The Ising model Summary of L12
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1 The Ising model Summary of L2 Aim: Study connections between macroscopic phenomena and the underlying microscopic world for a ferromagnet. How: Study the simplest possible model of a ferromagnet containing the essential physics: the Ising model. Objective: Gain qualitative understanding of the physics governing the phenomena and reveal possible universal behaviour. Collection of interacting spins s i = ±, i =, 2,..., N placed on a regular lattice of N sites r i. E si } = spin-spin interactions + spin-external field interactions = J s i s j H s i Ising model. ij Order parameter: The average magnetisation per spin mt, H) = p si }m si } = exp βe si })m si }, Z s i } s i } with m si } = N N s i and the partition function Z = s i } exp βe si }). The magnetisation in zero external field m T ) = lim H ± mt, H) for the Ising model in d 2. m T ) T/T c
2 The Ising model Summary of L3 Objective: Gain qualitative understanding of the phase transition in the Ising model: N lattice spins s i = ± with positive nearest-neighbour interaction J placed in an external field H, E si } = J ij s i s j H s i. In equilibrium, the spin system will minimise the free energy F = E T S. Assume zero external field H =. The order parameter is the average magnetisation per spin m T ) = lim H ± mt, H). When J/k B T ), the free energy is minimised by maximising the entropy. Spins are randomly orientated, m T ) =. When J/k B T ), the free energy is minimised by minimising the energy. Spins are aligning, m T ). m T ) H = H 2 > H 3 > H 2 T/T c The correlation length ξt, H) sets the scale of typical largest fluctuations away from the microstates with a) randomly orientated spins when T > T c b) fully aligned spins when T < T c. Trivially self-similar states with ξt, ) = at T = and T =. Non-trivially self-similar states with ξt c, ) = at T = T c.
3 The Ising model Summary of L4 Ising model in d = : Interacting spins s i = ± with pbc. E si } = J Z = s i } s i s i+ H s i e β [J N s i s i+ + H N 2 s i +s i+ )] = s i } T s s 2 T s2 s 3 T s3 s 4 T s4 s 5... T sn s N T sn s = λ N + + λ N Eigenvalues of T, i.e., T λi = F = k b T ln Z Nk b T ln λ + for N ) f sinhβh) mt, H) = =. H sinh 2 βh) + e 4Jβ T T 4 > T 3 mt, H).5 T 3 > T 2 T 2 > T T = - H χt, H) T 4 > T 3 T 3 > T 2 T 2 > T T = H
4 The Ising model Summary of L5 Ising model in d = : Interacting spins s i = ± with pbc. The spin-spin correlation function E = J s i s i+ H s i. witht the correlation length gr i, r j ) = s i s i ) s j s j ) ξ = ln[tanhβj)] = s i s j s i s j for T = = s i s i+r for T > for T = = exp r/ξ) for T >. for T 2 exp2βj) for T. The correlation function is related to the susceptibility per spin r j gr i, r j ) = k B T χ = M 2 2 M. N -domain 2-domains The free energy F = E T S = E k B T ln Ω so F 2-domains F -domain = 4J k B T ln NN ). A single domain of aligned spins is unstable against thermal fluctuations for finite T for large enough N since F 2-domains < F -domain. In the d = Ising model, T c =.
5 Ising Model Summary of L6 Mean-field approach ignores correlations between spins. E si } = NJzm 2 /2 Jzm + H) s i. Model of N noninteracting spins in an effective field Jzm + H, where each spin feels an average internal field Jzm from the z nearest neighbour spins in addition to the external field H. The partition function is readily calculated analytically Z = exp βnjzm 2 /2) [2 coshβjzm + βh)] N. The free energy per spin is a function of T and H, f = Jzm 2 /2 k B T ln [2 coshβjzm + βh)]. The magnetisation per spin minimises the free energy and satisfies m = tanhβjzm + βh) ). Letting T c = Jz/k B, Equation ) in zero external field reads m T ) = tanh T c T m T )) ). For T T c the solution m O T ) = is unique and stable. For T < T c the trivial solution becomes unstable but two new stable non-zero solutions appear for the first time, therefore for T T c m T ) = ± 3/T c T c T ) β for T Tc. The susceptibility per spin χt, ) = m/ H) T H= = Γ ± T T c γ± for T T ± c with exponents γ ± = and amplitudes Γ + = /k B, Γ = /2k B ). The magnetisation per spin at T = T c with ciritcal exponent δ = 3. mt c, H) signh) H /δ for H
6 The Ising model Summary of L7 Landau theory for the Ising model. powers of the order parameter m: Expanding the free energy per spin in f = f Hm + a 2 T T c )m 2 + a 4 m 4 a 2, a 4 >. The magnetisation m is determined by minimising the free energy, so it must T > T c T = T c T < T c satisfy the equation ) f m = implying T,H H + 2a 2 T T c )m + 4a 4 m 3 =. The magnetisation in zero external field β = /2) for T T c m T ) = ± a2 2a 4 T c T ) for T Tc. The susceptibility per spin in zero external field γ ± = ) ) m χt, ) = H = T T kb c) for T T c + T H= 2k T c T ) for T T B c. The magnetisation at T = T c in small external fields δ = 3) mt c, H) signh) H /δ for T = T c and H. The specific heat capacity in zero external field α ± = ) ) ɛ ct, ) = for T T c + T = 3 H H= 2 k B for T Tc.
