Mixed Order Phase Transitions in Physical Systems
Explore the phenomenon of mixed order phase transitions in physical systems as discussed by David Mukamel and Amir Bar. Examples include the Ising model, Poland-Scheraga model, and jamming transitions. Discover insights into the IDSI model and its exact solubility, as well as the extreme Thouless effect observed. This research is closely linked to the PS model of DNA denaturation.
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Mixed order phase transitions David Mukamel Amir Bar, DM (PRL, 122, 01570 (2014))
Phase transitions of mixed order (a) diverging length ? as in second order transitions ?~? ? or ?~?1/?? (b) discontinuous order parameter as in first order transitions
Examples: 1. 1d Ising model with long range interactions ?(?)~1/?2 non-soluble but many of its properties are known ?~?1/?? 2. Poland-Scheraga (PS) model of DNA denaturation ?~? ? 3. Jamming transition in kinetically constrained models ?~?1/?? Toninelli, Biroli, Fisher (2006) 4. Extraordinary transition in network rewiring Liu, Schmittmann, Zia (2012)
IDSI : Inverse Distance Square Ising model ? = ?>?? ? ? ??????= 1 ? ?(? ?)~ ? ??? = 2 For 1 < ? 2 the model has an ordering transition at finite T A simple argument: ? ++++++++++++-----------------++++++++++++++++++++ 1 L 2? 2? ? 1??1 ?+ ? 1 2 ?(?2 ? 1) ? ? = Anderson et al (1969, 1971); Dyson (1969, 1971); Thouless (1969); Aizenman et al (1988)
? = 2 model is special The magnetization m is discontinuous at ??( Thouless effect ) Thouless (1969), Aizenman et al (1988) 1 ?~exp KT type transition, Cardy (1981) ? ?? Phase diagram H T IDSI Fisher, Berker (1982)
Dyson hierarchical version of the model (1971) 1/26 1/24 1/22 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: +++++++++------------------------+++++++++++++++++--------------------- ? = ??? ????+1 ? ? ? ???? ?,? ??? ?? ??? ?????? 1 ? ?2 ? ??? ,? ? ? > 0 ?(? ?)~ The interaction is in fact not binary but rather many body.
Summery of the results ?with nonuniversal ? diverging correlation length at ?? ?~ ? ?? Extreme Thouless effect with ? = 0 ? = 1 Phase diagram H T The model is closely related to the PS model of DNA denaturation
The energy of a domain of length ? ? 1? ? 1 ??= ? ?2 = ?2 ~ ?? + ? ln? + ?????. ?>? ?=1 Interacting charges representation: ? ? a, ?, > 0 ? ??,? = ? ??+ ? ln??+ ? ?=1 ?=1 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 ???1 ? ? ????2 ???? ? ?,? = ..? ? ? ? ? ?1 ?2 ?? ? ?? ?1+ + ??= L Grand partition sum ???(?,?) ? ? = ? ? = ??2? 1 + ?2?2+ ?4?4+ ? Polylog function ????? ?? ? ? ?(???) ??2(?) 1 ?2?2(?) ?? ?? = ? ? = ?=1
Polylog function (???)? ??? ??2(?) 1 ?2?2(?) ? ?(???) ?? ?? = ? ? = ?=1 ? is the closest pole to the origin ? ?(?)~? ?? ?? ?,?~ ? ? ?? ? ? = ???(?) ? ? ,? = 1/? ? < 1, ferromagnetic coupling
? 0 ?(???) (???)? ?? ? ?? ????= ? ?(???) ?? 1 < 1/? ?=1 1 ? ??? 1 ??? ? e? 1 Phase transition: (? ???) = 1 , ?? ? ? Unlike the PS model the parameter c is not universal
Nature of the transition ?? ?~(???)? ?(?) =?(? ?) 1 ? Domain length distribution ?? ? ?(?) ? ? ~? ??? =? ?/? ? ?? ? Close to ??: ? = 1 ?? ?? ?? 1 characteristic length ? = ? ??? ? ?? ?? ?1 ?? ?1 ??? 1 1 < ? 2 ? > 2 ?1 ?? ? ? ??~ ??? 1 1 < ? 2 ? > 2 ?? 1 ? 1 1 < ? 2 ?~? ? ? = 1 ? > 2
Two order parameters 1. ? =? ? number of domains 2. ? =? ? magnetization 1. ? order parameter 1 ? ? = ?1 ???? Where at ?? ? ~ 1 for 1 < c 2 for ? ? is finite ? > 2 ? is continuous 1 < ? 2 ? is discontinuous in both cases ? ? > 2
2. ? order parameter ? = 1??? ? ?,?? = ?? ????+ ? ?(?,? = ?+ ? ) ? ?,? = ?,? ?? ?? ?(? 1 ?2? ?? ?(? ?) ? ?,? = ?) ? = ? ln ? (?) ln? is the magnetic field ? ln ? (? = 1) (? 1 ? > ?? ? = 0 ? < ?? either ?? = 1 or ? ? symmetry) ?= 1 ? = 1 Extreme Thouless effect
Phase diagram I n is continuous II and III n is discontinuous
,? phase diagram ? = 1 1 < ? < 1 ? ? = 1
Canonical analysis Free energy ? ?,?,? = ??(?,?) ? ?,? ???? ? ?,? = ?,? ???2(?) 1 ?2?2?2(?) ? ?,? = 1 ?(?,?) ??+1??+1 ? ?,? = 2??2 ???? ? ?,? = ? ???(?,?) ??(?)??+2 ???+1 ? ?,? = ?? ?? 4??? ??1(?,?) 4??
