Diploma Thesis - ``Analysis of Dynamics in Phase Space by … · 2014. 11. 30. · Diploma Thesis...

Metal-Insulator-TransitionHusimi FunctionMonge Distance

Dynamics of the Zeros

Diploma Thesis

�Analysis of Dynamics in Phase Spaceby means of the Zeros of the Husimi Function�

Martin Aulbach

University of Tokyo,

University of Augsburg

December 11, 2006

Martin Aulbach Diploma Thesis



Introduction

solid-state-physics: models for metals and insulators?

we consider the Anderson-Model and Aubry-André-Model

Phase Space give simultaneous access to location andmomentum information of conduction electrons

Husimi Function represents quantum state in phase space

How does the Husimi Function change during the metal-insulator-transition? How can these changes be described orquanti�ed?

⇒ qualitative and quantitative description of the dynamics bymeans of the zeros of the Husimi Function




Physical SystemAnderson-ModelAubry-André-ModelNumerical Approach

Metal-Insulator-Transition

How can the behaviour of metals and insulators be modelled?

Can one simulate a �phase transition� between them?

⇒ di�erent approaches: bonding model, doping, impurities, . . .

here: impurities in the crystal lattice potential are considered

=⇒ We expect a transition from metal to insulator when the�strength� of the impurities is increased.





How to model the transition?

⇒ use a simple model:

consider a single e− on an in�nite lattice

the e− can hop between neighboring lattice sites

impurities are modelled as irregularities in the potential

atmoic potential not relevant

for numerical calculations: restriction to the 1D case





Hamiltonian of the Anderson-Model (1958):

H = −∞∑

x =−∞( |x + 1〉〈x |+ |x〉〈x + 1| ) + W

∞∑x =−∞

vx |x〉〈x |

x denotes the position of the lattice site

|x + 1〉〈x | and |x〉〈x + 1| are the hopping matrix entries(transition probabilities between neighboring lattice sites)

|x〉〈x | stands for the potential energy at lattice site x

vx ∈ [−12 ,

12 ] denotes the relative disorder strength at x

(the vx are randomly chosen and then remain �xed)

W ≥ 0 is the global disorder strength and can be varied





Hamiltonian of the Aubry-André-Model (1980):

H = −∞∑

x =−∞( |x + 1〉〈x |+ |x〉〈x + 1| ) + λ

∞∑x =−∞

cos(2πβx) |x〉〈x |

single di�erence to Anderson: other disorder potential

here the potential is not random, but deterministic

β :=√5−12 is irrational

=⇒ cos(2πβx) is not a periodic, but quasiperiodic potential

λ ≥ 0 is the global disorder strength (equivalent to W ≥ 0)





Comparison of the potentials:

random potential

quasiperiodic potential

periodic potential





How to calculate these model systems numerically?

only �nite systems are computable:∞∑

x =−∞−→

L−1∑x = 0

for this, consider a periodic ring with L lattice sites

periodic boundary condition: |L〉 := |0〉

⇒ allowed values for the electron's position and momentum:

x ∈ [ 0, L) x = 0, 1, . . . , L− 1

k ∈ [−π, π) k = −π,−L−2Lπ, . . . , L−2

Lπ

the above values of k result from

the periodicity of |ψ〉the convention to choose k from the �rst Brillouin zone





for a system with L lattice sites, H is a L× L matrix

Hψ(x) = E ψ(x) is solved by diagonalization of HH is real and symmetric ⇒ fast routines with O(L3) available

Aubry-André-Model: quasiperiodicity allows H to bedecomposed into two real, symmetric and tridiagonalmatrices ⇒ especially fast solvable

upon diagonalization, L eigenvalues and eigenvectors are obtained




Phase Space in Quantum MechanicsHusimi FunctionZeros of the Husimi FunctionObtaining the Zeros Numerically

Husimi Function

Advantage of phase space representations in physics:

simultaneous information about position and momentum of objectsis available (as opposed to position and momentum representation)

classical physics: object's state is well de�ned at arbitrarytimes by a point in phase space

quantum physics: Heisenberg uncertainty principle ⇒quantum object �smeared out� in phase space

How to obtain a quantum phase space representation of a givenquantum state |ψ〉?





there is no unique phase space representation in QM

several useful representations have been proposed, e.g.

