Field Theory Approach to Equilibrium Critical Phenomena Uwe C. T - PowerPoint PPT Presentation

Field Theory Approach to Equilibrium Critical Phenomena Uwe C. T¨ auber Department of Physics (MC 0435), Virginia Tech Blacksburg, Virginia 24061, USA email: tauber@vt.edu http://www.phys.vt.edu/~tauber/utaeuber.html Renormalization Methods in Statistical Physics and Lattice Field Theories Montpellier, 24–28 August 2015

Lecture 1: Critical Scaling: Mean-Field Theory, Real-Space RG Ising model: mean-field theory Real-space renormalization group Landau theory for continuous phase transitions Scaling theory Lecture 2: Momentum Shell Renormalization Group Landau–Ginzburg–Wilson Hamiltonian Gaussian approximation Wilson’s momentum shell renormalization group Dimensional expansion and critical exponents Lecture 3: Field Theory Approach to Critical Phenomena Perturbation expansion and Feynman diagrams Ultraviolet and infrared divergences, renormalization Renormalization group equation and critical exponents Recent developments

Lecture 1: Critical Scaling: Mean-Field Theory, Real-Space RG

Ferromagnetic Ising model Principal task of statistical mechanics: understand macroscopic properties of matter (interacting many-particle systems): → thermodynamic phases and phase transitions Phase transitions at temperature T > 0 driven by competition between energy E minimization and entropy S maximization: minimize free energy F = E − T S Example: Ising model for N “spin” variables σ i = ± 1 with ferromagnetic exchange couplings J ij > 0 in external field h : N N ∑ ∑ H ( { σ i } ) = − 1 J ij σ i σ j − h σ i 2 i , j =1 i =1 Goal: partition function Z ( T , h , N ) = ∑ { σ i = ± 1 } e − H ( { σ i } ) / k B T , free energy F ( T , h , N ) = − k B T ln Z ( T , h , N ), thermal averages : ∑ ⟨ ⟩ 1 A ( { σ i } ) e − H ( { σ i } ) / k B T A ( { σ i } ) = Z ( T , h , N ) { σ i = ± 1 }

Curie–Weiss mean-field theory Mean-field approximation : replace effective local field with average: ∑ ∑ h eff , i = − ∂ H J ij σ j → h + � Jm , � = h + J = J ( x i ) , m = ⟨ σ i ⟩ ∂σ i j i More precisely: σ i = m + ( σ i − ⟨ σ i ⟩ ) → σ i σ j = m 2 + m ( σ i − ⟨ σ i ⟩ + σ j − ⟨ σ j ⟩ ) + ( σ i − ⟨ σ i ⟩ )( σ j − ⟨ σ j ⟩ ) Neglect fluctuations / spatial correlations → ( ) H ≈ Nm 2 � ) N ∑ 2 cosh h + � N ( J Jm σ i , Z ≈ e − Nm 2 � h + � J / 2 k B T − Jm 2 k B T i =1 yields Curie–Weiss equation of state ( ∂ F mf ) = tanh h + � m ( T , h ) = − 1 J m ( T , h ) N ∂ h k B T T , N ◮ Solution for large T : disordered, paramagnetic phase m = 0 ◮ T < T c = � J / k B : ordered, ferromagnetic phase m ̸ = 0 ◮ Spontaneous symmetry breaking at critical point T c , h = 0

Mean-field critical power laws Expand equation of state near T c : | τ | = | T − T c | ≪ 1 and h ≪ � J → | m | ≪ 1: T c ≈ τ m + m 3 h → k B T c 3 T = T c : h ≈ k B T c ◮ critical isotherm : m 3 3 ◮ coexistence curve : h = 0 , T < T c : m ≈ ± ( − 3 τ ) 1 / 2 ◮ isothermal susceptibility : ( ∂ m ) { 1 /τ 1 N 1 N τ > 0 χ T = N ≈ τ + m 2 ≈ 1 / 2 | τ | 1 τ < 0 ∂ h k B T c k B T c T → Power law singularities in the vicinity of the critical point Deficiencies of mean-field approximation: ◮ predicts transition in any spatial dimension d , but Ising model does not display long-range order at d = 1 for T > 0 ◮ experimental critical exponents differ from mean-field values ◮ origin: diverging susceptibility indicates strong fluctuations

