Chap.9 Fixed points and exponents

Transcription

1 Chap.9 Fixed points and exponents Youjin Deng

2 The fixed point and its neighborhood A point in the parameter space which is invariant under will be called a fixed point R s R μ* = μ * s μ * λ = We should carefully choose the value a in s or the equation will not have a solution There is no general theorem to tell us whether there is a discrete or a continuous set of fixed points, or any fixed point at all, let s assume now there is at least one fixed point s a

3 The fixed point and its neighborhood Critical surface μ All points in this subspace have the property Rs μ lim R μ = μ * s drives to a different point s As s increase, all points in the critical surface are eventually driven to μ * For sufficiently large s, R μ s will be in the immediate neighborhood of μ *

4 The fixed point and its neighborhood Neighborhood of fixed points μ μ * For a point near,we write μ = μ* + δμ μ ' = R The equation can be written as s μ L ' = Rs + O(( ) ) δμ δμ δμ Define a matrix L ( R ) s αβ μ ' α = μ β μ = μ*

5 The fixed point and its neighborhood Neighborhood of fixed points Then we can write μ ' α δμ ' = δμβ β μ β μ = μ* Next to determine the eigenvalues and L eigenvectors of Suppose the eigenvalues are found to be and the corresponding eigenvectors to be Since RRe = R s R e s s' j ss' j ρ j () s ρ j ( s) ρ ( s') = ρ ( ss') and ρ ( s) = s j j j j e j y

6 The fixed point and its neighborhood Neighborhood of fixed points Then we use the eigenvectors as a set of basis vectors and write δμ = = = j j te, δμ ' t ' e, t ' ts t j j j j j j j j ' If y j > 0, will grow as s increases If y j < 0, t j ' will diminish If y, will not change j = 0 t j ' The existence of one or more zero y j implies that there is a continuous set of fixed points. y

7 Large behavior of and critical exponents s R s RG: some transformation of probability distributions H [ σ ] Interpret P e in parameter space u In the language of Ising μ = ( Th, ) The critical point of the ferromagnet is given by T = T, c h= 0 Fundamental hypothesis linking RG to critical phenomena μ(,0) T c is a point on the critical surface of a fixed point lim R μ( T,0) = μ * s s c

8 Large behavior of and critical exponents s R s This hypothesis states that if we decrease the magnification by a sufficiently large amount, we shall not see any change if it is decreased further Restrict our discussion to h=0, that means there will be no odd power of σ x Let μ ( T ) denote μ( T,0) For sufficient small T T c, μ( T ) must be very close to the critical surface and Rsμ( T ) will move toward μ * as s increases However as s, Rsμ ( T ) will go away from μ * since is not quite on the critical surface μ( T )

9 Large behavior of and critical exponents s The manner Rsμ ( T ) goes away from μ * depends on the positive y j ' s,we call one of them, then y 1 For very large s, we have Expand R s δμ( T) = μ( t) μ* = tj( T) e L R ( T) μ* + R δμ( T) s s y1 y = μ * + ts e + O( s ) 1 1 t ( T) 1 t ( ) ( ) ( ) 1 T = A T T + B T T + c j c j

10 Large behavior of and critical exponents s R s Then for very small T T c R T A T T s e O s s y1 y ( ) μ * + ( ) + ( ) 1/ υ y = μ* + ( s/ ξ) + O( s ) Where we have defined 1/ υ =, = ( ) y1 ξ A T T c We apply this to examine the temperature and k dependence of Gk (, μ( T)) using the previous a d relation and λ = s s c 1 υ Gk (, μ) = λ sgsk (, μ') s

11 Large behavior of and critical exponents s We get R s Gk T s GskR T a+ d (, μ( )) = (, sμ( )) = ± + a+ d 1/ s G sk s e1 O s Since this is true for any value of s provided s is very large, we can set to obtain Gk t G k e O (, μ( )) ξ a+ d y = ( ξ, μ* ± + ( ξ )) ξ υ y (, μ* ( / ξ) ( )) a+ d s = ξ g( ξk) 1

