1. A Roadmap to Many Body Localization and Beyond
Changnan Peng
Ph 70c Popular Seminar

2. Recent Research 35 slides
Roadmap
Electrical Phenomena slide
Localization slides
Ergodicity slides
Simulation and Problem slides
Simplification and Question slides
Recent Research slides
Changnan Peng

3. Electrical Phenomena
Let’s get on the bus and start our trip. We begin with the electrical phenomena in our daily lives.
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4. e-
Let’s imagine the following situation: after a whole day’s class, you finally enter your lovely home. You press the switch on the wall and immediately the bulb gives you a warm welcome light. What’s hidden behind, is the copper wires that connect the switch, the bulb, and the power station.
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5. But if the light isn’t turned on, there might be some problem in the bulb or in the electric circuit. Then you might want to wear a pair of rubber gloves to check the bulb or the electric circuit. The reason you use rubber gloves is that rubber does not conduct current; it is an insulator.
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6. Conductors
Insulators
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7. e-
How the conductors conduct electricity? Classically, the valence electrons in the atoms move and carry electrical charges. Electrons are one of the fundamental particles that build up the world we live in. They have negative electrical charge. The static electricity that we are suffering during the dry days comes from these tiny little electrons.
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8. The conductors are not ideal
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9. The conductors are not ideal
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10. The conductors are not ideal
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11. One dimensional wave equation
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12. Tight-binding model e- e- e- e-
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13. e- Tight-binding model …… |0> |1> |2> …… |n> |n+1> ……
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14. Conductors
Insulators
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15. Conductors Insulators Semiconductors
Our beloved laptop and smartphones.
Insulators
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16.
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17. Defects and Disorder a) Interstitial impurity atom,
b) Edge dislocation,
c) Self interstitial atom,
d) Vacancy,
e) Precipitate of impurity atoms,
f) Vacancy type dislocation loop,
g) Interstitial type dislocation loop,
h) Substitutional impurity atom
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18. Defects and Disorder
disorder W
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19. Anderson Localization
Philip Warren Anderson (born Dec 13, 1923) (age 94 now)
Nobel Prize in Physics (1977)
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20. Anderson Localization
It is a conductor-insulator transition!
It is just a particle hopping on a lattice which has random potential on each site. The animation shows an one dimensional lattice, but the lattice can be any mathematical graph with vertices and edges linking them. I will show the lattice we use latter. Here in this example, if there is no random potential on the sites, the Bloch Theorem tells us that the wave function of the particle will be a Bloch wave. But if there is large enough random potential, because of the destructive interference between the waves scattered at these random potential, the wave function will be peaked at some certain site. In the intermediate range, the wave function will have a transition from the extended Bloch wave to the localized wave function. This transition is called the Anderson transition. The magnitude of the random potential is called the disorder, and is labeled by W.
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21. e-
However, the materials in the real world are more complicated than the theory which only includes disorder. The electrons move around and interact with each other. For example, in copper wires,
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22. Many Body Localization (MBL)
MBL system – many particles hopping on a lattice with random potential and interactions between particles
Violates Eigenstate Thermalization Hypothesis
we consider the system with many but not one particles, and with interactions between the particles. This is called a many body system.
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23. Eigenstate Thermalization Hypothesis
Since both are many body systems, why can’t we treat the MBL system as a box of gas, and use statistical physics?
Temperature, Pressure, Volume, etc.
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24. Eigenstate Thermalization Hypothesis
Classical system has thermalization through chaos
Information of the initial state is lost
Temperature, pressure, etc. are average effects, independent on any specific state
Ergodicity
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25. Eigenstate Thermalization Hypothesis
MBL system does not have thermalization by itself
Information of the initial state is kept
Might be used to build quantum computer!
Non-ergodicity
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26. Non-ergodicity e-
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27. Simulation Problem
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28. Simulation Anderson localization system
…… |0> |1> |2> …… |n> |n+1> …… ……
Memory needed = N
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29. Simulation MBL system …… |0> |1> |2> …… |n> |n+1> …………
Memory needed = 2^N
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30. Problem How large is 2^N? 2^1 = 2 2^10 = 1024 2^100 =
2^300 is larger than the total number of atoms in the universe
Impossible to perfectly simulate a large MBL system with today’s computers
1.26*10^30
Total number of atoms in the universe about 10^86
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31. Then? Simulate small MBL systems
Wait for the invention of a quantum computer
Or, use approximations to simplify the MBL model
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32. 1st Simplification
MBL system is equivalent to an Anderson localization system when a state is seen as a single electron
|0> |1> |2> …… |n> |n+1>
“hopping”
One way to solve many body systems is to turn back to Anderson localization system, which as I mentioned is easy to compute. However, it does not help much. The complexity of the interaction in many body system turns into the complexity of the high dimensional graphs. (Explain if have time.) It is still hard to solve. We need to simplify more.
|01> |02>
hopping
|0> |1> |2> …… |n> |n+1>
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33. 1st Simplification
0 particle: 1 particle: 2 particles: …… many particles: high-dimensional graph
|Ø>
|n-1>
|n>
|n+1>
|m(n+1)>
One way to solve many body systems is to turn back to Anderson localization system, which as I mentioned is easy to compute. However, it does not help much. The complexity of the interaction in many body system turns into the complexity of the high dimensional graphs. (Explain if have time.) It is still hard to solve. We need to simplify more.
|(m-1)n>
|mn>
|(m+1)n>
|m(n-1)>
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34. 2nd Simplification No loops!
We use upper triangle to represent a state which is viewed as a single electron.
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35. Now much easier to simulate!
2nd Simplification
High-dimensional graph  Tree graph
(Bethe Lattice)
Now much easier to simulate!
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36. Question Is there non-ergodicity in this simplified MBL model?
Answer not known yet
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37. Recent Research
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38. Maximally Tree-Like Graphs
Girth – The length of the smallest loop in the graph
E.g. some girth-5 graphs:
Soccer ball, dodecahedron, Petersen graph
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39. They all locally look like this tree graph
N = 60
N = 20
N = 10
Soccer ball, dodecahedron, Petersen graph
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40. Maximally Tree-Like Graphs
The quality of using finite graph to approximate infinite tree graph depends on the girth
Same girth, smallest N
Same N, largest girth
N = 10
Soccer ball, dodecahedron, Petersen graph
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41. Fractal Dimension
# of vertices that the wave function goes through at distance d is n^d
n<1: localized
d = 0 6 5
d = 1 4 2 3
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42. Fractal Dimension
# of vertices that the wave function goes through at distance d is n^d
n<1: localized
1≤n<2:non-ergodic
d = 0 6 5
d = 1 4 2 3
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43. Fractal Dimension
# of vertices that the wave function goes through at distance d is n^d
n<1: localized
1≤n<2:non-ergodic
n=2: ergodic
d = 0 6 5
d = 1 4 2 3
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44. Result
Ergodic
Non-ergodic
Localized
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45. Discussion
This recent research confirms the existence of non-ergodicity in the girth-16 maximally tree-like graph
The method can be extended to larger graphs to find out the result at the infinite tree limit
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46. Thank you!
Questions?
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