1549380323-Statistical Mechanics Theory and Molecular Simulation

624 Critical phenomena 16.6 Ising model in one dimension Solving the Ising model in one dimension is a relatively straightforwar ...

One-dimensional Ising model 625 Sinceσandσ′can be only 1 or -1, P is a 2×2 matrix with elements given by 〈 1 |P| 1 〉= eβ(J+h) 〈− ...

626 Critical phenomena g(h,T) =−kTlnλ+ =−J−kTln [ cosh(βh) + √ sinh^2 (βh) + e−^4 βJ ] . (16.6.11) From eqn. (16.6.11), the magn ...

Two-dimensional Ising model 627 ... ... ... ... ... 1 2 3 n n + 1 1 2 3 n n + 1 n 1 n 1 Fig. 16.9Two-dimensional Ising system su ...

628 Critical phenomena E(μj,μk) =J ∑n i=1 σi,jσi,k E(μj) =J ∑n i=1 σi,jσi+1,j+h ∑ i,j σi,j (16.7.4) in terms of which the Hamilt ...

Two-dimensional Ising model 629 an exact expression for the magnetization at zero field and showedthat whenT < Tc, whereTcis ...

630 Critical phenomena an unsolved problem. Should an exact solution emerge, the exponents for thed= 3, n= 1 universality class ...

Two-dimensional Ising model 631 Now consider just the first few terms of the product (i,j= 1,2), which contribute the following ...

632 Critical phenomena (1,1) (1,2) (2,1) (2,2) v (1,1) (1,2) (2,1) (2,2) ( (1,1) (1,2) (2,1) (2,2) (1,1) (1,2) (2,1) (2,2) ...

Spin correlations 633 r= 6 r= 6 r= 8 r= (^8) r= 8 Fig. 16.12Examples of graphs that contribute to the partition function of the ...

634 Critical phenomena correlation function should depend only onr. Heuristically,G(r) is assumed to decay exponentially accordi ...

Renormalization Group 635 Fig. 16.13Example of the block spin transformation on a 6×6 square lattice. The lattice on the right s ...

636 Critical phenomena T(σ′;σ 1 ,...,σ 9 ) =    1 σ′ ∑ 9 i=1σi>^0 0 otherwise . (16.9.2) This function ensures that when t ...

Renormalization Group 637 Requiring that the Hamiltonian itself remain unchanged under an RG transforma- tion is stronger than s ...

638 Critical phenomena temperature where local ordering is expected and the middle spin is likely to cause neighboring spins to ...

Renormalization Group 639 tanhK′= tanh^3 K K′= tanh−^1 [ tanh^3 K ] . (16.9.15) Eqn. (16.9.15) defines the RG transformation as ...

640 Critical phenomena .. Unstable Stable x =1 x = 0 K =^8 K = 0 Fig. 16.15RG flow for the one-dimensional Ising model. generate ...

Renormalization Group 641 Therefore, the correlation lengthξ(K)∼ 1 /ln(tanhK)−→ ∞asT−→0, so that at T= 0 the correlation length ...

642 Critical phenomena Fig. 16.16 shows a two-dimensional spin lattice and the interactions between two blocks, which are mediat ...

Linearized RG 643 ξ(K) =ξ 0 bn(K). (16.10.2) As the starting pointKis chosen closer and closer toKc, the number of iterations ne ...