Ising model and phase transitions

Transcription

1 Chapter 5 Ising model and phase transitions 05 by Alessandro Codello 5. Equilibrium statistical mechanics Aspinsystemisdescribedbyplacingaspinvariableσ i {, } at every site i of a given lattice. A microstate is defined by a spin configuration {σ i }, which specifies the values of the spins of every lattice site. The (canonical) probability to observe the system in the spin configuration {σ i } is given by the Gibbs Boltzmann weight: gibbs boltzmann distribution and partition function p {σi } = e βe {σ i } Z, (5.) where E {σi } is the energy of the given configuration. The (canonical) partition function Z appearing in (5.) is defined by: Z(β,B)= {σ i } e βe {σ i}, (5.) and in general depends on the inverse temperature β =/T and the external magnetic field B. Averages are computed as usual: O = {σ i } p {σi }O {σi }. We will measure the temperature in units of the Boltzmann constant, which can be reintroduced at any moment by the replacement T k B T. 6

2 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 63 As we will see in a moment, the knowledge of the partition function is the key to all thermodynamical quantities. We need now to specify the form of E {σi } for a given spin configuration. definition Nearest neighbor interactions are assumed so that the energy of a given of the spin configuration is given by ising model E {σi } = J ij σ i σ j σ i B i. (5.3) i,j i J ij is the spin interaction matrix strength and B i is the external applied magnetic field (it most cases it can be considered constant B i = B). If J>0 the interaction is ferromagnetic while it is antiferromagnetic if J<0. The energy (5.3) defines the model known as Ising model. The free energy F can be obtained from the partition function (5.) using free the following relation: energy, internal F (β,b)= log Z(β,B). (5.4) β The internal energy E and the entropy S are related to the free energy by the usual thermodynamic relation: F = E TS. (5.5) In particular, the internal energy and the specific heat can be obtained directly from the partition function (5.) by employing the following relations: E = β log Z = (βf) (5.6) β C = E T where used T = β β.thetotalmagnetization = β E β. (5.7) energy, entropy, specific heat, magnetization and susceptibility M = i σ i = N σ i Nm They are actually the more general, if we add a constant term, since the spins are fermions!

3 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 64 can also be computed from the partition function: while the susceptibility is determined/defined by: χ = N M = F B, (5.8) M B = N F B. (5.9) We will work with intensive variables, or equivalently, per spin variables: f = F N m = M N for which we have f = log Z and βn ɛ = β (βf) ɛ c = β β ɛ = E N m = f B c = C N, (5.0) χ = m B. (5.) From (5.5) one can find the entropy per spin s = ɛ f. Using = T T β T we can also write the internal energy and the specific heat as: ɛ = T ( ) T T f c = ɛ T. (5.) The computation of f,ɛ,c,m,χ and s is the main goal of the study of equilibrium spin systems. 5.. Fluctuations and correlations It s useful to define the generating function from which we find: W B W (β,b) = log Z(β,B)= βf(β,b), = β [ M M ] = β F B = βn f B. This last quantity is proportional to the magnetic susceptibility, susceptibility and specific heat as a measure of fluctuations χ = f B.

4 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 65 which plays thus the role of magnetic variance 3 since: χt = N [ M M ]. (5.3) Thus the susceptibility measures the relative size of magnetization fluctuations around the average. The specific heat is instead the energy variance: W β = E E = β E = β C, which measures the size of internal energy fluctuations around the average: ct = N [ E E ]. (5.4) From relation (5.3) and (5.4) we expect that at a phase transition, where fluctuations are large and on all scales, both the susceptibility and the specific heat will diverge. We will soon see that this is indeed the case. The two point spin connected correlation function is: C ij = σ i σ j σ i σ j = σ i σ j m = (σ i σ i )(σ j σ j ). (5.5) This correlation function can be obtained from the generating function W if we promote the magnetic field to be space dependent B B i : C ij = β W B i B j. (5.6) Obviously W then becomes the generating function of spin connected correlation functions. The two point connected correlation is obviously related to thesusceptibility since both are derived from W by derivatives with respect to the magnetic field. This relation is obtained as follows: C ij = [ σ i σ j σ i σ j ] ij ij = σ i σ j σ i σ j i j i j = M M = NTχ. (5.7) 3 The magnetic kurtosis is the so called Binder cumulant. correlation function