Wigner Function:

W (x , k) =12π

∫dx ′ e ikx

′ψ∗

(x+

x ′

2

)ψ

(x− x ′

2

)

Husimi Function:

Q(x , k) = 2∫dx ′ dk ′ exp

(− (x − x ′)2

2σ2− 2σ2(k − k ′)2

)W (x ′, k ′)

The Husimi Function can be obtained from the WignerFunction by �smearing� W with a Gauÿ Function.

σ is the ratio of the position and momentum smearings





Properties of the Husimi Function:

positive de�nite: 0 ≤ Q(x , k) ≤ 1 (contrary to the WignerFunction)

can be regarded as a probability distribution in phase space

equivalent de�nition as the projection of the quantum stateonto a coherent state:

Husimi Function (alternative de�nition):

Q(x , k) =

∣∣∣∣ ∫dx ′ 〈x , k |x ′〉〈x ′|ψ〉

∣∣∣∣2 = |〈x , k |ψ〉|2

with the wavefunction 〈x |ψ〉 = ψ(x) and the coherent state |x0, k0〉in position representation

〈x |x0, k0〉 =(2πσ2

)− 14 exp

(−(x − x0)

2

4σ2+ ik0x

)





numerically, the Husimi Function can be calculated from ψ(x)by performing L Fast Fourier Transformations with O(L2lnL)

−25

0

25

−25 0 25

−25

0

25

−25 0 25

−100

0

100

−100 0 100

Figure: Husimi Functions of the coherent state |0, 0〉 in phase space fordi�erent system sizes L and unsharpness ratio σ. Q is zero in the darkblue area and maximal in the red area. In the left picture σ isnonsymmetric while it is symmetric in the other two.





Visualizing the metal-insulator-transition in phase space:

W = λ = 0: electron describes a free wave=⇒ momentum fully determined, but position arbitrary

W , λ very large: electron is localized at one lattice site=⇒ momentum arbitrary, but position fully determined

intermediate regime; W , λ ≈ O(1): versatile behaviour=⇒ speci�c distribution patterns in phase space

Anderson-Model: avoided crossings in energy spectrum ⇒�jumping� between di�erent eigenstates at intermediate W

Aubry-André-Model: no avoided crossings ⇒ always the sameeigenstate

(next �gure: Husimi Function of the Anderson-Model (top) andAubry-André-Model (below) at di�erent disorder strengths)





a) W = 0 b) W = 1 c) W = 3 d) W = 15 e) W = 1000

−π

0

π

kk

0 L

xx

0 L

xx

0 L

xx

0 L

xx

0 L

xx

a) λ = 0 b) λ = 1 c) λ = 2 d) λ = 2,5 e) λ = 100

−π

0

π

kk

0 L

xx

0 L

xx

0 L

xx

0 L

xx

0 L

xx





The Husimi Function has been de�ned as Q(x , k) = |〈x , k |ψ〉|2where 〈x , k |ψ〉 is the projection of |ψ〉 onto a coherent state.

De�nition of the Bargmann Function:

B(x , k) = 〈x , k |ψ〉

Q(x , k) = |B(x , k)|2

Q is a real, positive-valued function while B is complex-valuedcorrespondences:

B ∼= ψ(x) (�wave function� in phase space)Q ∼= |ψ(x)|2 (�probabilitiy� in phase space)





Properties of the Bargmann Function:

B(z) with z = x + ik is periodic in phase space,but not analytic (i.e. complex di�erentiable)

however, one can de�ne B(z) = B(z) exp{πL

(|z|22 + ixk

)}B is an analytic, but not periodic function

results about B can be obtained with complex analysis

analogous de�nition of Husimi Function: Q := |B|2

Leb÷uf and Voros found that B has L zeros and can beentirely constructed from its zeros by means of theWeierstrass-Hadamard Factorization





Factorization of the Bargmann Function:

B(z) = Z exp

π L∑

j = 1

(zj − z0)

z

L∏

j = 1

B1(z + z0 − zj)

with the normalization factor Z , the zeros z1, z2, . . . , zL, and themiddle point in phase space z0.

