Thermalization of isolated quantum
systems
(+ comments on black holes)
M. Kruczenski
Purdue University
Based on arXiv:1312.4612
In collaboration w/
S.Khlebnikov (Purdue)
Aspen 2014
Summary
● Introduction and motivation
Thermalization of an isolated quantum system.
ETH (Eigenstate Thermalization Hypothesis).
Local observables and entropy generation.
● Numerical study of thermalization in an
interacting boson gas.
Numerical tests of ETH.
Generation of entanglement entropy.
Rapid initial growth:
● Discussion of black holes.
The metric is an ETH operator.
No hair theorem is the GR equivalent of ETH.
● Conclusions
Motivation to study thermalization
Cold atoms: thermalization (or not) of isolated
systems with few atoms (~100).
QGP formation in RHIC:
tth ~ 0.5 fm/c
Au
Foundations of statistical mechanics:
classical ergodic theorem  QM
Formation of black holes in quantum gravity.
Au
Reminder of relation to quantum gravity
Quantum gravity has two main problems:
Strongly coupled at large energies
(non-renormalizable  strings)
Information problem in black holes.
(Through a process where an initial state forms a
black hole and then evaporates, a pure state
seems to evolve into a thermal density matrix)
String theory and AdS/CFT give a partial answer.
Strings (quantum gravity) in asymptotically AdS
spaces is dual to a CFT (Maldacena, GubserKlebanov-Polyakov, Witten).
Gauge theory
λ small → gauge th.
e.g.
String theory
λ large → string th.
AdS5xS5
S5 : X12+X22+…+X62 = 1 AdS5: Y12+Y22+…-Y52-Y62 =-1
(sphere)
(hyperbolic space)6
Since gravity in AdS is dual to a QFT all the rules
of quantum mechanics apply. The theory is well
defined, finite and unitary
Black hole formation is
a standard thermalization
problem in the boundary.
The problem is the dictionary:
Where is information localized?
Is there any information in the
Hawking radiation?
Thermalization
Statistical mechanics is based on the premise
that a subsystem (of a larger macroscopic
system) eventually reaches thermal equilibrium
characterized by a density matrix with maximum
entropy for given conserved quantities (energy,
etc.). Local observables thermalize.
Example:
Expansion of
a gas
Entropy
S   T r ( ˆ ln ˆ )
ˆ :
• density matrix of the system. Von Neumann
entropy of the system. In this case S(t)=0, for all t.
• density matrix of subsystem A, ˆ A .
Entanglement entropy SAB, SAB(t=0)=0 but
increases with time.
• thermal density matrix for system or subsystem
thermodynamical entropy S(E,N).
1   H  N
Maximum S for given E,N.
ˆ 
e
Z
For thermalization, nothing is required from the state
of the total system other than it has a fixed total
energy, particle number, etc. In particular, in quantum
mechanics, if it is initially in a pure state it should
remain in a pure state and evolve according to
This creates an apparent problem:
Consider an observable
associated with the
subsystem. After a thermalization time it should
become time independent and equal to its canonical
average.
However we get:
Should be (approximately) constant in time and
independent of the
coefficients !
Requiring thermalization of a subsystem for any
initial state of the whole system leads to the ETH
hypothesis (Deutsch and Srednicki).
are small, they determine the size
of the time dependent fluctuations. In the initial
state, the coefficients
are chosen such that the
off-diagonal elements add up, but after some time
the coherence is lost (for most times) and the
contribution is negligible giving:
We still have to require that this is independent
of the
‘s.
We now require that the diagonal elements are a
smooth function of the total energy.
Since the total energy is fixed (in a narrow band)
then the mean value is fixed to a microcanonical
average that agrees with the canonical average
by standard arguments.
The result is a non-trivial statement for the matrix
elements of the local, or other ‘thermal’, operators
of a macroscopic system.
Is ETH true?
It can be tested numerically by exact
diagonalization of small systems.
Hard core lattice boson gas with near neighbors
interactions up to ~20,000 states in the basis.
“Thermalization and its mechanism for generic
isolated quantum systems”, Rigol, Dunjko,
Olshanii, Nature 452, 854 (2008).
Later larger systems were (Rigol et al.)
considered but such that the largest invariant
subspace was of dim ~30,000.
Alternative check. (S. Khlebnikov, M.K.)
+) Allows to consider systems up to dimension
~15,000,000 where ETH is more evident.
-) Test is done on a subspace of states.
Basic observation:
For a bounded H and fixed t, the series is
absolutely convergent and, for any given
accuracy we need to keep only a finite number of
terms ~ t. Truncate the Hilbert space to:
Projecting the state onto such subspace gives:
where the (Ritz) vectors
are the
eigenvectors of the projected Hamiltonian:
Therefore we can apply all the same reasoning to
the Ritz vectors instead of the exact eigenstates.
