Critical exponents for two-dimensional statistical

Transcription

1 California State University Department of Mathematics

2 Outline Two different type of Statistical Mechanics models in 2D: Spin Systems Coulomb Gas

3 Spin Systems

4 Ising Model Configuration: Place +1 or 1 at each site of Λ σ = {σ x = ±1 x Λ}

5 Ising Model Configuration: Place +1 or 1 at each site of Λ σ = {σ x = ±1 x Λ} Energy: given J (positive for definiteness) H(σ) = J x Λ j=0,1 σ xσ x+ej

6 Ising Model Configuration: Place +1 or 1 at each site of Λ σ = {σ x = ±1 x Λ} Energy: given J (positive for definiteness) H(σ) = J x Λ j=0,1 σ xσ x+ej Probability: given the inverse temperature β 0 P(σ) = 1 Z(Λ, β) e βh(σ) Z(Λ, β) = σ e βh(σ)

7 Ising Model Onsager s Exact Solution [1944] free energy 1 βf (β) := lim ln Z(Λ, β) Λ β Λ π dk π 0 dk 1 = π 2π π 2π for α(k) = 2 cos(k 0 ) cos(k 1 ) [ ( ] 2 log 1 sinh 2βJ) + α(k) sinh(2βj)

8 Ising Model Onsager s Exact Solution [1944] free energy 1 βf (β) := lim ln Z(Λ, β) Λ β Λ π dk π 0 dk 1 = π 2π π 2π for α(k) = 2 cos(k 0 ) cos(k 1 ) [ ( ] 2 log 1 sinh 2βJ) + α(k) sinh(2βj) specific heat C(β) := d2 1 [βf (β)] C log β βc βc = dβ2 2J log( 2 + 1)

9 Ising Model Onsager s Exact Solution [1944] correlations of the energy density G(x y) = O xo y O x O y O x = σ xσ x+ej j=0,1 for x y G(x y) C x y 2 if β = β c

10 Onsager-Kaufman s idea Kasteleyn 61; Schultz, Mattis Lieb 64 For ψ i,α Grassmann variables, { Z(Λ, β) = det M = DψD ψ exp α,β=1,2 i,j Λ ψ α,i M αβ ij ψ β,j }

11 Onsager-Kaufman s idea Kasteleyn 61; Schultz, Mattis Lieb 64 For ψ i,α Grassmann variables, { Z(Λ, β) = det M = DψD ψ exp α,β=1,2 i,j Λ Ising model= system of free fermions ψ α,i M αβ ij ψ β,j }

12 Rigorous results: RG Spencer, Pinson and Spencer (2000) Ising model with finite range perturbation: H(σ) = J x Λ j=0,1 σ xσ x+ej J 4 V (σ) For ε = J 4 /J small enough there exists β c(λ) = 1 2J log( 2 + 1) + O(λ) at which [i.e. the power 2 is unchanged!] G(x y) C x y 2

13 Rigorous results: RG Spencer, Pinson and Spencer (2000) Ising model with finite range perturbation: H(σ) = J x Λ j=0,1 σ xσ x+ej J 4 V (σ) For ε = J 4 /J small enough there exists β c(λ) = 1 2J log( 2 + 1) + O(λ) at which [i.e. the power 2 is unchanged!] G(x y) C x y 2 Method of the proof: Functional integral representation of the Ising model { Z(Λ, β) = DψD ψ exp ψmψ + λ } ( ψ ψ) λ J 4 /J Renormalization group approach for computing the critical exponent. based on RG approach for fermion system Feldman, Knörrer, Trubowitz, (1998)

14 Rigorous results: RG Spencer, Pinson and Spencer (2000) Ising model with finite range perturbation: H(σ) = J x Λ j=0,1 σ xσ x+ej J 4 V (σ) For ε = J 4 /J small enough there exists β c(λ) = 1 2J log( 2 + 1) + O(λ) at which [i.e. the power 2 is unchanged!] G(x y) C x y 2 Reviews and extensions: Mastropietro 08; Giuliani, Greenblatt, Mastropietro, 12

15 Double Ising Model Wu (1971), Kadanoff and Wegner (1971) Fan (1972) A configuration (σ, σ ) is the product of two configurations of spins σ = {σ x = ±1} x Λ and σ = {σ x = ±1} x Λ.

