Glass Phenomenology from the connection to spin glasses:

review and ideas

Z. Nussinov

Washington University

Conclusions (Moore)

• Mapping between glass to spin glass in a magnetic field

H = c2 (∇qαβ

α <β∑ )2 + t

2 qαβ2

α <β∑ −w1

6 Tr(q3)−w2

3 q3

α≠β∑

H =− J ijSii j∑ Sj −h Si

i∑

Magnetic field corresponds to non-linearities.

Conclusions continued (Moore)

This mapping qualitatively accounts for

(1) Seemingly divergent time scales (VFT)

(2) Kauzmann crisis

(3) stretched exponential

(4) Increasing length scale

More general idea/question:Is there a general framework that includesspin-glass/RFOT (random field ferromagnet)

/locally preferred structures (“icosahedrics”)/constrained dynamics/MCT/ many other approaches and facets as special cases?

Defects as generalized spins (or other non spin type degrees of freedom in a spin glass type action)?

RFOT on its own is remarkably potent and includes many ideas as subcases.

Outline

1. Qualitative success of simplest spin glass model (one dimensional Edwards Anderson model with field) in accounting for seminal glass features

(Tarzia and Moore)2. Non-orthogonality of local structure approaches to

RFOT (approaches complimentary)3. Possible interpretation of “spins” as local defects.4. General question based on symmetry grounds

One dimensional Edwards Anderson in a field

H =− J ijSii j∑ Sj −h Si

i∑

S =kBc(h)[T −TK ]

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Ising spin glass in a field. Spin glass and divergent correlation length only in zero field.

Spin glass transition is avoided for finite fields.

Microcrystalline order

Several glass forming systems poised close to microcrystalline phases.

E.g., several metallic glasses.

Local structure as well as entropic considerations may be important.

“Icosahedratics”- coexisting local energetics (and avoided transition) and entropic droplet arguments.

Ideal packing in a Lennard-Jones Liquid

minr

minr

minr

minr minr

We cannot keep going on forever

• The ideal packing can, however, be extended over a substantial volume if we consider the surface of a sphere embedded in d=4 dimensions.

• By endowing space with curvature we may remove the 7 degree void.

(J. Sethna, Phys. Rev. Lett. 51, 2198 (1983))

Landau-Ginzburg Expansion

μ= κ + +∑ 2 2n n n n

n

1F ( | D Q | r | Q | ) ...

2

μ μ μ= ∂ − κ nn 0 nD Q ( i L )Q

The theory: that of a system subjected to a non-Abelian background (magnetic field) (S. Sachdev & D. R. Nelson, Phys. Rev. B 32, 4592(1985))

An exponential number of metastable states in the theory of

icosahedral order

τ −K k~exp[DT / (T T )]

Extensive configurational entropy

Glassy dynamics (Vogel-Fulcher)

Physical origin: multitude of defect states against backdrop of geometrically incompatible locally preferred structures. Interaction between defects.

Z. Nussinov (cond-mat/0209292, PRB), G. Tarjus et al. (cond-mat/0509127, JPCM)

Unusual Equilibrium Thermodynamics:Avoided Phase Transitions in

icosahedral theory

0cT =

By performing a large n analysis (n=169 complex components in this theory) to second order in 1/n, deriving a generalized Mermin-Wagner theorem, and performing a thermal fluctuation analysis, we find that

The thermodynamic phase transition is avoided!

Z. Nussinov (cond-mat/0209292, PRB), G. Tarjus et al. (cond-mat/0509127, JPCM)

Tc(κ=0)

Tc(κ≠0)

κ

T

The generic phase diagram•For any finite curvature, the system is hot.

•Why is there no phase transition?

•The huge degeneracy and near degeneracy of the system makes it very susceptible to thermal fluctuations.

•All of this is true for all non-abelian fields.

•Avoided transition due to field Similar to spin glass in field.

Possible intuitive cause for spin glass behavior:

Elastic strains generated by other defects act on a given defect. If all motion is nearly frozen then to computethe leading order dynamics, we may regard the external defects as quenched when computing the elasticinteractions.

General long range interactions change phase.

Defects spoiling locally preferred structures

Simplest defects: Two level tunneling systems/Tunneling rotation impuritiesWe can envision defect modes against the backdrop of preferred structures (e.g., P. Harrowell’s and others).Integrating out phonons assuming generalcrystalline like environment [Schechter and Stamp, arxiv:0801.4944] obtained for two level defects.Generalizations to tunneling entities (Lubchenko and Wolynes)?

H =− [Djj∑ Sj

x + Δ jτ jx] +Veff

Veff = Uijzz

ij∑ Si

zSjz + Bi

i∑ Si

z + UijSτ

ij∑ Si

zτ jz

+ bi(η)

i∑ τ i

z + J ijij∑ τ i

zτ jz

Symmetry question for general quenched theories:

Forget particular derivation. Most general symmetry allowed Hamiltonian density functional that breaks time reversal

H = c2 (∇qαβ∑ )2 + t

2 qαβ2∑ −w1

6 Tr(q3)

−w2

3 q3∑ + 12 (∇M )2 + hM + ...

Allows for paramagnetic, ferromagnetic, and spin glassPhases. Includes replicas for quenched quantities.When no long length scales/very low frequency modespresent, all degrees of freedom must be present (as above).