1. Changnan Peng Mentors: Gil Refael and Samuel Savitz 2018.4.10
Continuous Unitary Transformation Flows of the Anderson Localization Hamiltonian on Maximally Tree-Like Cubic Graphs
Changnan Peng
Mentors: Gil Refael and Samuel Savitz
As you may notice, this three-line title contains lots of big words. I will try to make them clear in the introduction part.

2. Outline Introduction – what is the title saying? Methods Results
Anderson Localization Hamiltonian
Maximally Tree-Like Cubic Graphs
Continuous Unitary Transformation Flows
Methods
Results
Discussion
Acknowledgements
Here is the outline for today’s presentation. First I will introduce one by one the big words in the title. Then I will talk about the methods we use to approach the project. Finally I will show some results and discussion. Let’s start with the first line – what is the Anderson Localization? The model of the Anderson localization is pretty simple.
Changnan Peng

3. Introduction
Anderson localization – one particle hopping on a lattice with random potential
disorder W
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.
Changnan Peng

4. Introduction
Anderson Localization Hamiltonian where the on-site random potential ϵi is uniformly distributed in the interval [-W/2, W/2].
Here is the Hamiltonian for the model. The first term captures the hopping and the second term captures the on-site randomness. In some other research, the random potential might be a Gaussian distribution. Here our randomness is uniformly distributed.
Changnan Peng

5. Introduction
Anderson localization system …… |0> |1> |2> …… |n> |n+1> …… basis to represent the system: {|0>, |1>, |2>, ……, |N>}
Anderson localization is easy to simulate, because the dimension of the system is proportional to the number of nodes. The 1D Anderson localization does not contain many interesting features. Actually for any nonzero disorder, the system will be localized. To get more interesting features, and to be closed to the real word, (change slide) we consider the system with many but not one particles, and with interactions between the particles. This is called a many body system.
Changnan Peng

6. Introduction
Many body system – many particles hopping on a lattice with random potential and interactions between particles
we consider the system with many but not one particles, and with interactions between the particles. This is called a many body system.
Changnan Peng

7. Introduction Why are we interested in many body systems?
Metal-insulator transition
Violating Eigenstate Thermalization Hypothesis (ETH)
Non-ergodicity
Quantum Chaos
We are interested in many body systems also because (click) it shows a mechanism of metal-insulator transition: the extended state is conductive and the localized state is nonconductive. (click) It’s a system that violates the Eigenstate Thermolization Hypothesis, (click) which means that it has non-ergodic behavior. Here we demonstrate ergodicity and non-ergodicity with billiards platforms. After long enough time, the one on the right can cover every place on the platform, which is called ergodic. But the one on the left cannot, which is called non-ergodic. Many body systems behave like the left one. (click) It may also be further related to Quantum Chaos.
Changnan Peng

8. Introduction
Many body system – many particles hopping on a lattice with random potential and interactions between particles basis to represent the system: {|Ø>, |0>, |1>, |2>, ……, |N>, |00>, |01>, …, |0N>, |10>, …, |NN>, ……}
However, with these various features, many body systems are hard to solve. Because of the adding of particles, the dimension of the space increases exponentially, which brings us difficulties in do simulations.
Changnan Peng

9. Introduction
Many body system is equivalent to an Anderson localization system in the Fock space 0 particle: 1 particle: 2 particles: ……
|Ø>
|n-1>
|n>
|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(n+1)>
|(m-1)n>
|mn>
|(m+1)n>
|m(n-1)>
Changnan Peng

10. Introduction How to simplify?
High dimension  Low probability of loops
 No loops  Tree graph
The idea is that for high enough dimension, the particle is very unlikely to hop back to its original site and form a loop. (Explain by pointing.) So we can assume the lattice is a tree graph.
Changnan Peng

