Mixed order phase transitions David Mukamel Amir Bar, DM (PRL, 122, 01570 (2014))

Phase transitions of mixed order (a) diverging length 𝜉→∞ as in second order transitions 𝜉~ 𝑡 −𝜈 or 𝜉~ 𝑒 1/ 𝑡 𝜇 (b) discontinuous order parameter as in first order transitions

1d Ising model with long range interactions 𝐽(𝑟)~1/ 𝑟 2 Examples: 1d Ising model with long range interactions 𝐽(𝑟)~1/ 𝑟 2 non-soluble but many of its properties are known 2. Poland-Scheraga (PS) model of DNA denaturation 𝜉~ 𝑒 1/ 𝑡 𝜇 𝜉~ 𝑡 −𝜈 3. Jamming transition in kinetically constrained models 𝜉~ 𝑒 1/ 𝑡 𝜇 Toninelli, Biroli, Fisher (2006) 4. “Extraordinary transition” in network rewiring Liu, Schmittmann, Zia (2012)

IDSI : Inverse Distance Square Ising model 𝐻=− 𝑖>𝑗 𝐽 𝑖−𝑗 𝑆 𝑖 𝑆 𝑗 𝑆 𝑖 =±1 𝐽(𝑖−𝑗)~ 𝐽 𝑖−𝑗 𝛼 𝛼=2 For 1<𝛼≤2 the model has an ordering transition at finite T A simple argument: ++++++++++++-----------------++++++++++++++++++++ 1 L 𝑙 𝐸 𝑙 =− 2𝐽 𝛼−1 𝑙 𝐿 1−𝛼 + 2𝐽 𝛼−1 2−𝛼 ( 𝑙 2−𝛼 −1) Anderson et al (1969, 1971); Dyson (1969, 1971); Thouless (1969); Aizenman et al (1988)…

𝛼=2 model is special The magnetization m is discontinuous at 𝑇 𝑐 (“Thouless effect”) Thouless (1969), Aizenman et al (1988) 𝜉~exp 1 𝑇− 𝑇 𝑐 KT type transition, Cardy (1981) Phase diagram H T IDSI Fisher, Berker (1982)

Dyson hierarchical version of the model (1971) 1/ 2 6 1/ 2 4 1/ 2 2 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: +++++++++------------------------+++++++++++++++++--------------------- 𝐻=− 𝐽 𝑁𝑁 𝑖 𝑆 𝑖 𝑆 𝑖+1 − 𝑖,𝑗 𝑤𝑖𝑡ℎ𝑖𝑛 𝑒𝑎𝑐ℎ 𝑑𝑜𝑚𝑎𝑖𝑛 𝐽 𝑖−𝑗 𝑆 𝑖 𝑆 𝑗 𝐽(𝑖−𝑗)~ 1 𝑖−𝑗 2 𝐽 𝑁𝑁 , 𝐽 𝑖−𝑗 >0 The interaction is in fact not binary but rather many body.

Summery of the results diverging correlation length at 𝑇 𝑐 𝜉~ 𝑇− 𝑇 𝑐 −𝜈 with nonuniversal 𝜈 Extreme Thouless effect with 𝑚=0→𝑚= ±1 Phase diagram H T The model is closely related to the PS model of DNA denaturation

