MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � � � 8.334: Statistical Mechanics II Problem Set # 4 Due: 4/2/08 Transfer Matrices & Position space renormalization This problem set is partly intended to introduce the transfer matrix method, which is used to solve a variety of one-dimensional models with near-neighbor interactions. As an example, consider a linear chain of N Ising spins (σi = ±1), with a nearest–neighbor coupling K, and a magnetic field h. To simplify calculations, we assume that the chain is closed upon itself such that the first and last spins are also coupled (periodic boundary conditions), resulting in the Hamiltonian N −βH = K (σ1σ2 + σ2σ3 + · · · + σN−1σN + σNσ1) + hσi . (1) i=1 The corresponding parti tion function, obtained by summing over all states, can be ex-pressed as t he product of matrices, since N � � � � � � h Z = · · · exp Kσiσi+1 +(σi + σi+1)2 (2) σ1 =±1 σ2 =±1 σN =±1 i=1 ≡ tr [hσ1|T |σ2ihσ2|T |σ3i · · · hσN|T |σ1i] = tr � TN� ; where we have introduced the 2 × 2 transfer matrix T , w ith elements � � e K+h e−K h hσi|T |σji = exp Kσiσj + (σi + σj) , i.e. T = . (3) 2e−K K−h e The expression for trace of the matrix can be evaluated in the basis that diagonalizes T , in which case it can be writt en in terms of the two eigenvalues λ as ± Z = λN + λN = λN 1 + (λ /λ+)N ≈ λN (4) + + + .−−We have assumed that λ+ > λ , and since i n the limit o f N → ∞ the larger eigenvalue −dominates the sum, the free energy is βf = − ln Z/N = − l n λ+. (5) Solving t he characteristic equation, we find the eigenvalues λ = e K cosh h ± e2K sinh2 h + e−2K . (6) ±� � � � � � � � � � � � � � We shall leave a discussion of the singularities of the resulting free energy (at zero temper-ature) to the next section, and instead look at the averages and correlations in the l imit of h = 0. To calculat e the average of the spin at site i, we need to evaluate N1 hσii = · · · σi exp (Kσjσj+1)Z σ1 =±1 σ2 =±1 σN =±1 j=1 1 (7) ≡ tr [hσ1|T |σ2i · · · hσi−1|T |σiiσihσi|T |σi+1i · · · hσN|T |σ1i]Z TN−i+1 = 1 tr � Ti−1σˆz� = 1 tr � TNσˆz � ,Z Z 1 0 where have permuted the matrices inside the trace, and ˆσz = , is the usual 2 × 2 0 −1 Pauli matrix. One way to evaluate the final expression in Eq. (7) is to rotate to a basis where the matrix T is diagonal. For h = 0, this i s accomplished by the unitary matrix 1 1 U = √12 1 −1 , resulting in 1 �� λN 0 �� 0 1 �� 1 � 0 λN � hσii = tr + = + = 0. (8) Z 0 λN 1 0 ZλN 0− − 0 1 Note that under this t ransformation the Pauli matrix ˆσz is rotated into ˆσx = . 1 0 The vanishing of the magnetization at zero field is of course expected by symmetry. A more interesting quantity is the two-spin correlation function N1 hσiσi+ri = · · · σiσi+r exp (Kσjσj+1)Z σ1 =±1 σ2 =±1 σN =±1 j=1 (9) = 1 tr � Ti−1σˆzTrσˆzTN−i−r+1� = 1 tr � σˆzTrσˆzTN−r � . Z Z Once again rotating to the basis w here T is dia gonal simplifies the trace to �� �� �� �� �� 1 0 1 λr 0 0 1 λN−r 0 hσiσi+ri = tr ++ Z 1 0 0 λr 1 0 0 λN−r 1 � λN−rλr 0 − � λN−rλr + λN−rλ− r (10) = tr + − = + − − + . λN−rλr λN + λNZ 0 − ++ − Note that because of t he periodic boundary conditions, the above answer is invariant under r → ( N − r). We are interested in the limit of N ≫ r, for which ��r e−r/ξhσiσi+ri ≈ λ− ≡ , (11) λ+� � ��� � � � � � � with the correlation length λ+ −1 1 ξ = ln = − . (12) λ ln tanh K− The above transfer matrix a pproach can be generalized to any o ne dimensional chain with variables {si} and nearest neighbor interactions. The partition function can be written as NN Z = exp B(si, si+1) = e B(si,si+1), (13) {si} i=1 {si}i=1 where we have defined a transfer matrix T with elements, hsi|T |sji = e B(si,sj ). (14) In the case of periodic boundary conditions, we then obtain Z = tr � TN� ≈ λN . (15) maxNote that for N → ∞, the trace is dominated by the largest eigenvalue λmax. Quite generally the largest eigenvalue of the transfer matrix is related to the free energy, while the correlation lengths are obtained from ratios o f eigenvalues. Frobenius’ theorem states that for any finite matrix with finite positive elements, the largest eigenvalue is always non-degenerate. This i mplies that λmax and Z are analyti c functions of the parameters appearing in B, and that one dimensional models can exhibit singularities (and hence a phase transition) only at zero temperature (when some matrix elements become infinite). While the above formulation is framed in the language of discrete variables {si}, the method can also be applied to continuous va riables as illustrated in the following problems. As an example of the latter, let us consider three component unit spins �si = (six, siy, siz), with the Heisenberg model Hamiltonian N −βH = K �si · �si+1. (16) i=1 Summing over all spin configurations, the parti tion function can be written as �N i=1 ~si ~si Z = tr e K ~si·~si+1 = tr e K~s1·~s2 e K~s2·~s3 · · · e K~sN ·~s1 = tr TN , (17)� � � � � � � � � where h�s1|T |�s2i = eK~s1·~s2 is a transfer function. Quite generally we would l ike to bring T into the diagonal form α λα|αihα| (in Dirac notation), such that h�s1|T |�s2i = λα h�s1|αi hα|�s2i = λαfα(�s1)fα∗(�s2). (18) α α From studies of plane waves in quantum mechanics you may recall that the exponential of a dot product can be