12

13 Large behavior of and critical exponents s By the definition of,we see that η = a+ d a = ( η d) / Setting k=0 for sufficiently small T T c R s η G T G e O T T c y (0, μ( )) ξ (0, μ* ± + ( ξ )) η ν( η) 1 From the definition of we get a previously obtained scaling law γ γ = ν( η)

14 Large behavior of and critical exponents s R s h 0 Now we generalize the above arguments to The field h appears as an entry in representing μ the term d hb σ x The transformation of h under can be easily worked out d hsb ( ) λ σ = hs b x' x R s + a σ s x' x' x' h ' = hs 1 ( d η ) + 1

15 Large behavior of and critical exponents s R s While for free energy, it s not that straightforward to get the conclusion F( μ( Th, )) = s d FR ( μ( Th, )) + A( μ( Th, ), st ) + BsT ( ) s The reason is that Gk ( ) =< σ k > average values of long wavelength (small k) Fourier components of the spin configuration, On the other hand the free energy involves all Fourier components directly.

16 Chap.9 Gaussian fixed point and fixed point in 4- ε dimensions Youjin Deng

17 The Gaussian fixed point Ginzburg-Landau model it is more convenient to use the wave vector representation. 1 d 1 H = d x r + u + cδ 4 1 u = ( r + ck ) σ + L 8 Parameter space 4 ( 0σ σ σ) d 0 k k1 k k3 k1 k k3 k k, k, k 1 3 σ σ σ σ μ = ( r, u, c) 0

18 The Gaussian fixed point Ginzburg-Landau model μ ' = R RG transformation e d = δϑe H' AL H d AL : additive constant generated by dσ k This integration is easy when 0 while it becomes very difficult s μ σ s 1 η / a+ d = η δϑ = dσ k σ Λ / s< k<λ sk k u = u 0

19 The Gaussian fixed point Three fixed points for u = 0 u = 0 When the integration will be 1 H' = s ( ck + r ) σ AL k<λ/ s k ' <Λ η 0 sk 1 = ( rs + cs k' ) σ d = η η 0 k ' 1 π ln( ) r + ck Λ / s< k<λ 0

20 The Gaussian fixed point Three fixed points for H ' We can see is still a Ginzburg-Landau form with μ Set η = 0,we get Gaussian fixed point The value of c is arbitrary u = 0 ' ( r0', 0, c') ( r0s η η = =, 0, cs ) 0 μ * = (0,0, c) c d c H0* = d x( Δ ϕ) = k σ e H c k * = k e σ k k<λ k

21 The Gaussian fixed point Three fixed points for Set η = we obtain another fixed point μ * = ( r,0,0) The value of 0 if arbitrary as long as r0 d r0 H * = d xσx = σk For,we get 0 u = 0 r r > 0 0 k<λ r r H * e = e = e 0 0 x 0 0 σx σk r < μ * = ( + r0,0,0) e H r0 σ * x = e x k 4 uσ ( x)

22 The Gaussian fixed point Physical meaning of the three fixed points * μ : Each block spin is independent of all others Spin values tend to be zero because of the Gaussian potential This is the infinite temperature fixed points μ * : Each block spin is independent of all others Because of small u and negative r 0,spins tend to take a large value This is the zero temperature fixed points

23 The Gaussian fixed point Physical meaning of the three fixed points μ * 0 : cδϕ accounts for coupling between neighboring block spins Gaussian distributions for σ k

24 The linearized RG near the Gaussian fixed point Perturbation near fixed points H = H* +ΔH H' + AL = H* + A* L +Δ H' +ΔAL Then we have d d d e d H' AL H d = δϑe σ s k 1 η / (1 Δ ' Δ ) = (1 Δ ) H* + AL d H* e H AL δϑe H σ sk σ s k 1 η / σ sk

25 The linearized RG near the Gaussian fixed point Perturbation near fixed points Before preceding, we remark that in general one can break H into two pieces in any manner And obtain H = H + H 0 1 ( H δϑ ) 1 d H' + AL = ln e ( 1) η / H0 m = < > ln δϑe σ s k m σ sk H 1 c m= 1 m! σ s k 1 η / σ sk

