This document summarizes research on phase transitions in random regular graphs. It discusses several types of phase transitions, including metal-insulator transitions, wave function localization, and Anderson localization. Random matrix theory is also covered as a way to characterize the extended and localized phases. The document examines localization in both real and Fock space, and how properties like the density of states and level spacing statistics can indicate phase transitions. It was written by Amin Shahnazari Zazerani from the University of Isfahan under the guidance of Dr. Mohsen Amini.

1. A Survey of Statistical Properties of
Random Regular Graphs to Study Existing
Phase Transitions in this Systems
Amin Shahnazari Zazerani
Physics Department, University of Isfahan
amin.shahnazari@hotmail.com
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Under Guidance of:
Dr. Mohsen Amini

2. Table of Content
 Metal-Insulator Phase Transitions
 Wave Function Localization
 Thermalization
 Anderson Localization
 Random Matrix Theory
 Localization in Fock Space
 On The Computer!
 Statistical Analysis of Random Regular Graphs
 Acknowledgments
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3. Metal-Insulator Phase Transitions
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4. Metal-Insulator Phase Transitions
Good Metal Good Insulator
Si
14
Silicon
Ge
32
Germanium
Au
79
Gold
Cu
29
Copper
79
Phase Transition PointGood Conductivity Lake of Conductivity
The Question: Which parameter determines what will be where ?!
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5. Metal-Insulator Phase Transitions
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Familiar Answer:
Electronic Band Structure (Band Theory!)

6. Metal-Insulator Phase Transitions
Non-familiar Answers:
Electron Interactions (Mott Phase Transition)
Disorders and Complexity (Anderson Phase Transition)
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7. Metal-Insulator Phase Transitions
Nobel Prize 1977:
Fundamental theoretical investigations of the electronic
structure of magnetic and disordered systems
Philip Warren
Anderson
Prize share: 1/3
Sir Nevill
Francis Mott
Prize share: 1/3
John Hasbrouck
van Vleck
Prize share: 1/3
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8. Metal-Insulator Phase Transitions
Nobel Prize 1977:
Philip Warren
Anderson
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9. Wave Function Localization
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10. Wave Function Localization
Assumes following diffusion equation which demonstrate the scattering of wave
function from random impurities through the lattice:
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𝜕𝑝(𝑟, 𝑡)
𝜕𝑡
− 𝐷𝛻2 𝑝 𝑟, 𝑡 = 𝛿(𝑟)𝛿(𝑡)
𝑝 𝑟, 𝑡 = 𝜓(𝑟, 𝑡) 2 =
exp(−
𝑟2
2𝐷𝑡)
(2𝜋𝐷𝑡) 𝑑/2
𝐷 =
ℏ𝑘 𝑓 𝑙
2𝑚
𝑝 𝑟, 𝑡 Probability density
𝜓(𝑟, 𝑡) Wave function
𝐷 Diffusion coefficient
𝑘 𝑓 Fermi momentum
𝑚 Electron mass
𝑙 Mean free path length
𝑑 Diffusion dimension

11. Wave Function Localization
𝑘 𝑓 𝑙 ∶ ↑ ⟹ 𝐷: ↑
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𝑘 𝑓 𝑙 ∶ ↓ ⟹ 𝐷: ↓
Disorder: ↓
Disorder: ↑

12. Wave Function Localization
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If 𝑘 𝑓 𝑙 ≪ 1 ⟹ Localized Phase!
𝜓𝑙𝑜𝑐 ~𝐴𝑒𝑥𝑝(− 𝑟 − 𝑟𝑎 /𝜉)⟹
If 𝜓 𝑟, 𝑡 = 0 = 𝛿 𝑟,0 = σ 𝑛 𝐶 𝑛 𝜓 𝑛(𝑟) 𝜓 𝑟, 𝑡 = σ 𝑛 𝐶 𝑛 𝑒−𝑖𝐸 𝑛 𝑡 𝜓 𝑛(𝑟)
𝑟𝑎 Center of localization
𝜉 Localization length
𝜓 𝑛
(𝑟) 𝑛th Eigen State
𝐸𝑓 Fermi Energy
⟹
𝑝 𝑟, 𝑡 = 𝜓 𝑟, 𝑡 2 ≤ 𝑒
−
2𝑟
𝜉 𝑟 ≫ 𝜉⟹
DOS: 𝜌 𝐸 =
1
𝑁
σ 𝑛 𝛿(𝐸 − 𝐸 𝑛)
Conductivity: 𝜎0= 𝑒2 𝜌 𝐸𝑓 𝐷
If 𝑘 𝑓 𝑙 ≪ 1 ⟹ 𝐷 = 0 ⟹ 𝜎0= 0

