Exact results for quenched disorder at criticality Gesualdo Delfino - PowerPoint PPT Presentation

Quantissima in the Serenissima III Venice, 19-23 August 2019 Exact results for quenched disorder at criticality Gesualdo Delfino SISSA – Trieste

Based on: G. Delfino, Phys. Rev. Lett. 118 (2017) 250601 G. Delfino and E. Tartaglia, Phys. Rev. E 96 (2017) 042137; J. Stat. Mech. (2017) 123303 G. Delfino and N. Lamsen, JHEP 04 (2018) 077; J. Stat. Mech. (2019) 024001

Introduction quenched disorder: some degrees of freedom take too long to reach thermal equilibrium and can be considered as random variables disorder average is taken on the free energy F ( { J } ) � F = P ( { J } ) F ( { J } ) { J } with a probability distribution P ( { J } ) examples: impurities in magnets, spin glasses, ... numerics/experiments: exists “random” criticality with new critical exponents

theory (short range interactions): • perturbative results for exponents in very few cases for weak disorder only numerics for strong disorder • surprising absence of exact results in 2D (pure systems solved in ’80s) moreover in 2D bond disorder softens 1st order transitions [Aizen- man, Wehr ’89; Hui, Berker ’89] , normally in favor of 2nd order ones, making more room for conformal invariance

theory (short range interactions): • perturbative results for exponents in very few cases for weak disorder only numerics for strong disorder • surprising absence of exact results in 2D (pure systems solved in ’80s) moreover in 2D bond disorder softens 1st order transitions [Aizen- man, Wehr ’89; Hui, Berker ’89] , normally in favor of 2nd order ones, making more room for conformal invariance questions: • does quenched disorder generally fall within the field theoretical framework? • do random critical points possess conformal invariance? • if so, why the corresponding conformal field theories in 2D have never been found?

theory (short range interactions): • perturbative results for exponents in very few cases for weak disorder only numerics for strong disorder • surprising absence of exact results in 2D (pure systems solved in ’80s) moreover in 2D bond disorder softens 1st order transitions [Aizen- man, Wehr ’89; Hui, Berker ’89] , normally in favor of 2nd order ones, making more room for conformal invariance questions: • does quenched disorder generally fall within the field theoretical framework? • do random critical points possess conformal invariance? • if so, why the corresponding conformal field theories in 2D have never been found? historically, the Potts model received a special attention

2D random bond Potts model: disorder � H = − J ij δ s i ,s j s i = 1 , 2 , . . . , q � i,j � 0 2 q 4 • permutational invariance S q ; exists continuation to q real • transition of pure ferromagnet ( J ij = J > 0) 2nd order up to q = 4, 1st order after • bond disorder yields 2nd order transition extending to q = ∞ [Aizenman, Wehr ’89; Hui, Berker ’89] • perturbative random critical point for q → 2 [Ludwig ’90, Dotsenko et al ’95] • numerical hints of q -independent exponents [Chen et al ’92,’95; Domany, Wiseman ’95; Kardar et al ’95] . Superuniversality? • q -dependent exponents (weakly for ν ) [Cardy, Jacobsen ’97 (nu- merical transfer matrix)]

Random criticality from scale invariant scattering [GD ’17] • Euclidean field theory in 2D ← → relativistic quantum field the- ory in (1+1)D − → particles • at criticality ∞ -dimensional conformal symmetry makes scat- tering completely elastic p p 2 1 t p p x 1 2 • center of mass energy only relativistic invariant, dimensionful ⇒ constant amplitudes by scale invariance

O(N) model [GD,Lamsen ’18,’19] � H = − s i = ( s i, 1 , s i, 2 , . . . , s i,N ) J ij s i · s j , � i,j �

O(N) model [GD,Lamsen ’18,’19] � H = − s i = ( s i, 1 , s i, 2 , . . . , s i,N ) J ij s i · s j , � i,j � pure case: J ij = J particles: a = 1 , . . . , N scattering amplitudes: S 1 S 2 S 3

