finite size scaling and quantum criticality

8/8/2019 Finite Size Scaling and Quantum Criticality

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Finite Size Scaling andFinite Size Scaling and

Quantum CriticalityQuantum Criticality

Sabre Kais

Department of ChemistryBirck Nanotechnology Center

Purdue UniversityWest Lafayette, IN 47907


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Kais

Research GroupDepartment of ChemistryPurdue University

Finite Size Scalingin

Quantum Mechanics

Quantum Computingand

Quantum InformationTheory

Qi WeiWinton Moy

Dr. Pablo Serra

Zhen Huang

Hefeng WangXu Qing


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IntroductionIntroduction

Finite Size Scaling In Quantum MechanicsFinite Size Scaling In Quantum Mechanics

ApplicationsApplications

(A)(A) Stability of atoms, molecules and quantum dotsStability of atoms, molecules and quantum dots

(B)(B) Atomic and molecular stabilization in superintenseAtomic and molecular stabilization in superintenselaser fieldslaser fields


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Statistical Mechanics

Classical Quantum

Phase Transitions

Free EnergyF(Ki)=-KBT log(Z)

Phase Transitions

T 0Ground State :)(0 iE

Critical Phenomena

Correlation Length )(~ CTT

Critical Phenomena

Mass Gap of H )(~

1CE

Finite Size ScalingThermodynamic Limit

N

Finite Size ScalingNumber of Basis Functions

M

Applications Applications


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Phase Transitions

Classical:Classical: Classical phase transitions are driven by thermal energyClassical phase transitions are driven by thermal energyfluctuationsfluctuations

Like the melting of an ice cube:Like the melting of an ice cube:

Solid GasLiquid

P

T

SolidLiquid

Gas

CP

Quantum:Quantum: Quantum phase transitions, at T=0, are driven by theQuantum phase transitions, at T=0, are driven by theHeisenberg uncertainty principleHeisenberg uncertainty principle

Like the melting of a Wigner crystal: Two dimensional electLike the melting of a W igner crystal: Two dimensional electronron

layer trapped in a quantum w elllayer trapped in a quantum well

crystalWigner liquidFermi


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Quantum Phase Transitions

Transitions that take place at the absolute zero oftemperature, T=0, where crossing the phaseboundary means that the quantum ground state

energy , of the system changes in somefundamental way.

This is accomplished by changing some parameterin the Hamiltonian of the system .

We shall identify any point of non-analyticity in theground state energy , as a quantum phase

transition.

)(0 E

)(0 H

C =


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Phase Transition

Free energy: f[K]=-kBT log[Z]

Coupling Constants: {K1, K2; KD}

As a function of [K], f[K] is analytic almost everywhere

Possible non-analyticities of f[K] are points (DS=0),

lines (DS=1), planes (DS=2),

Regions of analyticity of f[K] are called phases

PartitionFunction


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Phase Transitions

First-order:

is discontinuous across a phase boundaryiKf

Continuous Phase Transition:

All are continuous across the phase boundary.But, second derivatives or higher derivatives arediscontinuous or divergent

iKf


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Phase Transitions

P

TCT

CP

SolidLiquid

Gas

CP

T

CT

Liquid

PhaseOne

Gas

PhasesTwo

327.0 =

+

CTT

CriticalExponentOrder

Parameter


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Critical Exponents

The critical exponents describe the nature of the singularities in various measurablequantities at the critical point ],,,,,[

In the limit CTT

Heat Capacity:

Order Parameter:

Susceptibility:

Equation of State:

Correlation Length:

Scaling Laws: Fisher:

Rushbrooke:

Widom:

Josephson:

)2( =

22 =++

)1( =

= 2d

CTTC~

CTTM ~

CTT~

1~

CTT~


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Critical Exponents],,,,,[

ExponentExponent THTH EXPTEXPT MFTMFT ISING2ISING2 ISING3ISING3 HEIS3HEIS3

00--0.140.14 00 00 0.120.12 --0.140.14

0.320.32--0.390.39 1/81/8 0.310.31 0.30.3

1.31.3--1.41.4 11 7/47/4 1.251.25 1.41.4

44--55 33 1515 55

0.60.6--0.70.7 11 0.640.64 0.70.7

0.050.05 00 0.050.05 0.040.04

22 2.002.000.010.01 22 22 22 22

11 0.930.930.080.08 11 11 11

11 1.021.020.050.05 11 11 11 11

11 4/d4/d 11 11 11

++ 2

)(

)2( d)2(

TH. Theoretical values (from scaling laws); EXPT. Experimental values (from a variety of

systems); MFT. Mean field theory; ISINGd. Ising model in d dimension; HEIS3. classicalHeisenberg model. D=3

Kenneth G. Wilson (1982)


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Renormalization GroupRenormalization Group

The Nobel Prize in Physics 1982The Nobel Prize in Physics 1982

" for his theory for critical phenomena inconnection w ith phase transitions"

Kenneth G. Wilson

Cornell University

Q t P titi F ti Z


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Z=Tr exp[-H]

[ ]N

ee NHH

==

andintervaltimeis

Quantum Partition Function Z

=

n mmmm

HNHHH

N

nemmemmemmenZ,...,,,

32211

321

)(

Has the form of a classical partition function, i.e. a sum over

configurations expressed in terms of a transfer matrix, if we think

about the imaging time as an additional spatial dimension

dimensions1)(dClassical

itdimensionsdQuantum

+

+

iT

ee iHTH

operator)evolution(Time:matrixdensityOperator

FeynmanPath-Integral


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Analogies

Classical (KT)Classical (KT) Quantum (T 0)Quantum (T 0)

(D+1) space dimension(D+1) space dimension D space, 1 timeD space, 1 time

Transfer MatrixTransfer Matrix Quantum HQuantum HPartition FunctionPartition Function Generating FunctionalGenerating Functional

EqualibriumEqualibrium StateState Ground StateGround State

Ensemble AverageEnsemble Average Expectation ValuesExpectation Values

Free EnergyFree Energy Ground State of HGround State of H

TemperatureTemperature Coupling ConstantCoupling ConstantCorrelation FunctionsCorrelation Functions PropagatorsPropagators

Correlation length ( )Correlation length ( ) Mass Gap of HMass Gap of H )( E1


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Finite Size Scaling

In statistical mechanics, the finite size scaling method provides a

systematic way to extrapolate information obtained from a finite

system to the thermodynamic limit

Importance

The existence of phase transitions is associated w ith singularities

of the free energy. These singularities occur only in the

thermodynamic limit.

Yan g an d Lee Phy s. Rev . 87 , 40 4 ( 19 52 )

The Nobel Prize inPhysics 1957

Fi it i ff t i


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Finite-size effects inStatistical Mechanics

Cv

TC

N'''

N

N'

N''

T


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In the present approach, the finite size corresponds not to the

spatial dimension, as in statistics, but to the number of elementsin a complete basis set used to expand the exact eigenfunction of a

given Hamiltonian.

Quantum Mechanics

=

==

M

n

nn

n

nn aa00

(Variational Calculations)

)(

Classical

N )(

Quantum

M

Kais et . a l Phys. Rev. Letters 79, 3142 (1997)


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Finite Size Scaling: Quantum Mechanics

VHH += 0

=)(

)()( )(NM

n

nN

nN

a

The FSS ansatz

CON

NFOO ~)(

)ln(

ln

),;(

)()(

NN

OO

NN

NN

O =

The curves intersect at the critical point

),;(),;( NNNN COCO =


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Short Range PotentialsYukawa Potential

reH

r

= 221)(

Basis Set

)()()2)(1(

1),( ,

)2(2/ ++

=

mllnr

n YrLelnln

r

Where is the Laguerre polynomial of degreen and order 2 and are the spherical harmonic

functions of the solid angle

)(, mlY

)()2( rL ln

Hamiltonian

Yukawa


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839903.0=C

Yukawa

=

VH

H 839908.0=exact

C

Phys. Rev. A 57, R1481 (1998)


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Gaussians Slater

Finite Size Scaling : Data


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Finite Size Scaling : DataCollapse