The energy of a domain of length 𝑙 𝐸 𝑙 =− 𝑖>𝑗 1 𝑖−𝑗 2 =− 𝑘=1 𝑙−1 𝑙−𝑘 𝑘 2 ~−𝑎𝑙+ 𝑐 ln 𝑙+𝑐𝑜𝑛𝑠𝑡. Interacting charges representation: 𝐻 𝑙 𝑖 , 𝑁 =−𝑎 𝑖=1 𝑁 𝑙 𝑖 + 𝑐 𝑖=1 𝑁 ln 𝑙 𝑖 +Δ𝑁 a, 𝑐 , Δ >0 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 𝑍 𝐿,𝑇 = 𝑘 𝑙 𝑖 𝑒 𝛽 𝑙 1 𝑙 1 𝛽 𝑐 𝐴 𝑒 𝛽 𝑙 2 𝑙 2 𝛽 𝑐 …..𝐴 𝑒 𝛽 𝑙 𝑘 𝑙 𝑘 𝛽 𝑐 𝑙 1 + …+ 𝑙 𝑘 =L Grand partition sum 𝑄 𝑧 = 𝐿 𝑧 𝐿 𝑍(𝐿, 𝑇) 𝑄 𝑧 =𝐴 𝑈 2 𝑧 1+ 𝐴 2 𝑈 2 + 𝐴 4 𝑈 4 +… Polylog function 𝑈 𝛽 𝑐 𝑧 = 𝑙=1 ∞ 𝑧 𝑙 𝑒 𝛽𝑙 𝑙 𝛽 𝑐 ≡ Φ 𝛽 𝑐 (𝑧 𝑒 𝛽 ) 𝑄 𝑧 = 𝐴 𝑈 2 (𝑧) 1− 𝐴 2 𝑈 2 (𝑧)

𝑧 ∗ is the closest pole to the origin 𝑈 𝛽 𝑐 𝑧 = 𝑙=1 ∞ (𝑧 𝑒 𝛽 ) 𝑙 𝑙 𝛽𝑐 ̅ ≡ Φ 𝛽 𝑐 (𝑧 𝑒 𝛽 ) 𝑄 𝑧 = 𝐴 𝑈 2 (𝑧) 1− 𝐴 2 𝑈 2 (𝑧) Polylog function 𝑧 ∗ is the closest pole to the origin 𝑄(𝑧)~ 𝑧 ∗𝐿 𝑒 −𝛽𝐹 𝐿,𝑇 ~ 𝑧 ∗ 𝑒 −𝛽𝑓 𝑇 𝐿 𝑧 ∗ = 𝑒 𝛽𝑓(𝑇) 𝑈 𝑧 ∗ ,𝑇 =1/𝐴 𝐴<1, ferromagnetic coupling

̇ ̇ 𝑈 𝛽 𝑐 𝑧 𝑒 𝛽 = 𝑙=1 ∞ (𝑧 𝑒 𝛽 ) 𝑙 𝑙 𝛽 𝑐 ≡ Φ 𝛽 𝑐 (𝑧 𝑒 𝛽 ) 1< 1/𝐴 𝑈(𝑧 𝑒 𝛽 ) 𝑧 𝑒 𝛽 1 1 𝛽→∞ 𝛽→0 ̇ 𝛽 𝑐 𝑈 𝛽 𝑐 𝑧 𝑒 𝛽 = 𝑙=1 ∞ (𝑧 𝑒 𝛽 ) 𝑙 𝑙 𝛽 𝑐 ≡ Φ 𝛽 𝑐 (𝑧 𝑒 𝛽 ) ̇ 𝑧 𝑒 𝛽 1 𝑧 ∗ e 𝛽 Phase transition: (𝑧 ∗ 𝑒 𝛽 𝑐 )=1 , 𝛽 𝑐 𝑐 ≡𝑐 Unlike the PS model the parameter c is not universal

Nature of the transition 𝑝(𝑙)= 𝑍(𝐿−𝑙) 𝑍(𝐿) 𝐴 1 𝑙 𝛽 𝑐 ~ (𝑧 𝑒 𝛽 ) 𝑙 𝑙 𝛽 𝑐 Domain length distribution 𝑝 𝑙 ~ 𝑒 −𝑙𝛿𝑧 𝑙 𝑐 = 𝑒 −𝑙/𝜉 𝑙 𝑐 𝑐≡ 𝛽 𝑐 𝑐 Close to 𝑇 𝑐 : 𝑧=1−𝛿𝑧 characteristic length 𝜉= 1 𝛿𝑧 𝜉→∞ 𝑓𝑜𝑟 𝑇→ 𝑇 𝑐 Φ 𝑐 1−𝛿𝑧 ≈ Φ 𝑐 1 −𝛿 𝑧 𝑐−1 1<𝑐≤2 Φ 𝑐 1 −𝛿𝑧 𝑐>2 𝑡≡𝑇− 𝑇 𝑐 ~ 𝛿 𝑧 𝑐−1 1<𝑐≤2 𝛿𝑧 𝑐>2 𝜉~ 𝑡 −𝜈 𝜈= 1 𝑐−1 1<𝑐≤2 1 𝑐>2