5 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 66 Changing variables to r = i j gives ij C ij = N r C(r) and we find the sum rule: χ = C(r). (5.8) T We will use relations (5.3) and (5.8) in the following to make precise quantitative statements about the strength/weakness of fluctuations and to derive scaling relations between critical exponents. 5. Mean field theory The mean field approach consists in discarding fluctuations, and thus, as we will see it is quantitatively correct only above the upper critical dimension. The assumption of the mean field approach is that fluctuations around the average are small. Stillitoftengivesaqualitativelycorrectpicture of the phase diagram. In the Ising model, the mean field approximation is implemented by linearizing the interaction between two near spins: σ i σ j = [σ i σ i + σ i ][σ j σ j + σ j ] r = σ i σ j +[σ i σ i ] σ j +[σ j σ j ] σ i + [σ i σ i ][σ j σ j ] }{{} small m i m j + m i [σ j m]+m j [σ i m i ] = m i m j + σ i m j + m i σ j, (5.9) where σ i = m i and we discarded the quadratic fluctuation term since we are assuming that the fluctuation are small. If the external magnetic field is constant then m i m for every i due to translation invariance and the the energy of a spin configuration becomes simply: mean field theory weak fluctuations E{σ MF i } = J [ m + m (σ i + σ j ) ] B i,j i σ i = m J Nz (Jzm + B) i σ i, (5.0) where we used the fact that there are Nz nearest neighbors on a lattice (z is the coordination number) and that i,j σ i = z i σ i. For a square lattice

6 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 67 we have z =d and in general the coordination number is proportional to the dimension. In this approximation the net effect of the spin interactionis to shift the external magnetic field to the value Jmz + B. The partition function is easily evaluated: Z MF = {σ i } e βemf {σ i } = e βm J Nz = e βm J Nz {σ i } {σ i } e β(jzm+b) i σ i i e β(jzm+b)σ i = e βm J Nz {cosh[β (Jzm + B)]} N. Thus the free energy per spin f MF = βn log ZMF is: mean field free energy f MF = m J z log cosh [β (Jzm + B)]. (5.) β Note that this relation is valid for any lattice in arbitrary dimension. 5.. Mean field phase diagram To find the equilibrium value for the magnetization per spin weneedtofind the minimum of the free energy: [ ] f MF Tc =0 m(t,b)=tanh m T m(t,b)+b, (5.) T where we defined the mean field critical temperature T c = Jz. For B =0 is easy to see that there is a phase transition at T = T c by studying the solutions of: m =tanh T c m, (5.3) T which can be found graphically as shown in Figure 5.. Near T c the magnetization is small and we can expand equation (5.0) to obtain 4 4 tanh x = x x m = T c T m T 3 c 3T 3 m3 + O ( m 4), mean field phase diagram: phase transition

7 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 68 Figure 5.: Graphical solution to equation (5.3) at B =0.FromtopT<T c, T = T c and T>T c.fort<t c the system has a spontaneous magnetization. and so 0 T T c m ± = ( ) 3 T T ± 3 c ± T c 3( t) / T T T c where we defined the reduced temperature t = T T c. Theorderparameter is m and at the second order phase transition behaves thus as: m ± ±( t) / T T c t 0. (5.4) in the mean field ap- Thus the magnetization critical exponent is β MF = proximation. At criticality the magnetization as a function of the external magnetic field can be written in the following way: m(t c,b) = m(t c,b)+ B T c 3 = m + B T c 3 m ( m(t c,b)+ B T c )

8 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 69 Figure 5.: Mean field per spin magnetization for the Ising model.