26 The linearized RG near the Gaussian fixed point Perturbation near fixed points Note that the subscript c means taking the cumulant m < H >=< ( H < H > ) > m 1 c 1 1 m < H1 >= δϑe First two cumulants are 1 c 1 H δϑe < H > =< H > 0 H H 0 m 1 < H >=< H > < H > 1 c 1 1

27 The linearized RG near the Gaussian fixed point Perturbation near fixed points d ΔH' Δ AL = ΔH Δ H δϑe H* δϑe ΔH H* σ s 1 η / The average is take over,i.e. over with Λ / s< k <Λ = One can take high-order powers of expansions ϑ k σ sk σ k

28 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point The Gaussian fixed point For GL model 1 d 1 4 Δ H = d x( r0 < σ >+ u< σ > ) 4 Separate 1 d c d xδϕ k H* e e e = = ϑ σ from by defining σ = σ ' + ϑ k σk

29 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point Where d/ σ ' = L d/ ϑ = L k<λ/ s Λ / s< k<λ σ e k k ik x σ e < σ >=< σ ' + σ ' ϑ+ ϑ > = σ ' +< ϑ > 4 < σ >=< ( σ ' + σ ' ϑ+ ϑ ) > 4 4 = σ ' + σ ' < ϑ >+ 4 < ( ϑ σ ') >+< ϑ > 4 4 = σ ' + 6 σ ' < ϑ >+< ϑ > ik x

30 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point d ϑ L σk Λ / s< k<λ < >= < > = = Λ d d ( π ) d k(1/ ck ) K c d Λ Λ / s = n s Λ / s dk k d c (1 ) d 3

31 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point Where K d d n = K Λ /[ c( d )] c K d d 1 = d+ 1 π d/ / Γ( d) is the surface area of a unit sphere in d- dimensional space divided by ( π ) d

32 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point 4 < ϑ >= < σ σ σ σ > let k k, k 1 3 k, k 1 3 = k k, k, k = 3 < σ σ σ σ > d c = 3 < σ σ >< σ σ > = 3 n (1 s ) k k k k k k k k k k k k k k

33 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point We get Then < σ >= σ ' + n (1 s ) d c < σ >= σ ' + 6 n (1 s ) + 3 n (1 s ) 4 4 d d c c 1 d 1 4 Δ H = d x( r0 < σ >+ u< σ > ) 4 1 3u 1 = d x( r + n (1 s ) σ ' + uσ ' +ΔAL 4 d 1 d 3 d Δ AL = [ r0 nc(1 s ) + unc (1 s ) ] 4 d d 4 d 0 c

34 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point Recall that the rescaling transformation ' = s y x x' x' σ λσ σ s d/ = s y+ d/ k sk sk σ λ σ σ For Gaussian fixed point, we require σ σ with η 0 k 1 η / s sk = so y = 1 d / σx' σ x' = x/ s x' s 1 d /

35 The linearized RG near the Gaussian fixed point GL model near Gaussian fixed point Near Gaussian fixed point 1 d 1 4 Δ H = d x'( r0 ' σ + u' σ ) +ΔAL σk sσsk 4 d =Δ H' +ΔAL 3 d r0' = s ( r0 + unc (1 s )) with 4 d u' = s u L R s This is the formula for within the parameter space near the Gaussian fixed point μ( r, u, c) 0 μ *(0,0, c) d

36 The linearized RG near the Gaussian fixed point Renormalization exponents Matrix representation of L R s r ' r 0 R L 0 = s u' u R L s 3 4 d s unc ( s s ) = 4 d 0 s

37 The linearized RG near the Gaussian fixed point Renormalization exponents Matrix representation of 3 Let B = un c,the eigenvectors are With eigenvalues L R s 1 e1 = = r0 0 B e = = Br0 + u 1 y1 t = s = s y = υ = 1 1 y t s s = = 4 d y 1/ = 4 d

