critical exponents for two-dimensional statistical models

Critical exponents for two-dimensional statistical

Pierluigi Falco

California State UniversityDepartment of Mathematics

Pierluigi Falco Critical exponents for two-dimensional statistical

Outline

Two different type of Statistical Mechanics models in 2D:

• Spin Systems

• Coulomb Gas

Spin Systems

Ising Model

Configuration: Place +1 or −1 at each site of Λ

σ = {σx = ±1 | x ∈ Λ}

Energy: given J (positive for definiteness)

H(σ) = −J∑x∈Λ

σxσx+ej

Probability: given the inverse temperature β ≥ 0

P(σ) =1

Z(Λ, β)e−βH(σ) Z(Λ, β) =

e−βH(σ)

Ising Model

σ = {σx = ±1 | x ∈ Λ}

H(σ) = −J∑x∈Λ

σxσx+ej

P(σ) =1

e−βH(σ)

Ising Model

σ = {σx = ±1 | x ∈ Λ}

H(σ) = −J∑x∈Λ

σxσx+ej

P(σ) =1

e−βH(σ)

Ising Model

Onsager’s Exact Solution [1944]

• free energy

βf (β) := − limΛ→∞

β|Λ|ln Z(Λ, β)

∫ π

2πlog

[(1− sinh 2βJ

)2+ α(k) sinh(2βJ)

]for α(k) = 2− cos(k0)− cos(k1)

• specific heat

C(β) :=d2

dβ2[βf (β)] ∼ C log |β − βc | βc =

2Jlog(√

2 + 1)

Ising Model

• free energy

βf (β) := − limΛ→∞

β|Λ|ln Z(Λ, β)

∫ π

2πlog

[(1− sinh 2βJ

)2+ α(k) sinh(2βJ)

]for α(k) = 2− cos(k0)− cos(k1)

• specific heat

C(β) :=d2

dβ2[βf (β)] ∼ C log |β − βc | βc =

2Jlog(√

2 + 1)

Ising Model

• correlations of the energy density

G(x− y) = 〈OxOy〉 − 〈Ox〉〈Oy〉 Ox =∑

σxσx+ej

for |x− y| → ∞

G(x− y) ∼C

|x− y|2if β = βc

Onsager-Kaufman’s idea

Kasteleyn ’61; Schultz, Mattis Lieb ’64

For ψi,α Grassmann variables,

Z(Λ, β) = det M =

∫DψDψ exp

{ ∑α,β=1,2

∑i,j∈Λ

ψα,i Mαβij ψβ,j

Ising model= system of free fermions

Onsager-Kaufman’s idea

Kasteleyn ’61; Schultz, Mattis Lieb ’64

For ψi,α Grassmann variables,

Z(Λ, β) = det M =

∫DψDψ exp

{ ∑α,β=1,2

∑i,j∈Λ

ψα,i Mαβij ψβ,j

}Ising model= system of free fermions

Rigorous results: RG

Spencer, Pinson and Spencer (2000)

Ising model with finite range perturbation:

H(σ) = −J∑x∈Λ

σxσx+ej − J4V (σ)

For ε = J4/J small enough

• there exists βc (λ) = 12J

log(√

2 + 1) + O(λ) at which

G(x− y) ∼C

|x− y|2

[i.e. the power 2 is unchanged!]

Method of the proof:

– Functional integral representation of the Ising model

Z(Λ, β) =

∫DψDψ exp

{∑ψMψ + λ

∑(ψ∂ψ)2 + . . .

}λ ∼ J4/J

– Renormalization group approach for computing the critical exponent.based on RG approach for fermion system Feldman, Knorrer, Trubowitz, (1998)

H(σ) = −J∑x∈Λ

log(√

G(x− y) ∼C

|x− y|2

– Functional integral representation of the Ising model

Z(Λ, β) =

∫DψDψ exp

{∑ψMψ + λ

∑(ψ∂ψ)2 + . . .

}λ ∼ J4/J

– Renormalization group approach for computing the critical exponent.based on RG approach for fermion system Feldman, Knorrer, Trubowitz, (1998)

H(σ) = −J∑x∈Λ

log(√

G(x− y) ∼C

|x− y|2

Reviews and extensions:Mastropietro ’08; Giuliani, Greenblatt, Mastropietro, ’12

Double Ising Model

Wu (1971),Kadanoff and Wegner (1971)Fan (1972)

A configuration (σ, σ′) is the product oftwo configurations of spinsσ = {σx = ±1}x∈Λ andσ′ = {σ′x = ±1}x∈Λ.

