Field theory of glass transition
Taka-H. Nishino and Hisao Hayakawa
(YITP, Kyoto University)
February 5, Molecule meeting in winter
Taka H. Nishino and HH, PRE68, 061502 (2008)
Contents
 Introduction : What is mode-coupling
theory?
 Earlier field theoretic approaches
 Our field theoretic analysis
 Action with TRS
 The derivation of MCT
 Numerical analysis
 Discussion and summary
I. Introduction: Glassy materials
(a) traffic jam (congestion) (b) sandcastle
(c) colloidal glass
Relationship between this talk and
complex eigenvalue problems
 The dynamics of glassy materials are
in principle described by the Liouville
equation.
 The conventional theory predicts the
ideal glass transition but actual
processes do not have.
 To escape the glass state we need to
have imaginary part of eigenvalues.
What is Mode-coupling theory?
 MCT can be derived by a reliable
basic equation (Chong’s talk).
 MCT captures many aspects liquid
side, but its description for glass
transitions has some defects
 Existence of non-ergodic transition.
 Existence of divergence of viscosity.
 Actual observation may not have such an
anomaly.
MCT equation and its prediction
Equation for density correlation function
Memory kernel
Vertex function
Quick derivation of MCT
 Start from Liouville equation
 Derive Zwanzig-Mori equation
 Use the decoupling equation for the
memory kernel
Success and failure of MCT
Non-ergodic part of f(k,t)
Ergodic transition: a complete
freezing in the low temperature region.
Purpose of this work
 To develop a systematic perturbation
which can go beyond 1st order.
 If this can be done, we may give the
theoretical basis of EMCT.
 The 1st order perturbation should
recover MCT.
 To clarify the validity and the
limitation of fluctuating
hydrodynamics.
II. Earlier field theoretic approaches
 Factorization approximation is a
totally uncontrolled.
 It is extremely hard to improve the
theory within the projection operator
method=>Chong’s talk.
 We need a systematic field theoretic
treatment on this problem.
The earlier field theoretic
formulation of glass transition (i)
 Das-Mazenko (1986): renormalized perturbation
method (RPM) for fluctuating hydrodynamics of the
density and the momentum.
 Cut-off mechanism (absence of ergodic transition)
 Shimitz, Dufty and De (SDD) (1993): support the
conclusion of Das-Mazenko based on a simple
argument.
 Their method does not preserve Galilean invariance.
 Kawasaki (1994) : indicated equivalency between the
fluctuating hydrodynamics and Dean-Kawasaki
equation. =>No role of momentum.
 Miyazaki and Reichman (2005): simple field theoretic
perturbations do not preserve FDR in order by order.
Earlier field theoretic analysis (ii)
 Andreanov, Biroli, and Lefevre (ABL)
(2006): indicated the importance of the
time-reversal symmetry (TRS) in the action.
 FDR directly follows from TRS.
 They introduced some auxially fields.
 They developed the perturbation of fluctuating
hydrodynamics, but the result is far from MCT.
 Kim and Kawasaki (2007,2008) starts from
Dean-Kawasaki equation and obtain an
equation similar to MCT.
 The role of momentum is underestimated.
III. Our field theoretic analysis
 We start from fluctuating
hydrodynamics.
g: momentum
MSR action
Using an integral representation of the delta function, we obtain
where
Introduction of auxiliary fields
 To satisfy TSR we
introduce new
variables. The
These choices ensure the separation
action is
between the linear part and the nonlinear
part.
Time reversal symmetry
The action is invariant under
We also note
Schwinger-Dyson equation
 We calculate the Schwinger-Dyson
equation
where the propagator is defined by
The structure factor is represented by
Self-energies, vertices and …
 The self-energy satisfies
in the first-order approximation, where
Non-Gaussian part
the vertex function is
Note that free-propagator satisfies
Gaussian part
First-order perturbation in the long
time limit
 We assume that the propagators
including the momentum decay faster
than the density correlation.