9 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 70 from which we find: ( ) 3 B 3 m(t c,b)=sign(b) sign(b) B δ. (5.5) This gives the mean field magnetic exponent δ MF =3. It s easy now to compute the susceptibility: χ(t,b)= m B = T T c cosh ( m Tc T Around the critical point we find using (5.4): { 4 T T c thus χ ± (t, 0) = T c T + B T T T + c T T c ) T c T. (5.6) χ ± (t, 0) χ ± t t 0 ±. (5.7) This gives the mean field susceptibility critical exponent γ MF =and the mean field amplitude ratio χ + /χ =.Notethatthefactthatthesusceptibility diverges at the critical point means that the magnetic variancediverges. Next we compute the energy per spin; we find simply: ɛ = (βf) β = Jz m. The specific heat per spin is not well defined in the mean field approximation (in particular it is not divergent at the critical point), but it usually assumed/defined that α MF = Correlation function If we want to compute the correlation function we need to relax theassumption of constant magnetic field in order to use (5.6). In this case the partition function turns out to be: Z MF = {σ i } e βj i,j [ m im j +σ i m j +m i σ j ]+β i B iσ i = e βj i,j m im j e βj i,j [σ im j +m i σ j ]+β i B iσ i. (5.8) {σ i } computing the mean correlation function

10 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 7 We need now to evaluate the sums: {σ i } e βj i,j [σimj+miσj]+βb i σi = {σ i } e βj i,j σ im j +β B i σ i = {σ i } e β i σ i[j j(i) m j+b i] = i coshβ J m j + B i, j(i) where j(i) are the nearest neighbors of i. Thefreeenergyisthus: f MF = J m i m j log cosh β J m j + B i. (5.9) N βn j(i) i,j Minimizing the free energy leads to: i m i =tanh β J m j + B i, (5.30) j(i) where we used: and m k δ jk = δ ik j(i) i,j m i m j = i,j m i δ jk = i m i δ jk = m k. The average magnetic field felt by a spin σ i is now J j(i) m j + B i ;equation (5.30) reduces to (5.) when m i is constant since j(i) =z. To compute the correlation function using (5.6) we need to determine the solutions of (5.30) to linear order in the external magnetic field. Since we areinterested in the critical region, we can assume that also the magnetization is small and expand the hyperbolic tangent to first order: j(i) βj j(i) m j + m i = βb i. (5.3)

11 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 7 To solve (5.3) we perform a Fourier transform: βjm k cosk a + m k = βb k, (5.3) a where a =,..., d runs over the basis vectors of the lattice. We find: which gives 5 : m k = m i =Ω BZ T J a cosk B k, a d d k (π) d e ixi k T J a cosk B k. a This expression is IR divergent when T = T c since: J a cosk a = Jz + J a k a +... = T c + J a k a +... Finally, since C ij = m i B j,thecorrelatornearthecriticalpointcanbewritten as (on a hyper-cubical lattice): C(r = r ) = d d k e ir k T c (π) d t + z a ( cos k a), (5.33) where t = T T c is the reduced temperature. At large r we can write 6 : C(r) T c = T c (π) d T c (π) d d d k e ir k (π) d t + k ( ) d t K d r ( tr) ( ) d t π r tr e tr. (5.34) The correlation length is the inverse lattice mass ξ =/m and thus from (5.34) we see that at the critical point it diverges like: 5 m i m(x i ). 6 K n (x) is a Bessel function. ξ = t / t ν. (5.35)

12 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 73 The mean field correlation length critical exponent is thus ν MF =. Exactly at the critical point we find instead: C(r) r, (5.36) d rd +η telling us that the mean field anomalous dimension is η MF =0. Finally the mean field critical exponents are: α MF =0 β MF = δ MF =3 ν MF = γ MF = η MF =0. (5.37) mean field critical exponents To understand the quantitative validity of the mean field approximation we need to understand the quantitative meaning of weak fluctuations assumption Ginzburg criterion The Ginzburg criterion establishes when the mean field analysis is valid, i.e. under which conditions fluctuations are small. We start from: M M M, which is equivalent to NTχ M. At criticality fluctuations are on all scales and we have (for t 0 ): the ginzburg criterion and upper critical dimension ξ t ν M ( t) β L d Nχ ( t) γ L d. Thus, setting ξ = L, wefind: T ( t) γ ξ d ( t) β ξ d, or T ( t) γ β+νd,

