1. No category

(Dynamical) quantum typicality: What is it and what are its
physical and computational implications ?
Jochen Gemmer
University of Osnabrück,
Kassel, May 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 should occur: We see it in system we assume to be closed.
Why it should not occur: 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
1 − Hˆ
1ˆ ˆ
e kT or ρˆeq ≈ δ(
H − E)
Z
Z
ρˆeq =
problem: invariance of Von Neuman-entropy
traditional cure: open quantum systems ⇒ this requires:
environment
special structure of structure of environment: large, broad band (oscillator)
bath
adequate weak coupling, 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 =
1ˆ ˆ
1 − Hˆ
e kT , ρˆeq = δ(
H − E)
Z
Z
evolves into
ˆ
≈ Tr{ˆ
ρeq A}
ˆ ............ 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(t)|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 expectation values eventually.
“dynamical typicality”
p
ˆ → B(t)
ˆ
|ψi := ρˆd |ωi: “taylored”, non-uniform random states, A
ˆ
ˆ ρ}
HA[hψ|B(t)|ψi]
= Tr{B(t)ˆ
ˆ
HV[hψ|B(t)|ψi]
≤
ˆ 2 (t)ˆ
Tr{B
ρ}
deff
deff : inverse of the largest eigenvalue of ρˆ, interpreted as “effective dimension”
⇒ Very many different pure states exhibit very similar dynamics of expectation
values
J.Gemmer
quantum typicality
Typicality in formulas
“correlation functions”
p
|ψi := ρˆd |ωi: “taylored”, non-uniform random states,
ˆ →C
ˆ (t) := 1 (B(t)
ˆ B
ˆ +B
ˆ B(t))
ˆ
A
2
ˆ (t)|ψi] = Tr{C
ˆ (t)ˆ
HA[hψ|C
ρ}
ˆ (t)|ψi] ≤
HV[hψ|C
ˆ 2 (t)ˆ
Tr{C
ρ}
deff
What to do practically?
ˆ
Compute |ψ(t)i := e −i Ht/~
p
ˆ
ˆ ρ}
hψ(t)|B|ψ(t)i
≈ Tr{B(t)ˆ
ˆ
ˆ
ρˆd |ωi , |ψ ′ (t)i := e −i Ht/~ B
p
ρˆd|ωi
ˆ ′ (t)i] ≈ Tr{ 1 (B(t)
ˆ B+
ˆ B
ˆ B(t))ˆ
ˆ
Re[hψ(t)|B|ψ
ρ}
2
ˆ
¯ 2
ˆ
¯ 2
ˆ
ˆ
If ρˆ is of “exponential form”, e.g., ρˆ ∝ e −β(H−E ) −α(B−B) or ρˆ ∝ e −β H−αB
than everything can be done based on “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 → Jˆ
here: infinite temperature, i.e., ρˆ ∝ ˆ
1, B
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 B
ˆ
ˆ
En ≈ Em → hn|B|ni
≈ hm|B|mi
Jochen’s formulation: “Eigenstates belong to the set of typical states”
If ETH applies:
- Expecation values from microncanonical ensembles are close to expecation
values of individual eigenstates
- Initial state independent equilibration:
X
ˆ
ρnm Bmn e i (En −Em ) t
hB(t)i
=
n,m
If the oscillating terms behave like “white noise” for τ large enough
ˆ )i =
hB(τ
X
ETH
ρnn Bnn ≈ Bnn
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
The End
more people involved in this: R. Steinigeweg, A. Khodja, H. Niemeyer, 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