Brief Review on ETH and MBL

0 写在前面

由于本人开始在家办公，故网站开发暂停。
不过这里姑且还是具有博客基本功能的个人网站，故把目前关注的话题的笔记放在这里。

本文是对Eigenstate Thermalization Hypothesis (ETH) 和Many-body Localization (MBL)的简要复习。
因为我学习参考的文献是英文，也没有读过中文的相关文献，故为免去术语翻译上可能的困难，以下将有选择性地使用英文。

1 What’s ETH

1.1 Closed-system many body mechanics

Here we focus on closed many-body quantum systems with short-range Hamiltonian,
and not onlyon low-lying states, but also on highly-excited states.
The states we are interested here also contain
mixed states.
Therefore density matrices are used to describe the states of the system.

The evolution of the density matrix with
Halmitonian obeys:

$$\rho(t)=e^{-\frac{i H t}{h}} \rho(0) e^{\frac{i H t}{h}} ; \quad i \hbar \frac{d \rho}{d t}=[H, \rho] ; \quad \operatorname{Tr}\{\rho\}=1$$

The expectation value of an operator is:

$$\langle O\rangle_t=\text{Tr}\{\hat O \rho(t)\}$$

1.2 What’s Thermalization

A quantum system
in thermal equilibrium is fully characterized by a small number of parameters
(temperature, chemical potential, etc.: one parameter for each extensive conserved quantity).
The process of going from an initial state to the
themal equilibrium suggests losing the information
contained in the initial state.
This seems to be a paradox because unitary time evolution cannot erase information.

Actually the information is still in the system
but is locally hidden when thermalized.
Spreading of quantum entanglement moves the information about the initial state
so that at long time it is inaccessible because recovering that information would require
measuring global operators.
This id the so-called “decoherence”.

We partition the full quantum system into a subsystem,
A, and its environment, B.