Double Ising Model

Wu (1971),Kadanoff and Wegner (1971)Fan (1972)

A configuration (σ, σ′) is the product oftwo configurations of spinsσ = {σx = ±1}x∈Λ andσ′ = {σ′x = ±1}x∈Λ.

Double Ising Models

The energy of (σ, σ′) is function of J, J′ and J4

H(σ, σ′) = −J∑x∈Λ

σxσx+ej − J′∑x∈Λ

σ′xσ′x+ej− J4V (σ, σ′)

where V quartic in σ and σ′:

V (σ, σ′) =∑x∈Λ

∑x′∈Λ

j′=0,1

vj−j′ (x− x′)σxσx+ej σ′x′σ′x′+ej′

for vj (x) a lattice function such that |vj (x)| ≤ ce−κ|x|.

Probability of a configuration (σ, σ′)

P(σ, σ′) =1

Ze−βH(σ,σ′) Z =

∑σ,σ′

e−βH(σ,σ′)

Double Ising Models

The energy of (σ, σ′) is function of J, J′ and J4

H(σ, σ′) = −J∑x∈Λ

σxσx+ej − J′∑x∈Λ

σ′xσ′x+ej− J4V (σ, σ′)

where V quartic in σ and σ′:

V (σ, σ′) =∑x∈Λ

∑x′∈Λ

j′=0,1

vj−j′ (x− x′)σxσx+ej σ′x′σ′x′+ej′

for vj (x) a lattice function such that |vj (x)| ≤ ce−κ|x|.

Probability of a configuration (σ, σ′)

P(σ, σ′) =1

Ze−βH(σ,σ′) Z =

∑σ,σ′

e−βH(σ,σ′)

Double Ising Models

Free energy

f (β) = − limΛ→∞

β|Λ|ln Z(Λ, β)

Energy Density - Crossover

Gε(x− y) = 〈Oεx Oε

y 〉 − 〈Oεx 〉〈Oε

∑j=0,1

σxσx+ej +∑

σ′xσ′x+ej

O−x =∑

σxσx+ej −∑

σ′xσ′x+ej

Double Ising Models

Free energy

f (β) = − limΛ→∞

β|Λ|ln Z(Λ, β)

Energy Density - Crossover

Gε(x− y) = 〈Oεx Oε

y 〉 − 〈Oεx 〉〈Oε

∑j=0,1

σxσx+ej +∑

σ′xσ′x+ej

O−x =∑

σxσx+ej −∑

σ′xσ′x+ej

Double Ising Models

Mastropietro (2004)

Case J′ = J and | J4J| � 1,

• one critical temperature βc ; correlation length

ξ ∼ |β − βc |ν

with convergent power series for βc ≡ βc (J4/J) and ν ≡ ν(J4/J)

• algebraic decay of correlations

Gε(x− y) ∼C

|x− y|2κε,

with convergent power series for κ+ ≡ κ+(J4/J) and κ− ≡ κ−(J4/J).

Non-Universality!

– Functional integral representation of the Double Ising model

– Renormalization group approach for fermion systemsGallavotti (1985); Gallavotti Nicolo (1985).crucial point: vanishing of the Beta functionBenfatto, Gallavotti, Procacci, Scoppola (1994); Benfatto, Mastropietro (2005)

Double Ising Models

Mastropietro (2004)

ξ ∼ |β − βc |ν

Gε(x− y) ∼C

|x− y|2κε,

Non-Universality!

Double Ising Models

Mastropietro (2004)

ξ ∼ |β − βc |ν

Gε(x− y) ∼C

|x− y|2κε,

Non-Universality!

Double Ising Models

Giuliani and Mastropietro (2005)

Case J 6= J′ but close; | J4J| � 1, | J4

J′ | � 1

• two critical temperatures βc (J4/J, J4/J′) and β′c (J4/J, J4/J′) and algebraicdecay of correlations

Gε(x− y) ∼C

|x− y|2,

Universality!

