Mixed order phase transitions David Mukamel Amir Bar

Mixed order phase transitions David Mukamel Amir Bar, DM (PRL, 122, 01570 (2014); ar. Xiv: 1406. 6219)

Phase transitions of mixed order

Examples: Equilibrium

Nonequilibrium: 3. Jamming transition in kinetically constrained models Toninelli, Biroli, Fisher (2006) 4. “Extraordinary transition” in network rewiring Liu, Schmittmann, Zia EPL, 100, 66007 (2012); JSTAT P 05021 (2014) Liu, Jolad, Schmittmann, Zia JSTAT P 8001 (2013)

5. No-enclaves percolation (NEP) bond percolation in 2 d where clusters surrounded by a larger cluster are absorbed in it. M. Sheinman, A. Sharma, F. C. Mac. Kintosh ar. Xiv: 1402. 0907

IDSI : Inverse Distance Square Ising model A simple argument: ++++++---------++++++++++ 1 L Anderson et al (1969, 1971); Dyson (1969, 1971); Thouless (1969); Aizenman et al (1988)…

KT type transition, Cardy (1981) Phase diagram H T IDSI Fisher, Berker (1982)

Dyson hierarchical version of the model (1971) Mean field interaction within each block The Dyson model is exactly soluble demonstrating the Thouless effect

Exactly soluble modification of the IDSI model microscopic configuration: +++++------------+++++++++---------- The interaction is in fact not binary but rather many body.

Summery of the results Phase diagram H T The model is closely related to the PS model of DNA denaturation

Interacting charges representation: Charges of alternating sign (attractive) on a line Attractive long-range nearest-neighbor interaction Chemical potential --suitable representation for RG analysis --similar to the PS model

Analysis of the model Grand partition sum Polylog function

Polylog function

Unlike the PS model the parameter c is not universal

Nature of the transition Domain length distribution using the properties of the polylog function one can show

Two order parameters

Extreme Thouless effect

Phase diagram magnetization m domains density n I II and III n is continuous n is discontinuous

Canonical analysis

saddle point:

c=2. 5

Finite L correction:

c=2. 5

Finite L corrections c=2. 5 L=1000

Renormalization group - charges representation + - y - fugacity a - short distance cutoff

Renormalization group equations

In the KT case (all charges interact with all other ones): + - Contribution of the dipole to the renormalized partition sum: renormalizes c. (Cardy 1981)

Kosterlitz-Thouless RG equations: compared with the those of the restricted model

Coarsening dynamics – Particles with n-n logarithmic interactions – Biased diffusion, annihilation and pair creation

Coarsening dynamics The coarsening is controlled by the T=0 (y=0) fixed point Like the dynamics of the T=0 Ising model

Coarsening dynamics Expected scaling form

z=2 z=1. 5

- Voter model (y=0, fixed c)

Summary Some models exhibiting mixed order transitions are discussed. A variant of the inverse distance square Ising model is studied and shown to have an extreme Thouless effect, even in the presence of a magnetic field Relation to the IDSI model is studies by comparing the renormalization group transformation of the two models. The model exhibits interesting coarsening dynamics at criticality. Distribution of the largest domain at the transition (with S. Majumdar)