16 Double Ising Model Wu (1971), Kadanoff and Wegner (1971) Fan (1972) A configuration (σ, σ ) is the product of two configurations of spins σ = {σ x = ±1} x Λ and σ = {σ x = ±1} x Λ.

17 Double Ising Models The energy of (σ, σ ) is function of J, J and J 4 H(σ, σ ) = J x Λ j=0,1 where V quartic in σ and σ : V (σ, σ ) = x Λ j=0,1 σ xσ x+ej J x Λ j=0,1 x Λ j =0,1 σ xσ x+e j J 4 V (σ, σ ) v j j (x x )σ xσ x+ej σ x σ x +e j for v j (x) a lattice function such that v j (x) ce κ x.

18 Double Ising Models The energy of (σ, σ ) is function of J, J and J 4 H(σ, σ ) = J x Λ j=0,1 where V quartic in σ and σ : V (σ, σ ) = x Λ j=0,1 σ xσ x+ej J x Λ j=0,1 x Λ j =0,1 σ xσ x+e j J 4 V (σ, σ ) v j j (x x )σ xσ x+ej σ x σ x +e j for v j (x) a lattice function such that v j (x) ce κ x. Probability of a configuration (σ, σ ) P(σ, σ ) = 1 Z e βh(σ,σ ) Z = σ,σ e βh(σ,σ )

19 Double Ising Models Free energy 1 f (β) = lim ln Z(Λ, β) Λ β Λ

20 Double Ising Models Free energy 1 f (β) = lim ln Z(Λ, β) Λ β Λ Energy Density - Crossover G ε(x y) = O ε x Oε y Oε x Oε y where O + x = σ xσ x+ej + σ xσ x+e j Ox = σ xσ x+ej σ xσ x+e j j=0,1 j=0,1 j=0,1 j=0,1

21 Double Ising Models Mastropietro (2004) Case J = J and J 4 J 1, one critical temperature β c; correlation length ξ β β c ν with convergent power series for β c β c(j 4 /J) and ν ν(j 4 /J)

22 Double Ising Models Mastropietro (2004) Case J = J and J 4 J 1, one critical temperature β c; correlation length ξ β β c ν with convergent power series for β c β c(j 4 /J) and ν ν(j 4 /J) algebraic decay of correlations G ε(x y) C x y 2κε, with convergent power series for κ + κ +(J 4 /J) and κ κ (J 4 /J). Non-Universality!

23 Double Ising Models Mastropietro (2004) Case J = J and J 4 J 1, one critical temperature β c; correlation length ξ β β c ν with convergent power series for β c β c(j 4 /J) and ν ν(j 4 /J) algebraic decay of correlations G ε(x y) C x y 2κε, with convergent power series for κ + κ +(J 4 /J) and κ κ (J 4 /J). Non-Universality! Method of the proof: Functional integral representation of the Double Ising model Renormalization group approach for fermion systems Gallavotti (1985); Gallavotti Nicolò (1985). crucial point: vanishing of the Beta function Benfatto, Gallavotti, Procacci, Scoppola (1994); Benfatto, Mastropietro (2005)

24 Double Ising Models Giuliani and Mastropietro (2005) Case J J but close; J 4 J 1, J 4 J 1 two critical temperatures β c(j 4 /J, J 4 /J ) and β c(j 4 /J, J 4 /J ) and algebraic decay of correlations C G ε(x y) x y 2, convergent power series for κ T s.t. Universality! β c β c J J κ T with convergent power series for κ T κ T (J 4 /J)

25 Universal formulas We have five critical exponents κ + κ κ T α ν all of them model-dependent (i.e. dependent upon J 4, J and also v j (x))

26 Universal formulas We have five critical exponents κ + κ κ T α ν all of them model-dependent (i.e. dependent upon J 4, J and also v j (x)) Kadanoff and Wegner (1971) Luther and Peschel (1975) 1 dν = 2 α ν = 2 κ +