38 The linearized RG near the Gaussian fixed point Renormalization exponents The conclusions concerning critical behavior are valid if y < i.e. d > d upper = 4 We say 0 μ μ μ * is stable for d > 4 * is unstable for d < 4 * is marginal for d = 4

39 The linearized RG near the Gaussian fixed point

40 Relevant, irrelevant and marginal parameters, scaling fields and crossover t The parameters t, h are relevant scaling fields d > = = d < 4 d u' us d y yh t' = ts, h' = hs ; y, y > 0 4 irrelevant 4 marginal 4 relevant Generally speaking, only the relevant scaling fields are relevant to critical behavior (not always right) t h

41 Relevant, irrelevant and marginal parameters, scaling fields and crossover The Hamiltonian near the fixed points can be expanded as H = H + t + O t Scaling parameter * iθi ( i ) i H θi = t i θ t = d d x σ θ h = d d x σ

42 Relevant, irrelevant and marginal parameters, scaling fields and crossover A larger parameter space for the GL model d 6 d x[ u σ + v ( σ ) + v σ ( σ ) ] 6 1 Here we have to extend the parameter space μ = ( rucu,,, v, v) 0, 6 1 The Gaussian fixed point remains the same and μ 0 * = (0,0, c,0,0,0) L R s By applying the same approximation procedure, the exponents of the linearized RG are (,4 d,0,6 d, d, d)

43 Relevant, irrelevant and marginal parameters, scaling fields and crossover A simple formula for term y i tdd d x i i d = (1 ) (power of σ in Di) + ( 1) (power of in Di ) + d σ The more powers of and, the more negative y i and therefore the more irrelevant the corresponding parameter

44 Relevant, irrelevant and marginal parameters, scaling fields and crossover A simple formula for term tdd d x i i d > 4 only r0 is relevant 3 > d > r0, u6 are relevant d < r0, u6, v1, v are relevant A fixed point is called unstable if there are one or more relevant parameters other than t ( T T c )

45 Critical exponents for d > 4 Near Gaussian fixed point, we have and y < 0 for d > 4 Scaling relations υ = 1/, η = 0 α = νd = d / ν 1 β = ( d + η) = ( d ) 4 γ = ν( η) = 1 d + η d + δ = = d + η d

46 Critical exponents for d > 4 And from the Gaussian approximation we get α = d / β = 1/ γ = 1 δ = 3 We expect that the Gaussian approximation is OK. Since the basic assumption is that u is small. Near the Gaussian fixed point u * u is indeed small 0,

47 Critical exponents for d > 4 What s the problem with the RG prediction? The RG arguments 1 d/ 1 + d/ 4 d mthu (,, ) = s mts (, hs, us ) Set s = t h= 1/ 1 and ( ) ( 4) 4 d d = 1 ± 1 mt (,0, u) t m( 1,0, u t ) The RG predictions implicitly assume m ± u t1 while the face is that m( ± 1,0, u) for small u is thus m u 1/ 1 ( d 4) ( 1,0, )

48 Critical exponents for d > 4 What s the problem with the RG prediction? d mt u t u t We see that 1 1 ( ) ( d 4) 4 (,0, ) 1 1 t 1 1 β = 1/ t1 = 0, s = h 1/ ( d /+ 1) If we set applying the same procedure we notice that m u 1/3 for t and 1 = 0 we find δ = 3 1

49 d = 4 ε O( ε ) The RG for and fixed points to The Gaussian fixed point becomes unstable for d<4, we expect another fixed point u * in addition to the Gaussian point u 0 * For d<4, performing the RG transformation, namely finding fixed points and linearized RG near it, becomes difficult Nevertheless, for d = 4 ε when ε is very small, we hope that it is sufficient to some lower orders of terms in perturbation expansions

50 d = 4 ε O( ε ) The RG for and fixed points to d < 4: we have tow fixed points u* and u0 * d 4 : u* and u0 * merge (degeneracy of fixed points) The fixed point u * should not be regarded as the continuation of the Gaussian fixed point. We expect that u * is close to u * 0,because the 4 d growth u' = s u = s ε u as s increases can be held back by nonlinear terms so far not included in this linearized equation For small the growth rate is very small ε