B1(z), the �factor function�, is related to a Jacobian ThetaFunction and has its only zero at z0B1(z) nearly radially symmetric ⇒ �generalization� of (z − z0)

by de�nition, B , B , Q and Q have the same zeros

=⇒ The Bargmann Function and equivalently the HusimiFunction are completely determined by their zeros!





How can the positions of the zeros be obtained in practice?

analytical calculation of Husimi Function is too complex

⇒ zeros have to be obtained from numerical results

we know L× L values of B (and thus of Q)

Two possible ways to extract the zeros:

1 Find the minima of the Husimi Function

2 Find the complex angle curls of the Bargmann Function





Minima of the Husimi Function:

Q(z) ≥ 0 for all z ∈ CQ(zi ) = 0 for all zeros zi

⇒ all zeros are minima

Are all minima zeros, too?

no proof for this, but all results indicate: �Yes�

composition of Q from elementary �radial� functions B1(z)makes additional minima seem rather unlikely

for a vast number of numerically examined Husimi Functionsall minima have been veri�ed as zeros (with the secondmethod)

visualization of the Husimi Function on a logarithmic scalereveals a very �good-natured� and smooth behaviour





−π

0

πkk

0 L

xx

0 L

xx

Figure: Left: Husimi Function on a linear scale. The positions of thezeros are indicated by crosses. Right: The same Husimi Function on alogarithmic scale where white points indicate the zeros. Note thesmoothness and the fact, that there are no bright spots in the rightpicture that aren't marked as zeros in the left picture.





−π

0

π

kk

0 5 10

xx

0 5 10

xx

Figure: Zeros that are close to each other may not be detectedseparately, if the resolution of the numerical data is too low.





Curls in the complex phase of the Bargmann Function:

B(z) is complex ⇒ B(z) = R(z) eiϕ(z)

consider only the complex phase ϕ(z) ∈ [0, 2π)

Weierstrass-Hadamard Factorization ⇒ ϕ(z) covers the wholecodomain near every zero

=⇒ zeros can be obtained reliably by �nding the curls of ϕ(z)

(next �gure: complex phase ϕ(z) of a Bargmann Function, drawn with acyclic color palette)





−π

0

π

kk

0 L

xxMartin Aulbach Diploma Thesis




numerical detection of curls:

check the monotonicity of 4 adjacent sample points

all zeros are detected (at least usually)

many false positive results

⇒ tighten the detection condition by requesting an additional�smoothness condition� for the curl

⇒ together, these methods allow for a reasonably good detectionof all L zeros

(next �gure: Left picture shows complex phase ϕ(z), upper right showsmany false positive detection results, lower right shows too strict�smoothness conditions�. In all pictures, the zeros are marked withcircles.)





−π

0

π

kk

0 L

xx




Monge ProblemLinear ProgrammingHungarian Method

Monge Distance

How to quantify the dynamical changes of the Husimi Function?

Idea: De�ne a measure of distance between two Husimi Fuctions!

1:1 correspondence between |ψ〉 and Q enables us to employ wellknown Density Operator Measures for this, e.g.

Trace Distance: Dtr(ρ1, ρ2) = tr√

(ρ1 − ρ2)2

Hilbert-Schmidt Distance: DHS(ρ1, ρ2) =

√tr

[(ρ1 − ρ2)

2]

Problem: They don't account for the shape of Husimi Functions.

