A Roadmap to Many Body Localization and Beyond

A Roadmap to Many Body Localization and Beyond Changnan Peng Ph 70 c Popular Seminar 2018. 5. 1

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 2

Electrical Phenomena Changnan Peng 3

Changnan Peng 4

Changnan Peng 5

Conductors Insulators Changnan Peng 6

e- Changnan Peng 7

The conductors are not ideal e- Changnan Peng 8

The conductors are not ideal e- Changnan Peng 9

The conductors are not ideal e- Changnan Peng 10

One dimensional wave equation Changnan Peng 11

Tight-binding model e e e Changnan Peng e 12

Tight-binding model e …… e e e |0> |1> |2> …… |n> |n+1> …… Changnan Peng 13

Conductors Insulators Changnan Peng 14

Conductors Semiconductors Insulators Changnan Peng 15

http: //amitngroup. blogspot. com/2014/06/doping-in-semiconductors. html Changnan Peng 16

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 17

Defects and Disorder disorder W Changnan Peng 18

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 19

Anderson Localization It is a conductor-insulator transition! Changnan Peng 20

e- Changnan Peng 21

Many Body Localization (MBL) � � MBL system – many particles hopping on a lattice with random potential and interactions between particles Violates Eigenstate Thermalization Hypothesis Changnan Peng 22

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 23

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 24

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 25

Non-ergodicity e- Changnan Peng 26

Simulation Problem Changnan Peng 27

Simulation � Anderson localization system …… |0> |1> |2> …… |n> |n+1> …… 0 1 2 3 4 …… Memory needed = N Changnan Peng 28

Simulation � MBL system …… |0> |1> |2> …… |n> |n+1> …… 0 1 10 2 20 21 210…… Memory needed = 2^N Changnan Peng 29

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 Changnan Peng 30

Then? � � � Simulate small MBL systems Wait for the invention of a quantum computer Or, use approximations to simplify the MBL model Changnan Peng 31

1 st 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 “hopping” |01> |02> |0> |1> |2> …… |n> |n+1> Changnan Peng 32

1 st Simplification |Ø> 0 particle: 1 particle: |n-1> |n+1> |m(n+1)> 2 particles: …… |(m-1)n> |mn> |(m+1)n> |m(n-1)> many particles: high-dimensional graph Changnan Peng 33

2 nd Simplification No loops! Changnan Peng 34

2 nd Simplification High-dimensional graph Tree graph (Bethe Lattice) � Now much easier to simulate! Changnan Peng 35

Question � � Is there non-ergodicity in this simplified MBL model? Answer not known yet Changnan Peng 36

Recent Research Changnan Peng 37

Maximally Tree-Like Graphs � � Girth – The length of the smallest loop in the graph E. g. some girth-5 graphs: Changnan Peng 38

They all locally look like this tree graph N = 60 N = 20 Changnan Peng N = 10 39

Maximally Tree-Like Graphs � � � The quality of using finite graph to approximate infinite tree graph depends on the girth Same girth, smallest N N = 10 Same N, largest girth Changnan Peng 40

Fractal Dimension � d = 0 d = 1 2 3 6 5 4 # of vertices that the wave function goes through at distance d is n^d n<1: localized Changnan Peng 41

Fractal Dimension � d = 0 d = 1 2 3 6 5 4 # of vertices that the wave function goes through at distance d is n^d n<1: localized 1≤n<2: non-ergodic Changnan Peng 42

Fractal Dimension � d = 0 d = 1 2 3 6 5 4 # 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 Changnan Peng 43

Result Ergodic Non-ergodic Localized Changnan Peng 44

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 45

Thank you! � Questions? Changnan Peng 46