Breakdown of the adiabatic approximation in low-dimensional gapless systems

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Breakdown of the adiabatic approximation in
low-dimensional gapless systems
Anatoli Polkovnikov, Boston University Vladimir
Gritsev Harvard University
AFOSR
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Adiabatic theorem for integrable systems.
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Adiabatic theorem in quantum mechanics
http//lab-neel.grenoble.cnrs.fr/themes/nano/fe8/1
5.gif
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Adiabatic theorem in QM suggests adiabatic
theorem in thermodynamics
Is there anything wrong with this picture?
Hint low dimensions. Similar to Landau expansion
in the order parameter.
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More specific reason.

• Equilibrium high density of low-energy states
  -gt
• destruction of the long-range order,
• strong quantum or thermal fluctuations,
• breakdown of mean-field descriptions.

Dynamics -gt population of the low-energy states
due to finite rate -gt breakdown of the adiabatic
approximation.
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This talk three regimes of response to the slow
ramp

1. Mean field (analytic)
2. Non-analytic
3. Non-adiabatic

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Example crossing a second order phase transition.
? ? ? t, ? ? 0
Gap vanishes at the transition. No true adiabatic
limit!
How does the number of excitations scale with ? ?
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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 Kibble-Zurek mechanism W. H.
Zurek, U. Dorner, Peter Zoller, 2005
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Possible breakdown of the Fermi-Golden rule
(linear response) scaling due to bunching of
bosonic excitations.
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Most divergent regime k0 0
Agrees with the linear response.
Assuming the system thermalizes
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Same at a finite temperature.
d1,2
d3
Artifact of the quadratic approximation or the
real result?
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Numerical verification (bosons on a lattice).
Expand dynamics in powers of U/Jn0 (Truncated
Wigner method more, very accurate for these
parameters.)
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Results (1d, L128)
Predictions
zero temperature
finite temperature
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T0.02
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2D, T0.2
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Conclusions.
Three generic regimes of a system response to a
slow ramp

1. Mean field (analytic)
2. Non-analytic
3. Non-adiabatic

There are interesting and open problems beyond
computing Z for various models.