Eigenstate thermalization hypothesis
ETH FETH & FP
Statistics
equilibrium: find const and do Gibbs Statistics
Problem. How to formulate non-equilibrium Statistics in a closed system without external bath?
Aside from the non-equilibrium case, quantum case is subtle enough.
In a quantum phase space, there is no real notion of trajectory.
We would talk about closed system, however without external bath, how can we define equilibrium?
Quantum Quench System
Problem. For , can this be equivalent to Gibbs ensemble?
such that
where is function of
Remark
Rabi oscillation without bath coupling would not thermalize.
this can only holds for local observables , since the whole system is pure. it can only be locally viewed as a thermal ensemble.
Remark
The large fraction of pure state could serve as the bath for small subsystem.
Remark
eigen state should be always thermalized.
Expectation value
We assume no accidental degeneracy.
we hope
for local observables.
Random Matrix Theory (RMT)
Wigner Matrix
This is not rotational invariant.
Rotation invariance
where is any unitary matrix.
Intersection of above two
Can check this is both WM and RI.
Interestingly,
is also both WM and RI.
Haar Random
I'm too lazy to type higher moments.
Back to Expectation value
If is RMT, we can average in the random matrix ensemble. , where is diagonalized in
Thus is independent of .
This is too strong, since the thermal state forgets all information from the initial state.
ETH
Definition 1
where , is smooth function of , and is somehow the dimension of local Hilbert space at energy .
Now that,