O(N) model [GD,Lamsen ’18,’19] � H = − J ij s i · s j , s i = ( s i, 1 , s i, 2 , . . . , s i,N ) � i,j � pure case: J ij = J particles: a = 1 , . . . , N scattering amplitudes: S 1 S 2 S 3 S 1 = S ∗ 3 ≡ ρ 1 e iφ S 2 = S ∗ crossing symmetry: 2 ≡ ρ 2 unitarity: ρ 2 1 + ρ 2 2 = 1 ρ 1 ρ 2 cos φ = 0 Nρ 2 1 + 2 ρ 1 ρ 2 cos φ + 2 ρ 2 1 cos 2 φ = 0 solutions are O ( N )-invariant RG fixed points

Solution cos φ N ρ 1 ρ 2 solutions: P 1 ± ( −∞ , ∞ ) 0 ± 1 - √ 2 − N ± 1 P 2 ± [ − 2 , 2] 1 0 2 � 1 − ρ 2 P 3 ± 2 [0 , 1] ± 0 1 P1 ± : free bosons/fermions P2 ± : dense/dilute self-avoiding loops P3 + : BKT phase

Solution N ρ 1 ρ 2 cos φ solutions: P 1 ± ( −∞ , ∞ ) 0 ± 1 - √ 2 − N ± 1 P 2 ± [ − 2 , 2] 1 0 2 � 1 − ρ 2 P 3 ± 2 [0 , 1] 0 ± 1 P1 ± : free bosons/fermions P2 ± : dense/dilute self-avoiding loops P3 + : BKT phase conformal data: Solution ∆ η ∆ ε ∆ s N c N 1 1 1 P 1 − ( −∞ , ∞ ) 2 2 2 16 P 1 + ( −∞ , ∞ ) N − 1 0 1 0 6 2 cos π P 2 − 1 − ∆ 2 , 1 ∆ 1 , 3 ∆ 1 2 , 0 p p ( p +1) ∆ m,n = [( p +1) m − pn ] 2 − 1 π 6 P 2 + 2 cos 1 − ∆ 1 , 2 ∆ 3 , 1 ∆ 0 , 1 4 p ( p +1) p +1 p ( p +1) 2 1 1 b 2 P 3 ± 2 1 4 b 2 16 b 2 NS 1 + S 2 + S 3 = e − 2 iπ ∆ η

energy-independent amplitudes are related to statistics right/left moving particles are created by chiral fields with spin s R/L = ± ∆ η . . . . = . . . . S = e − iπ ( s R − s L ) = e − 2 iπ ∆ η bosons/fermions for ∆ η =integer/half-integer; generalized statis- tics otherwhise

disordered case: Z n − 1 replica method: F = − ln Z = − lim n n → 0 n replicas coupled by average over disorder

disordered case: Z n − 1 replica method: F = − ln Z = − lim n n → 0 n replicas coupled by average over disorder particles: a i a = 1 , . . . , N i = 1 , . . . , n amplitudes: b j b i a i a i b i b j a i a i b i b i b j b j a i a i a i b i a i b i a i a i a i b j a i b j S 1 S 2 S 3 S 4 S 5 S 6

disordered case: Z n − 1 F = − ln Z = − lim replica method: n n → 0 n replicas coupled by average over disorder particles: a i a = 1 , . . . , N i = 1 , . . . , n amplitudes: b j b i a i a i b i b j a i a i b i b i b j b j a i a i a i b i a i b i a i a i a i b j a i b j S 1 S 2 S 3 S 4 S 5 S 6 S 1 = S ∗ 3 ≡ ρ 1 e iφ S 2 = S ∗ crossing symmetry: 2 ≡ ρ 2 S 4 = S ∗ 6 ≡ ρ 4 e iθ S 5 = S ∗ 5 ≡ ρ 5 ρ 2 1 + ρ 2 2 = 1 ρ 1 ρ 2 cos φ = 0 unitarity: ρ 2 4 + ρ 2 5 = 1 ρ 4 ρ 5 cos θ = 0 Nρ 2 1 + N ( n − 1) ρ 2 4 + 2 ρ 1 ρ 2 cos φ + 2 ρ 2 1 cos 2 φ = 0 2 Nρ 1 ρ 4 cos( φ − θ )+ N ( n − 2) ρ 2 4 +2 ρ 2 ρ 4 cos θ +2 ρ 1 ρ 4 cos( φ + θ ) = 0

