Density fluctuations and transport in normal and supercooled quantum liquids

1
Density fluctuations and transport in normal and
supercooled quantum liquids
Eran Rabani School of Chemistry Tel Aviv
University
2
Outline

• Quantum Mode-Coupling Theory
• Quantum generalized Langevin equation (QGLE)
• Quantum mode-coupling approximations
• Density fluctuations
• Classical liquids
• Quantum liquids
• Quantum supercooled liquids
• Quantum Kob-Andersen model
• Preliminary results
• Experimental realization

3
Literature

• E. Rabani and D.R. Reichman, Phys. Rev. E 65,
  036111 (2002)
• E. Rabani and D.R. Reichman, J. Chem. Phys. 116,
  6271 (2002)
• D.R. Reichman and E. Rabani, J. Chem. Phys. 116,
  6279 (2002)
• E. Rabani and D.R. Reichman, Europhys. Lett. 60,
  656 (2002)
• E. Rabani and D.R. Reichman, J. Chem. Phys. 120,
  1458 (2004)
• E. Rabani and D.R. Reichman, Ann. Rev. Phys.
  Chem. 56, 157-185 (2005)
• E. Rabani, K. Miyazaki, and D.R. Reichman, J.
  Chem. Phys. 122, 034502 (2005)

Collaboration Kunimasa Miyazaki, Columbia
University David Reichman, Columbia
University Roi Baer, Hebrew University Danie
l Neuhauser, UCLA Financial Support Israel
Science Foundation, EU-STERP, Ministry of
Science, US-Israel Binational Science
Foundation
4
Theory
5
Quantum mode coupling

• Step 1 Formulation of an exact quantum
  generalized Langevin equation (QGLE) using
  Zwanzig-Mori projection operator technique, for
  the Kubo transform of the dynamical variable of
  interest
• Step 2 Approximate memory kernel for the QGLE
  using a quantum mode-coupling theory.
• Step 3 Solution of the QGLE with the approximate
  memory kernel combined with exact static input
  generated from a suitable PIMC scheme.

