Memory Storage and Thermalization in Multi-Mode Systems

Memory Storage and Thermalization in Multi-Mode Systems

Oleg Kaikov

LMU Munich, Germany

Supervisor: Prof. Dr. Georgi Dvali Co-supervisor: Dr. Sebastian Zell

44 ^th IMPRS Workshop

MPP Munich, July 15 ^th 2019

Outline

Motivation

System with high entropy Specific sub-system Solution of Sub-System

Coupling sensitivity Stability

Equilibration

Thermalization

Outlook: ETH

Summary

Motivation Solution of Sub-System Outlook: ETH Summary

System with high entropy

Application to black holes

G. Dvali [Phys. Rev. D 97, 105005 (2018)]

(1) large number of micro-states

Ω ∼ e ^S

H = ˆ

 1 − n ˆ ₀

M



S

X

k=1

ε k n ˆ k

System with high entropy

Application to black holes

G. Dvali [Phys. Rev. D 97, 105005 (2018)]

(1) large number of micro-states

Ω ∼ e ^S

(2) all S modes ˆ n _k gapless to fit within infinitesimal energy gap H = ˆ

 1 − n ˆ ₀

M

 S

X

k=1

ε k n ˆ k

General system

Individual mass gaps

Interactions among ˆ a _j modes Interaction with external ˆ b 0 mode

H = ˆ X

j

 1 − n ˆ 0

M _j



ε j n ˆ j + X

j ,k j 6=k

A _jk

 ˆ

a ^† _j a ˆ _k + h.c.

 + λ

 ˆ

a ^† ₀ b ˆ 0 + h.c.



j , k = 1, . . . , D

Specific sub-system

Common mass gap

H = ˆ X

j

 1 − n ˆ ₀

M



εˆ n _j + X

j ,k j 6=k

A _jk  ˆ

a ^† _j a ˆ _k + h.c.  + λ 

ˆ

a ^† ₀ b ˆ ₀ + h.c. 

j , k = 1, . . . , D

Solution of sub-system

D = 2 solved

G. Dvali [arXiv:1810.02336]

Our generalization: arbitrary D

Observation: h ˆ n _j i (t) independent of (time-varying) common mass gap

Time-evolution

|ini = |N 1 , . . . , N D i ; N j ∈ [0, N] , P

j

N j = N

Figure: h ˆ n _j i (t) (solid) and h ˆ n _j i (t) (dashed) for j = 1, 2, 10 for D = 10, N = 10.

Coupling sensitivity

A jk ∈ [−1, +1]

Figure: h ˆ n j i (t) for j = 1, 2, 10 vs A 12 for D = 10, N = 10.

Stability

N

= 0, N

= 3

Figure: h ˆ n _j i (t) for j = 1, 2 for various systems A _i ∈ [−1, +1] ^D×D .

Equilibration

D = 10, N = 100 ; |ini = |N 1 , . . . , N D i

Figure: h ˆ n _j i (t) vs N j with slope k as “memory of |ini”.

Thermalization

“Memory of |ini” k:

• indep. of N

• lim

D→∞ k = 0

Figure: Mean of k over A _i ’s and |ini’s vs D at N = 10.

Thermalization

h ˆ Oi (t) = h ˆ Oi _mc

h ˆ Oi (t) = lim

T →∞

1 T

T

Z

0

h ˆ Oi (t) dt

h ˆ Oi _mc = 1 N _ε

X

|E −ε|<δ ε

hε| ˆ O |εi

Outlook: Eigenstate Thermalization Hypothesis (ETH)

J. M. Deutsch [Phys. Rev. A 43, 2046 (1991)], M. Srednicki [Phys. Rev. E 50, 888 (1994)]

1 of 3 conditions: ε _j ' ε k ⇒ hε j | ˆ O |ε j i ' hε k | ˆ O |ε k i

Figure: hε| ˆ n _j |εi vs eigenstate energy ε for D = 6, N = 18.

Summary

Analytic generalization of high S system to arbitrary dim. D

Thermalization for large systems

Indications for thermalization mechanism distinct from that of ETH

Any questions?

Appendix

Formulae (solutions)

h ˆ n j i (t) = X

k,l ,s

N s Q jl Q jk Q sl Q sk cos [(ω l − ω k )t]

ω j = Q ^T AQ

jj = X

i ,k

A ik Q ij Q kj

h ˆ n j i (t) = X

k,s

N s Q _jk ² Q _sk ²

h ˆ n _a

i (t) = N _a

+  λ ϕ

 ²

(N _b

− N _a

) sin ² (ϕt)

h ˆ n _b

i (t) = N _b

−  λ ϕ

 ²

(N _b

− N _a

) sin ² (ϕt)

ϕ = 1 2

v u u

t (2λ) ² + X

i

ε M N _i

! ²

Formulae (unitarity)

H ∼ A ˆ X

i ,j

ˆ

a _j ^† a ˆ i , N i ∼ N D

⇒ M _{i →j} = hN _j | ˆ H |N _i i ∼ A p N _i pN _j

⇒ Γ i =

D

X

j =1

 M i →j

2 ∼ A ² N ² D

∼ const !

⇒ A ∼ √ D/N

τ =

P

k,l ,s |N s Q jl Q jk Q sl Q sk | P

k,l ,s |N s Q jl Q jk Q sl Q sk (ω l − ω k )|

k vs D at N = 10

Figure: k vs D at N = 10.

k vs N at D = 10

Figure: k vs N at D = 10.

k vs D and N

Figure: Mean of k over A i ’s and |ini’s vs D and N.

τ vs D at N = 10

Figure: τ vs D at N = 10 for A jk ∈ h

− √

D/N, + √ D/N i

.

τ vs N at D = 10

Figure: τ vs N at D = 10 for A jk ∈ h

− √

D/N, + √ D/N i

.

Equilibration time scale τ

Figure: τ vs D and N for A _jk ∈ h

− √

D/N, + √ D/N i

.

ETH: scatter of hε| ˆ n j |εi

Figure: Scatter σ of hε| ˆ n _j |εi within appropriate energy bins vs D and N.

Integrability

“A system is quantum integrable:

(i) If it exhibits n physically meaningful mutually commuting quantities that are in some sense independent [...] or depend linearly on some parameter of the Hamiltonian. [...]”

[C. Gogolin, J. Eisert, Rep. Prog. Phys. 79, 056001 (2016)]