Slow dynamics in gapless low-dimensional systems

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Slow dynamics in gapless low-dimensional systems
Anatoli Polkovnikov, Boston University
Vladimir Gritsev Harvard Ehud Altman
- Weizmann Eugene Demler Harvard Bertrand
Halperin - Harvard Misha Lukin - Harvard
CMT Seminar, Yale, 11/08/2007
AFOSR
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Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
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Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
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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)
No thermalization during collisions of two
one-dimensional clouds of interacting
bosons. Fast thermalization if the clouds are
three dimensional.
Quantum analogue of the Fermi-Pasta-Ulam problem.
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Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
3. 12 Nonequilibrium thermodynamics?
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Adiabatic process.
Assume no first order phase transitions.
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Adiabatic theorem for integrable systems.
Density of excitations
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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 d.Hence expect

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
• strong quantum or thermal fluctuations,
• destruction of the long-range order,
• 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) high dimensions
2. Non-analytic low dimensions
3. Non-adiabatic lower dimensions

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Some examples.
1. Gapless critical phase (superfluid, magnet,
crystal, ).
LZ condition
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Second example crossing a QCP.
? ? ? t, ? ? 0
Gap vanishes at the transition. No true adiabatic
limit!
How does the number of excitations scale with ? ?
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Perturbation theory (linear response). (A.P. 2003)
Expand the wave-function in many-body basis.
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Uniform system can characterize excitations by
momentum
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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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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
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Finite temperatures.
Instead of wave function use density matrix
(Wigner form).
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Results
d1,2
Non-adiabatic regime!
Artifact of the quadratic approximation or the
real result?
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Numerical verification (bosons on a lattice).
Use the fact that quantum fluctuations are weak
and expand dynamics in the effective Plancks
constant (saddle point parameter)
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Classical limit use Gross-Pitaevskii equations
with initial conditions distributed according to
the thermal density matrix.
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How do we add quantum corrections?
Treat ? exactly, while expand in powers of ?.
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Results
Leading order in ? start from random initial
conditions distributed according to the Wigner
transform of the density matrix and propagate
them classically (truncated Wigner approximation)
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Results (1d, L128)
zero temperature
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T0.02
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Thermalization at long times.
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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

Open questions general fate of linear response
at low dimensions, non-uniform perturbations,
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M. Greiner et. al., Nature (02)
Adiabatic increase of lattice potential
What happens if there is a current in the
superfluid?
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Drive a slowly moving superfluid towards MI.
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Include quantum depletion.
Equilibrium
?
Current state
?
p
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Meanfield (Gutzwiller ansatzt) phase diagram
Is there current decay below the instability?
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Role of fluctuations
Phase slip
Below the mean field transition superfluid
current can decay via quantum tunneling or
thermal decay .
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1D System.
variational result
semiclassical parameter (plays the role of 1/ )
N1
Large N102-103
C.D. Fertig et. al., 2004
Fallani et. al., 2004
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Higher dimensions.
Longitudinal stiffness is much smaller than the
transverse.
r
Need to excite many chains in order to create a
phase slip.
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Phase slip tunneling is more expensive in higher
dimensions
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Current decay in the vicinity of the
superfluid-insulator transition
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Use the same steps as before to obtain the
asymptotics
Discontinuous change of the decay rate across the
meanfield transition. Phase diagram is well
defined in 3D!
Large broadening in one and two dimensions.
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Detecting equilibrium SF-IN transition boundary
in 3D.
p
Easy to detect nonequilibrium irreversible
transition!!
At nonzero current the SF-IN transition is
irreversible no restoration of current and
partial restoration of phase coherence in a
cyclic ramp.
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J. Mun, P. Medley, G. K. Campbell, L. G.
Marcassa, D. E. Pritchard, W. Ketterle, 2007
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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

Smooth connection between the classical dynamical
instability and the quantum superfluid-insulator
transition.
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Density of excitations