10. Interacting matter The grand potential is now Ω= k TVω(z,T ) 10.1. Classical real gas − B We take into account the mutual interactions of atoms and (molecules) p Ω The Hamiltonian operator is = = ω(z,T ) kBT −V N p2 N ∂ω(z,T ) H(N) = i + v(r ), r = r r . ρ = = z . 2m ij ij | i − j | V ∂z i=1 i Q = dr r e−βv(rij ) N 1 · · · N i 1 Permutations of particles do not change the values of the Denote z graphs, for example ξ = 3 . λT dr1dr2dr3f12f23 = dr1dr2dr3f13f23, The grand canonical partition function is now Z Z Z Z Z Z 1 N as we can see by exchanging the integration variables r Z(T,V,µ) = ξ QN 2 N! and r3. XN qµl 3 3 = l = νl νl!(l!) × 1 2 1 2 {νl} l X Y Example: 1 ξN N!δ N, lν Q 2 = N! l N l ! Q 2 = + X νl X q lνl Q 3 = + + + l l = ν ξ νl!(l!) l + + + + {ν } l Xl Y P 1 q νl = + 3 * + 3 * + = l ξl We classify the graphs: νl! l! {Xνl} Yl h i ∞ ν q in connected (coupled) graphs or a clusters one can 1 ql l l l ξl • = ξ = e l! . get from every black dot ( ) to every black dot νl! l! • l νl=0 l following a chain of lines ( ). Y X h i Y We end up with the cumulant expansion of the grand in unconnected (uncoupled) graphs there are parts canonical partition sum: • that are not connected by a line ( ). ∞ ∞ It is easy to see that an unconnected graph can be 1 N 1 l Z(T,V,µ)= ξ QN = exp ξ ql . factorized as a product of its connected parts. N! l! N=0 "l=0 # The sum of graphs of l connected points is called an X X l-cluster. graphs = exp[ connected graphs] We define q so that it is the sum of all l-clusters, e.g. l VirialP expansionP In the cumulant expansion every ql is proportional to the q = dr V 1 = volume V . We define the cluster integral bl depending Z only on temperature so that q = dr dr f 2 1 2 12 1 1 Z Z bl = ql l! V q3 = dr1dr2dr3(3f12f13 + f12f23f13). 1 1 Z Z Z = dr dr (1 + f ) . One can show that l! V 1 · · · l ij Z Z i 2 the function f(r ) deviates from zero only in the • iN Second virial coefficient range of the interaction. We assume that the term f(r )f(r ) can deviate from zero only if • iN jN the interaction has a hard core, i.e. the interaction is particles i and j are within the range of N. • strongly repulsive when r <σ. ∼ the probability that the particle j is in the range of • the particle i is 1 . on the average, the interaction is attractive when ∝ V • r >σ, but the temperature is so high that βv(r) 1 if the particle j is in the range of the particle i, then there.∼ ≪ • the integral over the variable r is 1, since the N ∝ Now function f has a short range. 0, when r <σ e−βv(r) ≈ 1 βv(r), when r∼>σ, We see, that − ∼ N−1 N−1 and 1 N 2 ∞ drN f(riN )f(rjN ) = . 2 −βv(r) ∝ V O V B2 = 2π dr r 1 e Z i 3 where F0(T,V,N) is the free energy of the ideal gas. The We can thus write equation of state is now δ Aˆα = χαβδaβ, 2 ∂F ∂F N k T β p = = 0 B dr f(r) D E X −∂V − ∂V − 2V 2 Z where Nk T 1 = B 1 ρ dr f(r) . V − 2 χ = β Aˆ Aˆ Aˆ Aˆ Z αβ α − α β − β D D E D EE Comparing with the virial expansion we see that the = β δAˆαδAˆβ . second virial coefficient is D E Notes: 1 1 −βv(r) B2 = dr f(r)= dr 1 e . −2 2 − χαβ tells how much the expectation value of the Z Z h i • observable Aˆα changes when the observable Aˆβ is influenced by one unit of disturbance. 10.2. Correlation functions The response functions are related to the correlations • Static linear response of the fluctuations of observables. The correlation ˆ ˆ Let Hˆ0 be the Hamiltonian of a system in the equilibrium CAB of the observables A and B is defined to be and 1 ˆ ˆ ˆ ρˆ = e−βH0 CAB = δAδB , 0 Z 0 D E the corresponding density operator. We disturb the where δAˆ = Aˆ Aˆ is the fluctuating part of the system with external time independent fields a , which − α observable Aˆ. D E couple to observables Aˆα of the system: The correlation functions δAˆαδAˆβ can be Hˆ = Hˆ0 Aˆαaα. • − calculated in the limit aα D=0 . TheE responses χαβ α X are determined, in the{ limit of} infinitesimal The corresponding density operator is disturbances, by the undisturbed density matrixρ ˆ0. 1 ˆ The theory of linear responses can be generalized for ρˆ = e−βH , • Z dynamic disturbances. where Despite of the possible incommutability of the ˆ ˆ −βHˆ −β(H0− Aαaα) • Z = Tr e = Tr e α . operators Aˆα, Aˆβ and Hˆ0 the results are exact in classical mechanics. In quantum mechanics the Now P commutation relations must be taken into accout. ∂Z −β(Hˆ 0− Aˆαaα) One can show that the response function can be = βTr Aˆ e α α written as ∂aα ˆ P ˜(Aβ ) ˆ χαβ = δAβ δAα , = βZ Aˆα D E D E where the operator A˜(Bˆ) is and ˆ 1 1 2 A˜(B) = Aˆ + A,ˆ Bˆ + A,ˆ Bˆ , Bˆ + . ∂ Z 2 −β(Hˆ0− Aˆαaα) = β Tr AˆαAˆβe α 2! 3! · · · ∂aα∂aβ h i hh i i P 2 = β Z AˆαAˆβ , Particle density D E Let rˆ1, rˆ2,..., rˆN be position operators, i.e. when we assume that Hˆ0, Aˆα and Aˆβ commute. We rˆiψ(r1,..., rN )= riψ(r1,..., rN ). define the static linear response function χαβ so that The number density operator is ˆ ∂ Aα 1 ∂ 1 ∂Z χαβ = = ρˆ(r)= δ(r rˆi). D∂aβ E β ∂aβ Z ∂aα − i 1 ∂2Z 1 ∂Z ∂Z X = 2 For example, in the pure state ψ(r , r ,..., r ) we get βZ ∂aα∂aβ − βZ ∂aα ∂aβ 1 2 N 1 ∂2 = ln Z ρˆ(r) = dr1 dri−1 dri+1 drN β ∂aα∂aβ h i · · · · · · i Z Z Z Z ˆ ˆ ˆ ˆ X 2 = β AαAβ β Aα Aβ . ψ(r ,..., r , r, r ,..., r ) . − | 1 i−1 i+1 N | D E D ED E 4 When the particles are identical bosons or fermions ψ 2 is Now r r | | 1 ∂ ln Z symmetric under permutations i j , so Nˆα = ↔ β ∂aα D E ρˆ(r) = N dr dr ψ(r, r ,..., r ) 2. and in the continuum limit h i 2 · · · N | 2 N | Z Z 1 δ ln Z ρ(r)= ρˆ(r) = . We assumed that the system is closed into the volume V h i β δa(r) and ψ normalized. Then We define the density-density response function χ so that ρˆ(r) dr = N dr dr ψ(r ,..., r ) 2 h i 1 · · · N | 1 N | ˆ ZV Z Z ∂ Nα = N. χαβ = β δNˆαδNˆβ D∂aβ E ≈ D E In general, we have and in the continuum limit N δ ρˆ(r) dr ρˆ(r) = dr δ(r rˆ ) χ(r, r′) = h i − i δa(r′) Z i=1 ZV X β δρˆ(r)δρˆ(r′) . N ≈ h i = 1= N, Here i=1 X δρˆ(r)=ˆρ(r) ρˆ(r) =ρ ˆ(r) ρ(r) so we can write − h i − is the fluctuation of the density. ˆ N = dr ρˆ(r). The approximativity of the last formulas is due to the Z non commutativity the Hamiltonian with the operators δρˆ(r) and δρˆ(r′). Density-density response function Pair correlation function We divide the volume V into elements ∆Vα. We restrict to homogeneous matter. Then V ρˆ(r) = ρ(r)= ρ. D V a h i Let us consider the function ρˆ(r)ˆρ(r′) = δ(r rˆ )δ(r′ rˆ ) h i h − i − i i Let i X ˆ ′ Nα = dr ρˆ(r)= Nα + δ(r rˆi)δ(r rˆj ) ∆V h − − i Z α i6=j X be the number of particles in the element ∆Vα. Let aα = δ(r r′) ρˆ(r) be a field coupling to Nˆα. The Hamiltonian of the system − h i + δ(r rˆ )δ(r′ rˆ ) . is h − i − j i Xi6=j Hˆ = Hˆ Nˆ a 0 − α α We define the pair correlation function g(r r′) so that α X − ˆ ′ ′ 2 ′ = H0 dr ρˆ(r)aα. r r r r r r − ρˆ( )ˆρ( ) = ρδ( )+ ρ g( ) α ∆Vα h i − − X Z or In the continuum limit we get ρ2g(r r′)= δ(r rˆ )δ(r′ rˆ ) . − h − i − j i i6=j Hˆ = Hˆ dr ρˆ(r)a(r) X 0 − It can be shown that in a homogenous pure state ZV ψ(r1,..., rN ) of N particles one has = Hˆ dr δ(r rˆ )a(r) 0 − − i i Z N(N 1) X g(r r′) = − = Hˆ a(rˆ ). − ρ2 × 0 − i i X dr dr ψ(r, r′, r ,..., r ) 2. 