Example: �semiclassical property� for coherent states not ful�lled:

D(|α1〉, |α2〉) 6= |α1 − α2|





Monge-Problem (1781):

How to move a pile of sand from one location to another with thesmallest possible e�ort?

x1

Qx2

Q1(x1, x2)Q2(x1, x2)

Mathematically: Find a map T : Q1 → Q2 between the grains ofsand of the two piles so that the transportation e�ort is minimized!





De�nition of the Monge Distance:

DM(Q1,Q2) = inf∫Rd

|x − T (x)|Q1(x) dnx

the map T �distorts� the plane so that Q1 is mapped to Q2

|x − T (x)| denotes the transportation distancein�mum is taken over all the T that map Q1 to Q2

x1

Qx2

DQ1

DQ2

T :R2→R2





How to evaluate this formula for two given Husimi Fuctions?

evaluation of integral is too complex!

simple-form analytical solutions exist only for 1D systems

⇒ make a simpli�cation:

replace the continuous Qi with a discrete approximation:

Qi −→N∑j=1

Qi (xj)δ(x − xj)

the sampling points {xj} are chosen to re�ect the shape of Qi

but: many sampling points needed for precise modeling of Qi





How to evaluate this formula for two given Husimi Fuctions?

evaluation of integral is too complex!

simple-form analytical solutions exist only for 1D systems

⇒ make a simpli�cation:

replace the continuous Qi with a discrete approximation:

Qi −→N∑j=1

Qi (xj)δ(x − xj)

the sampling points {xj} are chosen to re�ect the shape of Qi

but: many sampling points needed for precise modeling of Qi





⇒ make an even further simpli�cation:

�yczkowski et al.: for pure states the Husimi Function can beassociated with a discrete distribution consisting of the HusimiZeros:

|Ψ〉 −→ fΨ(x) :=1N

N∑i=1

δ(x − xi )

This yields the simpli�ed Monge Distance:

DsM(|Ψ〉, |Φ〉) := DM(fΨ, fΦ)

The Monge Problem has now been reduced to �nding the �shortestway�-map between the zeros of two Husimi Functions.





two zero distributions with L zeros each ⇒ L! possible mappings

simply trying out all possibilities requires exponential time

e�cient algorithms for �nding the optimal mapping do exist!

The theory of Linear Programming provides e�cient algorithmsfor many classes of problems. Two important classes are:

Assignment Problems:N items have to be transported from initial locations to �nallocations � how to transport them most e�cently?

Transportation Problems:Each one of A suppliers possesses ai items and each one of Bdestinations requires bi of the items.

Monge problem ∈ Assignment Problems ⊂ Transportation Problems





two zero distributions with L zeros each ⇒ L! possible mappings

simply trying out all possibilities requires exponential time

e�cient algorithms for �nding the optimal mapping do exist!

The theory of Linear Programming provides e�cient algorithmsfor many classes of problems. Two important classes are:

Assignment Problems:N items have to be transported from initial locations to �nallocations � how to transport them most e�cently?

Transportation Problems:Each one of A suppliers possesses ai items and each one of Bdestinations requires bi of the items.

Monge problem ∈ Assignment Problems ⊂ Transportation Problems





Two algorithms are suitable for the Monge Problem:

Simplex Algorithm

well known and well documentedfor Transportation Problems ⇒ very general algorithmusually fast, but exponential in the worst case

Hungarian Method

not well known and scarcely documentedfor Assigment Problems ⇒ specialised algorithmvery fast, run time never exceeds O(L3)

⇒ for speed and reliability, the Hungarian Method was chosen





Two algorithms are suitable for the Monge Problem:

Simplex Algorithm

well known and well documentedfor Transportation Problems ⇒ very general algorithmusually fast, but exponential in the worst case

Hungarian Method

not well known and scarcely documentedfor Assigment Problems ⇒ specialised algorithmvery fast, run time never exceeds O(L3)

⇒ for speed and reliability, the Hungarian Method was chosen





from the problem to the algorithm:

Example:

a23

a13

a12a11

a21a22

a33

a32a31

a13

a21

a32

two Husimi Functions with 3 zeros each

left �gure: shortest path from zero i of �rst distribution tozero j of second distribution is denoted by aij

right �gure: optimal map (the �Monge Plan�) is shown





Algorithmic approach:

determine the matrix of the distances {aij} with aij ≥ 0

Observations:

in every row and column, there is one entry that belongs tothe Monge Planchanging the values of a row or column by a constant doesn'tchange the Monge Plan

Basic idea: �nd the Monge Plan by appropriately subtractingconstants from rows and columns such that

as many matrix entries as possible become 0no matrix entry becomes negative

If one 0 can be chosen from every row and every column,then these 0's denote the Monge Plan!





Algorithmic approach:

determine the matrix of the distances {aij} with aij ≥ 0

Observations:

in every row and column, there is one entry that belongs tothe Monge Planchanging the values of a row or column by a constant doesn'tchange the Monge Plan

Basic idea: �nd the Monge Plan by appropriately subtractingconstants from rows and columns such that

as many matrix entries as possible become 0no matrix entry becomes negative

If one 0 can be chosen from every row and every column,then these 0's denote the Monge Plan!





The Hungarian Method is basically a prescription telling us whichvalues to subtract from which row/column, in order to �nd thedesired formation of 0's within O(L3) steps.

There are two common versions of the Hugarian Method in theliterature. One of the two is often stucked in an in�nite loop whenapplied to large matrices, thereby never terminating. . .

Gladly, the other version is sane and always terminates inpolynomial time. . .




Qualitative BehaviourNearest Neighbor StatisticsMonge Distance StatisticsConclusion


Until now:

physical system of interest and its models were presented

theory of Husimi Function and its zeros has been covered

numerical extraction of the Husimi Zeros has been discussed

a useful quantitative tool, the Monge Distance, was introduced

Now tackling:

Qualitative and quantitative analysis of the dynamics of theHusimi Zeros when varying the disorder potential.

similarities and di�erences between Anderson-Model andAubry-André-Model?

how are the dynamics of the Husimi Function and of theHusimi Zeros related?






Until now:

physical system of interest and its models were presented

theory of Husimi Function and its zeros has been covered

numerical extraction of the Husimi Zeros has been discussed

a useful quantitative tool, the Monge Distance, was introduced

Now tackling:

Qualitative and quantitative analysis of the dynamics of theHusimi Zeros when varying the disorder potential.

similarities and di�erences between Anderson-Model andAubry-André-Model?

how are the dynamics of the Husimi Function and of theHusimi Zeros related?





Analysis of the qualitative behaviour:

low disorder regime:

in the beginning all zeros are aligned along two horizontal lines

vertical movement of the zeros into phase space whenincreasing W , λ

at W , λ ≈ 1 nearly the whole phase space is covered

Aubry-André:

several movement linesall zeros lie at one of the movement lineszeros of one line are equidistant

Anderson:

only foremost movement lineszeros remaining behind the movement lines no longer move





top row: Aubry-André-Model bottom row: Anderson-Model

f) W = 10−12 g) W = 10−10 h) W = 10−6 i) W = 10−2 j) W = 100

a) λ = 10−12 b) λ = 10−10 c) λ = 10−6 d) λ = 10−2 e) λ = 100

−π

0

π

kk

0 L

xx

0 L

xx

0 L

xx

0 L

xx

0 L

xx

−π

0

π

kk

0 L 0 L 0 L 0 L 0 L





transition regime:

movement changes from vertical to horizontal

at one point, the entire phase space is covered by zeros

Aubry-André:

zeros possess a distinct ordering patterntransition from momentum to position localization occurs fast