13 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 74 which may happen when γ β + νd > 0 or: γ +β <d, ν which is the Ginzburg criterion. If we insert the mean field exponents we find d>4. Thus the mean field predictions are valid in dimensions greater than four, which for this reason is called upper critical dimension d c = Solving the Ising model In this section we explore the Ising model in dimensions below the upper critical dimension d c =4,wheretheGinzburgcriterionshowsthatthemean field approach is unable to describe the phase transition quantitatively. Fortunately in both d =and d =it is possible to solve the Ising model exactly. These solutions are invaluable models characterized by non mean field phase transitions d Ising model In one dimension the partition function for the Ising model can be rewritten as follows Z = {σ i } e βj N i= σ iσ i+ +βb N i= (σ i+σ i+ ) the transfer matrix approach = {σ i } N e β[jσ iσ i+ + B (σ i+σ i+ )] i= = {σ i } T σ σ...t σn σ = tr T N, (5.38) where we introduced the transfer matrix ( e β(j+b) e T = βj e βj e β(j B) ). (5.39) By diagonalizing (5.39) we can write the partition function in terms of the eigenvalues of the transfer matrix: ( ) N λ+ 0 Z = tr = λ N + + λ N, 0 λ

14 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 75 Figure 5.3: Ising spins on a one dimensional infinite lattice. where ) λ ± = e (cosh βj βb ± sinh βb + e 4βJ. Note that we have λ + >λ. The free energy is thus given by the following expression: { [ ) ]} N F = β log Z = β N log λ + + log + ( λ in the thermodynamic limit N only the contribution from λ + survives and we find the following free energy per spin: ) (cosh BT sinh + BT + e 4J/T f(t,b)= J T log In the case of zero magnetic field we have simply f(t,0) = J + T log ( +e J/T) = T log T log cosh T. We can easily calculate the magnetization per spin: m(t,b)= f B = λ + sinh B T sinh BT + e 4J/T. Since the hyperbolic sine is zero in the origin there is no spontaneous magnetization at B =0.Thesusceptibilityis χ(t,b)= m B = e 4J/T cosh B T T ( sinh B T + e 4J/T) 3/, while the energy per spin and the specific heat are: ɛ(t,b)= (βf) β = J tanh J T,.

15 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 76 Figure 5.4: Exact (per spin) thermodynamic functions for thed Ising model. From top: f,ɛ,c,s,m,χ; leftcolumnasafunctionoft,rightcolumnasa function of B.

16 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 77 c(t,b)= ɛ T β = J sech J T. T We can also calculate correlation functions. The magnetization is: σ i = σ i e βe {σ i } Z where {σ i } = T σ σ Z T σi σ i σ i T σi σ i+ T σn σ {σ i } = Z trti σ z T N i+ = Z trtn σ z, σ z = ( 0 0 When B =0the diagonalizing matrix is U = σ i = Z tr ( λ N λ N )( 0 0 ) ). ( = Z tr ( 0 λ N + λ N 0 ) and we find: ) =0, as expected. The correlation function can be calculated in a similar way: σ i σ i+r = Z trtn r σ z T r σ z = Z tr ( λ N r λ N r = λn r + λ r + λr + λn r λ N + + λ N )( 0 0 )( λ r λ r )( 0 0 ) Thus we have: ( λ λ + ) r. where the correlation length is: C(r) = σ i σ i+r = e r/ξ, ξ(t,0) = log tanh βj.

17 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 78 Figure 5.5: Exact (per spin) thermodynamic functions for thed Ising model. From top: f,ɛ,c,s,m,χ; thecolorcodeisconsistentwiththepreviousfigure.

18 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 79 We can also consider the limit r where C(r) 0, showing again that there is no spontaneous magnetization. The critical exponents for the one dimensional Ising model are: α d= = β d= =0 γ d= = δ d= = ν d= = η d= =. These clearly differ from the mean field ones. Note that the Ginzburg inequality is violated but saturated since γ+β ν =. critical exponents for the d = ising model 5.3. d Ising model The exact zero field free energy for the square lattice Ising model was found for the first time by L.Onsager using algebraic methods. The zero field free energy can be exactly computed for square, triangular and hexagonal lattices; the result is: f(t,0) = Tplog T where the argument of the logarithms is: P S (T,k) = cosh J T P T (T,k) = cosh 3 J P H (T,k) = sinh J T (cos k +cosk ) d k log P (T,k), (5.40) (π) T + J J sinh3 sinh T T (cos k +cosk +cosk 3 ) [ ] +cosh 3 J T J sinh T (cos k +cosk +cosk 3 ), for, respectively, square, triangular and hexagonal lattices. p is the number of sites in a unit cell of the lattice, which is one for the square and triangular lattices and two for the hexagonal lattice. onsager s solution The argument of the logarithm is always positive except for k =0in which a critical singularity occurs. This singularity corresponds to the phase transition, thus temperature the critical temperature is the solution of: P (T c, 0) = 0.