3 · · · N | 3 N | Thus the field a(r) is a 1-particle potential. Z Z − The state sum Z can be thought to be a function of We see that variables aα or a functional of the function a(r): { } g(r r′) is proportional to the probability for finding • − ′ Z = Z( aα ) Z[a(r)]. two different particles at the points r and r . { } continuum−→ 5 Because simultaneous events far away from each Now • other cannot be correlated we have † δρˆ(q)† = dr e−iq·rδρˆ(r) lim Aˆ(r)Bˆ(r′) Aˆ(r) Bˆ(r′) |r−r′|→∞ −→ Z D E D ED E = dr eiq·rδρˆ(r)= δρˆ( q), so that − lim g(r r′)=1. Z |r−r′|→∞ − so S(q) is a real and non negative function of the variable Note that for completely uncorrelated particles we q. have According to the definition of the response we have ′ ′ ′ 2 g(r r )=1, or ρˆ(r)ˆρ(r ) = ρδ(r r )+ ρ δ ρˆ(r) = dr′ χ(r r′)δa(r′). − h i − h i − Z The function Its Fourier transform is G(2)(r , r )= ρ2g(r r ) 1 2 1 − 2 δρ(q)= χ(q)δa(q). is called the pair distribution function. Distribution We assume that δa(r′) is constant. Then we can interpret functions of higher rank are defined analogously. In that particular, for a pure state the distribution function of δa(r′)= δµ rank (degree) n is is a change in the chemical potential, so (n) G (r1, r2,... rn)= δN N(N 1)(N 2) (N n + 1) δρ(r)= = δµ dr′ χ(r r′)= δµ lim χ(q). − − · · · − × V − q→0 Z dr dr ψ(r ,..., r , r ,..., r ) 2. n+1 · · · N | 1 n n+1 N | We see that Z Z The pair correlation function (like the higher rank 1 ∂N lim χ(q)= . functions) can be generalized for nonhomogenous q→0 V ∂µ T,V systems, for example, in a pure state we have It follows from Maxwell’s relations that ρ(r)ρ(r′)g(r, r′)= ∂N 1 2 ′ 2 = N κT , N(N 1) dr3 drN ψ(r, r , r3,..., rN ) . ∂µ V − · · · | | T,V Z Z where κT is the compressibility Compressibility 1 ∂V In the classical limit the density-density response function κ = . T −V ∂p is T,N ′ ′ Thus we obtain χ(r, r ) = β δρˆ(r)δρˆ(r ) 2 h i lim χ(q)= ρ κT = β (ˆρ(r) ρ)(ˆρ(r′) ρ) q→0 h − − i = β ρˆ(r)ˆρ(r′) βρ2 or h i− ′ 2 ′ 2 1+ ρ dr [g(r) 1] = ρk Tκ . = β ρδ(r r )+ ρ g(r r ) βρ − B T − − − Z or χ(r r′)= βρδ(r r′)+ βρ2[g(r r′) 1]. Fluctuation dissipation theorem − − − − We assume that fields a (t) are time dependent. Then Its Fourier transform α also the Hamiltonian −iq·r χ(q)= dr e χ(r) Hˆ (t)= Hˆ Aˆ a (t) 0 − α α Z α is X depends on time. χ(q)= βρ + βρ2 dr e−iq·r[g(r) 1]. − Let i Hˆ t − i Hˆ t Z Aˆ(t)= e h¯ 0 Aeˆ h¯ 0 The structure function S(q) is defined so that be the Heisenberg picture of the operator Aˆ. We use the 1 S(q) = δρˆ(q)δρˆ( q) notation N h − i 0 = Trˆρ h· · ·i 0 · · · = 1+ ρ dr e−iq·r[g(r) 1]. − for expectation values in undisturbed states. Z 6 It can be shown that in the linear limit one gets the left side, Cαβ(ω), describes spontaneous • fluctuations of the system. δA (t) Tr δρˆ(t)Aˆ α ≡ α ∞ it can be shown that an external field oscillating with = dt′ χ (t t′)a (t′), • αβ − β the frequency ω loses energy with the power β Z−∞ ′′ X ωχαβ(ω), i.e. the right side is associated with where dissipations. i 0 χ (t t′)= θ(t t′) Aˆ (t), Aˆ (t′) . skip until 10.4 αβ − ¯h − α β Dh iE Pair correlation function and equation of state Because We consider homogenous matter. According to the δA,ˆ δBˆ = Aˆ Aˆ , Bˆ Bˆ = A,ˆ Bˆ , definition of the pair correlation, − − h i h D E D Ei h i ρ2g(r r′)= δ(r rˆ )δ(r′ rˆ ) , we can in fact write − h − j − j i i6=j i 0 X χ (t t′)= θ(t t′) δAˆ (t),δAˆ (t′) , αβ − ¯h − α β we have in a classical system Dh iE i.e. the response function depends only on the fluctuating 2 ′ 1 ′ −βH parts of the operator. ρ g(r r ) = dΓδ(r ri)δ(r rj )e − ZN − − The Fourier transform with respect to time is defined as Xi6=j Z ∞ 1 iωt = dr1 dri−1 χαβ(ω)= dt e χαβ(t). QN · · · × i6=j Z Z Z−∞ X The inverse transform is then dr dr i+1 · · · j−1 × ∞ Z Z 1 −iωt χαβ(t)= dω e χαβ(ω). −β vkl 2π drj+1 drN e k 7 like or the internal energy is ′2 p i ′ HL(1+ǫ) = + v((1 + ǫ)rij ) 3 2 2 2m(1 + ǫ)2 E = NkBT +2πVρ dr r v(r)g(r). i i Since the new, primed, coordinates span the same range N(N 1) −β v(rij ) = − dr3 drN e i 8 (3) when r3 moves far from the points r1 ja r2. Since G is We divide the graphs in the expansion of B(r) into two symmetric with respect its arguments one can write classes: (3) G (r1, r2, r3)= nodal graphs are such diagrams that can be splitted • into two or more unconnected parts by cutting them 1 (2) (2) (2) G (r1, r2)G (r2, r3)G (r3, r1). at some black point. ρ3 This is know as the Kirkwood approximation or as the superposition approximation One can also derive diagram expansions for the pair correlation. Since g(r) is a non negative function it can be written as g(r)= e−βv(r)+B(r). We define the graphical elements: bridge or elementary diagrams cannot be separated in • parts by cutting them at any black point. r free variable r ◦ ⇔ r dr • ⇔ ′ Z r r h(r, r′)= g(r r′) 1. ⇔ − − ò d r 1 ) r ( h 1 r 1 1 ) ' r , We rewrite the pair correlation as h ( r , r r r ' g(r)= e−βv(r)+N(r)+E(r), ′ = dr1 h(r, r1)h(r , r1). where N(r) is the sum of the nodal diagrams E(r) the Z sum of the elementary diagrams. The HNC In the relevant graphs (HyperNetted Chain) approximation assumes that E(r) is ′ there are two white points r and r together with insignificant, i.e. • ◦ ◦ one or more black points, ri . • g(r) e−βv(r)+N(r). ≈ there is no direct link ( ) from one white point to • the other white point. It can be shown that the nodal diagrams