13. Wave Function Localization
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IPR as localization indicator:
𝑠𝑝 = < 𝜓 𝑟, 𝑡 = 0 |𝜓 𝑟, 𝑡 > 2 𝐼𝑃𝑅(𝜓) Inverse Participation Ratio
𝑠𝑝 Survival Probability
Return Probability𝑠𝑝𝑡→∞ = σ 𝑛 < 𝜓 𝑛|𝜓 0 > 4 = 𝐼𝑃𝑅(𝜓)
σ 𝑛 < 𝜓 𝑛|𝜓 0 > 2 = σ 𝑛 𝐶 𝑛
2 = 1
𝐼𝑃𝑅(𝜓) ≡ σ 𝑛 𝜓 𝑟 4 = σ 𝑛 < 𝜓 𝑛|𝜓 0 > 4
⟹ 𝐼𝑃𝑅 𝜓 ∼
1
𝜉2𝑑
∼ 𝜉−𝑑
If 𝐼𝑃𝑅 𝜓 > 0 , 𝑡 → ∞
If 𝐼𝑃𝑅 𝜓 = 0 , 𝑡 → ∞
Localized Phase!
Extended Phase!

14. Thermalization
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15. Thermalization
Thermalization is the process of
physical systems reaching to
equilibrium.
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Simple Definition:
When expectation value of specific
observable fluctuate around a
certain value.
(Equilibrium statistical mechanics)
Physical Definition:
Reaching system Von-Neuman
Entropy to its maximum value.
(Quantum thermalization )

16. Thermalization
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From Equilibrium statistical mechanics perspective
෡𝐻| ۧ𝜓 𝑛 = 𝐸 𝑛| ۧ𝜓 𝑛 | ۧ𝜓(0) = σ 𝑛=1
𝐷
𝐶 𝑛| ۧ𝜓 𝑛
| ۧ𝜓(𝑡) = σ 𝑛=1
𝐷
𝐶 𝑛 𝑒−𝑖𝐸 𝑛 𝑡/ℏ| ۧ𝜓 𝑛
𝐴 = 𝜓(𝑡) 𝐴 𝜓(𝑡)
𝐴 𝑡ℎ =
𝑇𝑟(𝑒−𝛽𝐻)
𝑍
, 𝛽 → 𝐻𝑡ℎ = ത𝐸
𝐴 𝑡ℎ − 𝐴 < 𝜀
System
𝐴
𝑡
𝐴 𝑡ℎ

17. Thermalization
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From Quantum thermalization perspective
𝜌 ≡ σ 𝑛 𝑤 𝑛| ۧ𝜓 𝑛 ‫ۦ‬ |𝜓 𝑛
𝜌 𝑡 = 𝑒−𝑖𝐻𝑡 𝜌 0 𝑒 𝑖𝐻𝑡
𝜌 𝐴 𝑡 = 𝑇𝑟𝐵{𝜌 𝑡 }
𝜌 𝑒𝑞 =
𝑒−𝛽𝐻
𝑍
A B
Density Operator:
𝑖
𝑑𝜌
𝑑𝑡
= 𝐻, 𝜌
Reduced density matrix for subsystem A:
Equilibrium Gibbs ensemble:
𝜌 𝐴
𝑒𝑞
𝑡 = 𝑇𝑟𝐵{𝜌 𝑒𝑞 𝑡 }
𝜌 𝐴 𝑡 = 𝜌 𝐴
𝑒𝑞
𝑇
𝑡 → ∞
The isolated system act
as it’s own reservoir.
Ergodic / Thermalization

18. Thermalization
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Von-Neuman Entropy
𝑆𝐴 ≡ 𝑇𝑟{𝜌 ln 𝜌}Von-Neuman Entropy: 𝑆𝐴 ∝ 𝐿 𝑑
A B
L