19 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 80 We find, respectively, the following equations: cosh 3 J T T c cosh J T S c + sinh 3 J T T c +cosh 3 J T H c sinh J T S c 3 sinh J T T c 3 sinh J T H c = 0 = 0 = 0, which give the following critical temperatures: Tc S = log(+ J ) =.699 J (5.4) Tc T = log J 3 = J (5.4) Tc H = log(+ J 3) =.5865 J, (5.43) which may be compared with the mean field result Tc MF = zj.thecritical temperature (5.4) was first found by Kramer and Wannier using duality arguments. Since the free energy diverges logarithmically we have α d= =0 in all three cases. We can now compute the per spin internal energy and the specificheatusing internal equation (5.): energy ɛ(t,0) = T d k P(T,k) (π) P (T,k) T (5.44) and c(t,0) = ɛ(t,0) + T [ ] d k P (T,k) P(T,k). T (π) P (T,k) T P (T,k) T (5.45) The three thermodynamic functions f,ɛ,c are shown in Figure 5.6. and specific heat Onsager and Yang were also able to determine the spontaneous magneti- magnetization zation exactly:

20 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 8 3 Square Lattice 3 Triangular Lattice 3 Hexagonal Lattice Figure 5.6: Per spin free energy (bottom), internal energy (middle) and specific heat (top) for the d Ising model on a square, triangular and hexagonal lattices.

21 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS m Figure 5.7: Spontaneous magnetization per spin for the d Ising model on a square (middle), triangular (right) and hexagonal (left) lattices. T m S (T,0) = m T (T,0) = m H (T,0) = [( +e 4J/T ) ( 6e 4J/T + e 8J/T) ( e 4J/T ) 4 [( 3 e 4J/T )( +e 4J/T) 3 ] 8 ( e 4J/T ) 3 (3 + e 4J/T ) [( ) +e 4J/T 3 ( +e 4J/T 4e J/T) ( e J/T ) 6 ( + e J/T ) ] 8 ] 8. These expressions are valid for T<T c and are shown in Figure 5.7. In the case of the square lattice we also have the equivalent simple form: first derived by Yang. m S (T,0) = [ sinh 4 J T ] 8, (5.46) The correlation length is the inverse of the lattice mass. P (T,k) plays the correlation role of the lattice propagator (it is the argument of the log in thefreeenergy) length

22 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS Ξ Figure 5.8: Correlation length for the d Ising model on a square (middle), triangular (right) and hexagonal (left) lattices. T as in the mean field case, thus the mass squared can be defined by writing: [ ] P (T,k) =Z(T ) m (T )+ ( cos k a ), (5.47) z where Z(T ) is a wave function renormalization. From this we can write: or m (T )= ξ(t )= P (T,0, 0) P (T,π/,π/, ) P (T,0, 0) a (5.48) P (T,π/,π/,) P (T,0, 0). (5.49) P (T,0, 0) For the lattices we are considering we find the following results: ξ S (T,0) = log. e J/T +e J/T Note that the argument of the log is e K D tanh K (= at the critical point) or something like this. At the critical point the correlation lengthdiverges

23 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 84 in the following way: giving ν d= =. ξ S ( T S c ( + t), 0 ) = 4 log ( + ) t + regular t + regular. ( ξ T T T c ( + t), 0 ) = 7Arctanh ( 3 ) t + regular ξ H ( T H c ( + t), 0 ) = t + regular 4Arctanh t + regular t + regular, We can expand the magnetization around the critical point finding: m S ( T S c ( + t), 0 ) = m T ( T T c ( + t), 0 ) = [ ( )] [( ) ( log + )] 8 ( t) 8 + regular.4 ( t) 8 + regular. [ ( Arctanh 3 )] ( t) 8 + regular ( m H T H c ( + t), 0 ) = ( + ) [( ( t) 8 + regular. ( t) 8 + regular ) Arccoth ] 8 3 ( t) 8 + regular We thus see that independently of the lattice the order parameter critical exponent is β d= = 8.Thisisaninstanceofuniversality. The knowledge of the two critical exponents α and β allows the determi- critical nation of all the other critical exponents if we assume the validity of the exponents scaling relations (to be proved in the next section): for the α d= =0 β d= = 8 γ d= = 7 4 δ d= =5 ν d= = η d= = 4. d = ising model

