Scaling Theory. Roger Herrigel Advisor: Helmut Katzgraber

Transcription

1 Scaling Theory Roger Herrigel Advisor: Helmut Katzgraber

2 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff construction (1966) Scaling for the correlation function Universality Finite size scaling Disordered systems / Harris criterion (1974) Polymer statistics

3 Critical exponents Summery of the critical exponents for a magnetic system t = T T c T c h = H k B T c Exponent Definition Description α C H t α specific heat at H = 0 β M t β magnetization at H = 0 γ χ t γ isothermal susceptibility at H = 0 δ M h 1 δ critical isotherm ν ξ t ν correlation length η G (r) r (d 2+η) correlation function Fundamental thermodynamics Exponent inequalities such as α + 2β + γ 2 (Rushbrooke)

4 Critical exponents The critical exponent λ of a function f (t) is λ = lim t 0 ln f (t) ln t The function f (t) near critical temperature T C (t 0) is dominated by t λ t λ describes f (t) at the transition.

5 The scaling hypothesis The singular part of the free energy density f is a homogeneous function near a second-order phase transition. Homogeneous functions 1 dim f (λr) = g (λ) f (r) n dim the scaling factor g is of the from g (λ) = λ p f (λ r) = g (λ) f ( r) Generalized homogeneous functions (i) λf (x, y) = f ( λ a x, λ b y ) (ii) λ c f (x, y) = f ( λ a x, λ b y )

6 The scaling hypothesis The singular part of the free energy density f is a homogeneous function near a second-order phase transition. The reduced temperature t and the order parameter h rescale by different factors. f (t, h) = b d f (b yt t, b y h h) If f is a homogeneous function then also its Legendre transform all thermodynamical potentials are homogeneous.

7 Derivation of the scaling relations I The scaling hypothesis postulates that the free energy is homogeneous f (t, h) = b d f (b yt t, b y h h) Let: b = t 1 yt f (t, h) = t d yt f (±1, t y h y t ) f (t, h) = t d yt φ ( t y h y t h ) h

8 Derivation of the scaling relations II

9 Derivation of the scaling relations III ) f (t, h) = t d yt φ ( t y h y t h 1. Magnetization: M t β M = 1 f k B T h h=0 = 1 d yh ( ) k B t yt φ t y h y t h T β = d y h y t t d y h yt 2. Critical Isotherm: M h 1 δ M should remain finite as t 0 φ (x) x d because then y h 1 M t d y h yt d y h h y h y h (d y h ) t y h yt δ = y h d y h

10 Derivation of the scaling relations IV ) f (t, h) = t d yt φ ( t y h y t h 3. Heat capacity: C t α C = 2 f t 2 h=0 t d yt 2 α = d y t 2 4. Magnetic susceptibility: χ t γ χ = 1 M k B T h = 1 h=0 t d 2yh h 2 yt γ = d 2y h h=0 (k B T ) 2 2 f y t

11 Hyperscaling I ( r ) G (r) = b 2(d yh) G b, byt t G (r) = t 2(d y h) yt 5. Correlation length: ξ t ν We have G e r ξ t also for t 0 Φ ( r t 1 yt ξ t 1 yt ν = 1 y t )

12 Hyperscaling II 6. Correlation function: G r (d 2 η) choose b = r and t = 0 ( r ) G (r) = b 2(d yh) G b, byt t G (r) r 2(d y h) η = d + 2 2y h

13 Scaling relations We got the following equations: α = d y t 2 β = d y h y t γ = d 2y h y t δ = y h d y h ν = 1 y t η = d + 2 2y h we can cancel the y h and y t which have no experimental relevance.

