Have you ever had a dream, Neo, that you seemed so sure it was real? But if were unable to wake up from that dream, how would you tell the difference between the dream world & the real world?

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## Long-RangeBose-HubbardThe 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. |

Page last modified on August 25, 2013, at 01:14 PM