Non-adiabatic dynamics and thermodynamics in isolated out of equilibrium systems.

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Non-adiabatic dynamics and thermodynamics in
isolated out of equilibrium systems.
Anatoli Polkovnikov, Boston University
Collaborators C. De Grandi, R. Barankov, BU V.
Gritsev, Fribourg
Trieste, Italy, July 2009
AFOSR
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" Basic facts from thermodynamics energy,
heat, work, first and second laws of
thermodynamics, fundamental thermodynamic
relation. Non-equilibrium work relations
(Jarzyisky equality). " Thermalization in
isolated systems eigenstate thermalization
hypothesis " Connection between quantum and
thermodynamic adiabatic theorems. " Heat,
entropy and defects generation in driven gapless
systems. " Scaling of various thermodynamic
quantities for fast and slow quenches near
quantum critical points. " Work
distribution for sudden quenches and the
Loschmidt echo.
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Electric oven heats food
Microwave oven does work on food
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Usual candidate for entropy
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Ergodic hypothesis
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In the continuum this system is equivalent to an
integrable KdV equation. The solution splits into
non-thermalizing solitons Kruskal and Zabusky
(1965 ).
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Qauntum Newton Craddle.(collisions in 1D
interecating Bose gas Lieb-Liniger model)
T. Kinoshita, T. R. Wenger and D. S. Weiss,
Nature 440, 900 903 (2006)
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Thermalization in Quantum systems.
Consider the time average of a certain observable
A in an isolated system after a quench.
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M. Rigol, V. Dunjko M. Olshanii, Nature 452,
854 (2008)
a, Two-dimensional lattice on which five
hard-core bosons propagate in time. b, The
corresponding relaxation dynamics of the central
component n(kx 0) of the marginal momentum
distribution, compared with the predictions of
the three ensembles c, Full momentum
distribution function in the initial state, after
relaxation, and in the different ensembles.
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Information about equilibrium is fully contained
in diagonal elements of the density matrix.
This is true for all thermodynamic observables
energy, pressure, magnetization, . (pick your
favorite). They all are linear in ?.
This is not true about von Neumann entropy!
Off-diagonal elements do not average to zero.
The usual way around coarse-grain density matrix
(remove by hand fast oscillating off-diagonal
elements of ?. Problem not a unique procedure,
explicitly violates time reversibility and
Hamiltonian dynamics.
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Von Neumann entropy always conserved in time (in
isolated systems). More generally it is invariant
under arbitrary unitary transfomations
If these two adiabatic theorems are related then
the entropy should only depend on ?nn.
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Thermodynamic adiabatic theorem.
In a cyclic adiabatic process the energy of the
system does not change no work done on the
system, no heating, and no entropy is generated .
General expectation
- is the rate of change of external parameter.
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Adiabatic theorem in quantum mechanics
Landau Zener process
In the limit ??0 transitions between different
energy levels are suppressed.
This, for example, implies reversibility (no work
done) in a cyclic process.
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Adiabatic theorem in QM suggests adiabatic
theorem in thermodynamics

1. Transitions are unavoidable in large gapless
   systems.
2. Phase space available for these transitions
   decreases with the rate. Hence expect

Low dimensions high density of low energy
states, breakdown of mean-field approaches in
equilibrium
Breakdown of Taylor expansion in low dimensions,
especially near singularities (phase transitions).
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Three regimes of response to the slow linear
ramp A.P. and V.Gritsev, Nature Physics 4, 477
(2008)

1. Mean field (analytic) high dimensions
2. Non-analytic low dimensions
3. Non-adiabatic low dimensions, bosonic
   excitations

In all three situations quantum and thermodynamic
adiabatic theorem are smoothly connected. The
adiabatic theorem in thermodynamics does follow
from the adiabatic theorem in quantum mechanics.
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Connection between two adiabatic theorems allows
us to define heat (A.P., Phys. Rev. Lett. 101,
220402, 2008 ).
Consider an arbitrary dynamical process and work
in the instantaneous energy basis (adiabatic
basis).

• Adiabatic energy is the function of the state.
• Heat is the function of the process.
• Heat vanishes in the adiabatic limit. Now this is
  not the postulate, this is a consequence of the
  Hamiltonian dynamics!