51 d The RG for and fixed points to We write a differential equation u' = du ' dl If we have a nonlinear term du ' = εu' gu' g constant dl 0 u* = ε / g If is small, then = 4 ε O( ε ) ε s u = εu' with l = ln s u * g = constant > O( ε )

52 The RG for and fixed points to Method d = 4 ε O( ε ) The basic method is to use perturbation expansions up to O( ε ) The RG transformation e d H' AL H d/ d/ k<λ/ s Λ / s< k<λ Λ / s< k<λ 1 η / σ = σ ' + ϑ δϑ = dσ σ ' = L ϑ = L = δϑe σ e k k σ s k ik x σ e ik x σ sk k

53 The RG for and fixed points to Method d The Hamiltonian m 1 And the cumulants are = 4 ε O( ε ) 0 0 H = H + H 0 1 ( 1) H' + A' L = H ' + A ' L + < H > < H >= m+ 1 d d m c m= 1 m! σ s δϑe H δϑe 1 c 1 H H < H > =< H > m 1 < H >=< H > < H >=< ( H < H > ) > 1 c k 1 η / σ sk

54 The RG for and fixed points to Method d = 4 ε O( ε ) The GL model 1 d 1 H0 = d x( r0 + c( ) ) = ( r0 + ck ) 1 u u 4 8 d 4 d H1 = d x σ = L k<λ/ s k ' <Λ σ σ σ k k<λ k, k, k H' = s ( ck + r ) σ η 0 sk 1 = ( r + s ck' ) σ η η 0 k ' σ σ σ σ k k k k k k

55 The RG for and fixed points to Method d The GL model = 4 ε O( ε ) 1 π A0 ' = ln Λ / s< k<λ r0 + ck u d 4 < H1 > c = 8 d x< σ > u d d 4 4 < H1 > c = d x d y ( < σ ( x) < σ( x) > > )( < σ ( y) < σ( y) > > ) 18 r u ( ) Note that 0 and are of order O ε and our aim is to evaluate < H1 > c and < H1 > c to an accuracy of O( ε )

56 The RG for and fixed points to Method 4 We need to evaluate < σ > to order O( ε ) And evaluate to order 1. Therefore we set Let d σ = σ ' + ϑ = 4 ε O( ε ) < σ >= σ ' + 6 σ ' < ϑ >+< ϑ > 4 4 ( < σ ( x) < σ( x) > > )( < σ ( y) < σ( y) > > ) d = 4 and r = 0 1 ( r0 + ck ) k k d k k e Λ> k>λ/ s Λ> k>λ/ s < ϑ >= < σ σ >= σ σ Λ> k>λ/ s Λ 1 d d Λ/ s 0 = ( r + ck ) = K dkk ( r + ck ) 0 σ k

57 The RG for and fixed points to Method d = 4 ε O( ε ) Since r O ε,we expand 0 ( ) ( r + ck ) = k / c ( k / c ) r Λ 1 d 3 d 5 K d dk c k c k Λ/ s < ϑ >= ( + ) = n (1 s ) K c r ln s+ O( ε ) c + ε d (1) u 4 < H1 >= d x( r0 σ ' + σ ' ) 4 (1) 3u 3u r0 = K4Λ (1 s ) + ucε r0 K 4ln s 4c c

58 The RG for and fixed points to Method d = 4 ε O( ε ) There is no term in < H 1 > proportional to That means c is unchanged up to O( ε ) Thus η = 0,in other words ( σ ') η = O( ε ) After tedious calculations, which can become simpler with the help of Feynman diagram, and after scale transformation σ s σ = sσ 1 η / k sk sk

59 The RG for and fixed points to Method d = 4 ε O( ε ) 3u 3u r0' = s [ r0 + KdΛ (1 s ) + ucε r0 K 4ln s+ u D] 4c 4c 3 + O( ε ) c' = c ε 9u 3 u' = s [ u K 4 ln s] + O( ε ) c

