundary• Rigid Phase – Lattice• Superconductivity (Meissner effect, vortices)• Laws of Hydrodynamics• Laws of Thermodynamics• Quantum sound• Quantum Hall resistance• Fermi and Bose Statistics of composite particles• … • …

8.258122

h

e

4

Mesoscopic systems

Emergence of a macroscopic phenomena with N.

52 1010~ N

Appearance of “finite size corrections” to familiar macroscopic phenomena in very small probes (quantum dots, quantum wells,quantum junctions, quantum wires).

T. P.Martin Physics Reports 273 (1966) 199-241

Emergence of cubic crystal structure in salt clusters

5

Abu

ndan

ce in

the

clus

ter

beam

fcc lattice: Close packing with translational symmetry

Icosahedra: Close packing with small surface

bulk

Ca clusters: the transition to the bulk is not smooth

T. P.Martin Physics Reports 273 (1966) 199-241 6

Abu

ndan

ce in

the

clus

ter

beam

Water - dramatic example

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8

Emergent phenomena - nuclei

• The nucleon liquid • Superfluidity,

superconductivity• Shell structure • Spatial orientation• Temperature• Phases and phase

transitions

Extrapolationto bulk

Finite nuclei

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Neutron stars

SGR 1806-20

TeslaB 810

Suprafluid, superconducting nuclear matter and more.

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Studying the scaling of clusters properties seems instructive, because these properties are well known for the bulk.

Astrophysics:What is the equation of state for nuclear matter?

Nuclei are only stable for A<300.

Clusters can be made for any N.

Liquid drop model:Volume + Surface energy

3/2NaNaE SVB

Transition to the bulk liquid

Neutral –one component

3/1 NaaN

ESV

B

Coulomb energy

The liquid drop model scaling law seems reliable.

8

Binding energy of K clusters

Ionization energy of Na clusters

3/1

)(

Naa

NEIE

Cb

coulombbulk

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Other quantitiesscale in the same way.

223/423/1 )( AZNaAZaAaaA

ESCSV

B

What is the bulk equation of state?

For example: compressibilityd

dE

How good is it? Symmetry energy ????

Nuclei: charged two-component liquid

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Strong correlation

Is there a term ?44)( AZN

Clusters mayprovide examplesfor scaling.

He droplets – getting really close to nuclei

clusters are most similar to nuclei.He3

Liquid at zero temperatureElectrical neutral: Limit N-> easily achieved.

Very hard to experiment with, because of small energy scale.

He3 clusters probably exist only for N>50

He4 produced for all N.

Strong zero point motion. Weakly bound nuclei

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Study of : theoryHe4

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Experiment?

Superconductivity/SuperfluidityDescribed by the Landau – Ginzburg equations forthe order parameter

Controlled by ( inside the superconductor)

coherence length (size of Cooper pair)

penetration depth of magnetic field

/0 Fv

G/)()( rr

/)/( 2/1220 GemcL

G, , Fermi energy , and critical Temperature related by BCS theory.cT

2/2FF mv

13

2|)(| r Density of Cooper pairs

T

H

normal

super

Phase diagram of a macroscopic type-I superconductor

32

Meissner effect

Type II superconductor

Solid state, liquid He:Calculation of very problematic – well protected.Take from experiment.cT

K

K

T

T

N

N

F

c

F510

1~~~

RmvF 15~/0 local

BCS very good

Nuclei: Calculation of not possible so far. Adjusted to even-odd mass differences.

fmRfmvF 5~40~/0 highly non-local

MeV

MeV

T

T

N

N

F

c

F 40

1~~~

BCS poor

How to extrapolate to stars?

Vortices, pinning of magnetic field?

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Mesoscopic regime

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Superfluidity

2

1

Intermediate state ofReduced viscosity

Atttractive interaction between Fermions generates Cooper pairs -> Superfluid

He3

rigid

Moments of inertia at low spin are well reproduced by cranking calculations including pair correlations.

irrotational

Non-local superfluidity: size of the Cooper pairs largerthan size of the nucleus.

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19

is superfluidat this T.

He4

is not superfluidat this T.

He3

6SF

Rotational spectrum of in a droplet He46SF

free

He 60 4 behave like a superfluid!

