FIELD THEORY OF ISING PERCOLATING CLUSTERS

UK Meeting on Integrable Models and Conformal Field Theory University of Kent, Canterbury

16-17 April 2010

FIELD THEORY OF ISING PERCOLATING CLUSTERS

Gesualdo Delfino SISSA-Trieste

Based on :

GD, Nucl.Phys.B 818 (2009) 196 (arXiv:0902.3339 [hep-th])

The Ising model is the basic example of ’magnetic’ phase transition: sponta- neous magnetization below T_c

Percolation is the basic example of ’geometric’ phase transition: infinite clus- ter above p_c

The Ising model, however, encodes itself a percolation problem for clusters of magnetically interacting spins

There 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

The issue can be resolved exactly in d = 2 (integrability, but not the Ising one...)

Ising percolation

−H_Ising = 1 T

X

hiji

σ_iσ_j + H X

i

σ_i , σ_i = ±1 (T ≥ 0)

No magnetic transition away from T ≤ T_c, H = 0

=+1

=−1 0

H

T_c T

P ≡ probability that a site belongs to an infinite cluster of + spins (percola- tive order parameter)

∀ T ≥ 0 ∃ H_c(T ) such that P > 0 for H > H_c(T )

H_c(0) = 0 e^Hc(∞)

2 cosh H_c(∞) = p⁰_c p⁰_c = critical point of random percolation (non-universal)

Three observations:

i) H_c(T ) monotonic

ii) spontaneous magnetization =⇒ infinite cluster (Coniglio et al ’77) iii) pc ≤ p⁰_c at H = 0 (interaction makes percolation easier)

Consequences:

a) p⁰_c > 1/2, i.e. H_c(∞) > 0

i), ii) =⇒ H_c(T ) = 0 for T ≤ T_c

b) p⁰_c < 1/2, i.e. H_c(∞) < 0 p = 1/2 for T ≥ T_c, H = 0 iii) =⇒ T_p < Tc

c

0 T T

H_c H

0 H

(T)

T H_c(T)

T_p

T_c P > 0

Ordinary lattices in d = 2 have p⁰_c ≥ 1/2 −→ a) d = 3 have p⁰_c < 1/2 −→ b)

Kasteleyn-Fortuin representation

−H_q = −H_Ising + J X

hiji

t_it_j (δ_si,sj − 1) , t_i = 1

2(σ_i + 1) = 0, 1, s_i = 1, . . . , q Z_q = X

{ti}

X

{si}

e^−Hq = (Murata ’79)

= X

{ti}

e^−HIsing q^Nv X

G

p^b_B(1 − p_B)^¯bq^Nc

N_v = # of empty sites

G = graph made of bonds between occupied sites b = # of bonds in G; ¯b = # of absent bonds

Nc = # 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

hXi_q→1 = Z_Ising⁻¹ X

{ti}

e^−HIsingX

G

X p^b_B(1 − p_B)^¯b dilute percolation average pure percolation for H = +∞

hXi_q→1 = hXi_Ising for X Ising magnetic observable

Example of percolative observable:

Mean cluster number in pure percolation (N_v = 0) = 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 + 

– Ising cluster properties at q = 1 + , J → +∞

RG analysis in d = 2 (Coniglio-Klein ’80 + CFT)

Look for fixed points of H_q→1(T, H, J ) = H_Ising(T, H)−J X

hiji

t_it_j(δ_si,sj−1) Need a magnetic fixed point to start with =⇒ T = T_c, H = 0. 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(T_c, 0): 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:

√q = 2 sin π(t − 1)

2(t + 1), m =







t, critical (H = +∞) : X_s = X^m−1

2 ,^m+1₂ , X_t1 = X_2,1 t + 1, tricritical: X_s = X^m

2,^m₂, X_t1 = X_1,2, X_t2 = X_1,3 q → 1 : m =







2, pure perc: c = 0, X_s = 5/48, X_t1 = 5/4

3, dilute perc: c = 1/2, X_s = 5/96, X_t1 = 1/8, X_t2 = 1

We found a fixed point of dilute percolation for J = J^∗ A trivial (purely magnetic) fixed point is at J = 0

J irrelevant at these two fixed points =⇒ ∃ a third one

0 2/T_c J^* J

−H_q|_{J =2/T} = 2 T

X

hiji

(δ_νi,νj − 1) + (ln q − 2H)X

i

δ_νi,0, ν_i = 0, 1, . . . , q

=⇒ fixed point at J = 2/T_c as q → 1, with X_s = X_σ = 1/8 (KF clusters with J = 2/T ) = 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