Two order parameters 1. 𝑛 order parameter 𝑛= 𝑁 𝐿 number of domains 𝑚= 𝑀 𝐿 magnetization 1. 𝑛 order parameter 𝑛= 1 𝑙 𝑙 ~ 1 ∞ 𝑙 1 𝑙 𝑐 𝑑𝑙 Where at 𝑇 𝑐 for 1<c≤2 𝑙 →∞ for 𝑐>2 𝑙 is finite 𝑛 is continuous 1<𝑐≤2 in both cases 𝜉→∞ 𝑛 is discontinuous 𝑐>2

2. 𝑚 order parameter 𝑍 𝐿,𝑀 𝐿= 𝑙 𝑖 𝑀= −1 𝑖 𝑙 𝑖 𝑍 𝐿,𝑀 𝐿= 𝑙 𝑖 𝑀= −1 𝑖 𝑙 𝑖 𝑄 𝑧,𝑟 = 𝐿,𝑀 𝑧 𝐿 𝑟 𝐿 + −𝐿 − 𝑍(𝐿,𝑀= 𝐿 + − 𝐿 − ) 𝑄 𝑧,𝑟 = 𝐴𝑈 𝑧𝑟 𝑈( 𝑧 𝑟 ) 1− 𝐴 2 𝑈 𝑧𝑟 𝑈( 𝑧 𝑟 ) 𝑚=− 𝜕 ln 𝑧 ∗ (𝑟) 𝜕 ln 𝑟 (𝑟=1) ℎ≡ ln 𝑟 is the magnetic field 𝑇> 𝑇 𝑐 𝑚=0 (𝑟→ 1 𝑟 symmetry) 𝑇< 𝑇 𝑐 either 𝑧𝑟=1 or 𝑧 𝑟 =1 𝑚=±1 Extreme Thouless effect

Phase diagram I n is continuous II and III n is discontinuous

ℎ,𝑇 phase diagram −1<𝑚<1 𝑚=1 𝑚=−1 ℎ 𝑇

Free energy 𝐹 𝐿,𝑁,𝑇 =𝐿𝑓(𝑛,𝑇) Canonical analysis Free energy 𝐹 𝐿,𝑁,𝑇 =𝐿𝑓(𝑛,𝑇) 𝑄 𝑧,𝑞 = 𝐿,𝑁 𝑍 𝐿,𝑁 𝑧 𝐿 𝑞 𝑁 𝑄 𝑧,𝑞 = 𝐴𝑞 𝑈 2 (𝑧) 1− 𝐴 2 𝑞 2 𝑈 2 (𝑧) 𝑍 𝐿,𝑁 = 1 2𝜋𝑖 2 𝑑𝑧𝑑𝑞 𝑄(𝑧,𝑞) 𝑧 𝐿+1 𝑞 𝑁+1 𝑍 𝐿,𝑁 = 𝑒 −𝛽𝐿𝑓(𝑛,𝑇) 𝑍 𝐿,𝑛 = 𝑑𝑧 4𝜋𝑖 𝐴𝑈(𝑧) 𝐿𝑛+2 𝐴 𝑧 𝐿+1 ≈ 𝑑𝑧 4𝜋𝑖 𝑒 −𝐿 𝑓 1 (𝑛,𝑧)