 Then we can obtain a closure of the
density correlation.
 The equation is reduced to the steady
MCT in the long time limit.
MCT from the field theory in the
long time limit
static structure factor
IV. Can we ignore the momentum
correlation? (Numerical check)
 Time evolution is not clear.
 The momentum correlation decays much
faster than the density correlation.
 A numerical calculation of fluctuating
hydrodynamics [Lust etal, PRE(1993)]
 However, from the strong non-linearity
and memory effect, the momentum
correlation might cause the ergodicrestoring.
We need to verify its effect by numerical calculation.
Time evolution equation of the density correlation
（derived from fluctuating hydrodynamics by field theory）
“2” Time scale
 Memory function Mi (1st loop)
Memory
function
 Model1: We ignore all correlations which
include momentum (same as MCT).
 Model 2: We include all terms except for the
assumption that the transverse mode can be
separated from other modes.
We calculate these types.
Outline of numerical method
 Mono-atomic hard sphere model
 We employ the algorithm by Fuchs et al. (J.
Phys.: Condens. Matter 1991)
 Each time step length is twice after some
steps.
 Static structure factor => Verlet-Weis.
 Momentum correlation
 We assume that the longitudinal mode can be
represented by the density correlation.
 We also assume that the transverse mode is
irrelevant.
Results of numerical calculation
 There is no momentum contribution.
 There exists the ideal glass transition.
V. Discussion (1):
Comparison with other works
 We followed ABL, but ABL derived several
unexpected? results.
 Choice of the auxiliary fields is crucial.
 We also use the similar argument by SDD.
 The violation of Galilean invariance by SDD is
crucial.
 We have obtained the essentially same
result as that by Kim and Kawasaki
 This is because we ignored the contribution from
momentum correlations. =>We have checked
that the momentum correlations are irrelevant.
 Ours is essentially reformulation of Kawasaki
(1994)
Discussion (2)
 Das-Mazenko suggested the existence of
cut-off mechanism but our conclusion
within the first-order perturbation is the
absence of the cutoff.
 Their calculation does not satisfy FDR in
each order.
 They introduced V=g/ρas another collective
variable.
 Their calculation captures some aspects of
non-perturbative calculation, but we cannot
understand the details of their paper.
Discussion (3)
From the precise evaluation at the first order,
we obtain the formal result
This is the cutoff mechanism that Das-Mazenko introduced.
Discussion (4)
 M_2 is not zero when the mass
conservation exists.
 Indeed Das-Mazenko cannot evaluate
underline nonlinear term
This can be rewritten as in our notation.
But, our method exactly satisfies the relation.
Discussion (5) : Perspective
 The second-order perturbation
 The naïve calculation leads to divergence
of diagrams.=>tough work…
 We still miss something to recover
ergodicity at the low temperature.
 Is ergodic restoring process similar to
Landau damping?=> A simple argument
only predicts an exponential decay of the
autocorrelation function.
 How can we derive EMCT?
Summary
 We have developed a FDR-preserving
field theoretical calculation for the
glass-transition.
 The equation for the density
correlation in the first-loop order is
reduced to MCT in the long-time limit.
 We get a tool beyond MCT in the next
step, but ….
 Reference: （PRE68, 061502 (2008) ）
Thank you for your attention.
Appendix
 Free energy consists of two part:
F=FK+FU.
 The model is analyzed by MSR
method
Coarse-grained free energy
 (r )
FU{}  kT  dr (r )[ln
 1]
0
Entropy term
kT

drdr ' (r )C (r  r ' )(r )


2
Direct correlation function
MSR action
Using an integral representation of the delta function, we obtain
where
Supplement
In the previous slide we have used
These new fields do not include any linear terms.
First-order perturbation in the long
time limit
 We assume that the propagators
including the momentum decay faster
than the density correlation.
 Then we can obtain a closure of the
density correlation.
 The time evolution equation is the
second order as
The right-hand side
In the first-loop order we can estimate the right-hand side
of the previous equation as