Field theory of glass transition

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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)
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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

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I. Introduction Glassy materials

• traffic jam (congestion) (b) sandcastle
• (c) colloidal glass

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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.

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What is Mode-coupling theory?

• MCT can be derived by a reliable basic equation
  (Chongs 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.

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MCT equation and its prediction
Equation for density correlation function
Memory kernel
Vertex function
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Quick derivation of MCT

• Start from Liouville equation
• Derive Zwanzig-Mori equation
• Use the decoupling equation for the memory kernel

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Success and failure of MCT
Non-ergodic part of f(k,t)
Ergodic transition a complete freezing in the
low temperature region.
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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.

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II. Earlier field theoretic approaches

• Factorization approximation is a totally
  uncontrolled.
• It is extremely hard to improve the theory within
  the projection operator methodgtChongs talk.
• We need a systematic field theoretic treatment on
  this problem.

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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. gtNo role of momentum.
• Miyazaki and Reichman (2005) simple field
  theoretic perturbations do not preserve FDR in
  order by order.

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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.

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III. Our field theoretic analysis

• We start from fluctuating hydrodynamics.

g momentum
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MSR action
Using an integral representation of the delta
function, we obtain
where
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Introduction of auxiliary fields

• To satisfy TSR we introduce new variables. The
  action is

These choices ensure the separation between the
linear part and the nonlinear part.
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Time reversal symmetry
The action is invariant under
We also note
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Schwinger-Dyson equation

• We calculate the Schwinger-Dyson equation
• where the propagator is defined by
• The structure factor is represented by

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Self-energies, vertices and

• The self-energy satisfies
• in the first-order approximation, where the
  vertex function is
• Note that free-propagator satisfies

Gaussian part
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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.

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MCT from the field theory in the long time limit
static structure factor
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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
  ergodic-restoring.

We need to verify its effect by numerical
calculation.
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Time evolution equation of the density correlation
(derived from fluctuating hydrodynamics by field
theory)
Memory function
2 Time scale

• Memory function Mi (1st loop)
• 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.
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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 gt 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.

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Results of numerical calculation

• There is no momentum contribution.
• There exists the ideal glass transition.

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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. gtWe have checked that the
  momentum correlations are irrelevant.
• Ours is essentially reformulation of Kawasaki
  (1994)

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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 Vg/?as another collective
  variable.
• Their calculation captures some aspects of
  non-perturbative calculation, but we cannot
  understand the details of their paper.

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Discussion (3)
From the precise evaluation at the first order,
we obtain the formal result
This is the cutoff mechanism that Das-Mazenko
introduced.
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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.
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Discussion (5) Perspective

• The second-order perturbation
• The naïve calculation leads to divergence of
  diagrams.gttough work
• We still miss something to recover ergodicity at
  the low temperature.
• Is ergodic restoring process similar to Landau
  damping?gt A simple argument only predicts an
  exponential decay of the autocorrelation
  function.
• How can we derive EMCT?

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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) )

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• Thank you for your attention.

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Appendix

• Free energy consists of two part FFKFU.
• The model is analyzed by MSR method

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Coarse-grained free energy
Entropy term
Direct correlation function
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MSR action
Using an integral representation of the delta
function, we obtain
where
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Supplement
In the previous slide we have used
These new fields do not include any linear terms.
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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

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The right-hand side
In the first-loop order we can estimate the
right-hand side of the previous equation as