19. Anderson Localization
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20. Anderson Localization
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An Example of Failing Eigenstates Thermalization Hypothesis!
Philip Warren
Anderson
Localized System:
“Area” law entropy
Extended System:
“Volume” law entropy 𝑆𝐴 ∝ 𝐿 𝑑−1

21. Anderson Localization
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Non-Interacting(single body) fermions moving in a random potential:
𝐻 = ෍
𝑖𝑗 𝜎
𝑡𝑖𝑗 𝑐𝑖𝜎
†
𝑐𝑗𝜎 + ℎ. 𝑐 + ෍
𝑖𝜎
𝜖𝑖 𝑐𝑖𝜎
†
𝑐𝑖𝜎 −
𝑤
2
≤ 𝜖𝑖≤
𝑤
2
𝑤 < 𝑤𝑐
𝑤 > 𝑤𝑐

22. Anderson Localization
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How to distinguish phase transition?
Density of States! Local Density of States!
𝜌 𝐸 =
1
𝑁
෍
𝑛
𝛿(𝐸 − 𝐸 𝑛) 𝜌 𝐸, 𝑖 =
1
𝑁
෍
𝑛
𝜓 𝑛(𝑖) 2 𝛿(𝐸 − 𝐸 𝑛)
Extended: Continues, Smooth
Localized: Continues
Extended: Continues, Smooth
Localized: Discrete
E E E E
DoS
DoS
LDoS
LDoS

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24. Random Matrix Theory
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25. Random Matrix Theory
Consider a 𝑁 × 𝑁 Hermitian random matrix:
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Eigenvalue sets: 𝐸1, 𝐸2, 𝐸3 , … 𝐸 𝑛
Mean Level Spacing: 𝛿1 = (𝐸 𝑛+1 − 𝐸 𝑛)
Level Spacing: 𝑠 =
𝐸 𝑛+1 − 𝐸 𝑛
𝛿1
Level Spacing distribution function: 𝑃(𝑠)
𝑃 𝑠 = 0 = 0
𝑃 𝑠 ≪ 1 ∝ 𝑠 𝛽
𝛽 = 1, 2, 4

26. Random Matrix Theory
Reasoning:
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𝐻 =
𝐻11 𝐻12
𝐻∗
21 𝐻22
When 𝑠 → 0 ∶ 𝑃 𝑠 = 0 because:
𝑃 𝑠 = 0 = 0
𝑃 𝑠 ≪ 1 ∝ 𝑠 𝛽
𝛽 = 1, 2, 4
𝐸2 − 𝐸1 = (𝐻22 − 𝐻11)2− 𝐻12
2
The assumption is that the matrix elements are
statistically independent. Therefore probability of two
levels to be degenerate vanishes.
Small SmallSmall

27. Random Matrix Theory
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Level Spacing Statistics:
𝛽 Matrix Elements Ensemble Realization
1 Real Orthogonal T-inverse
2 Complex Unitary None T-inverse
4 2*2 Matrices Symplectic T-inverse
Spin-Orbit coupling
Poisson Distribution: Localized Phase!

28. Random Matrix Theory
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Anderson Model as RMT (without interaction)
𝐻 = ෍
𝑖𝑗
𝑡 (| ۧ𝑗 ‫ۦ‬ |𝑖 + | ۧ𝑖 ‫ۦ‬ |𝑗 ) + ෍
𝑖
𝜀𝑖 | ۧ𝑖 ‫ۦ‬ |𝑖 −
𝑤
2
≤ 𝜀𝑖≤
𝑤
2
𝐻 =
𝜀1 𝑡
𝑡 𝜀2
… 0
… 0
… …
0 0
… …
𝑡 𝜀 𝑛
𝑊 < 𝑊𝑐
Extended -Metal
Wigner-Dyson distribution
𝑊 > 𝑊𝑐
Localized -Insulator
Poisson distribution

29. Random Matrix Theory
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How to distinguish phase Transition?

30. Localization in Fock Space
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31. Localization in Fock Space
Invariance of statistics (base independent)
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If Poissonian
Statistics ∃ A Base where the
Eigenfunctions are localized
If Wigner-Dyson
Statistics ∀ Basis the eigenfunction are
extended