11. Introduction Bethe lattice – infinite trivalent tree
Finite approximations:
Truncated Bethe lattice
too much leave nodes
Random regular graphs (RRGs)
extra disorder
small loops
Cages
Bethe pronounce as beta. Trivalent pronounce as try valent. This tree graph is called a Bethe lattice, which is an infinite graph and each vertex has three neighbors. (click) However, it is still impossible to simulate an infinite tree. So we need finite approximation. There are three ways to do. (click) First is to just truncate it. However this will leave lots of nodes at the boundary, which is not there in the original many body system. (click) Second is to randomly sample in all trivalent graphs. However, it will introduce a new source of disorder. Also, rrgs may have small loops in the graph, although for a large enough graph the possibility of small loops is small, we still want the system be as clean as possible. (click) So we use cages.
Changnan Peng

12. Introduction
Cages – smallest trivalent graphs with longest minimal loop (the length of minimal loop is girth)
E.g. girth-5 cage or Petersen graph
We use cages to be most efficient, and mostly get rid of the randomness from graphs.
Maximally Tree-Like Cubic Graphs
Cages are smallest graph that contains longest minimal loop, and the length is called the girth. For example, in this graph, the minimal loop has length five, and this is the smallest graph that has this property, so this is a girth-5 cage. In our project, we use cages to be clean in the model and to be most efficient. So here is what the second term, Maximally Tree-like Cubic Graphs, means.
Changnan Peng

13. Introduction
We would like to find the non-ergodicity in this simplified model.
Previous works by other people:
Exact solution to the localization transition point on Bethe lattice, Wc=17.5 (Abou-Chacra et. al. (1973))
Non-ergodic intermediate phase found by spectrum statistics on RRGs (Biroli et. al. (2012))
Confirmed the non-ergodicity with multifractal analysis on RRGs (De Luca et. al. (2014))
We confirmed the non-ergodicity with flow level repulsion metric Ξ on cages.
Many body systems have interesting non-ergodic behaviors. It is simplified into Anderson localization systems on cages to make it nice to compute. Are the interesting non-ergodic behaviors still in this simplified model? Now we can introduce the goal of our project: we would like to find the non-ergodicity in this simplified model. In 1973, a theoretical localization transition point of 17.5 was given with an infinite Bethe lattice. 5 years ago, the non-ergodic phase was reported on random regular graphs with spectrum statistics, and it was confirmed two years later with multifractal analysis. What we did is we confirmed the non-ergodicity with flow level repulsion metric Ξ on cages.
Changnan Peng

14. Introduction What is flow level repulsion metric Ξ?
Continuous Unitary Transformation Flows
We have a flow like this. Going through the fictional time, the Hamiltonian matrix will get diagonalized. In the process there is a value that characterized the level repulsion within two sites only. This value Xi implies the interaction between two nodes.
So what is flow level repulsion metric Ξ? (click) Here comes our third term: Continuous Unitary Transformation Flows. (click) Suppose we have a Hamiltonian matrix. (click) It is a flow of the Hamiltonian matrix on fictional time, (click) and the Hamiltonian will gradually get diagonalized.
Wegner-Wilson flow
Changnan Peng

15. Introduction What is flow level repulsion metric Ξ? Wegner-Wilson flow
Note that Xi is a matrix, each matrix element represents the repulsion of levels between the corresponding two nodes. In other words, Xi implies the interaction between the nodes.
During the flow, there is a matrix Xi that captured the interaction between the sites. If Xi is large between a pair of node, the interaction between them is large and they are in delocalized state. If Xi is small, the pair is in localized state. We will look into the entries in the Xi matrix to find out the transition from the delocalized state to the localized state.
Wegner-Wilson flow
Changnan Peng

16. Introduction Those are what the title
“Continuous Unitary Transformation Flows of the Anderson Localization Hamiltonian on Maximally Tree-Like Cubic Graphs”
is saying : )
So here is what this three-line long title means. I hope someone could understand a bit more about it from my introduction.
Changnan Peng

17. Methods Picked a cage. Wrote C++ code.
Ran Wigner-Wilson flow with Anderson localized Hamiltonian on the cage.
Collected the data of the flow level repulsion metric Ξ with distances between sites.
Analyzed the distribution of Ξ.
Behind the title, what we actually did is the following.
Changnan Peng

