On the Kertsz line: Thermody- namic versus Geometric phase - PDF document

On the Kertèsz line: Thermody- namic versus Geometric phase transitions Jean RUIZ Centre de Physique Théorique Marseille works with Ph. Blanchard, H. Satz (Bielefeld) D. Gandolfo (Marseille Toulon) L. Laanait (Rabat) 1

Summary • What is the Kertèsz ( 2 D –Ising model) • Some Results for Kertèsz line for Potts model (analytical and numerical) − Potts model and some known results − Various representations of Potts models − Results − Scheme of proofs • “Kertèsz line” for Potts model on the complete graph − Erdös–Renyi random graph − Known results for Curie–Weiss Potts model − Various representations − Known results for Fortuin–Kasteleyn representation − What’s we expect 2

The Kertèsz Line (Kertèsz 1989, Stauffer and Aharony, Percolation Theory: the trouble with Kertèsz) 2 − D Ising model i ∈ Λ ⊂ Z 2 σ i = ± 1 , − − + + + + + + − − + − − − − + + + + + − − + + + − − + − + Boltzmann weight β h 2 ( σ i σ j − 1) � 2 ( σ i − 1) � ω Ising ( σ ) = e e � i,j � i e β ( δ σi,σj − 1) � � e h ( δ σi, 1 − 1) = � i,j � i 1 β = k T , inverse temperature h external magnetic field product is over nearest neighbour pairs. 3

• h = 0 . Phase transition √ ( e β c − 1) 2 = 2 , β c = ln (1 + 2 ) − β > β c → positive spontaneous magnetization → Two translation invariant Gibbs states → Positive surface tension between the two states − β < β c → no spontaneous magnetization → Unique phase • h > 0 “Nothing happens” − For all β → Analytic free energy → Unique phase 4

Edwards–Sokal representation � [ e − β δ η ij , 0 + (1 − e − β ) δ η ij , 1 δ σ i ,σ j ] ω ES ( σ , η ) = � i,j � � e h ( δ σi, 1 − 1) × i where the RV η ij ∈ { 0 , 1 } : write for each edge � i, j � : e β ( δ σi,σj − 1) = e − β + (1 − e − β ) δ σ i ,σ j � e − β δ η ij , 0 + (1 − e − β ) δ σ i ,σ j δ η ij , 1 = η ij =0 , 1 + − + + + − + + − − + − − − + + + − + + − − + + − − − + + + • h = ∞ all σ i = + 1 � [ e − β δ η ij , 0 + (1 − e − β ) δ η ij , 1 ] ω ES ( η ) = � i,j � → Usual 2 − D bond Percolation problem with parameter e − β Percolation transition at e − β p = 1/2 , → β p = ln 2 5

β √ β c = ln(1 + 2) ✉ Percolation No percolation ✉ β p = ln 2 h 6

Potts model � , q } i ∈ Λ ⊂ Z d σ i ∈ { 1 , 1 1 3 6 7 2 4 1 1 4 4 3 2 6 3 3 2 1 3 4 2 6 5 4 3 4 4 1 2 2 Boltzmann weight e β ( δ σi,σj − 1) � � e h ( δ σi, 1 − 1) ω Potts ( σ ) = � i,j � i 7

• h = 0 . q large First order phase transition √ ) ( e β c − 1) 2 = q, β c = ln (1 + d = 2: q β c ≃ 1 d � 3: d ln q − β > β c → positive spontaneous magnetization → q ordered translation invariant Gibbs states → Positive surface tension between the states → Vanishing mass gap (exponential decrease of correlations) − β < β c → no spontaneous magnetization → Unique phase: disordered state → Positive mass gap (finite correlation length) − β = β c → discontinuity of mean energy and magnetiza- tion → q + 1 phases → Positive surface tension between the phases → Positive mass gap Kotecky Shlosman (1982),... 8

• h > 0 small q large : First order transition at some β c ( h ) − β > β c ( h ) → 1 ordered translation invariant Gibbs state − β < β c ( h ) → Unique phase: disordered state − β = β c ( h ) → discontinuity of mean energy → 2 phases → Positive surface tension between the phases Bakchich, Benyoussef, Laanait (1989),... 9

Various representations Sarting from ω ES ( σ , η ) 1. Fortuin-Kasteleyn in external field e hδ σi, 1 = 1 + ( e h − 1) δ σ i , 1 δ θ i , 0 + ( e h − 1) δ σ i , 1 δ θ i , 1 � θ i =0 , 1 Sum over spin variables ω ES − βδ ηij, 0 (1 − e − β ) δ ηij, 1 � ω FK ( η , θ ) = e � i,j � − hδ θi, 0 (1 − e − h ) δ θi, 1 q C ( η | θ ) � × e i 2. Sum directly over spins ω ES − βδ ηij, 0 (1 − e − β ) δ ηij, 1 � ω ( η ) = e � i,j � C ( η ) � (1 + ( q − 1) e − h ) S i ( η ) × i 10

� 1) Colored–Edwards–Sokal representation Write δ σ i ,σ j = χ ( σ i = σ j = 1) + χ ( σ i = σ j Replace edge variables η ij ∈ { 0 , 1 } by n ij ∈ { 0 , 1 , 2 } (white red blue). Then � 1) � � e − β δ n ij , 0 � ω CES ( σ , n ) = � i,j � + (1 − e − β ) δ n ij , 1 χ ( σ i = σ j = 1) + (1 − e − β ) δ n ij , 2 χ ( σ i = σ j � e hδ σi, 1 × i 1 1 3 6 7 2 4 1 1 4 4 3 2 6 3 3 2 1 3 4 2 6 5 4 3 4 4 1 2 2 11