can be summed. They satisfy the Ornstein-Zernike relation there is a path from every point to every other point, • i.e. they are connected. N(r)= ρ dr′[g( r r′ ) 1 N( r r′ )][g(r′) 1]. | − | − − | − | − Z One can show that B(r) is the sum of all these graphs. Via Fourier transformation we end up with the algebraic relation (S(k) 1)2 N˜(k)= − , S(k) where S(k) is the structure function and Thus this graph expansion of B(r) depends on the pair N˜(k)= ρ dr e−ik·rN(r). correlation g(r). Provided that we can sum the graph Z expansion, we can solve g iteratively: Jastrow’s theory 1. guess g(r). Although the previous cumulant expansion and 2. evaluate B(r) using the graph expansion. approximative methods for the pair correlation are valid only for a classical system it turns out that these methods 3. new g(r) is now are useful also in quantum mechanical systems. We consider the ground state of N identical particles (the −βv(r)+B(r) g(r)= e . temperature is T =0), so the system is in a pure quantum state Ψ. 4. if the new and old ones deviate from eachother We assume that, due to the interactions, the particles are remarkably we continue from 2. strongly correlated, i.e. the independent particle model is 9 not applicable. A good guess for the ground state wave we get for the expectation of the kinetic energy function is then the function, known as the Jastrow trial, N ¯h2 T = 2 h i − 2m ∇i Ψ(r1,..., rN )=Φ(r1,..., rN ) f( ri rj ). * i=1 + | − | X 1≤i 10 Since the functions N(r) and g(r) are related by the In the Jastrow theory the excited states are constructed Ornstein-Zernike relation explicitely. For example let every particle in the system have the momentum N(r)= ρ dr′ [g(r′) 1 N(r′)][g( r r′ ) 1], • − − | − | − ¯hk, i.e. Z k rˆ one can take the energy ǫ as a functional of the pair excite the particle i with the operator ei · i . • distribution g(r) only. It turns out that in fact a more every particle is excited with the same phase, i.e. convenient variable is g(r), so • if the ground state is Ψ0 the excited state is pǫ = ǫ[√g]. • N ik·rˆi As well known, the ground state wave function is that Ψ, Ψk = e Ψ0 "i=1 # whose expectation value X =ρ ˆ(k)Ψ0, 1 H = dr dr Ψ∗HΨ h i Ψ Ψ 1 · · · N where ρˆ(k) is the Fourier transform of the density h | i Z operator is minimized. We now seek the minimum of the N ρˆ(r)= δ(rˆ r). expectation of the Hamiltonian among all the functions of i − i=1 Jastrow form. Equivalently, find out such a √g, that the X energy ǫ[√g] attains its minimum. A condition for the One can show that for bosons this kind of collective existence of the extremum is that the variation excitation Ψk in the long wave length (small wave vector k) limit is energetically most favorable. δǫ = ǫ[√g + δ√g] ǫ[√g] − The excitation energy can be obtained evaluating the expectation vanishes up to linear order in δ√g. A straightforward calculation shows that Ψ H Ψ E = H = k k . k Ψk h | | i h i Ψk Ψk δǫ = dr L[ g(r)] δ g(r), h | i A straightforward calculation shows that Z p p where ¯h2k2 E = E + , k 0 2mS(k) ¯h2 L[ g(r)] = 2 g(r)+ v(r) g(r)+ W (r) g(r). −2m ∇ when E0 is the energy of the ground state Ψ0. The p p p p excitation energy is thus In order δǫ to vanish independent on the variation δ√g, the coefficient L must vanish, i.e. ¯h2k2 ǫ = E E = . k k − 0 2mS(k) ¯h2 2 g(r)+ v(r) g(r)+ W (r) g(r)=0. −2m ∇ These kind of excitations and corresponding excitation p p p energies are called Bijl-Feynman ecitations. The function W (r) is the so called induced potential. Its Fourier transform is 10.3. Density fluctuations and correlation length W (k) = ρ dr e−ik·rW (r) Let’s consider the canonical partition sum Z ˆ 2 2 2 −βFN −βH ¯h k (S 1) (2S + 1) ZN = e = TrN e , = − . − 2m S2 where FN is the free energy. We divide the volume into Although the above Euler-Lagrange equation for √g looks elements Vα, whose particle numbers are like a Schr¨odinger equation at 0 energy it is Nα =0, 1, 2,.... strongly nonlinear since the induced potential W Let δ(Nˆ ,N ) be an operator satisfying • depends (nonlinearly) on the structure factor S, α α which in turn depends via the (linear) Fourier N , if Nˆα N = Nα N 2 δ(Nˆα,Nα) N = | i | i | i transform on (√g) . | i 0, if Nˆ N = N N , α | i 6 α | i solvable only numerically. There are several solution i.e. δ(Nˆ ,N ) is the Kronecker delta function. The • methods but they all are iterative. α α identity operator operating in the volume element α can be written as an equation for the ground state only. Even if there ∞ • are more solutions the solutions associated with other Iˆα = δ(Nˆα,Nα). energies have no physical meaning. NXα=0 11 The identity operator of the whole system can be written, the reduced free energy is the volume integral of the • for example, as free energy density fN . ∞ the free energy density at every spatial point depends ˆ ˆ ˆ • I = Iα = δ(Nα,Nα) only on the local particle density and its low order α α " # Y Y NXα=0 derivatives at that particular point. ˆ = δ(Nα,Nα). Thus the energy functional of the system is α {XNα} Y ˜ Here stands for the summation over all possible FN [ρ]= dr fN [ρ(r), ρ(r), ρ(r),...]. {Nα} ∇ ∇∇ configurations, i.e. Z P If there is an external potential u(r), there is the ∞ ∞ additional term dr u(r)ρ(r) in the functional. As we [ ]= [ ]. · · · · · · · · · noted above the most probable configuration corresponds {Nα} N1=0 N2=0 R X X X to the minimum of the reduced free energy. We restrict to The partition sum is now homogenous systems so that the constant density ρ0 minimizes the functional F˜N [ρ]. Let −βHˆ ZN = TrN e −βHˆ δρ(r)= ρ(r) ρ0 = TrN δ(Nˆα,Nα)e − α be a small deviation from the constant density. The {XNα} Y −βHˆ simplest model for the variation of the energy functional is = Tr δ(Nˆα,Nα)e {Nα} α F˜ [δρ]= X Y N ˜ −βFN 1 1 = e . dr f + f (δρ(r))2 + f ( δρ(r))2 , 0 2 1 2 2 ∇ {XNα} Z Here where f0, f1 and f2 are constants independent on the r ˜ ˆ position (but can depend on the temperature and the −βFN ˆ −βH e = Tr δ(Nα,Nα)e constant density ρ0). In the expansion α Y −βHˆ there is no linear term in the variation δρ, since = Tr{N }e , • α according to the hypothesis ρ0 minimizes the energy. where Tr means that in the evaluation of the trace {Nα} due to the minimum condition the