Slow dynamics in gapless lowdimensional systems

1
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
AFOSR
2
Cold atoms (controlled and tunable Hamiltonians,
isolation from environment)
3. 12 Nonequilibrium thermodynamics?
3
Adiabatic process.
Assume no first order phase transitions.
4
Adiabatic theorem for integrable systems.
Density of excitations
5
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.
6
Adiabatic theorem in QM suggests adiabatic
theorem in thermodynamics

• Transitions are unavoidable in large gapless
  systems.
• 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.
7
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, e.g.
  Landau theory of phase transitions.

Dynamics -gt population of the low-energy states
due to finite rate -gt breakdown of the adiabatic
approximation.
8
This talk three regimes of response to the slow
ramp

• Mean field (analytic) high dimensions
• Non-analytic low dimensions
• Non-adiabatic lower dimensions

9
Example crossing a QCP.
? ? ? t, ? ? 0
Gap vanishes at the transition. No true adiabatic
limit!
How does the number of excitations scale with ? ?
10
Transverse field Ising model.
Phase transition at g1.
Critical exponents z?1 ? d?/(z? 1)1/2.
A. P., 2003
Linear Response
Interpretation as the Kibble-Zurek mechanism W.
H. Zurek, U. Dorner, Peter Zoller, 2005
11
Possible breakdown of the Fermi-Golden rule
(linear response) scaling due to bunching of
bosonic excitations.
We can view the response of the system on a slow
ramp as parametric amplification of quantum or
thermal fluctuations.
12
Most divergent regime k0 0
13
Finite temperatures.
Instead of wave function use density matrix
(Wigner form).
Real result or the artifact of the harmonic
approximation?
14
Numerical verification (bosons on a lattice).
15
T0.02
16
Thermalization at long times.
17
2D, T0.2
18
M. Greiner et. al., Nature (02)
Adiabatic increase of lattice potential
What happens if there is a current in the
superfluid?
19
Drive a slowly moving superfluid towards MI.
20
Include quantum depletion.
Equilibrium
?
Current state
?
p
21
Meanfield (Gutzwiller ansatzt) phase diagram
Is there current decay below the instability?
22
Role of fluctuations
Phase slip
Below the mean field transition superfluid
current can decay via quantum tunneling or
thermal decay .
23
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
24
Higher dimensions.
Longitudinal stiffness is much smaller than the
transverse.
r
Need to excite many chains in order to create a
phase slip.
25
Phase slip tunneling is more expensive in higher
dimensions
26
Current decay in the vicinity of the
superfluid-insulator transition
27
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.
28
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.
29
J. Mun, P. Medley, G. K. Campbell, L. G.
Marcassa, D. E. Pritchard, W. Ketterle, 2007
30
Conclusions.
Three generic regimes of a system response to a
slow ramp

• Mean field (analytic)
• Non-analytic
• Non-adiabatic

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