solutions with n = 0 extending to positive N : Solution N ρ 2 cos φ ρ 4 cos θ P 1 ± ( −∞ , ∞ ) ± 1 - 0 - √ 2 − N ± 1 P 2 ± [ − 2 , 2] 0 0 - 2 � 1 − ρ 2 P 3 ± 2 0 0 - ± 1 � √ 1 N − 1 N + 2 V 1 ± [ 2 − 1 , ∞ ) 0 ± 0 N + 1 N + 1 N ± 1 ± N 2 +2 N − 1 S 1 ± ( −∞ , ∞ ) 0 1 √ √ 2 2( N 2 +1) ± 1 ± 1 S 2 ± ( −∞ , ∞ ) 0 1 √ √ 2 2 ρ 4 = 0 yields decoupled replicas (pure case) − → ρ 4 ∼ disorder strength

S2 − S1 − V1 − disorder P2 P1 − + 0 N* 1 2 3 N • V 1 perturbatively accessible for N → 1 − √ • N ∗ = 2 − 1 = 0 . 414 .. (cfr numerical estimate N ∗ ≈ 0 . 5 of [Shimada,Jacobsen,Kamiya ’14]) • S1 − : Nishimori-like multicritical line • S2 − : zero temperature line • disordered polymers ( N = 0) renormalize on S2 −

Potts model [GD ’17; GD,Tartaglia ’17] � H = − J ij s i s j , s i = 1 , 2 , . . . , q � i,j � pure case: J ij = J particles: α | β α, β = 1 , . . . , q , α � = β δ γ δ γ amplitudes: α β α β α α α α γ γ γ γ S 0 S 1 S 2 S 3 S 0 = S ∗ S 1 = S ∗ 2 ≡ ρ e iϕ , S 3 = S ∗ crossing: 0 ≡ ρ 0 , 3 ≡ ρ 3 3 + ( q − 2) ρ 2 = 1 ρ 2 unitarity: 2 ρρ 3 cos ϕ + ( q − 3) ρ 2 = 0 ρ 2 + ( q − 3) ρ 2 0 = 1 2 ρ 0 ρ cos ϕ + ( q − 4) ρ 2 0 = 0

solutions: Solution Range ρ 0 ρ 2 cos ϕ ρ 3 I q = 3 0, 2 cos ϕ 1 [ − 2 , 2] 0 ±√ 3 − q ±√ 3 − q II ± q ∈ [ − 1 , 3] 0 1 √ 4 − q ±√ 4 − q III ± q ∈ [0 , 4] ± 1 ± (3 − q ) √ � � � q − 3 q − 4 q − 3 q ∈ [ 1 � IV ± 2 (7 − 17) , 3] ± ± (3 − q )(4 − q ) ± q 2 − 5 q +5 q 2 − 5 q +5 q 2 − 5 q +5 √ � � � q ∈ [4 , 1 q − 3 q − 4 q − 3 � V ± 2 (7 + 17)] ± ∓ (3 − q )(4 − q ) ± q 2 − 5 q +5 q 2 − 5 q +5 q 2 − 5 q +5 • solution III ending at q = 4: ferromagnet √ • solutions up to q max = 1 2 (7 + 17) = 5 . 56 .. (larger than previously expected value 4) • room for 2nd order transition in q=5 antiferromagnet (numerical candidate in [Deng et al ’11]) • lattice realization of solution I: q = 3 antiferromagnet on self-dual quad- rangulations [Lv et al ’18]

√ q Solution Potts c ∆ ε ∆ η ∆ σ III sin ϕ< 0 6 π 2 cos F critical 1 − ∆ 2 , 1 ∆ 1 , 3 ∆ 1 2 , 0 − ( p +1) p ( p +1) III sin ϕ> 0 2 cos π 6 F tricritical 1 − ∆ 1 , 2 ∆ 3 , 1 ∆ 0 , 1 − p p ( p +1) 2 III sin ϕ< 0 2( N − 1) π N − 1 2 N 2 cos AF square − ( N +2) N +2 N N +2 8( N +2) S 3 + ( q − 2) S 2 = e − 2 iπ ∆ η