coefficients f1 and the summation is over all microscopical degrees of • f2 must be positive. freedom keeping, however, the particle numbers Nα constant and fixing for the total number the gradient term ( ρ)2 favors slowly varying • densities, so the wave∇ lengthts of the density N = Nα. fluctuations cannot be arbitrary short. At points r α ′ X and r close to each other the deviations δρ(r) ja ′ The function δρ(r ) are roughly the same. physically the gradient term can be motivated by the F˜N = F˜N (T,V, Nα ) { } • tendency of the stochastic thermal motion to smooth is the free energy or the reduced free energy of the down the density differences in close by volume configuration Nα . elements. Thus the factor f2 depends on the { ˜} The quantity e−βFN is proportional to the probability for correlations of the particles in volumes close to each the configuration Nα . Thus the most probable other. configuration is such{ where} the reduced free free energy Since the particle number is constant we have F˜ (T,V, N ) attains its minimum. N { α} Density functional theory δN = dr δρ(r)= δρ(q =0)=0, In the continuum limit the configuration N is { α} Z described by the density ρ(r) and the reduced free energy and, with the help of the Fourier transform, the free will become a functional of the density: energy can be written as ˜ ˜ 1 FN = FN [ρ]. ˜ ˜0 ′ 2 FN = FN + (f1 + f2q )δρ(q)δρ( q), 2V q − Now all the microscopical degrees of the freedom are X reduced to the single density distribution. This kind of ′ where q means that the term q =0 is not to be model is call the density functional theory. Normally the summed. Since the variation δρ(r) is real its Fourier P reduced free energy cannot be calculated exactly. A transform satisfies phenomenological method is the Local Density Approximation, LDA): δρ( q)= δρ(q)∗, − 12 so Note: The results are characteristically qualitative δρ(q)δρ( q)= δρ(q) 2. because they are derived using a nonmicroscopic model. − | | The physical meaning of this term is that δρ(q)δρ( q) , Scattering in medium as we recall, describes density correlations.h − i We consider the scattering of photons or massive particles the probability for the fluctuation δρ is now in a medium. One can show that the intensity of the elastic scattering is proportional to the structure factor, ˜ P [δρ] e−βFN i.e. ∝ 1 1 I(k, q) S(q)= δρˆ(q) δρˆ( q) . exp ′ f + f q2) δρ(q) 2 . ∝ N h − i ∝ −2k T V 1 2 | | " B q # Here k is the wave vector of the incoming particle and q X its change due to the scattering, i.e. the wave vector of the scattered particle is Correlation length Since the distribution P [δρ] derived above is of Gaussian k′ = k q. shape one can directly read from it the correlation function − Since the scattering is elastic we have k T V q q B δρ( )δρ( ) = 2 ′ h − i f1 + f2q k = k . | | | | kBT V 1 = 2 2 , The intensity of the inelastic scattering in turn is f2 q + qc proportional to the dynamic structure factor: where 2 f1 I(k; q,ω) S(q,ω), qc = . ∝ f2 where q the change in wave vector and ¯hω in the energy. The density-density response χ(q) was defined so that When the temperature approaches the critical point from β above the isothermal compressibility κT diverges, i.e. an χ(q)= βρS(q)= δρ(q)δρ( q) , V h − i infinitesimal change in the pressure causes an finite change in the volume. Then when S(q) is the structure factor. So we get 1 f1 = 0. 1 1 ρ2κ critical−→ q T point χ( )= 2 2 . f2 q + qc On the other hand, there is no reason to assume that, for Its inverse Fourier transform is example, the correlations would become independent on 1 1 the wave vector at the critical point, as would happen if χ(r)= e−r/ξ. f2 4πr ξ2 f2 = 0. ρ2κ critical−→ The parameter T point 1 f ξ = = 2 That’s why we can assume the correlation length ξ qc sf1 diverges at the critical point. is the correlation length. Consider elastic scattering of light. When the scattering 2 Since we had limq→0 χ(q)= ρ κT we must have angle is θ the change in the wave vector is θ 1 q =2k sin , f1 = 2 , ρ κT 2 so the wave length being ξ2 f2 = 2 . 2π ρ κT λ = . k The pair correlation h(r)= g(r) 1 can be written with the help of the density-density response− (excluding the We see that the intensity is autocorrelation proportional to δ-function) as 1 1 I(θ) 2 2 . 1 ∝ f1 + f2q ∝ 2 h(r)= χ(r). sin θ + λ βρ2 2 4πξ We see that Then at the critical point 1 kBTκT 1 −r/ξ I(θ) , h(r)= 2 e . 2 θ ξ 4πr ∝ sin 2 13 i.e. the scattering intensity is strongly peaked at forward Heisenberg’s model directions and the total cross section ( dΩI(θ)) In the external field B0 the Hamiltonian according to the diverges. Thus the radiation cannot pass∝ through the Heisenberg model is R medium: in the vicinity of the critical point tranparent 1 matter becomes opaque. The phenomenon is called the H = Jij si sj γB0 si −2 · − · ij i critical opalescence. X X when the magnetic moment of the particles is 10.4. Discrete interaction models We first consider interaction between atomic spins in a µi = γsi. solid. Assuming that the atoms are bound to their lattice We use notation σi = s = s,s,...,s , Hspin = H(s1, s2,..., sN )= H( si ) | i | i { } i 1 −βH O = ln Tr s e . −β { i} where the spins at all lattice points are parallel is the ground state. Here Tr{si} means that the trace is evaluated keeping the Let z be the coordination number of the lattice (the spin configuration fixed. The total partition function is number of nearest neighbours at each lattice point). For example, in the 3-dim. cubic lattice z =6 and in 2-dim. Z = Trspine−βH({si}), square lattice z =4. It is easy to see that the ground state energy is z where now the trace is over spins. In the spin model E = N Js2. 