Tri–Clor–Edge representation Sum over spin variables − βδ nij, 0 (1 − e − β ) ( δ nij, 1 + δ nij, 2 ) � ω TER ( n ) = e � i,j � × e hS 1 ( n ) ( q − 1) C 2 ( n ) × ( q − 1 + e h ) | Λ |− S 1 ( n ) − S 2 ( n ) − S 1 ( n ) (resp. S 2 ( n ) ) denotes the number of sites that belong to edges of color 1 (resp. of color 2 ) − C 2 ( n ) denotes the number of connected com- ponents of the set of edges of color 2 − | Λ | is the number of sites of the box under consideration Geometric order parameter Let p Λ ( i ↔ j ) be the probability that the site i is connected to j by a path of edges of color 1 . mass–gap (inverse correlation length) 1 m ( β, h ) = − lim | i − j | ln lim Λ ↑ Z d p Λ ( i ↔ j ) | i − j |→∞ − i and j belong to some line parallel to an axis of the lattice. 12

Diagram of ground states • b a be the value of the Boltzmann weight (in the TER representation) of the ground state configuration of color a = 0 , 1 , 2 per unit site b 0 = e − βd ( q − 1 + e h ) b 1 = (1 − e − β ) d e h b 2 = (1 − e − β ) d β 2 1 β 0 ( h ) 0 h β 0 ( h ) = ln [1 + (1 + ( q − 1) e − h ) 1/ d ] All the ground states coexist at (0 , β 0 (0)) . Below β 0 ( h ) only the 0 –state dominates. Above β 0 ( h ) the 1 –state dominates: it coexists with the 0 –state on β 0 ( h ) and with the 2 – state on the line h = 0 , β ≥ β 0 (0) 13

Analytic results q is large enough and h not too large Using Pirogov–Sinaï theory • the phase diagram of the TER model mimics the diagram of ground state configurations • the model undergoes a thermodynamic first order phase transition in the sense that the derivative of its free energy with respect to β (or h ) is discontinuous at some β c ( h ) ∼ β 0 ( h ) • the model exhibits a geometric (first order) transition, in the sense that, on the critical line, the mass gap is discontinuous. Theorem 1. Assume d ≥ 2 , q and h such that c d (1 + ( q − 1) e − h ) − 1/2 d < 1 holds, where c d is a given number (depending only on the dimension), then there exists a unique β c ( h ) = β 0 ( h ) + O (1 + ( q − 1) e − h ) − 1/2 d ) such that m ( β, h ) > 0 for β ≤ β c ( h ) and m ( β, h ) = 0 for β > β c ( h ) . Since the free energies of Potts model and of the TER model are the same, the critical lines coincides with that of Potts model. 14

Numerical simulations Generalization of the Swendsen–Wang algorithm inherited from colored Edwards–Sokal model. 1.5 β 1.3 q = 10 1.1 5 0.9 4 2 0.7 h 0.5 0 1 2 3 4 5 d = 2 . 1. For q ≤ 4 : − a whole geometric transition line for which m ( β, h ) > 0 when β � β c ( h ) m ( β, h ) = 0 when β > β c ( h ) − The mass gap is continuous at β c ( h ) − For β � β c ( h ) the mean cluster sizes remain finite − For β > β c ( h ) the size of 1 –edge clus- ters diverges. − The mean energy as well as the mag- netisation do not show any singular behavior 15

1.5 β 1.3 q = 10 1.1 5 0.9 4 2 0.7 h 0.5 0 1 2 3 4 5 2. For q ≥ 5 : some critical h c appears − the transition becomes first order when h < h c in accordance with the previous analytic results − both the mass gap and the mean energy exhibit discontinuities at β c ( h > h c ). − However when h ≥ h c , the scenario is the same as for q ≤ 4 . Numerics are in accordance with theory for van- √ ) and ishing and infinite fields: β c (0) = ln (1 + q β c ( ∞ ) = ln 2 . 16

Description of the algorithm inherited from colored Edwards–Sokal model. � 1) � � e − β δ n ij , 0 ω CES ( σ , n ) = � � i,j � + (1 − e − β ) δ n ij , 1 χ ( σ i = σ j = 1) + (1 − e − β ) δ n ij , 2 χ ( σ i = σ j i e hδ σi, 1 × � 1. Given a spin configuration: put between any two neighbouring spins of the same color: - an edge colored 0 with probability e − β - w.p. 1 − e − β , an edge colored 1 if these spins are of colour 1 , and coloured 2 other- wise. When two neighbouring spins disagree, the corresponding edge is colored 0 . 2. Starting from an edge configuration, a spin configuration is constructed as follows. - Isolated sites (endpoints of 0 –bonds only) e h /( q − 1 + e h ) are coloured 1 w.p. and coloured c ∈ { 2 , ..., q } w.p. 1/( q − 1 + e h ) . - Non–isolated sites are colored 1 (w.p. 1) if they are endpoints of 1 –bonds and colored c ∈ { 2 , ..., q } w.p. 1/( q − 1) . 17