E R G O D I C I T Y, E I G E N S TAT E T H E R M A L I Z AT I O N , - - PowerPoint PPT Presentation

E R G O D I C I T Y, E I G E N S TAT E T H E R M A L I Z AT I O N , A N D T H E F O U N D AT I O N O F C L A S S I C A L A N D Q U A N T U M S TAT- M E C H

L O R E N Z O C A M P O S V E N U T I ( U S C )

D O L O M I T E S J U LY 2 0 1 9

Lorenzo Campos Venuti Lawrence Liu

arXiv:1904.02336

What is ergodicity here?

G E N E R A L Q U E S T I O N

Question: How do we build equilibrium statistical mechanics starting from the microscopic laws?

Classical laws (Newton) One possibility is Boltzmann’s ergodic hypothesis (to be proved in actual models) Quantum laws

?

One possibility is ETH [Deutsch91, Srednicki94, Berry77, Rigol-Dunjko- Olshanii08] (to be proved in actual models)

Large math literature: Shnirelman74, Zeldtich87, Sunada97, Rudnik&Sarnak94, Hassell08…

S E T T I N G T H E S TA G E : ( Q U A S I - ) I S O L AT E D Q S Y S T E M

H = X

n

EnΠn

Energy shell Energy shell’s Hilbert space

SV

Set of quantum states supported in HV

hAiV = Tr(AρV ), ✓ ρV = ΠV NV ◆

Microcanonical state

Energy constrained to V ⊂ σ(H)

V = {En| E ≤ En ≤ E + |∆}

e.g.

ΠV = X

En∈V

Πn

HV = Ran(ΠV )

Ideally:

Projector

NV = Tr(ΠV ) = dim(HV )

Dimension Eigendecomposition (finite dim)

ρV = X

En∈V

fnΠn

More generally ρV invariant ⇒

E I G E N S TAT E T H E R M A L I Z AT I O N H Y P O T H E S I S ( E T H )

Tr[A(t)ρ0] = Tr X

n

ΠnAΠnρ0 ! ' hAiV Tr(ρ0) = hAiV

ΠnAΠm ' hAiV Πnδn,m En, Em 2 V ΠnAΠn ' hAiV Πn

ETH: ETH-D (Diagonal)

ρ0 ∈ SV

hEn|A|Emi ' hAiV δn,m

For non-degenerate levels: Implications:

f := lim

T →∞

1 T Z T dtf(t) !

E T H

et al., 2016). While all known examples of thermalizing systems obey ETH, at present it is not clear if ETH is a necessary condition for thermalization.

Abanin, E. Altman, I. Bloch, and M. Serbyn arXiv:1804.11065 Ergodicity, Entanglement and Many-Body Localization:

Proposition 0 If observable A satisfies ETH-D then A thermalizes (for all initial states) Definition 1 Observable A thermalizes if Tr[A(t)ρ0] = hAiV

8ρ0 2 SV

ETH-D thermalization What about ? What about ergodicity?

Often taken as synonym for thermalization:

1 We note that in the context of quantum many-body systems

the term ergodicity is defined somewhat differently compared to classical mechanics. Our use of this term is synonymous with thermalization, as discussed in Section II.A.

E T H - I N T E R L U D E

E R G O D I C I T Y I N C L A S S I C A L D Y N A M I C A L S Y S T E M S

Theorem Let (M, gt, μ) be a dynamical system. M measure space, gt flow, μ 


gt-invariant measure (normalized). The following are equivalent:


• 1. For any (almost) invariant set X ⊆ M, either μ(X)=0 or μ(X)=1.
• 2. For any two functions f,g∈ L∞(M,μ) 


• 3. For any f∈ L1(M,μ)



 
 
 for almost any initial point x0

hf(t)giµ = hfiµhgiµ

lim

T →∞

1 T Z T dtf(gt(x0)) = hfiµ

aka metric indecomposability

Q U A N T U M D Y N A M I C A L S Y S T E M S

Definition Observable A is shell-ergodic if

hA(t)AiV = (hAiV )2

Proposition 1 Observable A is shell-ergodic if and only if A(t)ΠV = hAiV ΠV

(characterization 2.) (characterization 3.)