0 − 2 the most important interactions are between nearest Since the scalar product s s is invariant under • i j neighbours. rotations the Hamiltonian of· the system is also rotationally invariant. The ground state the interactions are associated with the links • connecting the lattice points. does not obey the symmetry of the Hamiltonian. It is • said that a spontaneous symmetry breaking has the spins associated with the lattice points are the occured. • dynamical variables. is degenerate: rotating all spins equally we end up • with a state with the same energy. s s i j Antiferromagnetic coupling J < 0 J i j The interaction favors neighbours with opposite spins. Assuming that opposite configurations were possible for all nearest neighbours the classical ground state energy were z E = N Js2. In some cases it is possible to construct spin models also 0 2 for continuum systems by discretizing the field variables. This kind state of alternating spins, We denote the lattice points by i,j,.... If the spin quantum number of the particles in the model is s, the σ = s = s, s,s,... , | i ± i | − i state sum is i O is, however, not a quantum mechanical eigenstate of the Z = e−βH({si}) operator {Xσi} s s H = J si sj = e−βH({si}). − · · · · 14 since the spin pairs are not connected to eigenstates of the that the energy of each link is minimized the last spin operator direction will differ from the one we started with. That’s 2 2 sij = (si + sj ) . why all interactions cannot be minimized simultaneously and the ground state cannot be determined. The correct eigenstate can be solved only in the one dimensional system (so called Bethe’s Ansatz method). ? 6 7 1 2 Ising’s model We simplify the Heisenberg model by restricting the spin 1 quantum number to the case si = 2 and taking into 5 4 3 account only the z components. Then H = J σ σ h σ , XY model − i j − i 15 We approximate the ice structure with two dimensional 11. Phase transitions square lattice. There are 6 possible link configuration for each lattice point. We have a so called 6 vertex model. 11.1. Lee-Yang theory A phase transition happens at an exactly determined temperature which depends on the density, pressure and other intensive properties of the system. Since the state variables behave differently on each side of the transition point the partition sum must be non analytic at the transition point. The energy spectrum E of finite { n} Let θi be the ice condition for the lattice point i: number particles in a finite volume is discrete so the partition function 1, condition satisfied θi = −βEn 0, condition not satisfied. ZN = e n A suitable Hamiltonian for the system is such that the X energy of the forbidden configurations is infinite, e.g. is a positive and, on the positive real axis β > 0 and in the neighbourhood of it, an analytic function of its argument β. In this kind of a system there can be no H = lim U(1 θi). U→∞ − i sharp phase transition point. Phase transition can thus X occur only in the thernodynamic limit where Now the energy of an allowed configuration is zero. One N can also associate different energy ǫk with each vertex V and N but ρ = constant. type k. The the total energy of the lattice, in an allowed → ∞ → ∞ V → configuration, is 6 The model by Lee and Yang explains how the analytic partition function develops toward non-analytic form E = Nkǫk. k=1 when we approach the thermodynamic limit. We consider X a system of hard spheres confined in the volume V . Let V0 Here Nk is the total number of the k type verteces. be the volume of one sphere. Then One can easily see that the state sum is V 6 −β Nkǫk Nm Z = e k=1 θi. ≈ V0 {σij } i X P Y is the maximum number of spheres. The partition function Nm N ZG(T,V,µ)= z Z(T,V,N) NX=0 is a polynomial of degree Nm of the fugacity z = eβµ. We use the shorthand notation Z(z)= ZG(T,V,µ). Let ξ1, ξ2,...,ξNm be the zeros of the polynomial Z(z). Since Z(0) = 1, we have, according to the fundamental theorem of algebra, N m z Z(z)= 1 . − ξ n=1 n Y Because Z(z) is real when z is real the zeros must occur as conjugate pairs, i.e. for every root ξn there must be the ∗ root ξn. When we approach the thermodynamic limit the number of zeros of the partition function Z(z) tends to infinity. One can assume that the real axis remains clean of the zeros excluding, maybe, some separate points. In the vicinity of those points the density of zeros is very high and the function Z(z) non-analytic. 16 Let’s assume that the zeroes of the partition function Z(z) We are thus dealing with a typical first order phase close to the real axis condense on the curve C. The transition where the density jump is function Z(z) is analytic on both sides of the curve but its 2πz analytic properties are different on different sides. When ∆ρ = 0 . the zeros lying on the curve C condense to continuum we v0 can write z ln Z(z) = ln 1 11.2. Ising model − ξ n n X Practically the only exactly solvable models are the one z and two dimensional models by Ising. dξ w(ξ) ln 1 . → − ξ We consider one dimensional chain of spins ZC Here 1↑2↑3↓4↑ N↓ , dn = dξ w(ξ) · · · is the number of zeroes on the arc dξ of the curve. The where we apply periodic boundary conditions, i.e. we set density w(ξ) is N V and so an extensive quantity. ∝ m ∝ From this expression for the partition function one can σN+1 = σ1. clearly see that Z(z) is not analytic if z happens to lie on the curve formed by the zeroes. As an example we The Hamiltonian operator of the system is then consider the partition function which in the vicinity of z0 behaves like N N H = J σiσi+1 h σi π − − Φ(z) i=1 i=1 Z(z) e cosh (z z0) , X X ≈ b − N−1 N h i = J σ σ Jσ σ h σ , where Φ(z) is analytic. The zeroes of the partition − i i+1 − N 1 − i i=1 i=1 function are then at the points X X where each spin variable can attain the values 1 ξn = z0 + ib n + , n =0, 1, 2,.... 