D =











91/48 = 1.89.. pure percolation 187/96 = 1.94.. Ising clusters 15/8 = 1.87.. Ising droplets

Field theory of Ising clusters

Ising field theory:

A_Ising = A^Ising_{CF T} − τ Z

d²x ε(x) − h Z

d²x σ(x), τ ∼ T − T_c, h ∼ H

Dilute Potts field theory:

A_q = A^tricr_{CF T} − g Z

d²x ϕ_1,3(x) − λ Z

d²x ϕ_1,2(x)

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 = 0 q degenerate vacua for λ < 0

λ

0 g

1

2 3

4 q

The q → 1 limit of the Potts critical surface is the Ising percolation transition:

1st order (massive) at T < Tc, 2nd order (massless) at T > Tc

A_q=1 = A_Ising, however, is purely massive: no transition above T_c We can take the limit analytically

The 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 p¹ = ±p⁰ Poles of two-particle amplitudes contained in S_−1/2(θ)/(S_1/2(θ) cosh ρ(iπ −θ))

Sγ(θ) =

∞

Y

n=0

Γ

1

2 + 2n + ³₂ − γ

ρ − ^ρθ_iπ

 Γ

1

2 + 2n + ¹₂ − γ

ρ + ^ρθ_iπ



Γ

1

2 + 2n + ³₂ − γ

ρ + ^ρθ

iπ

 Γ

1

2 + 2n + ¹₂ − γ

ρ − ^ρθ

iπ

^, ρ = 1/(m−1)

s = µ²e^θ :







Im θ ∈ (0, π) physical sheet Im θ ∈ (0, −π) second sheet No poles on physical sheet

Pole at θ0 = −iπ(m − 3)/2, on 2nd sheet for m ∈ (3, 5) q → 1⁺: resonance B with Im s0 ∝ (q − 1)µ²

s/_µ² 0 1

m=3 (q=1) m=4 (q=2)

• q = 1 +  :  massless particles A_k, one resonance B with lifetime ∝ 1/

• q = 1 : 0 massless particles, one stable particle B with mass µ

=⇒ percolative transition in absence of magnetic singularities above T_c

B S_q-singlet (survives at q = 1) =⇒ only S_q-invariant fields φ have non-zero correlations at q = 1:

lim

q→1hφ(r)φ(0)i = lim

q→1(q − 1) Z

dθ1dθ2|h0|φ(0)|A_k(θ1)A_q−k(θ2)i|² e^{−rE2,0(θ1,θ2)} + . . .

= lim

q→1(q − 1) Z

dβdθ |R_φ|²

(θ − θ0)(θ + θ0) e^−rE2,0((β+θ)/2,(β−θ)/2)

+ . . .

∝ Z

dβ e^{−rE1,µ(β)} + . . .

E_2,0(θ1, θ2) ≡ µ(e^θ1 + e^−θ2)/2, θ0 ∝ −i(q − 1), E_1,µ(β) ≡ µ cosh(β/2)

φ φ φ φ

q−k k B

Σ

k=1 A q 1

q−1

This is how the canonical space of fields [I] ⊕ [σ] ⊕ [] of Ising field theory is recovered at q = 1

Field theory of Ising droplets

Droplets are KF clusters with J = 2/T , i.e. p_B = 1− e^−2/T T = ∞ : p_B = 0, no percolation

H = +∞ : transition at T = −2/ ln(1 − p_c)

p_c = threshold of pure bond percolation

00 11

Tc

0 H

T

Kertesz line

Scaling limit : A˜_q = A^(q+1)_{CF T} −τ_q

Z

d²x ϕ_2,1(x)+2hq

Z

d²x δ_ν(x),0, ν(x) = 0, 1, . . . , q

RG invariant : η_q = τ_q/h^(2−X_q ^{2,1)/(2−Xs)} ( ˜A_q=1 = A_Ising, η1 ≡ 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_q^c

The Kert´esz line is the limit q → 1 of the flow from S_q+1 fixed point at h_q = 0 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 Kert´esz line: lattice data (Fortunato, Satz ’01) + lattice-continuum relations (Caselle, Grinza, Rago ’04) give η_K ≡ η_q→1^c ' 0.12

Summary

00 11

0 T H

T_c 0 τ

h

η ∼0.12K

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 sym- metry

Magnetic and percolative 1st order transitions have the same location in d = 2 The mechanism allowing a 2nd order percolation line above T_c despite a finite magnetic correlation length has been identified and exhibited analytically The percolative critical lines above T_c are massless trajectories flowing from c = 1/2 (Ising percolation) to c = 0 (pure percolation): η = +∞ for clusters, η ' 0.12 for droplets