24 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 85 AFM vs FM Figure 5.9: Comparison between ferromagnetic and antiferromagnetic thermodynamic quantities for the triangular lattice Ising model. 0.8 Entropy Figure 5.0: The frustrated antiferromagnetic ground state ofthetriangular lattice Ising model has non zero entropy at T =0.

25 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 86 These clearly differ from the mean field ones, in accord with the factthat the Ginzburg inequality is violated (but saturated) since γ+β ν = Anti ferromagnetism and frustration For antiferromagnetic coupling J<0 the triangular lattice Ising model becomes very interesting since it represents the simplest example of frustrated system. There is no phase transition at finite temperature and the model is always disordered, even at T =0where the ground state is frustrated and the system has a non zero entropy. The thermodynamic quantities and the entropy per spin are shown in Figure 5.0. anti ferromagnet and frustration 5.4 Scaling hypothesis A function of one variable is said to be homogeneous if there exist a scaling exponent a such that homogeneous functions f(λ a x)=λf(x) λ>0. It is easy to see that a necessary and sufficient condition for f(x) to be an homogeneous functions that it is a power law. Taking λ = x /a we find: as discussed in the first lecture. f(x) =f(±) x /a, Afunctionofn variables x, y, z,... is homogeneous if there exist scaling exponents a, b, c,.. such that: f(λ a x, λ b y, λ c z,...) =λf(x, y, z,...). In this case we can eliminate one variable by introducing the scaling functions F ± defined as before by setting λ = x /a : ( ) f(x, y, z,...) = x /a y f ±, x, z,... b/a x c/a ( ) y x /a F ± x, z,.... b/a x c/a

26 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 87 In the case of a two variable function the scaling function is afunctionofone data collapse rescaled variable: by plotting x /a f(x, y) as a function of y/ x b/a all data collapses to the graph of the scaling function F ±. Near the critical point the various thermodynamic functions exhibitascaling scaling behavior. The scaling hypothesis assumes that near to the phase transition the (non-analytical part of the) free energy per spin is an homogeneous function: hypothesis ( ) b f(t, b) = t α F ± t t 0 ± b 0, (5.50) where t = T Tc T c is the reduced temperature, b = βb and the exponents α< and are the first two critical exponents. Since derivatives of homogeneous functions are again homogeneous, the scaling ansatz (5.50) implies that all thermodynamic quantities are so near the phase transition. The magnetization per spin scales as: m(t, b) = β f b ( ) b = β t α F ± t ( ) b t α M ±. (5.5) t The exponents β and δ where defined by the relations: m(t, 0) ± t β m(0,b) sign(b) b /δ. (5.5) Comparing (5.5) to (5.5) gives the first two scaling relations: The susceptibility scales as follows: β = α = βδ. (5.53) χ(t, b) = β m b = β t α F ± t α χ ± ( b t ( ) b t ). (5.54)

27 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 88 The exponent γ is defined by the relation χ(t, 0) χ ± t γ and from (5.54) we find the scaling relation: γ = α (5.55) and the relation χ ± = χ ± (0). Thesingularpartofthespecificheatscalesas: c(t, b) = T f T = T ( ) ( α)( α) b t α F ± T c T c t ( ) b t α C ± ; (5.56) t thus at the phase transition the specific heat diverges like c(t, 0) C ± t α with C ± = C ± (0) Scaling relations Combining the relations (5.53) with (5.55) we can eliminate to obtain α +β + γ = β + γ = βδ. (5.57) scaling One also finds the scaling relation: γ = ν( η). (5.58) To derive (5.58) we start from (5.8) and transform the sum in an integral: χ L 0 dr r d C(r) L 0 dr r d r (d +η) = L 0 dr r η = L η η, (5.59) where we used (5.36) to express the correlation function at criticality. At the critical point ξ = L and thus χ ξ η.recallingthescalingbehaviorofboth the susceptibility and the correlation length this implies t γ t ν( η), which immediately gives (5.58). Note that this scaling relation is the same we found when we studied percolation. Note also that since χ>0 we must have η<, whichisthusarestrictiononthepossiblevaluesassumedbythe anomalous dimension. anomalous relations dimension scaling relation

