Critical exponents for two-dimensional statistical models
Transcript of 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
Pierluigi Falco Critical exponents for two-dimensional statistical
Spin Systems
Pierluigi Falco Critical exponents for two-dimensional statistical
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
j=0,1
σxσx+ej
Probability: given the inverse temperature β ≥ 0
P(σ) =1
Z(Λ, β)e−βH(σ) Z(Λ, β) =
∑σ
e−βH(σ)
Pierluigi Falco Critical exponents for two-dimensional statistical
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
j=0,1
σxσx+ej
Probability: given the inverse temperature β ≥ 0
P(σ) =1
Z(Λ, β)e−βH(σ) Z(Λ, β) =
∑σ
e−βH(σ)
Pierluigi Falco Critical exponents for two-dimensional statistical
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
j=0,1
σxσx+ej
Probability: given the inverse temperature β ≥ 0
P(σ) =1
Z(Λ, β)e−βH(σ) Z(Λ, β) =
∑σ
e−βH(σ)
Pierluigi Falco Critical exponents for two-dimensional statistical
Ising Model
Onsager’s Exact Solution [1944]
• free energy
βf (β) := − limΛ→∞
1
β|Λ|ln Z(Λ, β)
=
∫ π
−π
dk0
2π
∫ π
−π
dk1
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 =
1
2Jlog(√
2 + 1)
Pierluigi Falco Critical exponents for two-dimensional statistical
Ising Model
Onsager’s Exact Solution [1944]
• free energy
βf (β) := − limΛ→∞
1
β|Λ|ln Z(Λ, β)
=
∫ π
−π
dk0
2π
∫ π
−π
dk1
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 =
1
2Jlog(√
2 + 1)
Pierluigi Falco Critical exponents for two-dimensional statistical
Ising Model
Onsager’s Exact Solution [1944]
• correlations of the energy density
G(x− y) = 〈OxOy〉 − 〈Ox〉〈Oy〉 Ox =∑
j=0,1
σxσx+ej
for |x− y| → ∞
G(x− y) ∼C
|x− y|2if β = βc
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results: RG
Spencer, Pinson and Spencer (2000)
Ising model with finite range perturbation:
H(σ) = −J∑x∈Λ
j=0,1
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results: RG
Spencer, Pinson and Spencer (2000)
Ising model with finite range perturbation:
H(σ) = −J∑x∈Λ
j=0,1
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results: RG
Spencer, Pinson and Spencer (2000)
Ising model with finite range perturbation:
H(σ) = −J∑x∈Λ
j=0,1
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!]
Reviews and extensions:Mastropietro ’08; Giuliani, Greenblatt, Mastropietro, ’12
Pierluigi Falco Critical exponents for two-dimensional statistical
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.
Pierluigi Falco Critical exponents for two-dimensional statistical
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.
Pierluigi Falco Critical exponents for two-dimensional statistical
Double Ising Models
The energy of (σ, σ′) is function of J, J′ and J4
H(σ, σ′) = −J∑x∈Λ
j=0,1
σxσx+ej − J′∑x∈Λ
j=0,1
σ′xσ′x+ej− J4V (σ, σ′)
where V quartic in σ and σ′:
V (σ, σ′) =∑x∈Λ
j=0,1
∑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′)
Pierluigi Falco Critical exponents for two-dimensional statistical
Double Ising Models
The energy of (σ, σ′) is function of J, J′ and J4
H(σ, σ′) = −J∑x∈Λ
j=0,1
σxσx+ej − J′∑x∈Λ
j=0,1
σ′xσ′x+ej− J4V (σ, σ′)
where V quartic in σ and σ′:
V (σ, σ′) =∑x∈Λ
j=0,1
∑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′)
Pierluigi Falco Critical exponents for two-dimensional statistical
Double Ising Models
Free energy
f (β) = − limΛ→∞
1
β|Λ|ln Z(Λ, β)
Energy Density - Crossover
Gε(x− y) = 〈Oεx Oε
y 〉 − 〈Oεx 〉〈Oε
y 〉
where
O+x =
∑j=0,1
σxσx+ej +∑
j=0,1
σ′xσ′x+ej
O−x =∑
j=0,1
σxσx+ej −∑
j=0,1
σ′xσ′x+ej
Pierluigi Falco Critical exponents for two-dimensional statistical
Double Ising Models
Free energy
f (β) = − limΛ→∞
1
β|Λ|ln Z(Λ, β)
Energy Density - Crossover
Gε(x− y) = 〈Oεx Oε
y 〉 − 〈Oεx 〉〈Oε
y 〉
where
O+x =
∑j=0,1
σxσx+ej +∑
j=0,1
σ′xσ′x+ej
O−x =∑
j=0,1
σxσx+ej −∑
j=0,1
σ′xσ′x+ej
Pierluigi Falco Critical exponents for two-dimensional statistical
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!
Method of the proof:
– 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)
Pierluigi Falco Critical exponents for two-dimensional statistical
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!
