1. No category

Quantum typicality: what is it and what can be done with it?
Jochen Gemmer
University of Osnabrück,
LMU Muenchen, May, Friday 13th, 2014
J.Gemmer
quantum typicality
Outline
Thermal relaxation in closed quantum systems?
Typicality in a nutshell
Numerical experiment: model, observables and results
Typicality in formulas
Spin transport in the Heisenberg chain
Eigenstate thermalization hypothesis
J.Gemmer
quantum typicality
Thermal relaxation in closed quantum systems?
Why it exists: We see it in system we assume to be closed.
Why it does not exist: There are issues with the underlying theory:
Quantum Mechanics
(Non-eq.) Thermodynamics
autonomous dynamics of a few
macrovariables
attractive fixed point, equilibrium
often describable by master equations,
Fokker-Planck equations, stochastic
processes, etc.
autonomous dynamics of the wave
function (number of parameters:
insane)
no attractive fixed point (Schroedinger
equation)
Schroedinger equation is no rate
equation
Quantum systems that explicitly exhibit relaxation but are not of the “small
system + large bath” type appear to be rare in the literature.
To cut it short: Why and how do two cups of coffee thermalize each other?
J.Gemmer
quantum typicality
Typicality in a nutshell
The naive view on relaxation i.e. 2nd law of thermodynamcis:
QM:
ρˆ0 = |ψihψ|
evolves into
1ˆ ˆ
1 − Hˆ
e kT or ρˆeq ≈ δ(
H − E)
Z
Z
ρˆeq =
problem: invariance of Von Neuman-entropy
traditional cure: open quantum systems ⇒ this requires:
large, doable, broad band environment (usually oscillators), adequate weak
coupling (Van-Hove structure), applicability of projection techniques, specific
initial states: factorizing, thermal bath, etc.
Typicality:
ρˆ0 = |ψihψ|
but
does not evolve into
ˆ
hψ|A(t)|ψi
ρˆeq =
1 − Hˆ
1ˆ ˆ
e kT , ρˆeq = δ(
H − E)
Z
Z
evolves into
ˆ
≈ Tr{ˆ
ρeq A}
ˆ |ψi ............ Can this be true?
for many (all?) A,
J.Gemmer
quantum typicality
Numerical experiment: model and observables
R
L
spin-model
Heisenberg-type Hamiltonian: A ladder with anisotropic,
XXZ -type couplings which are strong along the legs and
weak along the rungs:
X
ˆ =
Jij (ˆ
σxi σ
ˆxj + σ
ˆyi σ
ˆyj + 0.6 σ
ˆzi σ
ˆzj ),
H
ij
Jij = 1 for solid lines, Jij = κ = 0.2 for dotted lines and
Jij = 0 otherwise. Total number of spins: NP= 32. The
z-component of total magnetization Sz = i σ
ˆzi is
conserved
We analyze: “magnetization difference” xˆ
!
X l X r
σ
ˆz
σ
ˆz −
xˆ =
l ∈L
r ∈R
eigenvalues of xˆ within the subspace of vanishing total
magnetization, Sz = 0: X = −16, −14, ....., +16.