28 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 89 The hyper-scaling relation is: α = νd, (5.60) hyperscaling relation which depends explicitly on d. Toseewhy(5.60)holdsconsiderthatthefree energy per spin is an intensive variable and thus scales as f(t, 0) N L d. But from the scaling ansatz (5.50) we also know that f(t, 0) t α, thus at criticality where ξ = L we must have t νd t α and so we get(5.60). Note that we have already met this relation when we studied the random walk; we called α = 3 the Hurst exponent and we had d =and d f = =. ν Note that the mean field critical exponents (5.37) and the one and two dimensional Ising model critical exponents indeed satisfy the scaling relations. Note that in the mean field case the hyper-scaling relation implies =d/ or d =4. Combining the previous relations we find: with only ν and η independent. α = νd β = ν d +η γ = ν( η) δ = d + η d +η, (5.6) there are only two independent critical exponents 5.5 RG for lattice models We discuss the RG approach to lattice models developed mostly by Kadanoff and Wilson RG for the d Ising model We construct the RG transformation by grouping the spins in pairs and by applying the decimation procedure where we sum over the second spin in

29 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 90 R C S Figure 5.: RG on a one dimensional lattice. each pair. By defining the coupling K = βb and K = βj we have that the parti- rg tion function becomes: trans- Z = e K i σ i+k i σ iσ i+. (5.6) {σ i } To perform the decimation, consider the sum over the second spin: e K (σ +σ 3 ) σ =± e K σ +K σ (σ +σ 3 ) = e K (σ +σ ) cosh[k + K (σ + σ 3 )]. e K 0 + K (σ +σ 3 )+K σ σ 3, (5.63) where we defined the renormalized couplings so to preserve the functional form of the partition function. From (5.63) we can extract three independent relations by evaluating it for the various configuration of the two spins, i.e. for σ = σ 3 =,forσ = σ 3 =and for σ = σ 3 =, whichgive respectively: e K cosh(k +K ) = e K 0 +K +K coshk = e K 0 K e K cosh(k K ) = e K 0 K +K. (5.64) Equation (5.64) defines (implicitly) the RG transformation K 0 K 0 R K = K (5.65) K K forma- tion for the d = ising model

30 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS Figure 5.: The RG flow for the coupling K. in the three dimensional theory space of all the free energies oftheform: βf(k 0,K,K )=K 0 + K σ i + K σ i σ i+. The fact that theory space is finite dimensional is a peculiarity of the one dimensional model, note that this implies that (5.65) is an exact RG. Let s consider for a moment the case where the magnetic field is zero.thus there are only parity invariant interactions, i.e. only the couplings K 0 and K : i i K 0 = log + log cosh K K = log cosh K. (5.66) The RG transformation (5.66) has a fixed point for K =0and K =. The second is attractive and corresponds to the ordered phase transition while the first represent the disordered high temperature phase. Note that K 0 converges in the IR to the free energy, since at every RG step we are integrating out some degree of freedom (spins). In terms of the variables u = e K,v= e K, w = e K 0 the solution of

31 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 9 v e ΒB u e ΒJ Figure 5.3: RG flow for the d Ising model in the (u, v) plane. (5.64) can be written as: u = v = w = ( v + v ) / ( u4 + u 4 + v + ( u 4 + v ) / v ) /4 u 4 + v ( v + ) / ( u 4 + v u + 4 v + ) /4, (5.67) v and the exact RG transformation becomes: u u R v w = v w In the (u, v) plane there is a repulsive fixed point (0,B), amixedfixedpoint (0, ) and a line of fixed points (,B), ascanbeseeninfigure General RG analysis The general case is much more complicated. The coarse graining is still implemented by decimation but for d>this procedure introduces new.