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Isolated systems. Initial stationary state.
Unitarity of the evolution
In general there is no detailed balance even for
cyclic processes (but within the Fremi-Golden
rule there is).
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What about entropy?

• Entropy should be related to heat (energy), which
  knows only about ?nn.
• Entropy does not change in the adiabatic limit,
  so it should depend only on ?nn.
• Ergodic hypothesis requires that all
  thermodynamic quantities (including entropy)
  should depend only on ?nn.
• In thermal equilibrium the statistical entropy
  should coincide with the von Neumanns entropy

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Properties of d-entropy (R. Barankov, A.
Polkovnikov, arXiv0806.2862. ).
Jensens inequality
Therefore if the initial density matrix is
stationary (diagonal) then
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yields
The statement is also true without the detailed
balance but the proof is more complicated
(Thirring, Quantum Mathematical Physics, Springer
1999).
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Classical systems.
Instead of energy levels we have orbits.
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Classic example freely expanding gas
Suddenly remove the wall
by Liouville theorem
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Example
Cartoon BCS model
Mapping to spin model (Anderson, 1958)
In the thermodynamic limit this model has a
transition to superconductor (XY-ferromagnet) at
g 1.
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Change g from g1 to g2.
Work with large N.
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Simulations N2000
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Entropy and reversibility.
?g 10-4
?g 10-5
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Nearly adiabatic dynamics in many-particle systems
Let us assume we are hanging some external
parameter in time according to the protocol
Can we say anything about system response in the
limit ??0?
Assume (for now) that we start in the ground
state.
Need to solve
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Convenient to work in the adiabatic (co-moving,
instantaneous) basis
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Perturbative analysis, keep only term with m0 in
the sum.
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Infinite integration limits
where ? is the complex root of
Finite integration limits
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Landau-Zener problem
Change coupling ? in the infinite range. Exact
solution
Perturbative solution. Eigenstates
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Spurious factor ?2/9
Now start ti??, tf finite (or vice versa)
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Gapless systems with quasi-particle excitations
Ramping in generic gapless regime (low energy
contribution)
Uniform system
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Low energy contribution
High dimensions high energies dominate
dissipation, low-dimensions low energies
dominate dissipation.
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Adiabatic crossing quantum critical points.
Relevant for adiabatic quantum computation,
adiabatic preparation of correlated states.
V ? ? t, ? ? 0
How
does the number of excitations (entropy, energy)
scale with ? ?
Use scaling arguments in the adiabatic
perturbation theory
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Transverse field Ising model.
There is a phase transition at g1.
This problem can be exactly solved using
Jordan-Wigner transformation
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Spectrum
Critical exponents z?1 ? d?/(z? 1)1/2.
Linear response (Fermi Golden Rule)
A. P., 2003
Interpretation as the Kibble-Zurek mechanism W.
H. Zurek, U. Dorner, Peter Zoller, 2005
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Optimal adiabatic passage through a QCP. (R.
Barankov and A. Polkovnikov, Phys. Rev. Lett.
101, 076801 (2008) )
Given the total time T, what is the optimal way
to cross the phase transition?
Need to slow down near the phase transition
? (? t)r, ? 1/T
optimal power
number of defects at optimal rate.
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Three regimes of response to the slow linear
ramp A.P. and V.Gritsev, Nature Physics 4, 477
(2008)

1. Mean field (analytic) high dimensions
2. Non-analytic low dimensions
3. Non-adiabatic low dimensions, bosonic
   excitations

In all three situations quantum and thermodynamic
adiabatic theorem are smoothly connected. The
adiabatic theorem in thermodynamics does follow
from the adiabatic theorem in quantum mechanics.
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Numerical verification (bosons on a lattice).
Nonintegrable model in all spatial dimensions,
expect thermalization.
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T0.02
Heat per site
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2D, T0.2
Heat per site
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Thermalization at long times (1D).
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Probing quasi-particle statistics in nonlinear
dynamical probes. (R. Barankov, C. De Grandi, V.
Gritsev, A. Polkovnikov, work in progress.)
T
0
1
K
massive fermions (hard core bosons)
massive bosons