14 Scaling relations From these six equations of the critical exponents one obtains: α + 2β + γ = 2 δ 1 = γ β Rushbrook s Identity Widom s Identity 2 α = dν Josephson s Identity γ = ν (2 η) There are only 2 independent Exponents

15 Onsager solution 1944 N H Ω = J s i s j H N ij i The exponents of the Onsager solution for the 2-dim Ising model. s i α = 0 β = 1 8 γ = 7 4 δ = 15 ν = 1 η = 1 4 equations values α + 2β + γ = = 2 δ 1 = γ β 15 1 = α = dν 2 0 = 2 1 γ = ν (2 η) 7 4 = 1 ( )

16 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff construction (1966) Scaling for the correlation function Universality Finite size scaling Disordered systems / Harris criterion (1974) Polymer statistics

17 Kadanoff construction 1966 Motivation: heuristic explanation Idea of Renormalization group Ising model: H Ω = J N s i s j H ij N i s i dimension d lattice of N sites with distance a spins s i = ±1 only nearest neighbor interactions For T T C ξ

18 Kadanoff construction Block spin transformation Partition the lattice into blocks of side ba Each block is associated with a new spin s. Set s to majority of spins in the block. cells cell length original lattice N a lattice of blocks N = b d N a = ba Each block contains b d sites of the original lattice

19 Kadanoff construction Assumption 1: Block spin interacts only with nearest neighbor block spin and an effective external field H Ω = J N s i s j H ij N i s i H Ω = J Nb d ij s i s j H Nb d Since Hamiltonians have the same structure free energy same with (different parameters) Nf (t, h) = Nb d f ( t, h) i s i

20 Kadanoff construction Nf (t, h) = Nb d f ( t, h) f (t, h) = b d f ( t, h) We expect that h = h (h, b) and t = t (t, b). From the above equation we conclude: h h t t Assumption 2: h = b y h h t = b yt t f (t, h) = b d f (b yt t, b y h h)

21 Scaling for the correlation function Consider Hamiltonian with non-uniform external field h not changing significantly over distances ba. Define: r = r/b βh Ω = βh Ω0 r h (r) s (r) β H Ω ( s) = β H Ω0 ( s) r h ( r) s ( r) If Z is the partition function the 2-point correlation function is given by G (r 1 r 2, H Ω ) = s (r 1 ) s (r 2 ) s (r 1 ) s (r 2 ) 2 = h (r 1 ) h (r 2 ) ln Z h(r)=0

22 Scaling for the correlation function 2 ( h) h ( r 1 ) h ( r 2 ) ln Z = LHS: G( r 1 r 2, H Ω ) 2 ln Z (h) h ( r 1 ) h ( r 2 ) RHS: If r = r 1 r 2 ba b 2y hb 2d G (r, H Ω ) ( r ) G b, H Ω = b 2d 2y h G (r, H Ω )

23 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff construction (1966) Scaling for the correlation function Universality Finite size scaling Disordered systems / Harris criterion (1974) Polymer statistics

24 Universality classes I Universality is a prediction of the renormalization group theory Definition: Systems whose properties near 2nd order phase transition are controlled by the same renormalization group fixed point are in the same universality class. Properties: have the same (relevant) critical exponents can have different transition temperatures 3D Heisenberg β Fe 0.34(4) Ni 0.378(4) CrB (5) EuO 0.36(1) Monte Carlo 0.364(4)

25 Universality classes II Universality classes are characterized by: spatial dimension symmetry of the order parameter range and symmetry of Hamiltonian - The details of the form and magnitude of interactions is not relevant. If the above properties of a system are the same of an other (well known) system, we already known its critical exponents!

26 Finite size scaling All numerical calculations use finite systems. Calculate quantities such as C, M,χ for different lattice size. Near the critical temperature. ) C L = L α ν C (L 1 ν t C is independent of lattice site but depends on Tc, α and ν. If these parameters are chosen correctly and we plot C L L α/ν against L 1 ν t the curve will collapse.

27 Disordered systems What happens if the system contains impurities? ordered phase is destroyed system remains ordered Harris criterion (1974) The critical behavior of a quenched disordered system does not differ form that of the pure system if dν > 2

28 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff construction (1966) Scaling for the correlation function Universality Finite size scaling Disordered systems / Harris criterion (1974) Polymer statistics

29 Polymers A polymer is a chain of monomers. Examples: Synthetic Polymers PVC PE PET Biopolymers Figure: protein T162 DNA RNA proteins

30 Random walk: a crude model for a polymer Random walk There is a starting point 0. Distance from one point to the next is a constant. Direction is chosen at random with equal probability. Self avoiding random walk same but no intersections

31 Random walk: a crude model for a polymer Random walk There is a starting point 0. Distance from one point to the next is a constant. Direction is chosen at random with equal probability. Self avoiding random walk same but no intersections

32 Random walk Properties of a random walk c N ( r) = number of distinct walks from 0 to r end-to-end vector r = n a n average square distance r 2 = a n a m + a n a m = Na 2 n=m n m }{{} =0 What is r 2 for a polymer (self-avoiding random walk)?