J.Gemmer
quantum typicality
Numerical experiment: results
Phys. Rev. E, 89, 042113, (2014)
xˆ: z-magnetization difference between legs
PX (t): probability to find a certain X
H. de Raedt, K. Michielsen
(Juelich))
time shifted hˆ
x (t)i
<x(t)>
12
1
0.8
0.6
<x(0)>=2
<x(0)>=4
<x(0)>=6
<x(0)>=8
<x(0)>=10
<x(0)>=12
8
4
0.4
100
Px(t) 0.2
80
0 15
0
0
60
10
5
40
0
-5
-15 0
|ψX (0)i = e
t
100
150
ˆ x P(S
ˆ z = 0)|ωi,
P
(remark: taking this picture took 6h on 65 000
CPU’s. Thanks to:
J.Gemmer
12
σ ² (t)
data from solving the Schroedinger equation
(N = 32) for two pure, partially random initial
states:
ˆ2
−αH
50
variances of x
20
-10
x
t
8
<x(0)>=0
<x(0)>=2
<x(0)>=4
<x(0)>=6
<x(0)>=8
<x(0)>=10
<x(0)>=12
typical variance
4
00
quantum typicality
50
t
100
150
Typicality in formulas
Phys. Rev. Lett., 102, 110403 (2009), Phys. Rev. Lett., 86, 1927 (2001)
“static typicality”
ˆ
< A >:= Tr{A}/d
: expectation value of the maximally mixed state
|ωi uniform random states sampled according to the unitary invariant measure
ˆ
HA[hω|A|ωi]
=< A >
ˆ
HV[hω|A|ωi]
=
1
< A2 > − < A >2
d +1
In a high dimensional Hilbert space almost all possible states feature very
similar expectation values for observables with bound spectra. ⇒ It is no
surprise to find these “equilibrium” expectation values overwhelmingly often.
“dynamical typicality”
p
|ψi := ρˆd |ωi: “taylored”, non-uniform random states,
ˆ
ˆ ρ}
hψ|A(t)|ψi
≈ Tr{A(t)ˆ
ˆ
The statisitcal variance of hψ|A(t)|ψi
decreases as 1/deff where the latter is the
inverse of the largest eigenvalue of ρˆ.
⇒ Very many different pure states exhibit dynamics of expectation values close
to those of corresponding mixed states
2
ˆ ¯ 2
ˆ
ˆ
ˆ
If ρˆ is of “exponential form”, e.g., ρˆ ∝ e −β(H−E ) −α(A−A0 ) or ρˆ ∝ e −β H−αA
than one may infer dynamics of mixed states from “pure state propagation”
J.Gemmer
quantum typicality
Spin transport in the anisotropic Heisenberg chain
2
(a) L=33 (β=0)
norm (a.u.)
0.05
0
(a) ∆=1.0 (β=0)
-1
[34,36]
[t1,t2]
0
100
tJ
-2
10
-1
(b) ∆=1.5 (β=0)
10
-2
10
[20,21]
0.05
[35]
0
1/30 1/20
1/10
1/L
2
-3
10
tJ
20
0.1
0.05
0
J.Gemmer
odd
fit
(β=0)
10 0
even
(b) Drude weight (β=0)
0
C(t1,t2) / J
C(t) / J
2
L
∆=0.5
∆=1.0
∆=1.5
2
L=18 (ED)
L=18
L=20
L=22
L=24
L=26
L=28
L=30 (k=0)
L=32 (k=0)
L=33 (k=0)
tDMRG
C(t1,t2) / J
C(t) / J
2
10
∆=0.5
∆=1.0
∆=1.5
0.1
C(t) / J
Linear response ⇒ conductivity from current
autocorrelation function
ˆ ,A
ˆ := Jˆ
here: infinite temperature, i.e., ρˆ := J/d
ˆ J}
ˆ
C (t) ∝ Tr{J(t)
(PRL., 112, 120601, (2014) )
0
quantum typicality
Bethe Ansatz
lower bound
ED, fit
RK, L=30
RK, L→∞
tDMRG
1
0.5
anisotropy ∆
1.5
Eigenstate thermalization hypothesis (ETH)
ˆ that are close in energy feature
ETH: Eigenstates of some Hamiltonian H
ˆ that are close to each other.
expectation values of some observable A
ˆ
ˆ
En ≈ Em → hn|A|ni
≈ hm|A|mi
Jochen’s formulation: “Eigenstates belong to the set of typical states”
If ETH applies:
- Expecation values from microncanonical ensembles are close to expectation
values of individual eigenstates
- Initial state independent (ISI) equilibration:
X
ˆ
ρnm Amn e i (En −Em ) t
hA(t)i
=
n,m
If the oscillating terms behave like “white noise” for τ large enough
ˆ )i =
hA(τ
X
ETH
ρnn Ann ≈ Ann
n
But does ETH apply? Have to check by exact diagonalization, or.......