7 The Ising model Summary of L8 Widom scaling ansatz for the magnetisation per spin mt, h) = t β M ± h/ t ) for t ± and h, where β and the so called gap exponent) are universal critical exponents and M ± universal scaling functions that must satisfy mt, h) = mt, h) mt, h) = ± t β M ± x) = M ± x) for t M ) = ±non-zero constant mt, h) = for t > M + ) = m, h) signh) h /δ M ± x) signx) x /δ for x ±, = βδ The susceptibility per spin so taking the limit h we find χt, h) = t β M ±h/ t ) = β + γ and M ±) = non-zero constants Widom scaling ansatz for the singular part of the free energy per spin f s t, h) = t 2 α F ± h/ t ) for t ± and h. and by taking derivatives with respect to the external field we find mt, h) t 2 α F ± h/ t ) for t ± and h χt, h) t 2 α 2 F ± h/ t ) for t ± and h. The Widom scaling ansatz for the free energy per spin and the correlation function see Exercise 2.5) implies scaling relations βδ = β + γ α + 2β + γ = 2 dν = 2 α γ = ν2 η) Widom scaling law Rushbrook scaling law Josephson scaling law Fisher scaling law. The exponents take the same value for t ±. There are only two independent critical exponents.
8 The Ising model Summary of L9 Defining the dimensionless reduced temperature t = T T c )/T c and external field h = H/k B T, the Widom scaling ansatz for the free energy per spin and the correlation function when t ±, h are f s t, h) = t 2 α F ± h/ t ) ) gr, t, h) = r d 2+η) G ± r / t ν, h/ t ). 2) The origin of scaling is intimately related to the existence of only one relevant length scale ξ which diverges at the critical point T c, ). Spins are correlated over scales up to ξ leading Kadanoff to introduce the idea of real-space transformation. Divide the system into blocks I each with b d spins. Coarse-grain system by replacing all spins in block I with a block spin S I. Rescale all length scales by factor b. The renormalisation implies N = b d N, t = b y tt, h = b y hh and ZN, t, h) = exp βe si }) = exp βe s I } ) = ZN, t, h ) si} si} s i } consistent with s I } The partition function is invariant but the free energy per spin satisfies ft, h) = b d ft, h ) = b d fb y t t, b y hh) for all b < ξ which by letting b = t /y t is equivalent with Equation ). Similarly, one can show the correlation function satisfies g r, t, h) = b 2β/ν g r, t, h ) = b 2β/ν g r /b, b y t t, b y h h) for all b < ξ which by letting b = t /y t is equivalent with Equation 2). Kadanoff block spin real-space renormalisation transformation gives heuristic explanation for the Widom scaling ansatz for the free energy and the correlation function.
9 The Ising model Summary of L2 Renormalisation in d =. The reduced energy for the Ising model βe si } = J k B T ij s i s j H k B T s i = K s i s j h ij s i. The partition function in zero external field, h = ) in d = : ZK, N) = N ) exp K s i s i+ ij = exp K [s s 2 + s 2 s 3 ]) exp K [s N s N + s N s ]) odd spins even spins = odd spins exp K + K s s 3 ) exp K + K s N s ) = expn K )ZK, N ), where the renormalised coupling constants and number of spins K = 2 cosh 2K K = 2 ln cosh 2K ) N = N/2. a) 2..5 K b) K = K In the renormalised lattice, nn spins couple with strength K < K. For < K <, the renormalisation induces a flow from the unstable fixed point K = spins fully aligned) towards the stable fixed point K = spins noninteracting): No phase transition in d = for t, h), ).
S i J <ij> h mf = h + Jzm (4) and m, the magnetisation per spin, is just the mean value of any given spin. S i = S k k (5) N.
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