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Mixed order phase transitions

Mixed order phase transitions. David Mukamel. Amir Bar, DM (PRL, 122, 01570 (2014); arXiv:1406.6219). Phase transitions of mixed order. (a) diverging length as in second order transitions or

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Mixed order phase transitions

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  1. Mixed order phase transitions David Mukamel Amir Bar, DM (PRL, 122, 01570 (2014); arXiv:1406.6219)

  2. Phase transitions of mixed order (a) diverging length as in second order transitions or (b) discontinuous order parameter as in first order transitions

  3. Examples: Equilibrium 1d Ising model with long range interactions non-soluble but many of its properties are known 2. Poland-Scheraga (PS) model of DNA denaturation

  4. Nonequilibrium: 3. Jamming transition in kinetically constrained models • Toninelli, Biroli, Fisher (2006) “Extraordinary transition” in network rewiring Liu, Schmittmann, Zia EPL, 100, 66007 (2012); JSTAT P05021 (2014) Liu, Jolad, Schmittmann, Zia JSTAT P8001 (2013)

  5. 5. No-enclaves percolation (NEP) bond percolation in 2d where clusters surrounded by a larger cluster are absorbed in it. discontinuous mass of the percolating cluster M. Sheinman, A. Sharma, F.C. MacKintosh arXiv:1402.0907

  6. IDSI : Inverse Distance Square Ising model For the model has an ordering transition at finite T A simple argument: ++++++++++++-----------------++++++++++++++++++++ 1 L Anderson et al (1969, 1971); Dyson (1969, 1971); Thouless (1969); Aizenman et al (1988)…

  7. model is special The magnetization m is discontinuous at (“Thouless effect”) Thouless (1969), Aizenman et al (1988) KT type transition, Cardy (1981) Phase diagram H T IDSI Fisher, Berker (1982)

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

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

  10. Summery of the results diverging correlation length at with nonuniversal Extreme Thouless effect with Phase diagram H T The model is closely related to the PS model of DNA denaturation

  11. The energy of a domain of length 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

  12. Analysis of the model Grand partition sum Polylog function

  13. Polylog function is the closest pole to the origin ferromagnetic coupling

  14. Phase transition: Unlike the PS model the parameter c is not universal

  15. Nature of the transition Domain length distribution Close to : using the properties of the polylog function one can show

  16. Two order parameters number of domains magnetization

  17. 1. order parameter Where at for for is finite is continuous is discontinuous in both cases

  18. 2. order parameter is the magnetic field symmetry) either or Extreme Thouless effect

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

  20. phase diagram

  21. Canonical analysis Free energy

  22. saddle point:

  23. c=2.5

  24. Finite L correction:

  25. c=2.5

  26. Finite L corrections c=2.5 L=1000

  27. Renormalization group - charges representation + - + - y - fugacity a - short distance cutoff Length rescaling This can be compensated by y rescaling

  28. + - + - The integral scales like hence it does not renormalize c . Rather it renormalizes y.

  29. Renormalization group equations

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

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

  32. Line of fixed points

  33. Coarsening dynamics • Particles with n-n logarithmic interactions • Biased diffusion, annihilation and pair creation

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

  35. Coarsening dynamics Expected scaling form - number of domains with

  36. L=5000 c=1.5 z=2 z=1.5

  37. with - Voter model (y=0, fixed c)

  38. 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)

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