• convergent power series for κT s.t.

|βc − β′c | ∼ |J − J′|κT

with convergent power series for κT ≡ κT (J4/J)

Universal formulas

We have five critical exponents

κ+ κ− κT α ν

all of them model-dependent (i.e. dependent upon J4, J and also vj (x))

Kadanoff and Wegner (1971)Luther and Peschel (1975)

dν = 2− α ν =1

2− κ+

Widom scaling relations:

• valid at criticality for any model in any dimension < 4;

• they don’t characterize classes of models

Universal formulas

κ+ κ− κT α ν

dν = 2− α ν =1

2− κ+

Universal formulas

κ+ κ− κT α ν

dν = 2− α ν =1

2− κ+

Universal formulas

Kadanoff (1977)

κ+ κ− = 1

Extended scaling relation:

• characterizes models with scaling limit given by Thirring Model

Universal formulas

Kadanoff (1977)

κ+ κ− = 1

Extended scaling relation:

• characterizes models with scaling limit given by Thirring Model

Thirring model

Thirring model (Thirring 1955) is a toy model of interacting, 2-dimensional, fermion,quantum field theory. The Action is∫

dx ψx 6∂ψx + λ

∫dx (ψxψx)2

ψx = (ψ1,x, ψ2,x) ψx =

(ψ1,x

)6∂ = 2× 2matrix

From the formal explicit solution of the Thirring model (Johnson 1961; Klaiber 1967;Hagen 1967)

κTh+ =

1− λ4π

1 + λ4π

κTh− =

1 + λ4π

1− λ4π

Thirring model

Thirring model (Thirring 1955) is a toy model of interacting, 2-dimensional, fermion,quantum field theory. The Action is∫

dx ψx 6∂ψx + λ

∫dx (ψxψx)2

ψx = (ψ1,x, ψ2,x) ψx =

(ψ1,x

)6∂ = 2× 2matrix

From the formal explicit solution of the Thirring model (Johnson 1961; Klaiber 1967;Hagen 1967)

κTh+ =

1− λ4π

1 + λ4π

κTh− =

1 + λ4π

1− λ4π

Thirring Model

Benfatto, Falco, Mastropietro (2007), (2009)

Thirring model for |λ| small enough:

• Existence of the theory (in the sense of the Osterwalder-Schrader)

• Proof of Hagen and Klaiber’s formula for correlations.

• Bosonization.

There was already an axiomatic proof of the existence of the interacting theory: not good for

scaling limit

Benfatto, Falco, Mastropietro (2009)

Double Ising model: for J4/J small enough

• proof of the universal formulas

2ν = 2− α ν =1

2− κ+κ+ κ− = 1

• a new scaling relation for the index κT

κT =2− κ+

2− κ−

Similar results for the XYZ quantum chain

Thirring Model

• Bosonization.

scaling limit

2ν = 2− α ν =1

2− κ+κ+ κ− = 1

κT =2− κ+

2− κ−

Thirring Model

• Bosonization.

scaling limit

2ν = 2− α ν =1

2− κ+κ+ κ− = 1

κT =2− κ+

2− κ−

Recapitulation

model lattice scaling limit

Ising (O 1944) free fermions free fermions

Ising + n.n.n. (PS 2000) interacting fermions free fermions

8V, AT, XYZ (BFM 2009) interacting fermions Thirring

in preparation:

(1 + 1)D Hubbard (BFM 2012) interacting fermions SU(2) Thirring

Open problems:

• Interacting dimers / 6V Model

• Four Coupled Ising / Two Coupled 8V

• q−States Potts / Completely Packed Loop /...equivalence with staggered 6-vertex, Temperley, Lieb (1971); Baxter, Kelland, Wu (1976)

Recapitulation

in preparation:

Open problems:

Recapitulation

in preparation:

Open problems:

Lattice Coulomb Gas

Coulomb Gas (sine-Gordon formulation)

Consider a Gaussian field {ϕx}x∈Λ with zero averageand covariance

E[ϕxϕy ] = β(−∆ + m2)−1(x − y)

where β > 0 is the inverse temperature [and m is a

temporary mass regularization because of the periodic b.c.].

This is the Gaussian Free Field.

