Matrix Product Toolkit | Main / Long-RangeBose-Hubbard

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The model bosehubbard-u1 now includes a term H_lambda, which represents a long-range exponentially decaying coulomb term. To use, specify a value for lambda (or xi) when constructing the model.

{% H_{\mbox{lambda}} = \sum_{i,j>i} \lambda^{j-i} n_i n_j %}

Note that this doesn't include an on-site Coulomb term - we still use H_U for that. Probably we want to set the prefactor of H_U and H_lambda the same, so the Hamiltonian we use is H_J + x*(H_U + H_lambda), where x is the coulomb interaction strength.

Also note that the usage of $\lambda$ here is different to how it is used in the Odintsov paper, where $\lambda$ refers to the screening length. I use $\lambda$ in this way because it has close similarities to how $\lambda$ is often used as an eigenvalue of transfer matrices etc, so for consistency throughout the toolkit I used it the same way here.

If you want to specify the screening length instead, set the xi parameter when constructing the model. This is equivalent to setting lambda to {$\lambda = \exp(-1/\xi)$}.

As an example, I construct a model with a screening length of 3, for a small 20 site lattice,

bosehubbard-u1 -L 20 --xi 3 -o lat

And try a dmrg calculation for 5 particles, and I set the prefactor of the Coulomb term to 2. (This will be the main parameter that we want to vary to find the phase transition boundaries).

mp-random -l lat -q 5 -o psi
mp-dmrg -H lat:"H_J + 2*(H_U + H_lambda)" -w psi -m 50 -s 10

This seems to converge pretty quickly - not surprising if the groundstate is localized.

We can easily calculate the local density of particles,

mp-localexpectation lat psi N 1 20

Or, as a shortcut, direct the output to a plotting program,

mp-localexpectation lat psi N 1 20 | xmgrace -pipe&

This gives the following density profile, which shows that the bosons are localized into 5 peaks,

Attach:bose-gs.png Δ

It is interesting to also consider excited states by forcing the wavefunction to be orthogonal to the groundstate. This is really a diversion, but probably no one has really looked at the excitation spectrum of these models before. Eg, the first excited state

cp psi psi2
mp-dmrg -H lat:"H_J + 2*(H_U + H_lambda)" -w psi2 --orthogonal psi -m 50 -s 10

This gives a density profile showing that the 5 bosons are now spread over 4 peaks,

Attach:bose-excited.png Δ

Similarly, we can obtain the 2nd excited state with

cp psi2 psi3
mp-dmrg -H lat:"H_J + 2*(H_U + H_lambda)" -w psi3 --orthogonal psi --orthogonal psi2 -m 50 -s 10

and this has 3 peaks.

Attach:bose-excited2.png Δ

We can also do the 3rd excited state, now the bosons are in two peaks at the edges. Note that when I first tried this, with U=3, I instead got back to a 4-peak structure, but with a different shape to the 4-peak structure of the first excited state. So there is some interesting behaviour in the excitation spectrum. Probably also structure in the distribution of the energy gaps.

Attach:bose-excited3.png Δ