Rotational spectrum of in a droplet HeHe-43OCSD

ensity of “normal” atom

sMoment of inertia largerTitle: SUPERFLUID HELIUM DROPLETS: AN ULTRACOLD NANOLABORATORY ,  By: Toennies, J. Peter, Vilesov, Andrej F., Whaley, K. Birgitta, Physics Today, 0031-9228, February 1, 2001, Vol. 54, Issue 2

Shell structure

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6/1

6/5 ,

NE

E

NENE

sh

sh

Fermions in spherical Potential

Clusters:More washed out.Dies out quicker.Not quantitativelyunderstood.

Nuclei: magnitude OK,damping with N and T OK.

canonical

400KT

Frauendorf,Pashkevich

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Explains the gross shell structure

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Clusters allow us to study shell structureover a much larger rangethan nuclei.

N-dependent factor multiplied for compensating the toorapid damping with N!

Supershell structure of Na clusters

Emergence of resistivity? 24

Imax>20

rgid

Currents causedby nucleons onperiodic orbits

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Nuclear moments of inertia at high spinPair correlations are quenched.

M. Deleplanque, S.F. et al. Phys. Rev. C 69 044309 (2004)

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RigV :lity susceptibi

mc

eBL 2

Larmor: System in Magnetic field behaveslike in rotating system(in linear order).

Emergence of thermodynamicsRegion of high level density: important for astrophysics, nuclear applications, …

dS

dETS

ature temperlnentropy

intervallenergy

states #density level average

Limits to predictability of quantal states: uncertainties in the Hamiltoniandeterministic chaos

Give up individual quantal states:

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Crossover phenomena

Phase transitions

exists N existnot does N

T=0 transitions betweendifferent symmetries innuclei.Spherical deformedIBA symmetries

Artificial limit by mean field approximation

solid-liquidsuperfluid-normalliquid-gas

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The Casten Triangle of IBM

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T

H

normal

super

Super-normal phase transition

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Grand canonicalCanonicalMicrocanonical

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Grandcanonical ensemble

Canonical ensemble

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Melting of Na clusters

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M. Schmitd et al.

Microcanonical

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dE

dS

T

1q latent heat

Transition from electronic to geometric shellsIn Na clusters

KT 250~

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More can be found in:S. Frauendorf, C. Guet, Ann. Rev. Nucl. Part. Sci. 51, 219 (2001)

Similar emergent phenomena in nuclear andnon-nuclear mesoscopic systems.

New principles of organization (+ parameters) – to be found.

Comparing different types mesoscopic systems is instructive.

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Studies are complementary: bulk limit accessible or not, energy scale, external heat bath, ….

More contact between the communities!

Emerge with increasing particle number, while calculatingthem microscopically becomes increasingly difficult.

Region where micro and makro calculations are possible.

Quantization of magnetic flux in type II superconductors

Magneto-optical images of vortices in a NbSe2 superconducting crystal at 4.3 K after cooling in magnetic field of 3 and 7 Oe.

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Emergence of orientationExample for spontaneous symmetry breaking:Weinberg’s chair

Hamiltonian rotational invariant

2|),(|),

:ondistributidensity

IM| :momentumangular

good of seigenstate

IMYρ(r,

Why do we see the chair shape?

Tiniest perturbation mixes |IM>states to a stable-oriented wavepacket: the symmetry broken state.

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Mesoscopic variant I: Molecules

3NH

1

2

3Can be kicked and turnedlike a chair.

Quantal states |IM>can be measured:Rotational bands

Classical moments of inertiaof arrangement ofpoint masses.

1618

HCl

)1()()1(

)1()(

IBIEIE

JIIBIIE

Microwave absorptionspectrum

Deformed potential aligns thepartially filled orbitals

Partially filled orbitals are highly tropic

Nucleus is oriented – rotational band

Well deformed Hf174 -90 0 90 180 2700.0

0.2

0.4

0.6

0.8

1.0

over

lap

Mesoscopic variant II: Nuclei

Symmetry broken state described by the mean field.

How is orientation generated?

Riley

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Dynucleus

medsuperdefor thefrom rays - 152

Pbnucleus

spherical thefrom rays - 199

E2 radiation

M1 radiation

deformed

Er163

spherical

Pb200

Spontaneous symmetry breaking Emergence

Finite N: Localization Shell structure, center of mass motion

Orientation rotational alignment, … rotational bands

Symmetry breaking

Periodic crystal structure rigidity, transverse sound

B

dB

dRig :lity susceptibi

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