What you know you can't explain, but you feel it. You've felt it your entire life, that there's something wrong with the world. You don't know what it is, but it's there, like a splinter in your mind, driving you mad.
What you know you can't explain, but you feel it. You've felt it your entire life, that there's something wrong with the world. You don't know what it is, but it's there, like a splinter in your mind, driving you mad.
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Main /
Long-RangeBose-Hubbard(redirected from Main.Long-rangeBose-hubbard) 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 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 As an example, I construct a model with a screening length of 3, for a small 20 site lattice,
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,
Or, as a shortcut, direct the output to a plotting program,
This gives the following density profile, which shows that the bosons are localized into 5 peaks, 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, 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. 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. |