60 The RG for and fixed points to Method d = 4 ε O( ε ) u' = u = u* and s ε = 1+ε ln s Let 9 u * ln s[ εu* K 4] = 0 c 0 Gaussian fixed point u* = c ε New fixed point 9K 4 Λ c r0 * = ε 6 Λ c c μ* ( ε, c, ε ) 6 9K 4

61 d = 4 O( ε ) The RG for ε and fixed points to Linearized formula near the new fixed point δ r0 = r0 r* δ u = u u* δr0' r L δ 0 = Rs δu' δu R L s y1 y1 y s B( s s ) = y 0 s 1 y1 = ε 1/ υ 3 where y = ε 3 B= Λ K4 4

62 d = 4 ( ) The RG for and fixed points to ε O ε y = ε is negative means the new fixed point is stable

63 d = 4 O( ε ) The RG for ε and fixed points to Critical exponents up to O( ε ) ε 1 α = 6 α = 6 1 ε 1 β = β α = = 6 3 β ε = d = 3 ε = 1 7 γ 1 = + γ = Numerical γ = δ 3 ε η = = + η = 0 η = 0 υ = υ = 1 ε 1 υ = + 1

64 Chap.9 Renormalization groups in selected models Youjin Deng

65 Definitions of the RG for discrete spins Some of the main contributors Niemeijer and van Leeuwen Nauenberg and Nienhuis Kadanoff and Haughton Wilson

66 Definitions of the RG for discrete spins Ising model on the triangular lattice The Hamiltonian reads H = ( K σσ + K 1 r r+ δ r r+ δ r δ= n. n δ= n. n. n + K σσ σ σ + 3 r r+ δ r+ δ' r+ δ '' + hσ + hσ σ σ + ) K 1 r r r+ δ r+ δ' σσ Where is the nearest-neighbor (n.n) coupling parameter, K is the next-nearest-neighbor (n.n.n) coupling parameter, K 3 the "4 spin" coupling parameter, h is the magnetic field, and h.. are higher odd spin coupling parameters.

67 Definitions of the RG for discrete spins Ising model on the triangular lattice This figure shows which spins are coupled in these terms.

68 Definitions of the RG for discrete spins Ising model on the triangular lattice A Kadanoff transformation can be defined by forming blocks each of which contain three spins as shown in this figure

69 Definitions of the RG for discrete spins Ising model on the triangular lattice 1 3 σ, σ, σ The mean of the three spins r' r' r' in the block r ' can take four values, namely ± 1, ± 1 / 3, instead of just ±1 One can force the four values into two values by defining the new block spin as σ σ σ σ 1 3 r ' = sgn( r ' + r ' + r ') 1 ( 1 ' ' 3 ' 1 ' ' 3 = σr + σr + σr σr σr σr ') Which of course is either +1 or -1. This is not quite the same as the coarse graining we are used to

70 Definitions of the RG for discrete spins The RG transformation The RG transformation R μ = μ is defined by e Where ρ[ σ ', σ] is a product taken over the new blocks The function p is a projector, explicitly We can see d 3 ' H[ σ '] AL H[ σ ] = ρσ [ ', σ] e σ 1 3 ρσ [ ', σ] = p( σr' '; σr', σr', σr' ) r ' p = [1 + σ ''( σ ' + σ ' + σ ' σ ' σ ' σ ' )] r r r r r r r p ( + 1; σ, σ, σ ) + p( 1; σ, σ, σ ) = 1

71 Definitions of the RG for discrete spins The RG transformation We need to shrink the new lattice by a factor 3 to obtain the same lattice spacing as the old one. The whole RG is defined by l R = ( R ), s= 3 l s 3 = 0,1, Note that there is no rescaling of the spin variable σ in contrast to the replacement σ 1 / d / s η in the earlier definition of the linear RG. Here the value of σ is always ± 1. No rescaling can be defined, nor is it necessary l/ σ

72 Definitions of the RG for discrete spins Transformation for various average values The transformation formulas for various average values are more complicated. e ' λr' σr' λr' tr' r' r' < > =< > t μ' e We can obviously see m( μ ') =< σ ' > =< t > μ' 1 = ( σ + σ + σ σ σ σ ) r' r' r' r' r' r' r' r' μ' r' Gr ( / 3, μ') =< σ ' σ ' > =< tt > etc. μ r' 0 μ r' 0 μ

73 Definitions of the RG for discrete spins Two-dimensional Ising on a square lattice This figure shows the formation of new blocks, each of which includes four spins.