The Coulomb Gas model is the perturbation of the Gaussian Free Field

〈 · 〉Λ :=lim

m→0E[e2z

∑x cosϕx ·

m→0E[e2z

∑x cosϕx

]Notation:

〈 · 〉 := limΛ→∞

〈 · 〉Λ

Pressure / Correlations

We want to study:

– the pressure

p(β, z) = limΛ→∞

β|Λ|ln

m→0E[e2z

∑x cosϕx

– the fractional charges correlations, i.e. correlations of the random variable e iqϕx ,the charge-q random variable, for q ∈ (0, 1):

〈e iq(ϕx−ϕy )〉

Pressure / Correlations

We want to study:

– the pressure

p(β, z) = limΛ→∞

β|Λ|ln

m→0E[e2z

∑x cosϕx

– the fractional charges correlations, i.e. correlations of the random variable e iqϕx ,the charge-q random variable, for q ∈ (0, 1):

〈e iq(ϕx−ϕy )〉

Expected phase diagram

Berezinskii(1971),Kosterlitz-Thouless (1973),Kosterlitz (1974),Frohlich-Spencer (1981)Giamarchi-Schulz (1988) 8π β0

• “dipole phase”: if β > βKT (z)

〈e iq(ϕx−ϕy )〉 ∼C(β, z)

|x − y |2κκ =

{βeff4π

(1− q)2 if q ∈ [ 12, 1)

βeff4π

q2 if q ∈ [0, 12

• “KT line”: β = βKT (z),

|x − y |2κ lnκ |x − y |κ =

{2(1− q)2 if q ∈ [ 1

2q2 if q ∈ [0, 12

• “plasma phase”: β < βKT (z), non-critical

|x − y |2κκ =

{βeff4π

(1− q)2 if q ∈ [ 12, 1)

βeff4π

q2 if q ∈ [0, 12

|x − y |2κ lnκ |x − y |κ =

{2(1− q)2 if q ∈ [ 1

2q2 if q ∈ [0, 12

|x − y |2κκ =

{βeff4π

(1− q)2 if q ∈ [ 12, 1)

βeff4π

q2 if q ∈ [0, 12

|x − y |2κ lnκ |x − y |κ =

{2(1− q)2 if q ∈ [ 1

2q2 if q ∈ [0, 12

|x − y |2κκ =

{βeff4π

(1− q)2 if q ∈ [ 12, 1)

βeff4π

q2 if q ∈ [0, 12

|x − y |2κ lnκ |x − y |κ =

{2(1− q)2 if q ∈ [ 1

2q2 if q ∈ [0, 12

Rigorous results

8π β0

1 Frohlich, Park (1978) existence of the thermodynamic limit for pressure andcorrelations (any β, any z, any q, free bc; Ginibre method)

2 Frohlich, Spencer (1981): power law decay for β ≥ β1(z)� βKT (z)(also other models; away from critical line, |q| < 1, no exact exponents)

3 Marchetti, Klein, Perez, Braga (1986-91) extended FS: β ≥ β2(z)� βKT (z) withβ2(0) = 8π (away from critical line, |q| < 1, no exact exponents)

Rigorous results

8π β0

Rigorous results

8π β0

Rigorous results

8π β0

4 Benfatto, Gallavotti, Nicolo (1986), Marchetti, Perez (1989), Dimock (1990),Kappeler, Pinn, Wieczerkowski (1991), Benfatto, Renn (1993), Guidi, Marchetti(2001): hierarchical case

5 Gallavotti, Nicolo (1985): multi-scale perturbation theory of the free energy(Gallavotti-Nicolo trees, non-convergent perturbation theory)

7 Nicolo, Perfetti (1989): renormalizability of dipole phase and KT line(Gallavotti-Nicolo trees, non-convergent perturbation theory)

8 Dimock and Hurd (1994): formula for the pressure for β ≥ β3(z)� βKT (z) withβ3(0) = 8π (Brydges-Yau method, no KT line, no correlations)

Rigorous results

8π β0

Rigorous results

8π β0

New Result

8π β0

F. (2011)

For |z| � 1 and β = βKT (z), convergent series for the pressure:

p(z, β) =∑j≥0

ej (z, β)

RG method of Brydges, Yau (1990) and Brydges (2007)’s Park City Lectures

New Result

8π β0

F. (2011)

For |z| � 1 and β = βKT (z), convergent series for the pressure:

p(z, β) =∑j≥0

ej (z, β)

RG method of Brydges, Yau (1990) and Brydges (2007)’s Park City Lectures

New Result

8π β0

F. (2012) (work in progress)

For |z| � 1, decay of the correlations with q ∈ (0, 1) along βKT (z).

|x − y |2κ lnκ |x − y |κ =

{2q2 for q ∈ (0, 1

2(1− q)2 for q ∈ ( 12, 1)

Is κ = κ ?

Interacting Fermion and Coulomb Gas

critical exponents for two-dimensional statistical models

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