Real-space renormalization group: Ising chain Partition sum for h = 0, K = J / k B T : a a´ K K K K K K ∑ e K ∑ N σ σ σ i =1 σ i σ i +1 ... ... Z ( K , N ) = i-1 i i+1 “ decimation ” of σ i , σ i +2 , . . . { σ i = ± 1 } { 2 cosh 2 K σ i − 1 σ i +1 = +1 } ∑ e K σ i ( σ i − 1 + σ i +1 ) = = e 2 g + K ′ σ i − 1 σ i +1 2 σ i − 1 σ i +1 = − 1 σ i = ± 1 ( ) K ′ = 1 2 ln cosh 2 K , N → Z ( K , N ) = Z 2 ℓ decimations: N ( ℓ ) = N / 2 ℓ , a ( ℓ ) = 2 ℓ a , RG recursion : K ( ℓ ) = 1 2 ln cosh 2 K ( ℓ − 1) Fixed points → phases, phase transition: ◮ K ∗ = 0 stable → T = ∞ , disordered ◮ K ∗ = ∞ unstable → T = 0, ordered T → 0: expand K ′− 1 ≈ K − 1 ( ) dK − 1 ( ℓ ) 1 + ln 2 ≈ ln 2 2 K − 1 ( ℓ ) 2 → ( ) 2 K d ℓ ℓ = ln(2 ξ/ a ) ≈ 0 → ξ ( T ) ≈ a 2 e 2 J / k B T Correlation length : K ln 2

Real-space RG for the Ising square lattice ∑ − β H ( { σ i } ) = K σ i σ j a´a n . n . ( i , j ) → − β H ′ ( { σ i } ) = A ′ + K ′ ∑ K L´ σ i σ j K´ K n . n . ( i , j ) + L ′ ∑ σ i σ j + M ′ ∑ σ i σ j σ k σ l n . n . n . ( i , j ) � ( i , j , k , l ) 2 cosh K ( σ 1 + σ 2 + σ 3 + σ 4 ) = = e A ′ + 1 2 K ′ ( σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 4 + σ 4 σ 1 )+ L ′ ( σ 1 σ 3 + σ 2 σ 4 )+ M ′ σ 1 σ 2 σ 3 σ 4 List possible configurations for four nearest neighbors of given spin: σ 1 σ 2 σ 3 σ 4 → 2 cosh 4 K = e A ′ +2 K ′ +2 L ′ + M ′ + + + + → 2 cosh 2 K = e A ′ − M ′ + + + − → 2 = e A ′ − 2 L ′ + M ′ + + − − + − + − → 2 = e A ′ − 2 K ′ +2 L ′ + M ′

RG recursion relations K ′ = 1 4 ln cosh 4 K ≈ 2 K 2 + O ( K 4 ) L ′ = K ′ 2 = 1 8 ln cosh 4 K ≈ K 2 A ′ = L ′ + 1 2 ln 4 cosh 2 K ≈ ln 2 + 2 K 2 M ′ = A ′ − ln 2 cosh 2 K ≈ 0 → drop [ K ( ℓ − 1) ] 2 + L ( ℓ − 1) , L ( ℓ ) ≈ [ K ( ℓ − 1) ] 2 → a ( ℓ ) = 2 ℓ/ 2 a , K ( ℓ ) ≈ 2 ◮ K ∗ = 0 = L ∗ stable → T = ∞ : disordered paramagnet ◮ K ∗ = ∞ = L ∗ stable → T = 0: ordered ferromagnet ◮ K ∗ c = 1 / 3, L ∗ c = 1 / 9 unstable: critical fixed point ( δ K ( ℓ ) = K ( ℓ ) − K ∗ ) ( 4 / 3 ) ( δ K ( ℓ − 1) ) 1 c Linearize RG flow: = δ L ( ℓ ) = L ( ℓ ) − L ∗ δ L ( ℓ − 1) 2 / 3 0 c √ ( ) with eigenvalues λ 1 / 2 = 1 2 ± 10 and associated eigenvectors: 3 ( K ( ℓ ) ) ( 1 / 3 ) ( ) ( ) 3 − 3 + c 1 λ ℓ √ + c 2 λ ℓ √ → ≈ 1 2 L ( ℓ ) 1 / 9 10 − 2 10 + 2