32. Localization in Fock Space
Many body Non-interacting Anderson Model:
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𝐻 = ෍
𝑖𝑗
𝑡 (| ۧ𝑗 ‫ۦ‬ |𝑖 + | ۧ𝑖 ‫ۦ‬ |𝑗 ) + ෍
𝑖
𝜀𝑖 | ۧ𝑖 ‫ۦ‬ |𝑖
 Lets fill eigenstates with N non-interacting
electron!
 Label eigenstates and eigenvalue with 𝛼
indicating occupation number 𝑛 𝛼.
 Now we have new base: 𝜇 = {𝑛 𝛼}
 Hamiltonian can be rewrite in new base.
 Energy level of each state equals to sum
of energies of filled state!
 Now we are in the Fock Space!!!
𝐻 = ෍
𝜇
𝐸𝜇 | ۧ𝜇 ‫ۦ‬ |𝜇
𝐸𝜇 = ෍
𝑘
𝜀 𝑘 = ෍
𝛼
𝜀 𝛼 𝑛 𝛼

33. Localization in Fock Space
Adding weak Interaction
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𝐻 = ෍
𝜇
𝐸𝜇 | ۧ𝑖 ‫ۦ‬ |𝑖 + ෍
𝜇,𝜇′
𝑡 𝜇,𝜇′ | ۧ𝜇′ ‫ۦ‬ |𝜇
 Interaction of electrons change the
order of eigenstates, thus we can
add interaction as a hopping
factor!!

34. Localization in Fock Space
Fock Space Structure: Cayley Tree of rank 4!
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Decay of exited interacting quasiparticles
in fermi liquid can produce 2 new
quasiparticle and one hole!

35. Localization in Fock Space
Random Regular Graphs (RRG)
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 A good alternate of Cayley Tree is RRG
which eliminate boundary effects!
 A Random Regular Graph is equal nodes
degree and random geometry!
 RRG Hamiltonian can be written using
Tight-Binding Model!

36. Localization in Fock Space
Random Regular Graphs (RRG)
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𝐻 = − ෍
𝑖𝑗
(𝑐𝑖
†
𝑐𝑗 + ℎ. 𝑐) + ෍
𝑖
𝜀𝑖 𝑐𝑖
†
𝑐𝑖

37. Localization in Fock Space
RRG indicate Tow Phase Transition!!!!
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Good Metal Good Insulator
Extended Extended, non-ergodic Localized
Disorder!

38. Localization in Fock Space
Multifractality as a property of non-ergodic phase!
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ε
Fractal Dimension
𝑀 𝑟 ∝ 𝑟 𝑑 𝑞 𝑑 𝑞 ∈ ℝ
𝐼𝑃𝑅 𝑞 = න 𝑑 𝑑 𝑟 𝜓(𝑟) 2𝑞 ∝ 𝐿 𝑑 𝑞

39. On The Computer!
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40. On The Computer!
Random Regular Graph Generation:
 NetworkX 1.11 (an Open Source Python Library)
Matrix Exact Diagonalization:
 Intel® Math Kernel Library 2013
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41. Statistical Analysis of Random Regular Graphs
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42. Statistical Analysis of Random Regular Graphs
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𝑠𝑝 𝑛(𝑡) = < 𝜓 0 |𝜓 𝑡 > 2
Survival Probability:
Extended:
Power-Law 𝛾 > 1
Non-ergodic, Extended:
Power-Law 0 < 𝛾 < 1
Localized:
Constant
𝑠𝑝 𝑛 𝑡 ~ 𝑡−𝛾

43. Statistical Analysis of Random Regular Graphs
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𝑠𝑝 𝑛(𝑡) = < 𝜓 0 |𝜓 𝑡 > 2
Shannon Entropy:
𝑆𝑠ℎ = ෍
𝑛
𝑠𝑝 𝑛 𝑡 (ln 𝑠𝑝 𝑛(𝑡))
Extended:
Power-Law
Non-ergodic, Extended:
Logarithmic
Localized:
Constant

44. Acknowledgments
Special Thanks to Dr. Amini
for his great supports,
kindly guidance
and his confidence in me.
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45. Thanks for your attention!!
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