6
Density fluctuations
To study density fluctuations we need to specify
the dynamical variable and the corresponding
correlation function (the intermediate scattering
function)
The exact QGLE for the Kubo transform of the
intermediate scattering function is given by
The formal expression for the memory kernel is
7
Binary portion
The formal expression for the memory kernel is
Short time expansion to second order in time
The time moments are given in terms of the
density moments
8
Mode coupling portion
The projected dynamics is replaced with the full
dynamics projected onto the slow decaying modes
In addition, four point correlation functions
are replaced by a product of two point
correlation functions
9
Total memory kernel
10
PIMC scheme
We need to calculate the following static
Kubo transforms
where
Using the coordinate representation of the
matrix element
We obtain (to lowest order in e using P
Trotter slices)
Our result looks similar to the Barker energy
estimator, however, it is numerically less noisy.
11
Classical liquids
12
Liquid lithium
The normalized intermediate scattering function
for liquid lithium. The red curves are results
obtained from molecular dynamics simulations and
the blue curves are results obtained from a
classical mode-coupling theory. The agreement
between the theory and simulations is remarkable
for all q values shown.
13
Quantum liquids
14
Quantum liquids p-H2
The normalized intermediate scattering function
for liquid para-hydrogen. The red curves are
results obtained from the QMCT and the green
curves are results obtained from an analytic
continuation approach (MaxEnt). Left panels show
the corresponding memory kernels computed from
the QMCT.
15
Dynamic structure factor p-H2
The normalized dynamic structure factor for
liquid para-hydrogen. Red QMCT. Green - MaxEnt.
Black QVM assuming a single relaxation time.
Blue circles - experimental results F. J.
Bermejo, B. Fak, S. M. Bennington, R.
Fernandez-Perea, C. Cabrillo,J. Dawidowski, M. T.
Fernandez-Diaz, and P. Verkerk, Phys. Rev. B 60,
15154 (1999).
16
Quantum liquids o-D2
The normalized intermediate scattering function
for liquid ortho-deuterium. The red curves are
results obtained from the QMCT and the green
curves are results obtained from an analytic
continuation approach (MaxEnt). Left panels show
the corresponding memory kernels computed from
the QMCT.
17
Dynamic structure factor o-D2
The normalized dynamic structure factor for
liquid ortho-deuterium. Red QMCT. Green -
MaxEnt. Black QVM assuming a single relaxation
time. Blue circles - experimental results from M.
Mukherjee, F. J. Bermejo, B. Fak, and S. M.
Bennington, Europhys. Lett. 40, 153 (1997).
18
Quantum Transport
19
QGLE for VACF
We need to obtain a QGLE for the velocity
autocorrelation function (v is the velocity of a
tagged liquid particle along an arbitrary
direction)
Following similar lines to those sketched for the
classical theory, we obtain an exact quantum
generalized Langevin equation (QGLE)
where we have used the following projection
operator
and the memory kernel is formally given by
20
Quantum Mode Coupling Theory
The Kernel is approximated by
Fast decaying quantum binary term
The slow decaying quantum mode-coupling term
The vertex
21
MC Memory Kernel for VACF
The slow decaying quantum mode-coupling term is
obtained using a set of approximations. The
projected dynamics is replaced with the full
dynamics projected onto the slow decaying modes
where the new projection operator is given
by
In addition, four point correlation functions
are replaced by a product of two point
correlation functions
22
Static input from PIMC
The static input for the memory kernel of the
velocity autocorrelation function generated from
a PIMC simulation method for liquid para-hydrogen
at T14K (red curve) and T25K (blue curve).
23
Velocity autocorrelation function
The normalized velocity autocorrelation function
calculated from the quantum mode-coupling theory
(blue curve) and from an analytic continuation
of imaginary-time PIMC data (blue curve) for
liquid para-hydrogen at T14K (lower panel) and
T25K (upper panel). The good agreement between
the two methods is a strong support for the
accuracy of the quantum mode-coupling approach
for liquid para-hydrogen.
24
Memory kernel for VACF
The Kubo transform of the memory kernel for the
velocity autocorrelation function for liquid
para-hydrogen at T14K (upper panel) and T25K
(lower panel). Shown are the fast-decaying binary
term (red curve), the slow-decaying mode-coupling
term (green curve) and the total memory kernel
(blue curve). The contribution of the slow
mode-coupling portion of the memory kernel is
significant at the low temperature, while at the
high temperature, the kernel can be approximated
by only the fast binary portion.
25
Self-Diffusion - Liquid para-H2
The frequency dependent diffusion constant for
liquid para-hydrogen at T14K and T25K. The
self-diffusion obtained from the Green-Kubo
relation is 0.30 and 1.69 (Å2/ps) for T14K and
T25K, respectively. These results are in good
agreement with the experimental results (0.40
and 1.60) and with the maximum entropy analytic
continuation method (0.28 and 1.47).
26
VACF Liquid ortho-D2
The normalized velocity autocorrelation function
and its Kubo transform calculated from the
quantum mode-coupling theory for liquid
ortho-deuterium (upper panel) and liquid
para-hydrogen (lower panel) at T20.7K
27
Self-Diffusion - Liquid ortho-D2
The frequency dependent diffusion constant for
liquid para-hydrogen (green curve) and
ortho-deuterium (red curve) at T20.7K. The
self-diffusion obtained from the Green-Kubo
relation is 0.49 and 0.64 (Å2/ps) for
para-hydrogen and ortho-deuterium, respectively.
The result for ortho-deuterium is in reasonable
agreement with the experimental results (0.36
Å2/ps).
28
Normal Liquid Helium
The normalized velocity autocorrelation function
calculated from the quantum mode-coupling theory
(blue curve), from an analytic continuation of
imaginary-time PIMC data (red curve), and from a
semiclassical approach (Makri green curve) for
liquid helium above the l transition.
29
Self-Diffusion Normal Liquid Helium
The frequency dependent diffusion constant for
normal liquid helium at T4K. The results shown
were calculated from the quantum mode-coupling
theory (blue curve), from an
analytic continuation of imaginary-time PIMC data
(red curve), from a semiclassical approach (Makri
green curve), and from the CMD method (black
curve)
30
Quantum glasses
31
Quantum glasses

• Can we form a structural quantum glass (onset of
  quantum fluctuations, super fluidity)?
• Are there any thermodynamic signatures that are
  different for a quantum glass?
• Are there any dynamic signatures that are
  different for a quantum glass?

