Heisenberg Model

B.Sc Project, Supervised by Prof. Abdollah Langari, Department of Physics, Sharif University of Technology, (Summer 2006).

In this research, we try to calculate analytically eigenvalues and eigenvectors of finite chain with 5 1⁄2-spin particles Heisenberg model. We drove eigenfuctions for closed chain. After that we renormalized the result for interaction between chains for zero temperature. Finally according to self similarity, fix points were found from effective renormalization energy.

1. Calculation of Eigenenergy and Eigenstates of Finite Chain

1-1. 2 Length Chain

There are 4 states for 2 length open chain:

Hamiltonian of this chain for open chain in XXY model is:

Now we operate the Hamiltonian in the states:

so the operational form of Hamiltonian is

Eigenvalues and eigenstates are:

1-2. 3 Length Chain

There are 8 states:

The Hamiltonian is:

Because the chain is open the Hamiltonian operational matrix is block diagonal:

For m = 3/2and m = −3/2 eigenvalue are Jand obviously eigenfunctions are ⎜+ + +⟩ and ⎜−−− ⟩. In other hand, for m = 1/2 and m = -1/2 are symmetric and they are same. So it is enough to calculate eigenstates for m = 1/2:

So we write matrix operation:

So eigenvalues and eigenstates are:

1-3. 5 Length Chain

The chain with 5 particles has 32 states and its Hamiltonian has block symmetry too. We want to find ground state, so it is enough to solving equation for m = 1/2 and m = −1/2 . Also these two have space symmetry, so we solve the equations for m = 1/2 then we extend the result to m = − 1/2 . The states which we have to solve are:

1-3-1. Open Chain

Hamiltonian for open chain is:

So Hamiltonian matrix is:

Calculation of Eigenvalue:

The Hamiltonian equation is

The determinant is expanded to 10 degree equation:

To solving this equation, we have to use vector form to finding eigenvectors:

Now we use the symmetry to determine eigenvector’s elements in term of each other.

To evaluate 5th element, the eigenvalue equations for 3rd and 8th elements are:

So,
means the solutions have been satisfy these conditions:
Four eigen values are arrived from
The graph of these four values is:
But for α = γ , there are six solution which will be arrived from a six degree education:
which should be solved by Mathematica. The graph of all eigenvalues in term of ∆ are scratched in this graph:

1-3-2. Closed Chain

For chain, we rewrite the Hamiltonian:

and Hamiltonian operational matrix become:
Calculate Eigenvalue:

In closed chain, the periodic condition makes a rotational symmetry. We call T the rotation operation. So all of states of will be categorize in two group. The states are shown in new form:

In other hand, the Hamiltonian and rotation are commutative. Easily we can reach to answers with solving Hamiltonian for only two states ⎜−+++−⟩ and ⎜+−+−+⟩:

with the knowledge of that momentum is invariant of Hamiltonian, eigenvalus and eigenstates are:

The eigenvalues for these states in interval of 1 and -1 in order of lower to higher level are written:

The graph of this energy states in term of ∆ is

2. Renormalization

Assume a chain with N particles. We can divide this chain to N/5 chain with 5 particle. So the Hamiltonian of system can be written with two part:

where HI0 is internal Hamiltonian of each chain with 5 particle and it HI0 is what we calculate in last part. HI,I+1 is interaction of Ith chain with next chain. In this project, the system is considered in zero temperature, so the ground state is what we are looking for. As an approximation, we consider each sub-chain (I) a closed chain because we know the analytical solutions of this model.

The ground state of HI0 , we have 4 degenerate state which two of them are m = 1/2 and two of them have m=−1/2, which m=−1/2 states are same as m=1/2 states with reverse spins.
In renormalization, the Hamiltonian of system will be written as effective Hamiltonian for image of system in eigenstates of state space of sub-chains and it will be done with P operator:

So effective Hamiltonian will be:

So for each sub-chain effective Hamiltonian is

So effective Hamiltonian for interaction between sub-chains is

For two neighbor sub-chain the interaction Hamiltonian is

and the effective interaction Hamiltonian will be:

Now we calculate each Pauli metrics:

Fixed points are where your system will decrease to finite chain system where the effective Hamiltonian is similar to finite chain Hamiltonian. In the next graph, f and g are scratch in term of ∆ and the value for fixed points are driven:

The values for fix points are =1.0000
=-1.1545
=0

which prove our expectation that the fix points are in the region of symmetry and this result shows this calculates are correct.