Proposition 2 Observable A thermalizes if and only if is shell-ergodic Proposition 3 Observable A is shell-ergodic if and only if it satisfies ETH-D ETH-D ergodicity thermalization

M E T R I C I N D E C O M P O S A B I L I T Y

What about characterization 1.? Heisenberg evolution:

Shell-ergodicity for all observables A

A(t)ΠV = hAiV ΠV

T (X) := E∗

t (ΠV XΠV ) = ΠV hXiV

T = |ΠV i ih hρV |

Moreover

T = T ∗ ⇒ ρV = ΠV NV =: ρMC

E∗

t : E∗ t (A) = eitHAe−itH

However: NOT possible unless HV is one-dimensional!

Thermalization only for some

• bservables

B E T T E R D E F I N I T I O N S

Definition 2’a Observable A is shell-ergodic with precision ε if

• Tr (A(t)⇢0) hAiV
•  ✏ kAk

Definition 1’ Observable A thermalizes with precision ε if Definition 2’b Observable A is strong shell-ergodic with precision ε if

• ⇣

A(t) hAiV ⌘ ΠV

•  ✏ kAk

Definition 3’ Observable A satisfies ETH-D to precision ε if

for all En ∈ V

k ΠnAΠn hAiV Πn k  ✏kAk

• hn|A|ni hAiV
•  ✏kAk

• hA(t)AiV (hAiV )2
•  ✏ kAk2

2

Non degeneracy ⟶

R E M A R K : W H AT A B O U T T H E S T E A D Y S TAT E ?

• Tr (A(t)⇢0) hAiV
•  ✏ kAk

Observable A thermalizes with precision ε

We can pick the ρV we prefer at the price of ε, e.g. ρV = ρMC

• Tr[AEt(⇢0)] hAiV
• =
• hAi0

V hAiV

•  ✏

B E T T E R P R O P O S I T I O N

Thermalization Shell-Ergodicity Strong
 Shell-Ergodicity ETH-D

✏

✏

✏

✏

✏

✏

✏

✏

q k⇢−1

V ΠV k✏

✏

H O W I S C L A S S I C A L S TAT- M E C H B U I LT ?

Possibility a) Fix energy exactly:

ME = {(q, p)| H(q, p) = E}

+ Metric indecomposable

f = hfiE

S(E) = kB ln(ω(E)) ω(E) = Z dqdp δ[H(q, p) − E]

Possibility b) Fix energy approximately:

VE,∆ := {(q, p)| E ≤ H(q, p) ≤ E + ∆}

S TAT- M E C H C O N T I N U E D hfiVE,∆ := R

VE,∆ dxf(x)

R

VE,∆ dx

= R

I dE !(E)hfiE

R

I dE !(E)

.

∀E ∈ I := [E, E + ∆]

Need metric indeomposable We want uniform measure on V:

f(x) = hfiVE,∆

What we really want: (for almost any x)

f = hfiE

∀E ∈ I := [E, E + ∆]

hfiE = hfiVE,∆

Obviously +

⇒

Classical ETH (ETH-C)

S TAT- M E C H I I I

When do we have ?

hfiVE,∆ = hfiE + ∆ 2 hfi0

E + O

• ∆2

∆ → 0

• hfiVE,∆ hfiE

hfiE

• . ∆

2✏f ,

• is ✏f = hfiE/
• hfi0

E

• .

to thermalize, we need

enient proxy is function ✏H = E and 1.

Hamiltonian function ✏ quirement ∆/E ⌧ 1. now go back to

For the Hamiltonian

Q U A N T U M S TAT- M E C H

Hamiltonian with non-degenerate spectrum in V ETH-D Shell-ergodicity Approach a) (fix H=E) not possible because:

• 1. Time-energy uncertainty (Landau)
• 2. Impossible to define meaningful

entropy Approach b) :

T H A N K Y O U