How to add couplings correctly for iDMRG?
Posted: 04 Apr 2024, 14:21
Hi everyone,
I have the following problem concerning iDMRG and in particular how to correctly add couplings.
Consider a 1D Hamiltonian like this \(H=-t\sum_i(b_i^\dagger b_{i+1}+\text{h.c.}) + \sum_i\sum_{r>0}^CV(r)n_in_{i+r}\), i.e. an extended (hard-core) Bose-Hubbard model with NN hopping and long-range interactions (up to some cutoff \(C\)). Let's say, I want to solve this by iDMRG using a unit cell of some (quite large) length \(L_\text{unit}\) (I simplify the model here, in my specific case, I have \(L_\text{unit}=24\) for reasons of translational symmetry and \(C=6\)).
Let us lable the sites with \(i=0,1,2,\dots,L_\text{unit}-1\). I want to do periodic boundary conditions. In TeNPy, do I add the terms of the Hamiltonian like
(1) \(-t\,b_{L_\text{unit}-1}^\dagger b_0\) and e.g. \(V(2)n_{L_\text{unit}-1}n_1\) or
(2) \(-t\,b_{L_\text{unit}-1}^\dagger b_{L_\text{unit}}\) and e.g. \(V(2)n_{L_\text{unit}-1}n_{L_\text{unit}+1}\)?
I.e. do I define my interactions directly in a PBC way or not? How does TeNPy handle this when adding unit cells in the iDMRG? Which way do I have to define the interactions, if I want TeNPy to "correctly implement" my Hamiltonian?
The way I understand it, method (1) creates uncoupled cells of \(L_\text{unit}\) which, in the iDMRG, are then repeated whereas case (2) creates the system the way "I really want it", i.e. the iDMRG simulates the thermodynamic limit of the system. What happens if I do a mixture? I add the interactions using (1) and hopping using (2). As hopping gives particle mobility, does this couple the cells? I am aware of the fact, that the interactions are different in the two cases, but this may be less relevant right now (as results for both cases are quite similar).
I am wondering about this, because my results do not converge (in some parameter region) using (1) for both interactions and hopping, while they converge and give reasonable results doing this mixture which are similar (but not the same!) to the results of (1) in some different parameter regime.
Thanks a lot in advance!
I have the following problem concerning iDMRG and in particular how to correctly add couplings.
Consider a 1D Hamiltonian like this \(H=-t\sum_i(b_i^\dagger b_{i+1}+\text{h.c.}) + \sum_i\sum_{r>0}^CV(r)n_in_{i+r}\), i.e. an extended (hard-core) Bose-Hubbard model with NN hopping and long-range interactions (up to some cutoff \(C\)). Let's say, I want to solve this by iDMRG using a unit cell of some (quite large) length \(L_\text{unit}\) (I simplify the model here, in my specific case, I have \(L_\text{unit}=24\) for reasons of translational symmetry and \(C=6\)).
Let us lable the sites with \(i=0,1,2,\dots,L_\text{unit}-1\). I want to do periodic boundary conditions. In TeNPy, do I add the terms of the Hamiltonian like
(1) \(-t\,b_{L_\text{unit}-1}^\dagger b_0\) and e.g. \(V(2)n_{L_\text{unit}-1}n_1\) or
(2) \(-t\,b_{L_\text{unit}-1}^\dagger b_{L_\text{unit}}\) and e.g. \(V(2)n_{L_\text{unit}-1}n_{L_\text{unit}+1}\)?
I.e. do I define my interactions directly in a PBC way or not? How does TeNPy handle this when adding unit cells in the iDMRG? Which way do I have to define the interactions, if I want TeNPy to "correctly implement" my Hamiltonian?
The way I understand it, method (1) creates uncoupled cells of \(L_\text{unit}\) which, in the iDMRG, are then repeated whereas case (2) creates the system the way "I really want it", i.e. the iDMRG simulates the thermodynamic limit of the system. What happens if I do a mixture? I add the interactions using (1) and hopping using (2). As hopping gives particle mobility, does this couple the cells? I am aware of the fact, that the interactions are different in the two cases, but this may be less relevant right now (as results for both cases are quite similar).
I am wondering about this, because my results do not converge (in some parameter region) using (1) for both interactions and hopping, while they converge and give reasonable results doing this mixture which are similar (but not the same!) to the results of (1) in some different parameter regime.
Thanks a lot in advance!