??(?)??+2 ???+1 ? ?,? = ?? ?? 4??? ??1(?,?) 4?? saddle point: ?(? ) ? ? (? ) ? = ? ? > ?? = ??? ??? ??(? ) ? ? < ?? = ???? ??? ??(??)
Finite L correction: ? = 2.5 ? = 1000
Finite L corrections c=2.5 L=1000
Renormalization group - charges representation + - + - ? ? ??? ? ??+1 ?? ? ?? ? ?,? = (??+1 ?? ?) ?=0 ?=1 y - fugacity a - short distance cutoff Length rescaling ? ??? This can be compensated by y rescaling ? ???(1 ?)
??+1 ?? ??? ??+1 ?? ? a??(??+1 ?? ?) ? ? ? + - + - ? ? 2? 1 ?2?? ?? ??? ? ?? ? The integral scales like 1/?? hence it does not renormalize c . Rather it renormalizes y.
Renormalization group equations ?? ??= ?? + ?2 ? 1 ? ?? ??= 0 compared with the Kosterlitz-Thouless model: ?? ??= ?? ?? ??= ?2
In the KT case: ? ? ? + - + - ? Contribution of the dipole to the renormalized partition sum: ? ? ? ?? ? ? ? ? ? ??1 + ?2?? ? ?? ? + ? ? 1 ? ? ? ? ? ? ??1 + ?2???? renormalizes c. (Cardy 1981)
?? ??= ?? + ?2 ?? ??= ?? ? 1 ? ?? ??= 0 ?? ??= ?2 Line of fixed points ? = ? 1 ?~?1/ ? ?? ? ? = ?~ ? ?? ? 1
Coarsening dynamics Particles with n-n logarithmic interactions Biased diffusion, annihilation and pair creation ?1 ?2 ?3 ?4 + + ??3,?4 ??1,?2 ??5,?6 ?5 ?6 ? 1 ?? = 1 + ??,? ? 1 ? 1 ? 1 ?? = 1 + ? 2 ??,? ?+?+1
Coarsening dynamics ? = ? < ?? The coarsening is controlled by the T=0 (y=0) fixed point =?2?(?,?) ??(?,?) ?? ??2+? ??(? ?? ?,? ) 1 ??(? ? ?,? = ?) Like the dynamics of the T=0 Ising model
Coarsening dynamics ? = ? = ?? ? ?,? =1 ? Expected scaling form ???( ?1/?) ?(? 1)~? ? ? ? 1 = 1 1 ? = <?> - number of domains ?~? ?(?) with ? ? =2 ? ?
L=5000 c=1.5 ? = ?? ? ?,? ?1.5 z=2 z=1.5 ?/?0.5 ?/?0.66 ? ?,? ~1 ? ??? ?1/?
?~??(?) with ? ? =2? ? - 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.
Domain representation of the 1 (Fortuin-Kasteleyn representation) ?2 Ising model H = ??,?(?? ,??) ??= ????(??,??) 1 ??,? ? ? = ? ?,?(1 + ?? ?) = ? ? ?,??? ? ??,?= 0,1 defines a graph on the vertices 1, ,? The sum is over all graphs E
A graph can be represented as composed of sub-graphs separated by breaking points ??= ????(??,??) 1 One has to calculate ? ? - the probability that the distance between adjacent breaking points is ?.