33 Solution by mapping to a O(n) vector model Goal: Find a connection of self-avoiding random walks and the O(n) n 0 model. Model: Hypercubic lattice of dimension d Spin ( on each lattice site has n components. S = S 1 i, Si 2,.., S i n ) Normalization: n ( ) 2 S α = n only nearest neighbor interactions Hamiltonian: H Ω = K Si α Sj α α ij,α

34 Moment Theorem Theorem: Let... 0 be the average over all spin orientation For n 0 we have S α S β 0 = δ αβ and all other momentum are 0. Examples: S α S β S γ 0 = 0 S α S α S α S β S γ 0 = 0

35 Mapping to O(n) n 0 The exponent of the Hamiltonian can be written as: exp( βh Ω ) = exp( βk Si α Sj α ) = exp( βk Si α Sj α ) ij,α ij α = (1 βk Si α Sj α (βk)2 ( Si α Sj α ) 2...) ij ij,α ij,α Therefor the partition function Z is given by Z = Tr exp ( βh Ω ) = k dω k exp ( βh Ω ) 0 = Ω ij (1 βk α S α i S α j (βk)2 ( α S α i S α j ) 2 )... 0 where Ω = i dωi

36 Diagram Representation Z = Ω ij (1 βk α S α i S α j (βk)2 ( α S α i S α j ) 2 ) 0 We expand the product over ij choose 1 do nothing choose βk α S i α Sj α a line form i to j draw choose 1 2 (βk)2 ( α S α i S α j ) 2 draw smallest loop 3 B diagrams

37 Diagram Representation Z = Ω ij (1 βk α S α i S α j (βk)2 ( α S α i S α j ) 2 ) 0 Taking the average only vertices with 2 lines survive S α i S α i S α i 0 = 0 S α i S α i S α i S α i 0 = 0 index of the spins must be the same less than 3 B diagrams

38 Diagram Representation Z = Ω ij (1 βk α S α i S α j (βk)2 ( α S α i S α j ) 2 ) 0 Taking the average only vertices with 2 lines survive index of the spins must be the same less than 3 B diagrams

39 Mapping to O(n) n 0 Taking the average Si α Sj α S β j S β k... S qs θ i ι 0 = δ αβ δ βγ...δ αι = αβ... αβ... n 1 = n α It follows: Z = Ω loop conf For n 0 we obtain Z = Ω. n number of loops number of bounds (βk)

40 Mapping to O(n) n 0 Correlation function G (i, j) = Si 1 Sj 1 = Z 1 Tr ij S 1 i S 1 j (1 βk α S α i S α j (βk)2 ( α S α i S α j ) 2 ) like before surviving diagrams have single line (self-avoiding walk) form i to j N c N (r) βk = lim n 0 G (r, βk) This is the important relation which connects a self avoiding random walk to the O(n) n 0 model.

41 Critical behavior Scaling of c N Define: x = βk c N x N = N N c N = r c N (r) c N (r) x N = lim r n 0 r G (r, x) = χ x x c γ Ansatz: c N xc N N γ 1 Ansatz is correct since: c N x N N 0 = ln ( ) x N ( x dnn γ 1 ln x c x c ) γ ( 1 + (x x ) γ c) x x c γ x c

42 Critical behavior Scaling of r 2 Similar calculations r 2 = r r 2 c N(r) c N r 2 G (r, x) = r r c N (r) x N r 2 = N N c N (r) r 2 x N r G (r, x) r (d 2+η) e r ξ r c N (r) r 2 x c x γ 2ν r 2 x N C N2ν+γ 1 N 2ν x N C Nγ 1

43 Summary Near the critical point of a second-order phase transition the thermodynamic potentials are assumed to be homogeneous functions f (t, h) = b d f (b yt t, b y h h) only 2 independent exponents Systems can be grouped into universality classes Harris criterion for quenched disordered systems dν > 2 Random walk r 2 N Self-avoiding random walk r 2 N 2ν