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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
q
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Journal of Chemical Physics, 120, 8412 (2004)
qc=3.9 a.u
++ 2,,2
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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