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Thermalization, Entanglement, Many-body systems, and all that jazz

1. Entanglement within many-body quantum states drives subsystems to thermalization although the full state remain pure and of zero entropy.

2. Closed many-body systems maybe integrable or not, and it is believed that non-integrable systems with very few conserved laws rapidly thermalize, and the mechanism is the “ETH”.

3. Information diffusion within many-body systems could be exponentially fast and lead to “scrambling”, “butterfly effect” and ...

Entanglement in two coupled kicked rotors

q⁰₁ = q₁ + p₁⁰ p⁰₁ = p₁ + K₁

2π sin(2πq₁) + b

2π sin(2π(q₁+q₂)) q⁰₂ = q₂ + p₂⁰

p⁰₂ = p₂ + K2

2π sin(2πq₂) + b

2π sin(2π(q₁+q₂)).

Quantize this to get a Unitary operatorU = (U₁⊗U₂)U_int(b).

What are the entanglement properties of the eigenstates ofU?

What connection does it have to classical chaos?

Use K₁ = 0.1,K₂ = 0.15. N = 1/h.

Left: N = 15,20,25: spectral avg. entanglement vs coupling

Right: N = 40,b= 2.0: Entanglement across a spectrum.

(Entangling power of quantum chaos, AL: Phys. Rev. E, 2001)

Two Coupled Quantum Chaotic Tops

QC time evolution leads to near maximum entanglement

Two Coupled Quantum Chaotic Tops

Quantum chaos leads to universal bipartite pure state

entanglement: Marcenko-Pastur distribution of eigenvalues of RDM

The entanglement is nearly maximal and S = ln(γN). For N =M, S = ln(N)−1/2. AsM → ∞, γ →1.

(J. Bandyopadhyay, AL Phys. Rev. Lett., 2002)

Ising Model in a tilted magnetic field

H(J,B, θ) =J

L−1

X

n=1

σ^z_nσ^z_n+1+B

L

X

n=1

(sin(θ)σ_n^x+ cos(θ)σ^z_n)

Symmetry reduce.

L

Y

i=1

⊗σ_i^y

!

H(J,B, θ)

L

Y

i=1

⊗σ_i^y

!

= −H(−J,B, θ).

B|s1s2. . .sNi=|sN. . .s2s1i, [H, B] = 0

Quantum Chaos: tilted field . J = B = 1

0 1 2 3 4 5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Energy

Angle of tilt

0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3

P(s)

Spacing s θ=99π/200

θ=15π/32 θ=7π/16 θ=π/6 Poisson Wigner

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Tilt angle 10 <τ >

SL/2

0 0.05 0.1 0.15 0.2

1.2 1.3 1.4 1.5 KS statistic

10 <τ >

SL/2

0.5 1 1.5 2 2.5 3 3.5 4

1 2 3 4 5 6

Sl

l θ=99π/200

θ=15π/32 θ=5π/8 θ= π/6

(J. Karthik, A. Sharma, AL Phys. Rev. A. (2007))

Spin chains: Many-body localization

(a)

(b)

Ising Spins with next nearest neighbor interactions

H =−

L−1

X

i=1

J_iσ_i^zσ_i+1^z +J₂

L−2

X

i=1

σ_i^zσ_i+2^z +h

L

X

i=1

σ_i^x

(Kjall, Bardarsson, Pollmann, PRL, 2014).

Experiments: 1. Bose Hubbard

Experiments: 2. Three globally coupled spins

Kicked tops to many-body spins

H = ~p

τ

J_y + ~κ

2j

J_z²

∞

X

n=−∞

δ(t −nτ)

(Kus, Scharf, Haake, 1987; Haake’s book.)

J^x,y,z =P2j

l=1σ^x,y,z/2, the unitary or Floquet operator:

U = exp −i κ 8j

2j

X

l6=l⁰=1

σ^z_lσ^z_l0

!

exp −iπ 4

2j

X

l=1

σ^y_l

! ,

(Wang, Ghose, Sanders, and Hu, 2004)

Thermodynamic limit is also classical limtj → ∞.

Classical Map: X

_n²

+ Y

_n²

+ Z

_n²

= 1

X_n+1 =Z_ncos(κX_n) +Y_nsin(κX_n) Y_n+1 =−Z_nsin(κX_n) +Y_ncos(κX_n) Zn+1 =−Xn.

Results from the 3-Qubit experiment

Results from the 3-Qubit experiment

Results from the 3-Qubit experiment

Eigenstate Thermalization Hypothesis

|ψ(t)i=e^−iHt|ψ(0)i=X

α

e^−iHt|E_αihE_α|ψ(0)i

=X

α

e^−iEαtC_α|E_αi, C_α=hE_α|ψ(0)i.

ρ(t) =|ψ(t)ihψ(t)|=X

α,β

e^{−i(Eα−Eβ)t}C_αC_β^∗|E_αihE_β|

ρ(t) =X

α

|C_α|²|E_αihE_α| ≡ρ_diag

A(t) = Tr(ρdiagA)? =?Amicro = 1 N_∆E

X

|E−Eα|<∆E

hEα|A|Eαi.

ETH: Deutsch, Srednicki, Rigol ...

hE_α|A|E_αi=hAi_micro,Eα + ∆_α. h∆_αi= 0,h∆²_αi ∼ hA²i_microe^−S(E).

(From Rigol et. al., Nature 2008)

ETH: Deutsch, Srednicki, Rigol ...

hEα|A|Eαi=hAimicro,Eα + ∆α. h∆_αi= 0,h∆²_αi ∼ hA²i_microe^−S(E).

(From J. Deutsch, arXiv:1805.01616)