Anderson:

no distinct ordering pattern recognizablechaotic shu�ing of the zeros due to avoided crossingstransition takes place over broader interval of disorder strength






f) W = 0, 1 g) W = 1 h) W = 2 i) W = 4 j) W = 10

a) λ = 0, 1 b) λ = 1 c) λ = 2 d) λ = 4 e) λ = 10

−π

0

π

kk

0 L

xx

0 L

xx

0 L

xx

0 L

xx

0 L

xx

−π

0

π

kk

0 L 0 L 0 L 0 L 0 L





high disorder regime:

zeros form vertical lines in the high disorder limit

Aubry-André:

position localization quickly �nished

Anderson:

position localization takes long timenot all zeros move to vertical lines; some remain scatteredscattering changes permanently and in a nonpredictable way






f) W = 2 g) W = 10 h) W = 20 i) W = 100 j) W = 1000

a) λ = 2 b) λ = 2, 5 c) λ = 3 d) λ = 6 e) λ = 10

−π

0

π

kk

0 L

xx

0 L

xx

0 L

xx

0 L

xx

0 L

xx

−π

0

π

kk

0 L 0 L 0 L 0 L 0 L





relationship between Husimi Function and Husimi Zeros:

Zeros change on a logarithmic scale while Q changes on alinear scale

similarly widespread zeros during the transition in both models,but very di�erent shapes of Husimi Function

Anderson: rapid scattering of zeros for big W doesn't a�ectshape of Husimi Function





Nearest Neighbor Statistics:

nearest neighbors of all zeros can be quickly calculated: O(L2)

distribution P(x) of nearest neighbor distances is obtained

integrated distribution: I (x) = 1L

x∫0

dx ′ P(x ′)

I (x) ∈ [0, 1] can be used to visualize the dynamics of thenearest neighbor statstics

(next �gure: I (x) of a Husimi Function of the Anderson-Model and ofthe Aubry-André-Model. I (x) is color-coded: I (x) = 0 for red andI (x) = 1 for blue.)





Anderson-Modell

0

5

10

15

20

25

Nachster-N

achbar-A

bstand

Nachster-N

achbar-A

bstand

10−8

10−4

100

104

108

Potenzialstarke WPotenzialstarke W

Aubry-Andre-Modell

0

5

10

15

20

25

10−8 10−4 100 104 108

Potenzialstarke λPotenzialstarke λ





λ = 0, 5 λ = 9

λ = 1, 3 λ = 2

0

5

10

15

20

Nachster-N

achbar-A

bstand

Nachster-N

achbar-A

bstand

0.5 1 2 5 10






Monge Distance Statistics:

determine the Husimi Zeros for equidistant λ on thelogarithmic scale

�nd the Monge Distance between all pairs of adjacent samplevalues of λ

plotting the results, one can see how fast the Husimi Zeroschange at di�erent λ





0

2500

5000

7500

Monge-

Dis

tanz

Monge-

Dis

tanz

10−8 10−4 100 104 108

Potenzialstarke WPotenzialstarke W

0

500

1000

1500

Monge-

Dis

tanz

Monge-

Dis

tanz

10−8 10−4 100 104 108


Figure: Monge Distance plot of the Husimi Zeros of the Anderson-Model(left) and of the Aubry-André-Model (right). The large and rapidlychanging values on the left are reminescent of the avoided crossings.Also, the fuzzy curve for high W is a result of the rapid scattering ofsome zeros in the Anderson-Model.





0

50

100

150

200

250

Monge-

Dis

tanz

Monge-

Dis

tanz

0.5 1 2 5 10


Figure: Transition area λ ≈ 2 of the same Husimi Function of theAubry-André-Model as in the last �gure.





Conclusion

close relationship between periodic B and analytic B ⇒complex analysis yields results about Bargmann Function

Q is completely determined by its zeros ⇒�the zeros of Q contain the physics of Q itself�

numerically obtained minima always veri�ed as zeros⇒ Are all minima of Q zeros? How to proof this?

nearest-neighbor-distribution eludicates the phase transition

Monge Distance is an excellent tool for detecting the phasetransition of the Aubry-André-Model


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Diploma Thesis - ``Analysis of Dynamics in Phase Space by … · 2014. 11. 30. · Diploma Thesis...

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