一维伊辛模型精确解计算

The Exact Solution of One-Dimensional Ising Model

Here we want to describe the process of working out the one-dimensional Ising model which have an exact solution. During this process mathematics of the quantum mechanics would be used, some notion such as eigenvector would also be involved.

In the first place, we introduce the definition of the Ising Model. Consider a two-dimensional square lattice composed of N = L ×L sites. Every site is occupied by a so-called spin, s i . In the magnetic material, the spins mean the magnetic dipoles positioned on the crystal structure lattice. In uniaxial magnetic materials, the magnetic dipole interactions constrain the spins to point parallel or anti-parallel along a given direction. Therefore, for simplicity, we assume that the spins can only be in one of two states, ether spin-up, s i = +1, or spin-down, s i = －1. By convention, the upwards direction is defined as positive direction. It is obvious that the spins interact with each other. A pair of parallel spins has an interaction energy of －J, while a pair of anti-parallel spins has an interaction energy of +J. We get the total internal interaction energy

where J ij are known as the coupling constants between spins s i and s j and the sum runs over all distinct pairs of spins. If we impose a uniform external field, H, which acts upon every spin, there would be an external energy. So the total energy of the entire model is

In the one-dimensional Ising model, the interaction energy of one spin s i has been simplified to a sum running over all distinct nearest-neighbor pairs. The new form of the total energy of 1D Ising model is

In addition, we apply periodic boundary conditions such that s N+1 = s i .

According to equilibrium statistical mechanics, one can only measure any thermodynamic quantity of interest based on the partition function of the ensemble. The partition function can be written :

where we have used the fact that for a periodic lattice. Now we introduce a 2×2 matrix :

whose matrix elements are defined as

∑-=ij

j i ij int s s J E {}∑∑=--=+=N i i ij j i ij ext int s s H s s J E E E i 1

{}∑∑==+--=N i i N i i i s s H s s J E i 1

11()()∑∑

∑±==++±=⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛++=11111211N s N i i i i i s N s s H s Js exp H ,T Z β ∑∑=+=+=N

i i i N i i )s s (s 11121⎪⎪⎭⎫ ⎝⎛=---+)H J (J J )H J (e e e e ββββP ()⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛++=+++11121i i i i i i s s H s Js exp s s βP

So the partition function can be written

where

are the eigenvalues of the matrix P . To find the eigenvalues of P , we set the determinant to zero and get the solution :

So with the help of the dirac mark, we define the operator P , the bra and the ket .We can simplify the process of solving the partition function to a process of solving eigenvalues of a 2×2 matrix. After getting the exact solution of the partition function, the thermodynamic quantities can be calculated. We can thoroughly understand the 1D Ising Model, especially its phase transition. ()∑∑±=±==1

1322111N s N s N s s s s s s H ,T Z P P P ∑±==1111s s s IP PIPI []N N N s N Tr s s -

+±=+===∑λλP P 1111±λ()()()()()J exp H sinh H cosh J exp ββββλ42-+±=±()0=-I P λdet ()()()[]()()0222=--+-+-J exp J exp H exp H exp J exp ββλβββλs s