## finite open, finite periodic and infinite periodic chains

How do I use this algorithm? What does that parameter do?
Qottmann
Posts: 9
Joined: 27 Mar 2019, 09:11
Location: ICFO Barcelona

### finite open, finite periodic and infinite periodic chains

Hello,

is there a way to have a finite periodic chain (so a ring) for the BoseHubbard model?
As far as I understand setting bc_MPS to "finite" gives a finite chain with open boundary conditions and "infinite" is basically performing iDMRG - or am I already getting this part wrong?

Best regards,
Korbinian

Johannes
Posts: 127
Joined: 21 Jul 2018, 12:52
Location: UC Berkeley

### Re: finite open, finite periodic and infinite periodic chains

Qottmann wrote:
14 Oct 2019, 10:19
is there a way to have a finite periodic chain (so a ring) for the BoseHubbard model?
Kind of. We do not support finite MPS with periodic boundaries for the MPS itself i.e. writing
$| \Psi\rangle = \sum Tr(A^{\sigma_1} \cdots A^{\sigma_L}) | \sigma_1 \cdots \sigma_L\rangle$
This is a form which is usefull analytically (e.g. if you write the ground state of the AKLT model on a finite ring), but not numerically, because it doesn't allow to define a canonical form.

However, you can define a bc_MPS="finite" and still have periodic couplings in your Hamiltonian. The price you pay for this is that the coupling over the "boundary" of the MPS is long range afterwards. Hence you can't use TEBD, and if you use DMRG, you should definitely use the mixer.
To do that, take a look at the parameters documented in init_lattice, i.e. use bc_MPS=True, bc_x="periodic".

Ian McCulloch once suggested that one can also use a different "enumeration" or "order" of the MPS compared to the physical sites in such a case to avoid having the terribly long range:
$| \Psi\rangle = \sum A^{\sigma_1} A^{\sigma_2} A^{\sigma_3} \cdots A^{\sigma_L}) | \sigma_1 \cdots \sigma_L\rangle$
$| \Psi\rangle = \sum A^{\sigma_L} A^{\sigma_1} A^{\sigma_{L-1}} A^{\sigma_2} A^{\sigma_{L-2}} A^{\sigma_3} \cdots A^{\sigma_{L/2}} | \sigma_1 \cdots \sigma_L\rangle$