)(~

CC

O +

CON NFOO ~

/

0~)(

xxFx

/~ NO

NN

)(~ /1/ CN NGNO

)(~/1/

0 CNGNE

Data Collapse ( )~/ GNE ( ) /1N


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Data Collapse

1,2 ==

( ) /1NC

Chem. Phys. Letters 319, 273 (2000)

( )~0 GNE ( ) NC


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-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.82 0.86 0.90 0.94 0.98

840.0=c

)Re(

)Im(

56

4=N

11

Yukawa 4=N

Kais et al. Chem. Phys. Letters 423, 45 (2006)

Quantum Mechanics


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Quantum Mechanics

ComplexAngular

Momentum

ComplexTime

ComplexEnergy

ComplexCharge

Regge

Theory

Instanton

Method

Rotation

ComplexCoordinates

Variation

Method

ScatteringAmplitude

Tunneling ResonancesStability

ofIons

Two Electrons Atoms


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1221

2

2

2

1

11

2

1

2

1)(

rrrH

+=

2112 rrr = r12

r2

r1

e-

e-

Z

ZC=0.911

ET=-0.5

He like system

)(gsE

H like +e-

ZC=0.911

H- He Z

Z

1=

Fi it Si S li d


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Finite Size Scaling procedure

Hamiltonian:

Basis Set:

Hamiltonian Matrix:

Renormalization Equation:

ZrrrH

1;

11

2

1

2

1

1221

2

2

2

1 =+=

( ) +=kji

krrijrrjikji rerrerrC

,,

12

21

21,, 2

12121

Nkji ++

10 E,Es,EigenvalueLeading

1

0

1

0

1

)1()1(

)()(

+

++=

NN

NENE

NENE


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097.1=C

Kais et . a l , Phys. Rev. Letters 80, 5293 (1998)

Conditional Probability


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C>

C=

C2.5D

c=0.655a.u

FSS for Critical Conditions forS bl Q d l d


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Stable Quadrupole Bound Anions

Hamiltonian: consists of a charge q at the origin and two charges q/2 atz = +1 & -1

Where is the variational parameter used to optimize the numerical results

and is the Legendre polynomial of order l.

Slater Basis Set:

)(2 lP

q q=QR


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Journal of Chemical Physics, 120, 8412 (2004)

qc=1.46 a.u

+

2,,

2

qq

q


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Journal of Chemical Physics, 120, 8412 (2004)

qc=3.9 a.u

++ 2,,2

qq

q

KCl2- Qzz=10 a.u

K2Cl- Qyy=27 a.u

(BeO)2- (13,13,0.5)

CS2- Qzz=4 a.u

Hexagonal Lattice of Quantum Dots


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Hexagonal Lattice of Quantum Dots

Hubbard Model


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Hubbard Model

> denotes

the nearest-neighbor pairs.

)( ii cc+

iii ccn+=


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Experimentally, one can prepare quantum dots with:

Different metal and this primarily determines the chemical

potential

Different sizes and this primarily determines U

Different packing distance and this primarily determines t

So the experimental Hamiltonian has 3 different parameters,

, U and t, each of which can be independently varied. Theessential physics are embodied in the Hubbard Hamiltonian

Why do we need to useRG method?

Why do we need to useRG method?


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RG method?RG method?

210 States

N=7, 784 states in total

14 States420 States 140 States

N=19, 2 billion states

RG ProceduresRG Procedures


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1. Divide the N-site lattice into appropriate ns -site blocks

labled by p (p=1,2,,N/ns) and separate the HamiltonianH

into intrablock partHIB and interblockHB ,

2. Solve HB exactly to get the eigenvalues, the

eigenfunctions of

3. Treat each block as one site on a new lattice and

the correlations between blocks as hopping

interactions

>

+',

',pp

pp

p

pIBB VHHHH

)4,....2,1( snpi iE =

Triangular lattice withHexagonal Block

Triangular lattice withHexagonal Block


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Hexagonal BlockHexagonal Block

Calculation Results for Metal-Insulator Transition

Calculation Results for Metal-Insulator Transition


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0 5 10 15 200

2

4

6

8

10

Metal Insulator

(U/t)c=12.5

U/t

p

/t

Charge gap:)(2)1()1( NENENEg ++=

Phys. Rev. B66, 081101 (2002)