27 Universal formulas We have five critical exponents κ + κ κ T α ν all of them model-dependent (i.e. dependent upon J 4, J and also v j (x)) Kadanoff and Wegner (1971) Luther and Peschel (1975) 1 dν = 2 α ν = 2 κ + Widom scaling relations: valid at criticality for any model in any dimension < 4; they don t characterize classes of models

28 Universal formulas Kadanoff (1977) κ + κ = 1

29 Universal formulas Kadanoff (1977) κ + κ = 1 Extended scaling relation: characterizes models with scaling limit given by Thirring Model

30 Thirring model Thirring model (Thirring 1955) is a toy model of interacting, 2-dimensional, fermion, quantum field theory. The Action is dx ψ x ψ x + λ dx ( ψ xψ x) 2 for ( ) ψ1,x ψ x = (ψ 1,x, ψ 2,x ) ψx = ψ 2,x = 2 2matrix

31 Thirring model Thirring model (Thirring 1955) is a toy model of interacting, 2-dimensional, fermion, quantum field theory. The Action is dx ψ x ψ x + λ dx ( ψ xψ x) 2 for ( ) ψ1,x ψ x = (ψ 1,x, ψ 2,x ) ψx = ψ 2,x = 2 2matrix From the formal explicit solution of the Thirring model (Johnson 1961; Klaiber 1967; Hagen 1967) κ Th + = 1 λ 4π 1 + λ κ Th = 1 + λ 4π 1 λ 4π 4π

32 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.

33 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

34 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 J 4 /J small enough proof of the universal formulas 2ν = 2 α ν = a new scaling relation for the index κ T 1 2 κ + κ + κ = 1 κ T = 2 κ+ 2 κ Similar results for the XYZ quantum chain

35 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

36 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

37 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)

38 Lattice Coulomb Gas

39 Coulomb Gas (sine-gordon formulation) Consider a Gaussian field {ϕ x } x Λ with zero average and covariance E[ϕ x ϕ y ] = β( + m 2 ) 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 [ lim E e 2z ] x cos ϕx m 0 Λ := [ lim E e 2z ] x cos ϕx m 0 Notation: := lim Λ Λ

40 Pressure / Correlations We want to study: the pressure { 1 [ p(β, z) = lim Λ β Λ ln lim E e 2z ] } x cos ϕx m 0

41 Pressure / Correlations We want to study: the pressure { 1 [ p(β, z) = lim Λ β Λ ln lim E e 2z ] } x cos ϕx m 0 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 )

42 Expected phase diagram Berezinskii(1971), Kosterlitz-Thouless (1973), z Kosterlitz (1974), Fröhlich-Spencer (1981) Giamarchi-Schulz (1988) 0 8π β

43 Expected phase diagram Berezinskii(1971), Kosterlitz-Thouless (1973), z Kosterlitz (1974), Fröhlich-Spencer (1981) Giamarchi-Schulz (1988) 0 8π β dipole phase : if β > β KT (z) e iq(ϕx ϕy ) C(β, z) x y 2κ κ = { βeff 4π (1 q)2 if q [ 1 2, 1) β eff 4π q2 if q [0, 1 2 ]

44 Expected phase diagram Berezinskii(1971), Kosterlitz-Thouless (1973), z Kosterlitz (1974), Fröhlich-Spencer (1981) Giamarchi-Schulz (1988) 0 8π β dipole phase : if β > β KT (z) e iq(ϕx ϕy ) C(β, z) x y 2κ κ = { βeff 4π (1 q)2 if q [ 1 2, 1) β eff 4π q2 if q [0, 1 2 ] KT line : β = β KT (z), e iq(ϕx ϕy ) C(β, z) x y 2κ ln κ x y κ = { 2(1 q) 2 if q [ 1 2, 1) 2q 2 if q [0, 1 2 ]