2 ± ± σ = 1. i ± Since π The partition function is ln Z(z)=Φ(z)+ lncosh (z z ) b − 0 N N h i βJ σiσi+1+βh σi is extensive the argument π/b(z z ) must be extensive. Z = e i=1 i=1 0 · · · − σ σ σ The only possibility is that 1/b V . We denote X1 X2 XN P P ∝ N v βJσ σ + 1 βh(σ +σ ) b = 0 , = e i i+1 2 i i+1 . · · · V σ σ σN i=1 X1 X2 X Y so in the thermodynamic limit We define the 2 2 transition matrix T so that × V or b 0 but v0 = constant βJσσ′+ 1 βh(σ+σ′) → ∞ → Tσσ′ = e 2 , we get where σ, σ′ = 1. The partition function can now be ± 1 1 1 πV written as ln Z = Φ(z)+ ln cosh (z z ) V V V v − 0 0 1 1 1 π Z = Tσ1σ2 Tσ2σ3 TσN σ1 Φ(z)+ ln + (z0 z), z 17 are real. Let S be a orthoganal matrix diagonalizing T In the weak field limit h 0 we get then (composed of the eigenvectors of T ), i.e. → ∂σ 1 2J χ = = e kB T . + ∂h k T −1 λ 0 h=0 B T = S − S. 0 λ When the coupling is ferromagnetic (J > 0) the system Now magnetizes strongly at low temperatures. When the (λ+)N 0 external field is removed the system returns to the T N = S−1 S − N disordered state σ =0: there is no spontaneous symmetry 0 (λ ) ! break. and, due to the cyclic property of the trace, If the coupling is antiferromagnetic (J < 0) the polarization is damped exponentially. + N N −1 (λ ) 0 The one dimensional Ising chain is thus a paramagnetic Z = Tr T = Tr SS − N 0 (λ ) ! system without any phase transitions. However, since it N N does not obey Curie’s law it is not a Curie paramagnet. = λ+ + λ− . Two dimensional Ising model can be solved exactly The logarithm of the partition function is generalizing the transition matrix method (Onsager, 1944). It turns out that in this case there is a phase N N ln Z = ln λ+ + λ− transition at the temperature h i N 2J λ− T = 2.269 J. = N ln λ+ + ln 1+ . c √ ≈ λ+ ln(1 + 2) " # The specific heat diverges logarithmically at the critical Since in the thermodynamic limit, N , → ∞ point T = Tc and the phase transition is continuous. λ− N Monte Carlo methods 0, λ+ → Because, in general, interacting systems can not be solved analytically numerical methods are of great value. An holds we get important class of numerical methods, Monte Carlo methods, handles interacting systems using stochastic 1 λ− N lim ln Z N ln λ+ + N ln λ+. simulations. Suitable simulations for continuum systems, N→∞ → N λ+ N→→∞ like 3He- 4He-liquids and electron gas, are mostly based " # on Green’s function Monte Carlo. In discretized systems Just like in the free spin system the free energy is one can often apply Metropolis’ Monte Carlo method: interpreted as the magnetic Gibbs function. Its value per Let the possible configurations of the system be spin is • G k T j J = 1, 2,...,K = B ln Z ∈ { } N − N and E(j) the corresponding energies. = J − Form a chain j1, j2,...,jn of configurations. k T ln cosh(βh)+ sinh2(βh)+ e−4βJ . • − B Choose the next configuration, (n +1)’th, in the q • The equilibrium values of other thermodynamic variables chain drawing randomly from the set J of the possible ′ can be calculated from the Gibbs function. In particular, configurations. The drawed configuration, j , will be the average of the spin variable is – accepted if ∆E = E(j′) E(j ) < 0. − n 1 ∂ ln Z ∂G/N – accepted with the probability e−β ∆E if σ σ = = ∝ ≡ h ii Nβ ∂h − ∂h ∆E > 0. sinh(βh) When the length N of the chain j increases = . • { n} sinh2(βh)+ e−4βJ (N ) the probability for each configuration j approaches→ ∞ q The expectation value σ is an order parameter of the P (j) e−βE(j). ∝ system: σ =0 corresponds to completely stochastically The chain is thus a canonical ensemble which can be oriented spins whereas σ =1 corresponds to the case • where all spins are ordered| | themselves parallely. The used to evaluate expectation values. order parameter σ is analogous to the magnetization M of Note: The method assumes that the energy eigenstates the free spin system when h corresponds to the magnetic of the system are known. So it can be applied for handling field H. The susceptivity is analogically of e.g. Ising models and all classical systems. ∂M ∂σ If the energy states are unknown the quantization must be χ = . included in the simulation. ∂H ⇔ ∂h 18 12. Critical phenomena δ tells how the order parameter depends on the • In a second order phase transition the system normally external field h at the critical temperature T = Tc: goes from a higher temperature phase to a lower 1/δ temperature phase with less symmetry. We say that a m(Tc,h)= Kh . symmetry is broken spontaneously. For example, ferromagnetic material will get polarized below a certain ν determines the dependence of the correlation length critical temperature. The spin rotation symmetry is • on the temperature, broken. The amount of the symmetry break is described by an order parameter, which is usually assigned to the ξ = K τ −ν . expectation value of some observable of the system. In the | | ferromagnetic system a suitable order parameter is the The index ν is not actually a thermodynamic quantity magnetization m. In the symmetric phase m =0 and in since it is related with the microscopical parameter ξ. the ordered, i.e. symmetry breaking phase m =0. Let the order parameter be m and h the external6 field coupling to the corresponding observable. We consider the 12.2. Scaling theory system close to the critical point T = Tc. When we denote A scalable equation is such that it remains invariant under the scale transformations if the units of τ = T T , − c measurements are selected by scaling them properly. As the critical point is at origin of the (τ,h)-plane. an example we consider Navier-Stokes’ equation of flow: Since the critical point is a singular point of ∂v 1 thermodynamic potentials we divide them into regular and + (v )v = f p + ν 2v, singular parts. For example, we write ∂t · ∇ − mρ ∇ ∇ F (T,m) = F0(T,m)+ Fs(T,m) where v is the velocity, ρ the density, p the pressure and f G(T,h) = G (T,h)+ G (T,h)= F hm, the force. The coefficient 0 s − η where the functions F0 and G0 are regular at the vicinity ν = of the point (τ =0,h = 0) whereas the functions Fs and mρ G are singular there. Their differentials are s is the kinematic viscosity and η the viscosity. Let T , L, V dF (T,m) = S dT + h dm and M be the dimensional units of the time, length, − dG(T,h) = S dT m dh. velocity and mass. With the help of the corresponding − − measures t′, r′, v′ and m′ (for example the mass is m = m′M) the Navier-Stokes equation takes the form 12.1. Critical exponents Close to the critical point the singular parts are (with ∂v′ 1 1 + (v′ ′)v′ = f′ ′p′ + ′2v′. great accuracy) proprtional to some powers of the ∂t′ · ∇ − m′ρ′ ∇ R ∇ thermodynamical quantities τ and h. The critical exponents or the critical indeces are defined as follows: The parameter R is the dimensionless Reynolds number ′ α, α determine the singular part of the specific heat L2 VL mρV L • R = = = , so that T ν ν η ∂2G Ch = T < − ∂T 2 which characterizes the flow: if R 10 . . . 100, the flow is h > ∼ −α usually laminar and if R 10 . . . 