𝑓 𝑛> 𝑛 𝑐 =𝑙𝑛 𝑧 ∗ −𝑛𝑙𝑛 𝐴𝑈( 𝑧 ∗ ) 𝑍 𝐿,𝑛 = 𝑑𝑧 4𝜋𝑖 𝐴𝑈(𝑧) 𝐿𝑛+2 𝐴 𝑧 𝐿+1 ≈ 𝑑𝑧 4𝜋𝑖 𝑒 −𝐿 𝑓 1 (𝑛,𝑧) saddle point: 𝑛= 𝑈( 𝑧 ∗ ) 𝑧 ∗ 𝑈′( 𝑧 ∗ ) 𝑓 𝑛> 𝑛 𝑐 =𝑙𝑛 𝑧 ∗ −𝑛𝑙𝑛 𝐴𝑈( 𝑧 ∗ ) 𝑓 𝑛< 𝑛 𝑐 =𝑙𝑛 𝑧 𝑐 −𝑛𝑙𝑛 𝐴𝑈( 𝑧 𝑐 )

c=2.5

Finite L correction: 𝑐=2.5 𝐿=1000

c=2.5

Finite L corrections c=2.5 L=1000

Renormalization group - charges representation + - + - 𝑍 𝑦,𝑐 = 𝑁=0 ∞ 𝑦 𝑁 𝑖=1 𝑁 𝑑 𝑟 𝑖 𝑎 𝑟 𝑖+1 − 𝑟 𝑖 𝑎 −𝑐 Ɵ( 𝑟 𝑖+1 − 𝑟 𝑖 −𝑎) y - fugacity a - short distance cutoff Length rescaling 𝑎→𝑎 𝑒 𝜅 This can be compensated by y rescaling 𝑦→ 𝑦 𝑒 𝜅(1−𝑐)

Ɵ 𝑟 𝑖+1 − 𝑟 𝑖 −𝑎 𝑒 𝜅 ≈Ɵ 𝑟 𝑖+1 − 𝑟 𝑖 −𝑎 −a𝜅𝛿( 𝑟 𝑖+1 − 𝑟 𝑖 −𝑎) + - + - 𝑠 𝑙−𝑠 𝑙 − 𝑦 2 𝑎𝜅 𝑎 𝑙−2𝑎 𝑑𝑠 1 𝑠 𝑐 𝑙−𝑠−𝑎 𝑐 The integral scales like 1/ 𝑙 𝑐 hence it does not renormalize c . Rather it renormalizes y.

Renormalization group equations 𝑑𝑦 𝑑𝜅 =𝑥𝑦+ 𝑦 2 𝑑𝑥 𝑑𝜅 =0 𝑥≡1−𝑐 compared with the Kosterlitz-Thouless model: 𝑑𝑦 𝑑𝜅 =𝑥𝑦 𝑑𝑥 𝑑𝜅 = 𝑦 2

Contribution of the dipole to the renormalized partition sum: In the KT case: + - + - 𝑠 𝑙−𝑠 𝑙 Contribution of the dipole to the renormalized partition sum: 𝑦 𝑙 𝑐 1+ 𝑦 2 𝑎𝜅 𝑎 𝑙−𝑎 𝑑𝑠 𝑠 𝑠+𝑎 𝑐 𝑙−𝑠−𝑎 𝑙−𝑠 𝑐 1− 𝑎 𝑠 − 𝑎 𝑙−𝑠 𝑦 𝑙 𝑐 1+ 𝑦 2 𝜅𝑙𝑛𝑙 renormalizes c. (Cardy 1981)

𝜉~ 𝑒 1/ 𝑇− 𝑇 𝑐 𝑑𝑦 𝑑𝜅 =𝑥𝑦+ 𝑦 2 𝑑𝑦 𝑑𝜅 =𝑥𝑦 𝑑𝑥 𝑑𝜅 =0 𝑑𝑥 𝑑𝜅 = 𝑦 2 𝑥≡1−𝑐 Line of fixed points 𝑦 ∗ =− 𝑥 ∗ 𝜉~ 𝑇− 𝑇 𝑐 −𝜈 𝜈= 1 𝑐−1 𝜉~ 𝑒 1/ 𝑇− 𝑇 𝑐

