A Roadmap to Many Body Localization and Beyond Changnan Peng Ph 70c Popular Seminar 2018.5.1

Recent Research 35 slides Roadmap Electrical Phenomena 1 slide Localization 17 slides Ergodicity 21 slides Simulation and Problem 25 slides Simplification and Question 29 slides Recent Research 35 slides Changnan Peng

Electrical Phenomena Let’s get on the bus and start our trip. We begin with the electrical phenomena in our daily lives. Changnan Peng

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. Changnan Peng

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. Changnan Peng

Conductors Insulators Changnan Peng

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. Changnan Peng

The conductors are not ideal Changnan Peng

The conductors are not ideal Changnan Peng

The conductors are not ideal Changnan Peng

One dimensional wave equation Changnan Peng

Tight-binding model e- e- e- e- Changnan Peng

e- Tight-binding model …… |0> |1> |2> …… |n> |n+1> …… Changnan Peng

Conductors Insulators Changnan Peng

Conductors Insulators Semiconductors Our beloved laptop and smartphones. Insulators Changnan Peng

http://amitngroup.blogspot.com/2014/06/doping-in-semiconductors.html Changnan Peng

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 https://www.tf.uni-kiel.de/matwis/amat/def_en/ Changnan Peng

Defects and Disorder disorder W Changnan Peng

Anderson Localization Philip Warren Anderson (born Dec 13, 1923) (age 94 now) Nobel Prize in Physics (1977) https://en.wikipedia.org/wiki/Philip_Warren_Anderson Changnan Peng

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. Changnan Peng

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, Changnan Peng

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. Changnan Peng

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. Changnan Peng

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 Changnan Peng

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 Changnan Peng

Non-ergodicity e- Changnan Peng

Simulation Problem Changnan Peng

Simulation Anderson localization system …… |0> |1> |2> …… |n> |n+1> …… 0 1 2 3 4 …… Memory needed = N Changnan Peng

Simulation MBL system …… |0> |1> |2> …… |n> |n+1> …… 0 1 10 2 20 21 210…… Memory needed = 2^N Changnan Peng

Problem How large is 2^N? 2^1 = 2 2^10 = 1024 2^100 = 1267650600228229401496703205376 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 Changnan Peng

Then? Simulate small MBL systems Wait for the invention of a quantum computer Or, use approximations to simplify the MBL model Changnan Peng

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> Changnan Peng

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)> Changnan Peng

2nd Simplification No loops! We use upper triangle to represent a state which is viewed as a single electron. Changnan Peng

Now much easier to simulate! 2nd Simplification High-dimensional graph  Tree graph (Bethe Lattice) Now much easier to simulate! Changnan Peng

Question Is there non-ergodicity in this simplified MBL model? Answer not known yet Changnan Peng

Recent Research Changnan Peng

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 Changnan Peng

They all locally look like this tree graph N = 60 N = 20 N = 10 Soccer ball, dodecahedron, Petersen graph Changnan Peng

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 Changnan Peng

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 Changnan Peng

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 Changnan Peng

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 Changnan Peng

Result Ergodic Non-ergodic Localized Changnan Peng

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 Changnan Peng

Thank you! Questions? Changnan Peng