15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Eigenstate thermalization hypothesis | 2/5 | https://en.wikipedia.org/wiki/Eigenstate_thermalization_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:23.990681+00:00 | kb-cron |
A
α
β
≡
⟨
E
α
|
A
^
|
E
β
⟩
.
{\displaystyle A_{\alpha \beta }\equiv \langle E_{\alpha }|{\hat {A}}|E_{\beta }\rangle .}
We now imagine that we prepare our system in an initial state for which the expectation value of
A
^
{\displaystyle {\hat {A}}}
is far from its value predicted in a microcanonical ensemble appropriate to the energy scale in question (we assume that our initial state is some superposition of energy eigenstates which are all sufficiently "close" in energy). The eigenstate thermalization hypothesis says that for an arbitrary initial state, the expectation value of
A
^
{\displaystyle {\hat {A}}}
will ultimately evolve in time to its value predicted by a microcanonical ensemble, and thereafter will exhibit only small fluctuations around that value, provided that the following two conditions are met:
The diagonal matrix elements
A
α
α
{\displaystyle A_{\alpha \alpha }}
vary smoothly as a function of energy, with the difference between neighboring values,
A
α
+
1
,
α
+
1
−
A
α
,
α
{\displaystyle A_{\alpha +1,\alpha +1}-A_{\alpha ,\alpha }}
, becoming exponentially small in the system size. The off-diagonal matrix elements
A
α
β
{\displaystyle A_{\alpha \beta }}
, with
α
≠
β
{\displaystyle \alpha \neq \beta }
, are much smaller than the diagonal matrix elements, and in particular are themselves exponentially small in the system size. These conditions can be written as
A
α
β
≃
A
¯
δ
α
β
+
A
2
¯
D
R
α
β
,
{\displaystyle A_{\alpha \beta }\simeq {\overline {A}}\delta _{\alpha \beta }+{\sqrt {\frac {\overline {A^{2}}}{\mathcal {D}}}}R_{\alpha \beta },}
where
A
¯
=
A
¯
(
E
α
)
{\displaystyle {\overline {A}}={\overline {A}}(E_{\alpha })}
and
A
2
¯
=
A
2
¯
(
E
α
,
E
β
)
{\displaystyle {\overline {A^{2}}}={\overline {A^{2}}}(E_{\alpha },E_{\beta })}
are smooth functions of energy,
D
=
e
s
V
{\displaystyle {\mathcal {D}}=e^{sV}}
is the many-body Hilbert space dimension, and
R
α
β
{\displaystyle R_{\alpha \beta }}
is a random variable with zero mean and unit variance. Conversely if a quantum many-body system satisfies the ETH, the matrix representation of any local operator in the energy eigen basis is expected to follow the above ansatz.
== Equivalence of the diagonal and microcanonical ensembles == We can define a long-time average of the expectation value of the operator
A
^
{\displaystyle {\hat {A}}}
according to the expression
A
¯
≡
lim
τ
→
∞
1
τ
∫
0
τ
⟨
Ψ
(
t
)
|
A
^
|
Ψ
(
t
)
⟩
d
t
.
{\displaystyle {\overline {A}}\equiv \lim _{\tau \to \infty }{\frac {1}{\tau }}\int _{0}^{\tau }\langle \Psi (t)|{\hat {A}}|\Psi (t)\rangle ~dt.}
If we use the explicit expression for the time evolution of this expectation value, we can write
A
¯
=
lim
τ
→
∞
1
τ
∫
0
τ
[
∑
α
,
β
=
1
D
c
α
∗
c
β
A
α
β
e
−
i
(
E
β
−
E
α
)
t
/
ℏ
]
d
t
.
{\displaystyle {\overline {A}}=\lim _{\tau \to \infty }{\frac {1}{\tau }}\int _{0}^{\tau }\left[\sum _{\alpha ,\beta =1}^{D}c_{\alpha }^{*}c_{\beta }A_{\alpha \beta }e^{-i\left(E_{\beta }-E_{\alpha }\right)t/\hbar }\right]~dt.}
The integration in this expression can be performed explicitly, and the result is
A
¯
=
∑
α
=
1
D
|
c
α
|
2
A
α
α
+
i
ℏ
lim
τ
→
∞
[
∑
α
≠
β
D
c
α
∗
c
β
A
α
β
E
β
−
E
α
(
e
−
i
(
E
β
−
E
α
)
τ
/
ℏ
−
1
τ
)
]
.
{\displaystyle {\overline {A}}=\sum _{\alpha =1}^{D}|c_{\alpha }|^{2}A_{\alpha \alpha }+i\hbar \lim _{\tau \to \infty }\left[\sum _{\alpha \neq \beta }^{D}{\frac {c_{\alpha }^{*}c_{\beta }A_{\alpha \beta }}{E_{\beta }-E_{\alpha }}}\left({\frac {e^{-i\left(E_{\beta }-E_{\alpha }\right)\tau /\hbar }-1}{\tau }}\right)\right].}
Each of the terms in the second sum will become smaller as the limit is taken to infinity. Assuming that the phase coherence between the different exponential terms in the second sum does not ever become large enough to rival this decay, the second sum will go to zero, and we find that the long-time average of the expectation value is given by
A
¯
=
∑
α
=
1
D
|
c
α
|
2
A
α
α
.
{\displaystyle {\overline {A}}=\sum _{\alpha =1}^{D}|c_{\alpha }|^{2}A_{\alpha \alpha }.}
This prediction for the time-average of the observable
A
^
{\displaystyle {\hat {A}}}
is referred to as its predicted value in the diagonal ensemble, The most important aspect of the diagonal ensemble is that it depends explicitly on the initial state of the system, and so would appear to retain all of the information regarding the preparation of the system. In contrast, the predicted value in the microcanonical ensemble is given by the equally-weighted average over all energy eigenstates within some energy window centered around the mean energy of the system
⟨
A
⟩
mc
=
1
N
∑
α
′
=
1
N
A
α
′
α
′
,
{\displaystyle \langle A\rangle _{\text{mc}}={\frac {1}{\mathcal {N}}}\sum _{\alpha '=1}^{\mathcal {N}}A_{\alpha '\alpha '},}
where
N
{\displaystyle {\mathcal {N}}}
is the number of states in the appropriate energy window, and the prime on the sum indices indicates that the summation is restricted to this appropriate microcanonical window. This prediction makes absolutely no reference to the initial state of the system, unlike the diagonal ensemble. Because of this, it is not clear why the microcanonical ensemble should provide such an accurate description of the long-time averages of observables in such a wide variety of physical systems. However, suppose that the matrix elements
A
α
α
{\displaystyle A_{\alpha \alpha }}
are effectively constant over the relevant energy window, with fluctuations that are sufficiently small. If this is true, this one constant value A can be effectively pulled out of the sum, and the prediction of the diagonal ensemble is simply equal to this value,