Two-site Hubbard model eigenstate-problem

[Suske]

Homework Statement

L.S.,

I'm breaking my head over this problem! To anyone who can help me out: thanks a lot!

Consider a two-point grid (point a en point b) and two electrons occupying those points, (they can both occupy one point at the same time), both wit spin-up or spin-down. Now, they state that you write the state of the spin-electrons as a product of the state of spin and the state of place: X * Psi (I don't understand why you'd do this and what this means, for starters..)

Now the problem is to show that we talk about 6-dimensional Hilbert space with basis:

\frac{1}{\sqrt{2}} X_{up, up}*(Psi_{a, b} - Psi_{b, a})

\frac{1}{2}(X_{up, down} + X_{down, up})*(Psi_{a, b} - Psi_{b, a})

\frac{1}{\sqrt{2}}(X_{down, down})*(Psi_{a, b} - Psi_{b, a})

(spin-triplet)

and

\frac{1}{2\sqrt{2}}(X_{up, down} - X_{down, up})*(Psi_{a, a})

\frac{1}{2}(X_{up, down} - X_{down, up})*(Psi_{a, b} + Psi_{b, a})

\frac{1}{2\sqrt{2}}(X_{up, down} - X_{down, up})*(Psi_b,b)

The Attempt at a Solution

I understand it is 6-dimensional, (2³ - the two combinations that cannot exist because of the Pauli principal)
Further, I have no clue how to derive this basis!

Would someone please help me with this problem? I thought I understood this chapter well, but now my brains are vaporizing..

Thanks!

 

[Mute]

It would help if you posted the Hubbard Hamiltonian you are working with. For now I'll assume you are working with

$$\mathcal H = -t \sum_{\langle i, j\rangle, \sigma } (c^\dagger_{i,\sigma}c_{j,\sigma} + h.c) + U\sum_{i=1}^N n_{i\uparrow} n_{i\downarrow}.$$

Homework Statement

L.S.,

I'm breaking my head over this problem! To anyone who can help me out: thanks a lot!

Consider a two-point grid (point a en point b) and two electrons occupying those points, (they can both occupy one point at the same time), both wit spin-up or spin-down. Now, they state that you write the state of the spin-electrons as a product of the state of spin and the state of place: X * Psi (I don't understand why you'd do this and what this means, for starters..)

The state of the electron is labelled by two properties: the electron's spin (up or down) and the electron's position. The wavefunction of the electron can thus be factorized into a wavefunction corresponding to the position and a wavefunction corresponding to the spin.

So, your basic wavefunction for a single electron is $|i,\sigma\rangle = |i\rangle |\sigma\rangle$, where i labels the position and $\sigma$ labels the spin. Are you familiar with Dirac notation? (the pointy brackets). If not, in your notation, it looks like $|i\rangle \sim \psi_i$, $|\sigma\rangle \sim X_\sigma$. The fact that the single electron state factors into a product, $|i,\sigma\rangle = |i\rangle |\sigma\rangle \sim X_\sigma \psi_i$ is just a statement that the 'spin' and 'position' properties of the electron live in different Hilbert spaces - basically, the two properties are entirely separate and aren't entangled in any way.

Since you have two electrons, the total wavefunction encodes both of their spins and positions,

$$|\Psi\rangle = |i_1,\sigma_1;i_2,\sigma_2\rangle$$

These are the 'natural' basis states, corresponding to the positions and spins of each electron. Based on the Pauli exclusion principle, you can figure out how many possible states there are. These 'natural' states are not the eigenstates of the system, however. The "basis" states that your problem wants you to derive are the eigenstates, which are just linear combinations of the 'natural' basis states.

Your question seems to be, how do you derive the eigenstates your problem wants?

Well, do you know how to convert the Hubbard Hamiltonian into matrix form? The eigenstates are just the eigenvectors of that matrix. Do you remember how to find the eigenvectors of a matrix?

 

[Suske]

First of all, Thanks a lot for your help!

I know understand the product of states. I also am familiar with the Dirac-notation.
I also know how to find the eigenvectors of a matrix (eigenvalues first, then solve for matrix - eigenvalue*I = 0 )

The only thing is that I don't know how to put the Hamiltonian in matrix form! My book doesn't say a lot about it either..
Could you hint/help me? I think I could solve the problem if I'd be able to do that!

Last thing: my problem doesn't state a Hamiltonian.. Is the one you gave me the standard hamiltonian for the Hubbard-model? Or are there more?

Thanks a lot!

 

[Mute]

First of all, Thanks a lot for your help!

I know understand the product of states. I also am familiar with the Dirac-notation.
I also know how to find the eigenvectors of a matrix (eigenvalues first, then solve for matrix - eigenvalue*I = 0 )

The only thing is that I don't know how to put the Hamiltonian in matrix form! My book doesn't say a lot about it either..
Could you hint/help me? I think I could solve the problem if I'd be able to do that!

Last thing: my problem doesn't state a Hamiltonian.. Is the one you gave me the standard hamiltonian for the Hubbard-model? Or are there more?

Thanks a lot!

Suppose you have a set of states $|e_j\rangle$, where you've chosen some ordering of the states corresponding to j=0,1,2,... . To write the Hamiltonian in matrix form in this basis, you simply compute $\mathcal\langle e_i| \mathcal H |e_j \rangle$; this will be the $i,j$th element of your matrix.

For example, suppose $\mathcal H = \sigma_+$ and you want to write this in the basis $|\pm z\rangle$. So, let's order the states by $|e_0 \rangle = |+z\rangle$, $|e_1\rangle = |-z\rangle$. Then, $\mathcal H_{00} = \langle +z| \sigma_+ |+z\rangle = 0$, $\mathcal H_{01} = \langle +z| \sigma_+ |-z\rangle = \langle +z|+z\rangle =1$, $\mathcal H_{10} = \langle -z| \sigma_+ |+z\rangle = 0$ and $\mathcal H_{11} = \langle -z| \sigma_+ |-z\rangle = \langle -z|+z\rangle = 0$. So, the matrix representation of $\mathcal H$ is just

$$\mathcal H = \left(\begin{array}{c c}0 & 1 \\ 0 & 0\end{array}\right).$$

If your problem doesn't state the Hamiltonian, is it listed somewhere else in the book? The hamiltonian I wrote is from the wikipedia article, but the problem might use specific parameter choices or variations of the Hamiltonian - for example, the problem could take U = 0, t = 1.

In the absence of a Hamiltonian, perhaps all the problem wants you to do is explain why the eigenspace is six-dimensional (which you already did) and double-check that the basis it gives you is orthonormal.

Otherwise, maybe try the Hubbard model Hamiltonian I wrote down with t = U = 1? It will be a learning exercise, at the very least. =P