Method of the proof:
– 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)
Pierluigi Falco Critical exponents for two-dimensional statistical
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!
Method of the proof:
– 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)
Pierluigi Falco Critical exponents for two-dimensional statistical
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
Universal formulas
Kadanoff (1977)
κ+ κ− = 1
Extended scaling relation:
• characterizes models with scaling limit given by Thirring Model
Pierluigi Falco Critical exponents for two-dimensional statistical
Universal formulas
Kadanoff (1977)
κ+ κ− = 1
Extended scaling relation:
• characterizes models with scaling limit given by Thirring Model
Pierluigi Falco Critical exponents for two-dimensional statistical
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
for
ψx = (ψ1,x, ψ2,x) ψx =
(ψ1,x
ψ2,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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
for
ψx = (ψ1,x, ψ2,x) ψx =
(ψ1,x
ψ2,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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
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
Pierluigi Falco Critical exponents for two-dimensional statistical
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Lattice Coulomb Gas
Pierluigi Falco Critical exponents for two-dimensional statistical
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 ·
]lim
m→0E[e2z
∑x cosϕx
]Notation:
〈 · 〉 := limΛ→∞
〈 · 〉Λ
Pierluigi Falco Critical exponents for two-dimensional statistical
Pressure / Correlations
We want to study:
– the pressure
p(β, z) = limΛ→∞
1
β|Λ|ln
{lim
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 )〉
Pierluigi Falco Critical exponents for two-dimensional statistical
Pressure / Correlations
We want to study:
– the pressure
p(β, z) = limΛ→∞
1
β|Λ|ln
{lim
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 )〉
Pierluigi Falco Critical exponents for two-dimensional statistical
Expected phase diagram
Berezinskii(1971),Kosterlitz-Thouless (1973),Kosterlitz (1974),Frohlich-Spencer (1981)Giamarchi-Schulz (1988) 8π β0
z
• “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),
〈e iq(ϕx−ϕy )〉 ∼C(β, z)
|x − y |2κ lnκ |x − y |κ =
{2(1− q)2 if q ∈ [ 1
2, 1)
2q2 if q ∈ [0, 12
]
• “plasma phase”: β < βKT (z), non-critical
Pierluigi Falco Critical exponents for two-dimensional statistical
Expected phase diagram
Berezinskii(1971),Kosterlitz-Thouless (1973),Kosterlitz (1974),Frohlich-Spencer (1981)Giamarchi-Schulz (1988) 8π β0
z
• “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),
〈e iq(ϕx−ϕy )〉 ∼C(β, z)
|x − y |2κ lnκ |x − y |κ =
{2(1− q)2 if q ∈ [ 1
2, 1)
2q2 if q ∈ [0, 12
]
• “plasma phase”: β < βKT (z), non-critical
Pierluigi Falco Critical exponents for two-dimensional statistical
Expected phase diagram
Berezinskii(1971),Kosterlitz-Thouless (1973),Kosterlitz (1974),Frohlich-Spencer (1981)Giamarchi-Schulz (1988) 8π β0
z
• “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),
〈e iq(ϕx−ϕy )〉 ∼C(β, z)
|x − y |2κ lnκ |x − y |κ =
{2(1− q)2 if q ∈ [ 1
2, 1)
2q2 if q ∈ [0, 12
]
• “plasma phase”: β < βKT (z), non-critical
Pierluigi Falco Critical exponents for two-dimensional statistical
Expected phase diagram
Berezinskii(1971),Kosterlitz-Thouless (1973),Kosterlitz (1974),Frohlich-Spencer (1981)Giamarchi-Schulz (1988) 8π β0
z
• “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),
〈e iq(ϕx−ϕy )〉 ∼C(β, z)
|x − y |2κ lnκ |x − y |κ =
{2(1− q)2 if q ∈ [ 1
2, 1)
2q2 if q ∈ [0, 12
]
• “plasma phase”: β < βKT (z), non-critical
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results
8π β0
z
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results
8π β0
z
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results
8π β0
z
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results
8π β0
z
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results
8π β0
z
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
Rigorous results
8π β0
z
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)
Pierluigi Falco Critical exponents for two-dimensional statistical
New Result
8π β0
z
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
Pierluigi Falco Critical exponents for two-dimensional statistical
New Result
8π β0
z
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
Pierluigi Falco Critical exponents for two-dimensional statistical
New Result
8π β0
z
F. (2012) (work in progress)
For |z| � 1, decay of the correlations with q ∈ (0, 1) along βKT (z).
〈e iq(ϕx−ϕy )〉 ∼C(β, z)
|x − y |2κ lnκ |x − y |κ =
{2q2 for q ∈ (0, 1
2]
2(1− q)2 for q ∈ ( 12, 1)
Is κ = κ ?
Pierluigi Falco Critical exponents for two-dimensional statistical
Interacting Fermion and Coulomb Gas
Pierluigi Falco Critical exponents for two-dimensional statistical