J.Gemmer
quantum typicality
ETH check for the Heisenberg ladder
(Phys. Rev. Lett., 112, 130403, (2014))
ˆ := xˆ2 : width of the magnetization distribution
A
X
2
ˆ
ˆ
¯ :=
ρnn Ann ≈ hψ|A|ψi
ρˆ ∝ e −α(H−U)
A
n
¯ 2 :=
Σ +A
2
X
ˆ ′ (τ )i]
≈ Re[hψ(τ )|A|ψ
ρnn A2nn
n
(a) L=7
(d)
ED
3
2
Ann
6
Ann
L=7
1
J||
0
J
0
0.03
0.02
-1/2
(deff)
L=10
0.01
0
ETH applies!
L
A
-4
L=9
L=8
(c) L=10
3
0
6
t1
0
(b) L=9
3
6
180
Σ
0
t J||
0
3
γ(t)
Ann
6
-2
U / J||
0
A±Σ
d=2
2
J.Gemmer
L
2L
ˆ was a random
Σ scales approx like H
matrix.
quantum typicality
Is ISI equilibration always driven by ETH?
Back to the two cups of coffee: Temperature differences between macroscopic
object always equilibrate, regardless of the objects ⇒ ETH always fulfilled? We
ˆ=H
ˆL − H
ˆ R between all sorts of coupled
investigate energy differences A
“spin-objects”
(a)
(b)
(a) coupling strength
γ=0.36
(c)
0
J
L
J
J
L
κ
γ=0.38
J
L
R
γ=0.41
γ=0.39
-2
10
R
R
10
∆=0.1
∆=0.6
∆=0.9
γ=0.29
Σ’
J
(b) anisotropy
(c) disorder
γ=0.41 γ=0.33
0
10
γ=0.32
(d) structure
W=0.0
W=0.2
W=2.0
1D
2D
γ=0.39
Σ’
κ
κ=0.1
κ=0.3
κ=0.6
γ=0.51
γ=0.40
-2
10
1
10
3
10
deff
5
10
1
10
3
10
deff
Seems like the ETH is indeed always fulfilled, but what about “integrable”
systems ?
J.Gemmer
quantum typicality
5
10
Is ISI equilibration always driven by ETH?
|ωi
There appears to be ISI equilibration
for large systems
Check scaling of Σ
The ETH is violated w.r.t. the bare
Σ.
Define “scaled” ETH parameter Σ/δ:
D(t) / D(0)
2
ˆ 2 −β(A−N
ˆ
L) )
|ψ(0)i ∝ e −αH
0.4
0.2
0
0
100
0.2
2
2
ˆ 1 e −αHˆ }2
ˆ 2 1 e −αAˆ }−Tr{A
δ = Tr{A
Z
Z
0
0
200
400
600
0.1
Σ, (Σ/δ)
2
t
0
×10
Σ
2
(Σ/δ)
0
0.1
0.2
1/N
(erratum : D ⇒ A)
J.Gemmer
200
D(0)>0
D(0)<0
(b) |D(0)|=8
2
The scaled ETH parameter appears
to vanish in the limit of large
systems
D(0)>0
D(0)<0
(a) |D(0)|=4
0.4
D(t) / D(0)
Consider energy difference dynamics
in a clean Heisenberg chain w.r.t.
1:2 partition using initial states like
quantum typicality
0.3
The End
People who truly did the work: R. Steinigeweg, H. de Raedt, K. Michielsen A.
Khodja, D. Schmidtke, C. Gogolin ....
Thank you for your attention!
The talk itself as well as the mentioned papers may be found on our webpage.
J.Gemmer
quantum typicality