We call this KETH, Krylov ETH.
Since ETH  KETH, checking KETH provides a
test of the more stringent ETH.
In the cases we study, a Krylov subspace of
order 1,000 is enough to follow the evolution until
thermalization. Given that, we can study much
larger Hilbert spaces, up to order 15,000,000.
(states still have 15,000,000 components).
It is a different type of test because it does not
imply that ETH is valid in the whole Hilbert space.
However, we checked that increasing the size of
the Krylov subspace (even up to the full Hilbert
space if possible) does not change the results.
Comments on numerics:
Time evolution is done by truncating the
expansion

e
 iH t
|  0   J 0 (t ) |  0   2  ( i ) J n (t )T n ( H ) |  0 
n
n 1
to a finite number of terms.
The Ritz vectors (eigenstates of H in the
reduced subspace) are found using the Lanczos
method with re-orthogonalization.
System considered: (Hard core bosons)
Results
Thermalization of site occupation numbers:
Nevertheless the evolution is completely
reversible. (Plus checks with different methods).
Thermalization of site occupation numbers:
Krylov ETH for site occupation numbers
Free fermions instead
KETH for 1-particle eigenstates occupation #’s
For free particles they are integers instead
Off-diagonal elements for site occupation #’s
 E l | n i | E l   as a function of energy difference
n k (Úk )
compared to Bose-Einstein distribution
for two energy eigenstates.
Agrees for the excited state
Why does this happen? (Deutsch)
Consider the free case where
are integers.
The interaction mixes a large
number of states that should be
considered degenerate.
If the
are essentially random, then the
average
fluctuates little between i ’s. It still
fluctuates, as usual, between QM measurements.
Summary
For a given Hamiltonian, ETH is a property
obeyed by certain operators that is equivalent to
the statement that their averages thermalize.
For free particles, local operators and single
particle occupation numbers do not obey ETH.
However they do if a small an essentially random
interaction is turned on. Mathematically this is a
consequence of the central limit theorem.
We tested these statement numerically for a
particular system using the Krylov subspace
method (allowing a large number of states in the
basis).
Generation of entanglement entropy.
For the system consider we can naturally study
the entanglement entropy between systems A
and B. Initially the entropy grows and then
decays.
Initial behavior follows from Taylor expansion
Derivation:
|  (t )  e
 iHt
|  (0 )
 A (t ) ;  A (0 )  t 
p
( p)
A
implies analyticity of ρA
(0 )  
Initially the eigenvalues of ρA are 1,0,0,0,0,0…
So, using perturbation theory they are now
1 + tp ρ(p)00 , tp ρ(p)aa implying
S ;  t ln(t )  
p
p
( p)
aa
 p
(p)
00
p
t ln t
a0
In our case p=2 and ρ00=-ΔE2
or
S~t
p
ETH property of entanglement entropy
SAB as a function of eigenstate energy
Summary
Entanglement entropy is generated by streaming
particles into the vacuum and rises fast until it
reaches the limit set by the thermodynamical
entropy. In that way the subsystem reaches a (quasi)
thermal state.
The growth of subsystem entanglement entropy can
be thought as the physical reason for thermalization
of local observables.
After that, the subsystem keeps loosing energy and
the entropy decreases until it reaches its true
equilibrium value.
Application to black holes. (ETH = no-hair)
A black hole has an entropy proportional to the
horizon area in Planck units and therefore a large
number of quantum microstates. From generic initial
conditions, the metric evolves and after some time it
becomes the black hole metric independently of the
initial state (no hair theorem implies unique metric).
● For that reason it is natural to suggest that the
metric is a “thermal” operator in the ETH sense.
Therefore it should have the same expectation value
in all microstates.
● This suggests that the no-hair theorem is the GR
version of the ETH property. (no-hair = ETH).
That is, both ETH and no-hair imply that the metric is
completely determined by the total energy.
● Since the metric is well defined and the same for
all microstates:
● It has no information on the microstate.
The information is nowhere in space.
● Anything computed from the metric (e.g.
Hawking radiation) ignores the microstate
information.
● Seems to disagree with fuzzball proposal.
● The behavior of the entropy generation and decay
is in agreement with the proposal of Page for the
entanglement of the black hole with the Hawking
radiation.
Conclusions
● We tested ETH for a particular system using a
Krylov subspace method that allows us to study
much larger systems than before.
● For the same system we studied the generation of
entanglement entropy, derived an expression for the
short time increase and showed that it is capped by
the thermal entropy at large times.
● Regarding black holes, we proposed that the
metric is a thermal operator in the ETH sense
implying that all microstates have a well defined
metric equal to the Schwarzschild metric.