45 Expected phase diagram Berezinskii(1971), Kosterlitz-Thouless (1973), z Kosterlitz (1974), Fröhlich-Spencer (1981) Giamarchi-Schulz (1988) 0 8π β dipole phase : if β > β KT (z) e iq(ϕx ϕy ) C(β, z) x y 2κ κ = { βeff 4π (1 q)2 if q [ 1 2, 1) β eff 4π q2 if q [0, 1 2 ] KT line : β = β KT (z), e iq(ϕx ϕy ) C(β, z) x y 2κ ln κ x y κ = { 2(1 q) 2 if q [ 1 2, 1) 2q 2 if q [0, 1 2 ] plasma phase : β < β KT (z), non-critical

46 Rigorous results z 0 8π β 1 Fröhlich, Park (1978) existence of the thermodynamic limit for pressure and correlations (any β, any z, any q, free bc; Ginibre method)

47 Rigorous results z 0 8π β 1 Fröhlich, Park (1978) existence of the thermodynamic limit for pressure and correlations (any β, any z, any q, free bc; Ginibre method) 2 Fröhlich, Spencer (1981): power law decay for β β 1 (z) β KT (z) (also other models; away from critical line, q < 1, no exact exponents)

48 Rigorous results z 0 8π β 1 Fröhlich, Park (1978) existence of the thermodynamic limit for pressure and correlations (any β, any z, any q, free bc; Ginibre method) 2 Fröhlich, Spencer (1981): power law decay for β β 1 (z) β KT (z) (also other models; away from critical line, q < 1, no exact exponents) 3 Marchetti, Klein, Perez, Braga ( ) extended FS: β β 2 (z) β KT (z) with β 2 (0) = 8π (away from critical line, q < 1, no exact exponents)

49 Rigorous results z 0 8π β 4 Benfatto, Gallavotti, Nicolò (1986), Marchetti, Perez (1989), Dimock (1990), Kappeler, Pinn, Wieczerkowski (1991), Benfatto, Renn (1993), Guidi, Marchetti (2001): hierarchical case

50 Rigorous results z 0 8π β 4 Benfatto, Gallavotti, Nicolò (1986), Marchetti, Perez (1989), Dimock (1990), Kappeler, Pinn, Wieczerkowski (1991), Benfatto, Renn (1993), Guidi, Marchetti (2001): hierarchical case 5 Gallavotti, Nicolò (1985): multi-scale perturbation theory of the free energy (Gallavotti-Nicolò trees, non-convergent perturbation theory) 7 Nicolò, Perfetti (1989): renormalizability of dipole phase and KT line (Gallavotti-Nicolò trees, non-convergent perturbation theory)

51 Rigorous results z 0 8π β 4 Benfatto, Gallavotti, Nicolò (1986), Marchetti, Perez (1989), Dimock (1990), Kappeler, Pinn, Wieczerkowski (1991), Benfatto, Renn (1993), Guidi, Marchetti (2001): hierarchical case 5 Gallavotti, Nicolò (1985): multi-scale perturbation theory of the free energy (Gallavotti-Nicolò trees, non-convergent perturbation theory) 7 Nicolò, Perfetti (1989): renormalizability of dipole phase and KT line (Gallavotti-Nicolò trees, non-convergent perturbation theory) 8 Dimock and Hurd (1994): formula for the pressure for β β 3 (z) β KT (z) with β 3 (0) = 8π (Brydges-Yau method, no KT line, no correlations)

52 New Result z 0 8π β F. (2011) For z 1 and β = β KT (z), convergent series for the pressure: p(z, β) = e j (z, β) j 0

53 New Result z 0 8π β F. (2011) For z 1 and β = β KT (z), convergent series for the pressure: p(z, β) = e j (z, β) j 0 RG method of Brydges, Yau (1990) and Brydges (2007) s Park City Lectures

54 New Result z 0 8π β F. (2012) (work in progress) For z 1, decay of the correlations with q (0, 1) along β KT (z). e iq(ϕx ϕy ) C(β, z) x y 2κ ln κ x y κ = { 2q 2 for q (0, 1 2 ] 2(1 q) 2 for q ( 1 2, 1) Is κ = κ?

55 Interacting Fermion and Coulomb Gas