32 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 93 C S R Figure 5.4: RG transformation for the two dimensional Ising model. The coarse graining is performed by summing over any second spin, while the similarity has scale factor λ = and a rotation of π. 4 couplings at every iteration of the RG transformation. The full complexity is already present in the two dimensional squarelat- rg for tice case. The RG transformation is illustrated in the Figure. Since now the d = ising e K σ 5 (σ +σ +σ 3 +σ 4 ) =cosh[k (σ + σ + σ 3 + σ 4 )] model σ 5 =± = e K 0 + K (σ σ +σ σ 4 +σ σ 3 +σ 3 σ 4 )+K 3 (σ σ 3 +σ σ 4 )+K 4 σ σ σ 3 σ 4, (5.68) new next nearest neighbor interactions K 3 and quadruple interactions K 4 are generated already after the first RG step. At every iteration the number of coupling constants grows and finally the RG flow takes place in the infinite dimensional theory space parametrized by the couplings K = (K 0,K,K,K 3,K 4,...): RK = K. (5.69) The most general free energy has now the form: βf(k) =K 0 +K σ i +K σ i σ j +K 3 σ i σ j +K 4 σ i σ j σ k σ l +... We go on with the theory. i i,j i,j (5.70) Phase transitions correspond to fixed points: RK = K, (5.7) fixed points

33 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 94 and their domain of attraction define a universality class. Linear stability. point: We can linearize the RG transformation around a fixed linear stability RK = RK + L(K K )+... (5.7) where the linear RG transformation is now the stability matrix: L ij = K i K j. (5.73) In general there is no reason whyl should be symmetric, but for ease of discussion we will assume it. Thus the eigenvalues and eigenvectors are all real: Lv n = λ n v n λ n = λ θn, (5.74) where we introduced, as customary, the RG eigenvalues θ n. The couplings in the direction of the eigenvectors are called the scaling fields or couplings and denoted with s i. Since we are interested in the Ising fixed point which has two relevant directions we will set t = s and b = s. We have seen the the Ginzburg criterium tells us that the mean field critical deriving exponents are correct in d>4. We have the critical exponents in d =, from the exact solutions. We now see that we can use the RG to compute the the scaling form critical exponents for every d. NotethatbytheRGdecimationprocedurewe and the are not changing the partition function (since we are considering the most critical general for the energy or free energy): exponents Z(K) = e βe {σ i }(K) {σ i } = {σ i } = {σ i } {σ dec i } e βe {σ i,σ i dec } (K) e βe {σ i }(K ) = Z(K ), (5.75)

34 CHAPTER 5. ISING MODEL AND PHASE TRANSITIONS 95 thus the free energy transforms as: Since N = λ d N we find: f(k) = log Z(K) βn = βn log Z(K ) = N N f(k ). (5.76) f(k) =λ d f(rk). (5.77) As we said, the coupling K 0 converges to the total free energy since at every step we integrate out half of the spins. Around a fixed point we can diagonalize the stability matrix and express the coupling vector intermsofthe scaling fieldsk =(t, b, s 3,s 4,...) and we find: f(k) = λ nd f(r n K) = λ nd f(λ nθt t, λ nθ b b, λ nθ 3 s 3,λ nθ 4 s 4,...) n λnd f(λ nθt t, λ nθ b b, 0, 0,...), (5.78) if we now choose λ = t /nθt,wefindthescalingformforthefreeenergy: ( ) b f(t, b) t d/θt f ±, t θ b/θ t, 0, 0,... t 0 b 0, (5.79) with: α = d θ t = θ b θ t F ± (x) =f(±, x,...). (5.80) The RG analysis has thus proved the scaling hypothesis and showed that the critical exponents are related to the RG eigenvalues! The first relation is just the hyperscaling relation (5.60) and thus ν = θ t, (5.8) which immediately tells us how to compute the correlation length critical exponent. Spin block transformations as defined in this section are not an exact way to implement the coarse grain C and are thus of of limited practical/quantitative use, even if conceptually very illuminating. In the next lecture we will develop a better formalism to construct the transformation R exactly.

35 Bibliography [] H. Maris and L. Kadanoff, Teaching the Renormalization Group, Am. J. Phys. 46(6) (978) 65. [] K. Wilson, The Renormalization Group and Critical Phenomena. Rev. Mod. Phys. 55(3) (983) 583. [3] E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Rev.Mod.Phys.7()(999)S358 S