32
Kob-Andersen model
The Kob-Andersen model is based on a binary
mixture of Lennard-Jones (BMLJ) particles with
the following parameters
The system undergoes an ergodic-to-nonergodic
transition at T0.435. Classical mode-coupling
theory predicts a transition at about
T0.92. Kob and Andersen, Phys. Rev. E 51, 4626
(1995) Kob and Andersen, Phys. Rev. E 52, 4134
(1995) Nauroth and Kob, Phys. Rev. E 55, 657
(1997)
33
Classical results for the Kob-Andersen model
34
Intermediate Scattering Function
MCT predictions
Self intermediate scattering function versus
time for A and B particles at two wave length for
several temperatures. Kob and Andersen, Phys.
Rev. E 52, 4134 (1995).
35
Self-diffusion
Left Mean square displacement versus time for A
particles at different temperatures. Right
Diffusion constant versus temperature for A and B
particles. Solid lines are best fits to power-law
and dashed lines are best fits to Vogel-Fulcher
law.
36
Quantum results for theKob-Andersen model
37
Average potential energy
Average potential energy per particle for the
quantum Kob-Andersen model. Simulation were done
for N500 and P100. There is a clear change near
T1 (similar to the classical case). Is there any
slowing down near this temperature?
38
Intermediate scattering function
Intermediate scattering function at r1.2 for
several values of the temperature at qqmax. No
significant slowing down is observed far below
the classical Tc0.92.
39
Static structure factor
40
Pair correlation function
41
Low wave vector results
Intermediate scattering function at r1.2 for
several values of the temperature at qqmax/2.
The A particles show coherent fluctuation not
observed classically at this value of q.
42
And even lower
Intermediate scattering function at r1.2 for
several values of the temperature at qqmax/4.
43
Experimental realization
44
Mixtures of p-H2 and o-D2
45
Mixtures of p-H2 and o-D2
46
Centroid configurations
At the triple-point (TP) density the mixture
freezes into an ordered crystals below T10. But
at a slightly lower temperature, the system can
be supercooled and even at T6 is still
disordered.
47
(No Transcript)
48
Classical Results Kob-Andersen
Time dependence of the coherent and incoherent
intermediate scattering function for two wave
vectors at T 2. The dashed line with the
symbols are the results from the simulation and
the solid lines are the prediction of the
classical MCT theory. From Kob and collaborators
(Phys. Rev. E 55, 657 (1997), J. Non-Cryst.
Solids 307, 181 (2002)).
49
Classical Results Kob-Andersen
Time dependence of the coherent and incoherent
intermediate scattering function for two wave
vectors at T 0.466. The dashed line with the
symbols are the results from the simulation and
the solid lines are the prediction of the
classical MCT theory. From Kob and collaborators
(Phys. Rev. E 55, 657 (1997), J. Non-Cryst.
Solids 307, 181 (2002)).
50
Simple Example - VACF
Now, lets make a simple Gaussian approximation
to the memory kernel
Even this simple approximation (short time
expansion) captures some of the hallmarks of
normal monoatomic liquids. The reason is that
the approximation is done at the level of the
memory kernel, and thus better results are
obtained for the correlation function
itself. However, this approximation completely
neglects the long time decay of the memory kernel.
51
Structure factor and nonergodic parameter
52

• E. Rabani and D.R. Reichman, J. Phys. Chem. B
  105, 6550 (2001)
• D.R. Reichman and E. Rabani, Phys. Rev. Lett. 87,
  265702 (2001)
• E. Rabani and D.R. Reichman, Phys. Rev. E 65,
  036111 (2002)
• E. Rabani and D.R. Reichman, J. Chem. Phys. 116,
  6271 (2002)
• D.R. Reichman and E. Rabani, J. Chem. Phys. 116,
  6279 (2002)
• E. Rabani, D.R. Reichman, G. Krilov, and B.J.
  Berne, PNAS USA 99, 1129 (2002)
• E. Rabani and D.R. Reichman, Europhys. Lett. 60,
  656 (2002)
• E. Rabani, in Proceedings of "The Monte Carlo
  Method in the Physical Sciences Celebrating the
  50th anniversary of the Metropolis algorithm",
  AIP Conference Proceedings, vol. 690, 281 (2003)
• E. Rabani and D.R. Reichman, J. Chem. Phys. 120,
  1458 (2004)
• E. Rabani and D.R. Reichman, Ann. Rev. Phys.
  Chem. 56, 157-185 (2005)
• E. Rabani, K. Miyazaki, and D.R. Reichman, J.
  Chem. Phys. 122, 034502 (2005)