100, it’s turbulent. K τ , when T >Tc ∼ = ′ Looking at the Navier-Stokes equation written using the K′ ( τ)−α , when T 19 If we want the systems to be similar the Reynolds number The scaling dimension of the correlation length is then must stay invarinat. For example obviously dξ = 1. Further, we see that the scaling dimension of the− quatity AxBy is L2 L2 s2L2 · · · R = = 1 = , T ν T ν s T ν d(AxBy ···) = xdA + ydB + . 1 1 T · · · so the scaling factor of the time sT must be Because the critical index ν was defined so that s = s2. ξ τ −ν , T ∝ | | Let A and A1 be some dimensional units of measure the scaling dimension of the temperature is corresponding to similar systems S and S . It turns out 1 d = ν−1. that all scaling laws are of the form τ The scaling dimensions of the most important quantities A = s−da A, 1 are where is da is a rational number. Correlation length ξ: We saw above that • Scaling hypothesis d = 1. In dense matter (liquid, solid, . . .) ξ − the microscopical length scale is determined by the Temperature τ: We had • • distance between atoms or molecules. 1 dτ = . when macroscopic properties are considered the ν • microscopic structure is invisible. Length ℓ: Since the correlation length determines the • the only macroscopically essential parameter related length scale of the system ℓ scales like ξ i.e. • to microscopical properties is the correlation length ξ, d = 1. because in the vicinity of the critical point it grows ℓ − macroscopically large. Wave vector q: The wave vector is inversely We can thus assume that when we approach the critical • proportional to the length so point the classical similarity will hold: dq =1. Consider two systems of same material with • correlation lengths ξ and ξ1. Order parameter m: The index β was defined so that • β The correlation length tells the scale of the m ( τ) , so • ∝ − fluctuations, i.e. the scale of structure of the matter β d = βd = . (provided that we cannot observe the atomic m τ ν structure). Free energy G: Scale transformation do not affect the When the systems are observed using such • • free energy of the system, so magnifications that ξ and ξ1 seem to be of equal length (and possibly adjusting sampling frequencies) dG =0. no differences between the systems can be found. Free energy per volume element g: Since g = G/V , Since the correlation length at the critical point is infinite • we have all sizes of fluctuations related to the order parameter are d = d d = dd = d, present. Except the atomic scale, there is no natural g G − V − ℓ measure of length in the system. Thus the system looks when d is the spatial dimension. similar no matter what scale is used, the system is self Specific heat c: Since the specific heat (density) is similar. The self similarity assumption is formulated • mathematically as the scaling hypothesis: ∂2g c Tc , The singular parts of all thermodynamic potentials ≈− ∂τ 2 • scale as exponential functions of the correlation the scaling dimension dc satisfies the condition length ξ only. 2 d = d 2d = d . The quantity A behaves in the vicinity of the critical c g − τ − ν • point like A ξ−dA , Thus we have for the specific heat ∝ where d is the scaling dimension of A. c ξ(2/ν)−d τ dν−2 A ∝ ∝ | | 20 and, according to the definition of the critical index where we have again separated the regular and singular α, parts from eachother at the singular point (τ =0,h = 0). c τ −α. Its differential is ∝ | | Hence the critical indeces are related via the scaling dg = s dτ m dh, law − − α =2 νd. when s = S/N is the entropy per primitive cell (or − particle). Since the order parameter m is zero at Field h: In an equilibrium the order parameter m is temperatures above the critical point the critical exponents • related to it must come from the singular function gs. ∂g m = , The functio f is a generalized homogenous function if it −∂h satisfies the condition so f(λα1 x , λα2 x ,...)= λf(x , x ,...). d = d d . 1 2 1 2 m − h For the field h we thus have According to Widom’s hypothesis the function gs behaves in the vicinity of the critical point like a generalized β homogenous function, i.e. it scales like dh = d dm = d . − − ν p q gs(λ τ, λ h)= λgs(τ,h), The suceptivity obeys according to the definition of the when λ> 0 and p and q are system independent index γ the relation exponents. ∂m Because the scaling equation holds for all positive values χ = τ −γ , −1/q q ∂h ∝ | | of λ it holds when λ = h , in which case λ h =1. Thus we can write the scaling hypothesis as so τ dm dh = γdτ . g (τ,h) = h1/qg , 1 − − s s hp/q Comparing the dimension obtained from this for d with τ h = h1/qφ . our earlier result we end up with the scaling law hp/q γ = νd 2β. Here we have defined − φ(x)= g (x, 1). Further, the index δ was defined so that at the critical s temperature The critical indeces can be obtained as follows m h1/δ. ∝ β. Take the derivative of the scaling equation Then we have • dh p q d = , gs(λ τ, λ h)= λgs(τ,h), m δ from which, using our earlier result, we get the scaling law with respect to the field h and recall that the order parameter is νd δ = 1. ∂g (τ,h) β − m(τ,h)= s , − ∂h The hard to measure index ν, which is related to the so we have the scaling condition microscopical correlation length, can be eliminated from the three scaling laws derived above. We are left with λqm(λpτ, λqh)= λm(τ,h). ( ) scaling laws relating thermodynamic indeces: ∗ Since the order parameter is supposed to be m =0, α +2β + γ = 2 we have τ < 0. Then we can choose λ so that 6 β(δ 1) = γ. λpτ = 1. Setting h =0 we get − − Note: Although the scaling laws derived above are based m(τ, 0)=( τ)(1−q)/pm( 1, 0). − − on phenomenological arguments they are valid within experimental accuracy. According to the definition we have m(τ, 0) ( τ)β , Widom scaling ∝ − We consider the Gibbs function per primitive cell (or so 1 q particle) β = − . g(τ,h)= g0(τ,h)+ gs(τ,h), p 21 δ. We set in the equation ( ) τ =0 and λq =1/h, so Combine the original microscopical state variables • ∗ • blockwise to block variables. m(0,h)= h1/q−1m(0, 1). Determine the efective interactions between the According to the definition m(0,h) h1/δ, so • ∝ blocks. This coarsing of the system is called the block q transform. δ = . 