FIELD THEORY OF ISING PERCOLATING CLUSTERS

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1 UK Meeting on Integrable Models and Conformal Field heory University of Kent, Canterbury April 21 FIELD HEORY OF ISING PERCOLAING CLUSERS Gesualdo Delfino SISSA-rieste Based on : GD, Nucl.Phys.B 818 (29) 196 (arxiv: [hep-th])

OF ISING PERCOLAING CLUSERS Gesualdo Delfino SISSA-rieste

2 he Ising model is the basic example of magnetic phase transition: spontaneous magnetization below c Percolation is the basic example of geometric phase transition: infinite cluster above p c he Ising model, however, encodes itself a percolation problem for clusters of magnetically interacting spins here is no interaction other than the magnetic one Naive expectation: cluster criticality is associated to magnetic criticality. Not true Mean cluster size can diverge while magnetic correlation length is finite he issue can be resolved exactly in d = 2 (integrability, but not the Ising one...)

spins here is no interaction other than the magnetic one Naive expectation: cluster criticality is associated to magnetic criticality.

3 Ising percolation H Ising = 1 σ i σ j + H i ij σ i, σ i = ±1 ( ) No magnetic transition away from c, H = H =+1 c = 1 P probability that a site belongs to an infinite cluster of + spins (percolative order parameter) H c ( ) such that P > for H > H c ( ) H c () = e H c( ) 2 cosh H c ( ) = p c p c = critical point of random percolation (non-universal)

of + spins (percolative order parameter) H c ( ) such that P > for H > H c ( ) H c () =

4 hree observations: i) H c ( ) monotonic ii) spontaneous magnetization = infinite cluster (Coniglio et al 77) iii) p c p c at H = (interaction makes percolation easier) Consequences: H a) p c > 1/2, i.e. H c( ) > P > H c () i), ii) = H c ( ) = for c c b) p c < 1/2, i.e. H c( ) < H p = 1/2 for c, H = iii) = p < c p c H c () Ordinary lattices in d = 2 have p c 1/2 a) d = 3 have p c < 1/2 b)

5 Kasteleyn-Fortuin representation H q = H Ising + J ij t i t j (δ si,s j 1), t i = 1 2 (σ i + 1) =, 1, s i = 1,..., q Z q = {t i } = {t i } e H q {s i } e H Ising qn v = (Murata 79) p b B (1 pb) b q N c G N v = # of empty sites G = graph made of bonds between occupied sites b = # of bonds in G; b = # of absent bonds N c = # of connected components in G (KF clusters) p B 1 e J prob that a bond is present (KF cl=ising cl for p B = 1) q continuous parameter X q 1 = Z 1 Ising {t i } e H Ising X p b B (1 p B) b G dilute percolation average pure percolation for H = +

between occupied sites b = # of bonds in G; b = # of absent bonds N c = # of connected components in G (KF clusters) p B 1 e J prob that a

6 X q 1 = X Ising for X Ising magnetic observable Example of percolative observable: Mean cluster number in pure percolation (N v = ) = d ln Z q dq H=+, q=1 f q ln Z q = f Ising + (q 1)F + O((q 1) 2 ) Dilute Potts model yields Ising magnetic properties at q = 1 KF cluster properties at q = 1 + ɛ Ising cluster properties at q = 1 + ɛ, J +

f Ising + (q 1)F + O((q 1) 2 ) Dilute Potts model yields Ising magnetic properties

7 RG analysis in d = 2 (Coniglio-Klein 8 + CF) Look for fixed points of H q 1 (, H, J) = H Ising (, H) J ij t i t j (δ si,s j 1) Need a magnetic fixed point to start with = = c, H =. J? d=2 fixed points characterized by central charge c = 1 6/[m(m + 1)] primary fields ϕ r,s with scaling dimensions X r,s = [(m+1)r ms]2 1 2m(m+1) H Ising ( c, ): m = 3, c = 1/2, X σ = X 1,2 = 1/8, X ε = X 1,3 = 1 H q possesses two critical lines as functions of q: π(t 1) t, critical (H = + ) : X s = X m 1,, X m+1 t1 = X 2,1 q = 2 sin 2(t + 1), m = 2 2 t + 1, tricritical: X s = X m, m, X t = X 1,2, X t2 = X 1,3 2, pure perc: c =, X s = 5/48, X t1 = 5/4 q 1 : m = 3, dilute perc: c = 1/2, X s = 5/96, X t1 = 1/8, X t2 = 1

1/2, X σ = X 1,2 = 1/8, X ε = X 1,3 = 1 H q possesses two critical lines as functions of q: π(t 1) t, critical (H = + ) : X s = X m 1,, X m+1 t1 = X 2,1 q = 2 sin 2(t + 1), m =

8 We found a fixed point of dilute percolation for J = J A trivial (purely magnetic) fixed point is at J = J irrelevant at these two fixed points = a third one 2/ c J * J H q J=2/ = 2 (δ νi,ν j 1) + (ln q 2H) δ νi,, ν i =, 1,..., q ij i = fixed point at J = 2/ c as q 1, with X s = X σ = 1/8 (KF clusters with J = 2/ ) = Ising droplets RG flows among fixed points with c = 1/2 (c-theorem does not apply) Critical behavior of Ising clusters is ruled by J (agrees with numerics) cluster size (linear extension) D D = d X s fractal dimension 91/48 = pure percolation D = 187/96 = Ising clusters 15/8 = Ising droplets

$(c-theorem does not apply) Critical behavior of Ising clusters is ruled by J (agrees with numerics) cluster size (linear extension) D D = d X s fractal$