Finite-size Scaling for Hubbardmodel on triangular lattice

Finite-size Scaling for Hubbardmodel on triangular lattice


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6 8 10 12 14 16 18 20 2210

-3

10-2

10-1

100

101

102

103

104

(U/t)c=12.52

N=7,72,7

3,7

4,7

5,7

6,7

7

U/t

g

N0.4

05/t

-6000 -3000 0 3000 6000 900010

-3

10-2

10-1

100

10

1

102

103

104

(U-U c)N0.5

/t

g

N0.4

05/t

model on triangular latticeode o t a gu a att ce

J.X.Wang, S. Kais, Phys.Rev.B66,081101(2002)

Atomic & Molecular Stabilizationby


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by

Superintense Laser FieldswithProf. Dudley Herschbach

Harvard University

The Nobel Prize in Chemistry 1986


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Multiply charged negativeatomic ions

in superintense laser fields

The peak-power ofpulsed lasers hasincreased by 12 ordersf it d d i

Superintense Laser Fields (I > a.u.)


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of magnitude during

the past 4 decades.

QUESTION: What is thehighest-level intensitypresently possible?

ANSWER: The highestpossible focused laserintensity =1019 W/cm2

This level of intensity canbe achieved withfemtosecond laser basedon Chirped PulseAmplification (CPA).

The electric field of a monochromatic plane waveb

Laser Atom Interaction


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can be written

)sintancos()( 0 teteEtE yx +=

ti

rrtrp

N

i

i

j jii

i=

+

++

=

=1

1

1

2 1

)(

1

2

1rr

rr

With ex and ey orthogonal to each other and to thepropagation direction

)()()( 00 tEEtwherer

r

=

0=E0/2, whereE0 and are the amplitude and frequency ofthe laser field.

Moving frame of reference which follows the quiver motion of the classical electron

on.polarizaticirculartoscorrespond/4

onpolarizatilineartoscorrespond0

=

=

High Frequency Floquet Theory


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Vo is the dressed Coulomb potential, is the time average of the shiftedCoulomb over one period of the laser

++=

2

0 000 )sintancos(2

1),(

yx eear

draV

v

Hamiltonian of atomic system in super-intense laser fields

( ) =

=

++=

N

i

i

j ji

ioirr

arVpH1

1

1

02 1

,2

1

e-e termmlaser terermcoulomb trmkinetic teH +

frequency.theisandintensitybeamaveraged-timetheisIwhere,2 Io =

)(EH 0=Structure Equation:

Hydrogen atom in super-intense linear laser fields


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J. H. Eberly, et alScience 262, 1229 (1993)M. Pont, et al. Phys. Rev. Lett. 61, 939 (1988)


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The electronic orbitals for the ground states. The presentation is given in a plane

passing through the axis of the field taken as polar axis z


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Do atoms in superintense laser fields behavelike diatomic molecules ?


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Summary

Symmetry breaking of electronic structure


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Symmetry breaking of electronic structure

configurations resemble classical phase transitions.Phys. Rev. Letters 77, 466 (1996)

J. Chem. Phys. 120, 2199 (2004)

Phys. Rev. Letters 79, 3142 (1997)

J. Chem. Phys. 120, 8412 (2004)

Phys. Rev. Letters 80, 5293 (1998)Adv. Chem. Phys. 125, 1-100 (2003)

Finite Size Scaling method can be used to obtaincritical parameters for the quantum Hamiltonian:Critical charges, critical dipole and quadrupole-bound

anions,

Quantum phase transitions can be used to explainand predict the stability of atoms, molecules andquantum dots.


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AtomicAtomic dianionsdianions are unstable in the gas phaseare unstable in the gas phase

Multiply charged negative ions are stable inMultiply charged negative ions are stable in

superintense laser fieldssuperintense laser fields

(Intensity > 1(Intensity > 1 a.ua.u. ~. ~ 1016 W/cm2)

J. Chem. Phys. 124, 201108 (2006)


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Thank You!Thank You!

Question 1Question 1


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Question 2Question 2


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finite size scaling and quantum criticality

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