- #1
Fightfish
- 953
- 118
So, I was examining the ground state of a Bose-Hubbard dimer in the negligible interaction limit, which essentially amounts to constructing and diagonalizing a two-site hopping matrix that has the form
<br /> H_{i,i+1}^{(n)} = H_{i+1,i}^{(n)} = - \sqrt{i}\sqrt{n-i+1},<br />
with all other elements zero. The superscript n refers to the fixed number of particles present on the dimer, and the dimension of the matrix is given by n+1.
Essentially this gives rise to a hollow centrosymmetric tridiagonal matrix. Explicitly, we have:
<br /> H^{(2)} =<br /> \left(<br /> \begin{array}{ccc}<br /> 0 & -\sqrt{2} & 0 \\<br /> -\sqrt{2} & 0 & -\sqrt{2} \\<br /> 0 & -\sqrt{2} & 0 \\<br /> \end{array}<br /> \right)<br /><br /> H^{(3)} =<br /> \left(<br /> \begin{array}{cccc}<br /> 0 & -\sqrt{3} & 0 & 0 \\<br /> -\sqrt{3} & 0 & -2 & 0 \\<br /> 0 & -2 & 0 & -\sqrt{3} \\<br /> 0 & 0 & -\sqrt{3} & 0 \\<br /> \end{array}<br /> \right)<br /><br /> H^{(4)} =<br /> \left(<br /> \begin{array}{ccccc}<br /> 0 & -2 & 0 & 0 & 0 \\<br /> -2 & 0 & -\sqrt{6} & 0 & 0 \\<br /> 0 & -\sqrt{6} & 0 & -\sqrt{6} & 0 \\<br /> 0 & 0 & -\sqrt{6} & 0 & -2 \\<br /> 0 & 0 & 0 & -2 & 0 \\<br /> \end{array}<br /> \right)<br /> and so on.
In examining the unnormalized eigenstate with the lowest (most negative) eigenvalue, there seems to exist a Pascal-triangle-like sequence:
|\psi_{g}^{(1)}\rangle= [1,1]|\psi_{g}^{(2)}\rangle= [1,\sqrt{2},1]|\psi_{g}^{(2)}\rangle= [1,\sqrt{3},\sqrt{3},1]|\psi_{g}^{(3)}\rangle= [1,\sqrt{4},\sqrt{6},\sqrt{4},1]|\psi_{g}^{(4)}\rangle= [1,\sqrt{5},\sqrt{10},\sqrt{10},\sqrt{5},1]
This is highly suggestive that some sort of recurrence relation or mapping to binomial expansion exists; however thus far I have not been successful in trying to extract it. Might some one be able to shed some light on this?
<br /> H_{i,i+1}^{(n)} = H_{i+1,i}^{(n)} = - \sqrt{i}\sqrt{n-i+1},<br />
with all other elements zero. The superscript n refers to the fixed number of particles present on the dimer, and the dimension of the matrix is given by n+1.
Essentially this gives rise to a hollow centrosymmetric tridiagonal matrix. Explicitly, we have:
<br /> H^{(2)} =<br /> \left(<br /> \begin{array}{ccc}<br /> 0 & -\sqrt{2} & 0 \\<br /> -\sqrt{2} & 0 & -\sqrt{2} \\<br /> 0 & -\sqrt{2} & 0 \\<br /> \end{array}<br /> \right)<br /><br /> H^{(3)} =<br /> \left(<br /> \begin{array}{cccc}<br /> 0 & -\sqrt{3} & 0 & 0 \\<br /> -\sqrt{3} & 0 & -2 & 0 \\<br /> 0 & -2 & 0 & -\sqrt{3} \\<br /> 0 & 0 & -\sqrt{3} & 0 \\<br /> \end{array}<br /> \right)<br /><br /> H^{(4)} =<br /> \left(<br /> \begin{array}{ccccc}<br /> 0 & -2 & 0 & 0 & 0 \\<br /> -2 & 0 & -\sqrt{6} & 0 & 0 \\<br /> 0 & -\sqrt{6} & 0 & -\sqrt{6} & 0 \\<br /> 0 & 0 & -\sqrt{6} & 0 & -2 \\<br /> 0 & 0 & 0 & -2 & 0 \\<br /> \end{array}<br /> \right)<br /> and so on.
In examining the unnormalized eigenstate with the lowest (most negative) eigenvalue, there seems to exist a Pascal-triangle-like sequence:
|\psi_{g}^{(1)}\rangle= [1,1]|\psi_{g}^{(2)}\rangle= [1,\sqrt{2},1]|\psi_{g}^{(2)}\rangle= [1,\sqrt{3},\sqrt{3},1]|\psi_{g}^{(3)}\rangle= [1,\sqrt{4},\sqrt{6},\sqrt{4},1]|\psi_{g}^{(4)}\rangle= [1,\sqrt{5},\sqrt{10},\sqrt{10},\sqrt{5},1]
This is highly suggestive that some sort of recurrence relation or mapping to binomial expansion exists; however thus far I have not been successful in trying to extract it. Might some one be able to shed some light on this?