1 q − The block transforms form a semigroup, so called • renormalization group. One can perform transforms γ,γ′. According to the definition the susceptivity is • sequentially. ∂m(τ,h) χ(τ,h)= , ∂h Because in a system at the critical point there is no • natural length scale the transformed systems look which close to the critical point behaves like copies of eachother. Thus the critical point τ −γ , when τ > 0 corresponds to a fixed point of the transformations. χ ′ ∝ ( τ)−γ , when τ < 0. − We apply the method to the d-dimensional Ising spin Differentiating ( ) with respect to the field h we get system. ∗ 2q p q Block transform λ χ(λ τ, λ h)= λχ(τ,h). We denote by i,j,... the origininal lattice points and the Setting h =0 and λpτ = 1 we have blocks obtained by combining them by indeces I,J,.... ′ ± The block spin σI of the block I is defined so that χ(τ, 0) = τ −(2q−1)/pχ( 1, 0). | | ± ′ σI = σi. From this we can read for γ and γ′ Xi∈I 2q 1 γ = γ′ = − . If we end up with the blocks I by scaling the length p measure by the factor L, each block has Ld spins. Because in the Ising model each spin can get the values σi = 1 α, α′. The specific heat is ± • the block spin can get the values 2 ∂ g ′ d d d ch . σ = L , L +2,...,L ∝ ∂τ 2 I − − d Differentiating the scaling equation i.e. alltogether L +1 different values. Let H be the original Hamiltonian. We denote p q gs(λ τ, λ h)= λgs(τ,h) [σ ]= βH = K σ σ h σ . H i − i j − i twice with respect to τ we get i Xhiji X 2p p q λ ch(λ τ, λ h)= λch(τ,h). The state sum is We set h =0 and λpτ = 1 and compare the result Z = e−G = Tr e−H[σi] = e−H[σi], with the definitions of α and± α′: {Xσi} τ −α, kun τ > 0 ch ′ where = βG, G is the Gibbs function. We divide the ∝ ( τ)−α , kun τ < 0. G − trace summation into two parts We see that 1 Z = e−H[σi] α = α′ =2 . − p {σi} X It is easy to verify that the Widom scaling hypothesis = e−H[σi] δ σ′ , σ leads to the scaling laws I i {σ′ } {σ } I i∈I ! XI Xi Y X ′ α +2β + γ = 2 = e−H[σI ]. β(δ 1) = γ. {σ′ } − XI Here we have defined Kadanoff scaling theory ′ −H[σ ] −H[σi] Unlike the Widom scaling hypothesis the method developed e I = Tr ′ e {σI } by Kadanoff (1966) is based on the microscopic properties of matter. −H[σi] ′ = e δ σI , σi . Outlines of Kadanoffs method: ! {Xσi} YI Xi∈I 22 As can be seen from the definition variations of the scaled parameters are proportional to the relative variation of the scale, i.e. ′ −H[σi] [σI ] = ln Tr{σ′ }e H I δ∆K δL L = x the Hamilton block function [σ′ ] is actually the reduced ∆K L H I L free energy. Thus it can be written as δh δL L = y , hL L ′ [σ ]= [σi] TS ′ , I ′ {σI } H H {σI } − | where x and y are constants. When the change in the ratio is infinitesimal we get the where [σi] is the expectation value of the energy differential equations H {σ′ } I (the internal energy) evaluated in the block configuration ∂ ln ∆KL σ′ . Hence the Hamiltonian block function contains the x = { I } ∂ ln L internal entropy related to the internal variables of the ∂ ln h y = L , blocks. ∂ ln L Close to the critical point, due to the scale invariance, we assume the reduced free energy to take approximately the which after integration give same form as the original Hamiltonian. To achieve this x we scale the range of the block spins so that KL = Kc + L (K Kc) y − hL = L h. ′ σI = zσI , To obtain the same energy from the original and scaled where σI = 1. Because the maximum of the block spin is Hamiltonians the coupling to the external field must Ld we must± have z Ld. satisfy the condition ≤ Critical exponents ′ h σi = hσI = hzσI = hLσI , According to Kadanoff effectively the most important i∈I values of the block variable are z. We denote X ± so the field scales like hL = zh. We see that L[σI ]= [zσI ]. H H z = Ly and y d. ≤ We let the new Hamiltonian L to be of the same form as the original : H The same reasoning allows us to assume that the relative H deviation of the temperature from the critical point, L[σI ]= KL σI σJ hL σI . H − − T Tc hIJi I τ = − , X X Tc The parameters KL and hL depend now on the scale L. behaves like the relative deviation of the coupling constant Let the values of the parameters at the critical point to be K from the critical value, i.e. K = Kc x τL = L τ. h = hc =0. Thus the Gibbs function per spin unit scales like Since at the critical point nothing changes while scaling x y d we must also have g(τL,hL)= g(L τ,L h)= L g(τ,h), where the factor Ld is due to the fact that the new block KL = Kc contains Ld old spins. Writing hL = hc =0. x = pd We consider the neighbourhood of the critical point. We suppose that h =0, so also the corresponding scaled field y = qd satisfies h =06 . We write the original coupling constant L we end up with the Widom scaling. as 6 K = Kc + ∆K Renormalization group Let us suppose that the Hamiltonian depends on the and the corresponding scaled coupling constant as parameters H µ = (µ1, µ2,...), KL = Kc + ∆KL. For example µ = (K,h), as above. Block transforms are We now vary the scale factor L by a (small) amount δL. now mappings in the parameter space The relative variation of the scale is then δL/L. we can assume (as a good approximation) that the relative µ µ . −→ L 23 Let RL be the operator corresponding to the block transform, i.e. µ µ = R µ. −→ L L Since the block transform is a change in the scale we must have RLRL′ = RLL′ . Furthermore, it does not matter in which order the scale transforms are performed: RLRL′ = RL′ RL. In the block transform we loose information, for example the detailed knowledge of the values of the original spin variables. Thus it is impossible to return to the original system by scaling: the operation RL′ has no inverse transformation among the operations RL . We see that the operations { } = R R { L} form a commutative semigroup which is called the renormalization group. A point µ∗ which satisfies the condition µ∗ = Rµ∗ R , ∀ ∈ R is called a fixed point. A system corresponding to the parameter values µ∗ is at the critical point since any transformation of that systems results an exactly identical system. The set of those points that after sequential block transforms lead to the fixed point µ∗ is called the critical surface. If the system is on the critical surface, but not at the critical point, it is at a critical point of the phase transition. However, one can still observe it using the scale where microscopical details are visible. 24