9 Field theory of Ising clusters Ising field theory: A Ising = A Ising CF τ d 2 x ε(x) h d 2 x σ(x), τ c, h H Dilute Potts field theory: A q = A tricr CF g d 2 x ϕ 1,3 (x) λ d 2 x ϕ 1,2 (x) g A q=1 = A Ising (g = τ, λ = h) A q integrable for g and/or λ equal zero S q symmetry breaks spontaneously at λ = q degenerate vacua for λ < λ q he q 1 limit of the Potts critical surface is the Ising percolation transition: 1st order (massive) at < c, 2nd order (massless) at > c A q=1 = A Ising, however, is purely massive: no transition above c We can take the limit analytically

spontaneously at λ = q degenerate vacua for λ < λ 1 2 3 4 q he q 1 limit of the Potts critical surface is the Ising percolation transition: 1st

10 he massless surface of A q is an integrable field theory (Fendley, Saleur, Zamolodchikov 93, in RSOS basis) Fundamental particles: right/left movers A k, k = 1,..., q 1 with p 1 = ±p Poles of two-particle amplitudes contained in S 1/2 (θ)/(s 1/2 (θ) cosh ρ(iπ θ)) ( 1 Γ 2 + ( ) ( 2n γ) ρ ρθ 1 Γ iπ 2 + ( ) 2n γ) ρ + ρθ iπ S γ (θ) = ( 1 n= Γ 2 + ( ) ( 2n γ) ρ + ρθ 1 Γ iπ 2 + ( ), ρ = 1/(m 1) 2n γ) ρ ρθ iπ Im θ (, π) physical sheet s = µ 2 e θ : Im θ (, π) second sheet No poles on physical sheet 1 s/µ 2 Pole at θ = iπ(m 3)/2, on 2nd sheet for m (3, 5) q 1 + : resonance B with Im s (q 1)µ 2 m=3 (q=1) m=4 (q=2) q = 1 + ɛ : ɛ massless particles A k, one resonance B with lifetime 1/ɛ q = 1 : massless particles, one stable particle B with mass µ = percolative transition in absence of magnetic singularities above c

( 2n + 3 2 γ) ρ + ρθ 1 Γ iπ 2 + ( ), ρ = 1/(m 1) 2n + 1 2 γ) ρ ρθ iπ Im θ (, π) physical sheet s = µ 2 e θ : Im θ (, π) second sheet No poles on physical sheet 1 s/µ 2 Pole at θ = iπ(m 3)/2, on 2nd

11 B S q -singlet (survives at q = 1) = only S q -invariant fields φ have non-zero correlations at q = 1: lim φ(r)φ() = lim dθ 1 dθ 2 φ() A k (θ 1 )A q k (θ 2 ) 2 e re 2,(θ 1,θ ) q 1 q 1 R φ 2 = lim (q 1) dβdθ q 1 (θ θ )(θ + θ ) e re 2,((β+θ)/2,(β θ)/2) +... dβ e re1,µ(β) +... E 2, (θ 1, θ 2 ) µ(e θ 1 + e θ 2 )/2, θ i(q 1), E 1,µ (β) µ cosh(β/2) q 1 Σ A k φ φ φ B φ A k=1 q 1 q k his is how the canonical space of fields [I] [σ] [ɛ] of Ising field theory is recovered at q = 1

.. q 1 q 1 R φ 2 = lim (q 1) dβdθ q 1 (θ θ )(θ + θ ) e re 2,((β+θ)/2,(β θ)/2) +... dβ e re1,µ(β) +.

12 Field theory of Ising droplets Droplets are KF clusters with J = 2/, i.e. p B = 1 e 2/ = : p B =, no percolation H = + : transition at = 2/ ln(1 p c ) H Kertesz line p c = threshold of pure bond percolation 1 1 c Scaling limit : Ã q = A (q+1) CF τ q d 2 x ϕ 2,1 (x)+2h q d 2 x δ ν(x),, ν(x) =, 1,..., q RG invariant : η q = τ q /h (2 X 2,1)/(2 X s ) q (Ã q=1 = A Ising, η 1 η = τ/h 8/15 ) h q breaks S q+1 into S q ; for q > 1 S q breaks spontaneously at η q = η c q he Kertész line is the limit q 1 of the flow from S q+1 fixed point at h q = to S q fixed point at h q = + Again, no transition at q = 1; resonance mechanism most likely, but no integrability in this case Universal limit of the Kertész line: lattice data (Fortunato, Satz 1) + lattice-continuum relations (Caselle, Grinza, Rago 4) give η K η c q 1.12

.., q RG invariant : η q = τ q /h (2 X 2,1)/(2 X s ) q (Ã q=1 = A Ising, η 1 η = τ/h 8/15 ) h q breaks S q+1 into S q ; for q > 1 S q breaks spontaneously at η q = η c q he Kertész line is the limit

13 H Summary h η.12 K 1 1 c τ Ising universal percolative properties are not described by Ising field theory but by its embedding into a S q invariant theory, with q 1 Percolative transitions are mapped onto spontaneous breakdown of S q symmetry Magnetic and percolative 1st order transitions have the same location in d = 2 he mechanism allowing a 2nd order percolation line above c despite a finite magnetic correlation length has been identified and exhibited analytically he percolative critical lines above c are massless trajectories flowing from c = 1/2 (Ising percolation) to c = (pure percolation): η = + for clusters, η.12 for droplets

Percolative transitions are mapped onto spontaneous breakdown of S q symmetry Magnetic and percolative 1st order transitions have the same location in d = 2 he

2. Illustration of the Nikkei 225 option data

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