1. Changnan Peng Mentors: Gil Refael and Samuel Savitz 2018.5.22
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 Models Methods Results Discussion Acknowledgements
Anderson Localization Hamiltonian
Maximally Tree-Like Cubic Graphs
Continuous Unitary Transformation Flows
Methods
Results
Discussion
Acknowledgements
Here is the outline for today’s presentation. In this project, we combine three parts together: the localization system in condensed matter physics, the cages, which is the maximally tree-like cubic graphs, in mathematics, and the continuous unitary transformation flows in numeric computation. Each of you might be an expert in some of these parts, but probably you might not be familiar with all the three parts. So I will use some time introduce the ideas of these three big terms.
Changnan Peng

3. Models
Problem – Many body localization system needs exponentially large memory. Hard to simulate a large MBL system.
Solution –
Anderson localization in the Fock space.
High-dimensional Fock space approximates to a tree graph.
Infinite tree graphs (the Bethe lattices) approximates to cages.
Let’s first come directly to the problem we want to solve in this project. We know that in order to simulate an MBL system, we need to specify the amplitude on each basis. For an MBL system of size N, the number of basis is 2^N. Therefore, we need exponentially large memory to do the simulation.
Changnan Peng

4. Models
Goal – Simulate the Anderson localization system on cages, and see whether there is non-ergodicity as in the MBL system.
Simulation – Use a continuous unitary transformation flow (the Wegner-Wilson flow, or WWF) to diagonalize the Hamiltonian matrix.
Seeing – Large level repulsion during the flow means extended wave function. Do statistics on the level repulsions.
This is the frame of our project. Now I will come to a brief introduction of the parts I mentioned in the beginning.
Changnan Peng

5. MBL vs. Anderson Localization
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.
…… |0> |1> |2> …… |n> |n+1> ……
Changnan Peng

6. MBL vs. Anderson Localization
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”
Instead of each site representing a basis state, in an MBL system each configuration represents a basis state. Exponentially large. However, also hint that the MBL system is equivalent to an Anderson localization system in the Fock space. The complexity of the interaction in many body system turns into the complexity of the high dimensional graphs. The equivalence is not exact. The randomness might be not uniform. This is an assumption.
|01> |02>
hopping
|0> |1> |2> …… |n> |n+1>
Changnan Peng

7. MBL vs. Anderson Localization
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)>
Changnan Peng

8. MBL vs. Anderson Localization
Anderson localization on high-dimensional Fock space.
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

9. Tree Graphs vs. Cages 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

10. Tree Graphs vs. Cages
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.
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

11. Others' Work vs. Our Work
We would like to find the non-ergodicity in the Anderson localization systems on cages.
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

12. The Wegner-Wilson Flow
The Hamiltonian evolves with the virtue “flow time” τ by
with the flow generator
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.
Changnan Peng

13. The Wegner-Wilson Flow
The two-state limit and where
Final diagonalized Hamiltonian
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.
Changnan Peng

14. Level Repulsion Metric Ξ
The two-state limit
In general
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.
Changnan Peng

15. The Wegner-Wilson Flow
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.
Changnan Peng

16. Level Repulsion Metric Ξ
Note that Xi is a matrix, each matrix element represents the repulsion of levels between the corresponding two sites. In other words, Xi implies the interaction between the sites.
During the flow, there is a matrix Xi that captured the interaction between the sites. If Xi is large between a pair of sites, 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.
Changnan Peng

17. Methods Picked cages. C++ & ArrayFire GPU acceleration.
girth-16, 1008 vertices, arc-transitive
girth-18, 2560 vertices
C++ & ArrayFire GPU acceleration.
Ran Wegner-Wilson flow with Anderson localized Hamiltonian on the cage.
total machine time 240 days
Collected the level repulsion metric Ξ.
Behind the title, what we actually did is the following.
Changnan Peng

18. 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

19. Methods Log-normal, evenly spaced, evenly spreading. Why?
The central limit theorem
Thus, we assume p(ln Ξd) ~ exp(-(ln Ξd - μd)2 / 2σd2) μd ~ M d σd2 ~ S2 d
d = 0
d = 1 2 3 4 5 6
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

20. 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

21. 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

22. Methods
# 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
Changnan Peng

23. Methods
# 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
Changnan Peng

24. Methods
# 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
Changnan Peng

25. Results
Changnan Peng

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. Thank you!
Questions?
Changnan Peng