Field theory of glass transition

1. 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)

2. 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

3. I. Introduction: Glassy materials • traffic jam (congestion) (b) sandcastle • (c) colloidal glass

4. 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.

5. 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.

6. MCT equation and its prediction Equation for density correlation function Memory kernel Vertex function

7. Quick derivation of MCT • Start from Liouville equation • Derive Zwanzig-Mori equation • Use the decoupling equation for the memory kernel

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

9. 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.

10. 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.

11. 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.

12. 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.

13. III. Our field theoretic analysis • We start from fluctuating hydrodynamics. g: momentum

14. MSR action Using an integral representation of the delta function, we obtain where

15. 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.

16. Time reversal symmetry The action is invariant under We also note

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

18. Non-Gaussian part 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

19. 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.

20. MCT from the field theory in the long time limit static structure factor

21. 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.

22. Time evolution equation of the density correlation （derived from fluctuating hydrodynamics by field theory） • 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. Memory function “2” Time scale We calculate these types.

23. 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.

24. Results of numerical calculation • There is no momentum contribution. • There exists the ideal glass transition.

25. 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)

26. 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.

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

28. 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.

29. 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?

30. 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) ）

31. Thank you for your attention.

32. Appendix • Free energy consists of two part: F=FK+FU. • The model is analyzed by MSR method

33. Coarse-grained free energy Entropy term Direct correlation function

34. MSR action Using an integral representation of the delta function, we obtain where

35. Supplement In the previous slide we have used These new fields do not include any linear terms.

36. 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

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