18. Methods (During SURF2017) Used girth-16 cage with 1008 vertices.
Used GPU acceleration.
RAM LEAKED!!! : (
(Update)
Used girth-18 cage with 2560 vertices.
CPU only, total machine time 240 days.
During the SURF last summer, we picked the girth-16 cage with 1008 vertices. We used GPU acceleration and found that it was about 20 times faster than CPU. However we encountered ram leaking later and did not find a good way to fix it. So we turned back to use CPU only. We recently got a powerful computer with 16 cores and 256GB of ram. We were able to use 2 weeks on that computer to calculate a even larger cage with girth-18 and 2560 vertices.
Changnan Peng

19. Methods Picked a cage. Wrote C++ code.
Ran Wigner-Wilson flow with Anderson localized Hamiltonian on the cage.
Collected the data of the flow level repulsion metric Ξ with distances between sites.
Analyzed the distribution of Ξ.
Now I will show the plot from the analysis in the last step, the distribution of the entries in the Xi matrix.
Changnan Peng

20. Methods
Here is the distribution of the Xi with respect to different distances between the pairs. Three interesting features in this figure. Smooth curves like normal distribution. Moving left as distance increasing. Spreading as distance moving left.
disorder
W=16
Changnan Peng

21. Methods Log-normal, evenly spaced, evenly spreading. Why?
Thus, we assume p(ln Ξd) ~ exp(-(ln Ξd - μd)2 / 2σd2) μd ~ M d σd2 ~ S2 d
It is reasonable because the central limit theorem tells us that if d is a multiplication of some distribution at each step, say y, then at large d the distribution of d would be log-normal.
These three features are reasonable because if we think that the Xi distribution is accumulated by a certain distribution as the distance increase by one. The center limit theorem tells us the distribution of Xi will be log-normal. And we assume it is log-normal.
Changnan Peng

22. Methods
Assume there is a critical level repulsion Ξdc Ξd>Ξdc : delocalized pair (large interaction) Ξd<Ξdc : localized pair (small interaction)
Guess Ξdc ~ 2-αd
Number of delocalized pairs at distance d Nd ~ 2d p(Ξd>Ξdc) ~ nd
n = exp(ln 2 – (M + α ln 2)2 / 2S2)
Less details. We have this math formula. Then we get this result. When disorder decreases, the system cross the boundary of the phase diagram.
Changnan Peng

23. Methods n = exp(ln 2 – (M + α ln 2)2 / 2S2)
Number of delocalized pairs ~ nd
Three possible phases:
n=2: ergodic delocalized (delocalized pairs are dominant)
1≤n<2: non-ergodic delocalized (delocalized pairs are fractal)
n<1: localized (no delocalized pairs)
Changnan Peng

24. Results
Changnan Peng

25. Results Non-ergodic delocalized Localized
The ergodic-nonergodic transition happens at around W=1, and the localization transition happens at around W=16.
Localized
Changnan Peng

26. Discussion
We used flow level repulsion metric Ξ to find the non-ergodicity of Anderson localization model on cages.
So as a brief conclusion of the project, we used flow level repulsion metric Xi to find the non-ergodicity of Anderson localization model on cages.
Changnan Peng

27. Discussion
We got the localization transition point at Wc~16, which is not far from the theoretical value 17.5.
Reasons for the difference:
Finite graph.
Averaging on all the energy levels.
Energy band. The Anderson localization transition happens first at the band edge, last at the band center. The 17.5 is for the band center.
The first issue is harder to solve because of the limit computational power we have got. The second issue is easier to deal with. We can do the analysis only for the states that are at the band center.
Changnan Peng

28. Discussion Non-ergodic delocalized W = 17 (band center)
W = 17 (average all)
Here is an example for the improved analysis. We can see that originally the W=17 data point was located in the localized region. Now only analyzing with the band center we found the new data point located in the delocalized region, which agrees with the theory.
Localized
Changnan Peng

29. Discussion
Future work: change cage size and look for the scaling relation. ?
For the issue of the finite graph, a possible solution could be repeating the method on different sized cages, and to extrapolate the result into infinity size. So one of our possible future works is to change the cage size and look for the scaling relation. ?
Changnan Peng

30. Acknowledgements
Thanks to Evert van Nieuwenburg and Marcus Bintz for many fruitful discussion. Thanks to Christopher White for the help with the computational devices.
Changnan Peng

31. Thank you!
Questions?
Changnan Peng