74 Definitions of the RG for discrete spins Two-dimensional Ising on a square lattice New block spins can be defined as σ ' = sgn( σ + σ + σ + σ ) r' r' r' r' r' Except that special attention has to be paid to the cases where the four spins sum to zero. One can arbitrarily assign σ ' for the r ' = 1 configurations shown in the next figure (b), and σ for the inverse of these configurations r ' ' = 1 as shown in the next figure (c)

75 Definitions of the RG for discrete spins Two-dimensional Ising on a square lattice Two different cases where the four spins sum to zero.

76 Homeworks 1, A brief survey of percolation. Please include the definition of the percolation model, its applications, exact solution on Cayley tree, exact solution of the percolation threshold on the square lattice.,exact solution of the mean-field Blume-Capel model. K H = ss + D s ( s = 0, ± 1) N N i j i N i j = 1 k = 1 Please locate the line of the phase transitions, which is consisted of a line of critical point, a tricritical point, and a segment of 1st order transition. Summarize the behavior of the energy density, the specific heat, the magnetization density, the susceptibility, the vacancy density, the compressibility near the transition line, and as a function of temperature 1/K. Derive critical exponents.

77 Homeworks 3, Ising model is an amazingly beautiful toy model in statistical mechanics. Please describe the exact solution of the Ising model in 1D (including partition sum and all observable quantities), the exact solution of the critical point on the square lattice. Summarize the critical exponent for d>1. Summarize its generalization to the Potts model (ordinary and chiral). Can you locate the critical point of the ordinary Potts model on the square lattice (using the duality relation)? 4, Quantum Ising model plays an important role in quantum statistics. Its Hamiltonian is H = t σ σ σ ij z z x i j k k where sigma is the Pauli matrix. Please describe its phase transition. Using the path-integral language (Suzuki-Trotter formula), a d- dimensional quantum Ising model can be mapped onto a (d+1)- dimensional quantum Ising model. Please derive this mapping.

78 Homeworks 5, The classical XY model is frequently used to describe the universal behavior in the phase transition between Mott-insulator and superfluidity (or superconductivity). Please give reasoning why this is possible. Summarize the phase transition of the classical XY model in 1D, D, 3D, and more. Give the reasoning why the D XY model does not have long-ranged ordering at non-zero temperature (as rigorously as possible) Mermin-Wagner theorem. 6. In real worlds, we have bosons and fermions. By interchanging two of such particles, the phase of the wave function accumulates 0 and PI, respectively, for bosons and fermions. Their statistical behavior obeys the Bose-Einstein and the Fermi-Dirac statistics, respectively. Please derive them. In some D systems with strong interactions, however, the excitons psedo-particles do not obey either of the statistics. Interchanging two excitons can lead to a phase change of any value between 0 and Pi. Such pesudo-particles are named anyons. They are used explain the quantum Hall effects as well as the recently discovered quantum Spin-Hall effects, and are found to have profound implication in quantum computation. Two beautiful models for anyons are the Kitaev model and the Wen model (arxiv: quant-ph/ ). Please give a survey of anyons and these two models.

79 Homeworks 7, Define the renormalization group (RG) in the language of the Ginzburg-Landau model. Explain the universality by RG. Derive the Gaussian the fixed point and the associated critical exponents. Derive the new fixed point in d=4-epsilon dimensions and the associated critical exponents. 8, Monte Carlo method is an important tool in researches, engineering, as well as in industries. In particular, Markovian-chain Monte Carlo (MCMC) method are found extensive applications in statistical physics. Please give a survey of MCMC. Design your own program for the D Ising model, calculate energy density, specific heat, magnetization density, and susceptibility. Locate the critical point, and calculate the associated critical exponents.