Critical point scaling Utilize linearized RG flow to analyze critical behavior: ◮ λ 1 > 1 → relevant direction; | λ 2 | < 1 → irrelevant direction ◮ Critical line : c 1 = 0, set L c = 0 (n.n. Ising model), ℓ = 0 ( K c ) ( 1 / 3 ) ( ) − 3 √ − 1 ≈ + c 2 → c 2 = √ 0 1 / 9 10 + 2 9( 10+2) → K c ≈ 0 . 3979; mean-field: K c = 0 . 25; exact: K c = 0 . 4406 ◮ Relevant eigenvalue determines critical exponent : 2 → 0, δ K ( ℓ ) ≈ e ℓ ln λ 1 ( K − K c ) ℓ ≫ 1: λ ℓ correlations : ξ ( ℓ ) = 2 − ℓ/ 2 ξ → ξ = ξ ( ℓ ) � � � δ K ( ℓ ) � ln 2 / 2 ln λ 1 K − K c ln 2 ξ ( ℓ ) ≈ a → ξ ( T ) ∝ | T − T c | − ν , ν = ≈ 0 . 6385 √ 2 ln 2+ 10 3 compare mean-field theory: ν = 1 2 ; exact (L. Onsager): ν = 1 Real-space renormalization group approach: ◮ difficult to improve systematically, no small parameter ◮ successful applications to critical disordered systems

General mean-field theory: Landau expansion Expand free energy (density) in terms of order parameter (scalar field) ϕ near a continuous (second-order) phase transition at T c : f ( ϕ ) = r 2 ϕ 2 + u 4! ϕ 4 + . . . − h ϕ r > 0 f r = 0 r = a ( T − T c ), u > 0; conjugate field h breaks Z (2) symmetry ϕ → − ϕ r < 0 f ′ ( ϕ ) = 0 → equation of state : φ φ φ 0 - + h ( T , ϕ ) = r ( T ) ϕ + u 6 ϕ 3 φ 2 ϕ 2 > 0 Stability: f ′′ ( ϕ ) = r + u h > 0 ◮ Critical isotherm at T = T c : 0 T T c h ( T c , ϕ ) = u 6 ϕ 3 h < 0 ◮ Spontaneous order parameter for r < 0: ϕ ± = ± (6 | r | / u ) 1 / 2

Thermodynamic singularities at critical point ◮ Isothermal order parameter susceptibility : ( ∂ h ) { 1 / r 1 = r + u 2 ϕ 2 → χ T r > 0 V χ − 1 T = V = 1 / 2 | r | 1 r < 0 ∂ϕ T → divergence at T c , amplitude ratio 2 χ T C h=0 0 T c 0 T c T T ◮ Free energy and specific heat vanish for T ≥ T c ; for T < T c : ( ∂ 2 f ) ± = − 3 r 2 = VT 3 a 2 f ( ϕ ± ) = r 4 ϕ 2 2 u , C h =0 = − VT ∂ T 2 u h =0 → discontinuity at T c

Scaling hypothesis for free energy Postulate: (sing.) free energy generalized homogeneous function : ( h ) , τ = T − T c f sing ( τ, h ) = | τ | 2 − α ˆ f ± | τ | ∆ T c two-parameter scaling, with scaling functions ˆ f ± , ˆ f ± (0) = const . Landau theory: critical exponents α = 0, ∆ = 3 2 ◮ Specific heat : ( ∂ 2 f sing ) C h =0 = − VT = C ± | τ | − α T 2 ∂τ 2 c h =0 ◮ Equation of state : ( ∂ f sing ) ( h ) = −| τ | 2 − α − ∆ ˆ f ′ ϕ ( τ, h ) = − ± | τ | ∆ ∂ h τ ◮ Coexistence line h = 0, τ < 0: ϕ ( τ, 0) = −| τ | 2 − α − ∆ ˆ − (0) ∝ | τ | β , β = 2 − α − ∆ f ′

Scaling relations ◮ Critical isotherm : τ dependence in ˆ f ′ ± must cancel prefactor, as x → ∞ : ˆ f ′ ± ( x ) ∝ x (2 − α − ∆) / ∆ → ϕ (0 , h ) ∝ h (2 − α − ∆) / ∆ = h 1 /δ , δ = ∆ β ◮ Isothermal susceptibility : ( ∂ϕ ) χ τ = χ ± | τ | − γ , γ = α + 2(∆ − 1) V = ∂ h τ, h =0 Eliminate ∆ → scaling relations : ∆ = β δ , α + β (1 + δ ) = 2 = α + 2 β + γ , γ = β ( δ − 1) → only two independent (static) critical exponents Mean-field: α = 0, β = 1 2 , γ = 1, δ = 3, ∆ = 3 2 (dim. analysis) Experimental exponent values different, but still universal : depend only on symmetry, dimension . . . , not microscopic details