Coarsening dynamics Particles with n-n logarithmic interactions Biased diffusion, annihilation and pair creation 𝑝 𝑙 1 , 𝑙 2 + 𝑞 𝑙 3 , 𝑙 4 − 𝑞 𝑙 5 , 𝑙 6 + 𝑙 1 𝑙 2 𝑙 3 𝑙 4 𝑙 5 𝑙 6 𝑝 𝑙,𝑘 ± = 1+ 𝑙𝑘 𝑙±1 𝑘∓1 −𝑐 −1 𝑞 𝑙,𝑘 ± = 1+ 𝑦 ∓2 𝑙𝑘 𝑙+𝑘+1 ±𝑐 −1

𝜕𝑃(𝑙,𝑡) 𝜕𝑡 = 𝜕 2 𝑃(𝑙,𝑡) 𝜕 𝑙 2 + 𝜕 𝜕𝑙 ( 𝑐 𝑙 𝑃 𝑙,𝑡 ) Coarsening dynamics 𝑇=∞→𝑇< 𝑇 𝑐 The coarsening is controlled by the T=0 (y=0) fixed point 𝜕𝑃(𝑙,𝑡) 𝜕𝑡 = 𝜕 2 𝑃(𝑙,𝑡) 𝜕 𝑙 2 + 𝜕 𝜕𝑙 ( 𝑐 𝑙 𝑃 𝑙,𝑡 ) 𝑃 𝑙,𝑡 = 1 𝑡 𝑔( 𝑙 𝑡 ) Like the dynamics of the T=0 Ising model

𝑛= 1 <𝑙> - number of domains Coarsening dynamics 𝑇=∞→𝑇= 𝑇 𝑐 𝑃 𝑙,𝑡 = 1 𝑙 𝑐 𝑔( 𝑙 𝑡 1/𝑧 ) Expected scaling form 𝑔 𝑥≪1 =1 𝑔(𝑥≫1)~ 𝑒 −𝑥 𝑛= 1 <𝑙> - number of domains 𝑛~ 𝑡 −𝜖(𝑐) with 𝜖 𝑐 = 2−𝑐 𝑧

L=5000 c=1.5 𝑇= 𝑇 𝑐 𝑃 𝑙,𝑡 𝑡 1.5 z=2 z=1.5 𝑙/ 𝑡 0.5 𝑙/ 𝑡 0.66 𝑃 𝑙,𝑡 ~ 1 𝑙 𝑐 𝑓 𝑙 𝑡 1/𝑧

- Voter model (y=0, fixed c) 𝑛~ 𝑡 −𝜖(𝑐) with 𝜖 𝑐 = 2−𝑐 𝑧

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 𝑟 2 Ising model (Fortuin-Kasteleyn representation) H=− 𝐽 𝑖,𝑗 ( 𝜎 𝑖 ,𝜎 𝑗 ) 𝜆 𝑟 = 𝑒 𝛽 𝐽 𝑟 ( 𝜎 𝑖 , 𝜎 𝑗 ) −1 𝑍 𝐿 = 𝜎 𝑖,𝑗 (1+ 𝜆 𝑖−𝑗 ) = 𝜎 𝐸 𝑖,𝑗 𝜆 𝑖−𝑗 𝐸 𝑖,𝑗 𝐸 𝑖,𝑗 =0,1 defines a graph on the vertices 1,…,𝐿 The sum is over all graphs E

A graph can be represented as composed of sub-graphs separated by “breaking points” 𝜆 𝑟 = 𝑒 𝛽 𝐽 𝑟 ( 𝜎 𝑖 , 𝜎 𝑗 ) −1 One has to calculate 𝑃 𝑙 - the probability that the distance between adjacent breaking points is 𝑙.