I'm looking into the extened Bose Hubbard model with filling n =1. My problem is that I cannot get the correct symmetry-broken ground state.
When the neaest-neighbot interaction V is strong, I would expect that the ground state looks like {202020....2020}, while the one I got is {2020...20020202.....0202}; there is always a kink in the middle.
I believe that this is due to the construction of the warm-up process. Can we give some initial guess to the ground state serach, so that we are not trapped in this local minimum?
Many thanks,
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Right now, you have the possibility to overwrite the initial guess for the quantum numbers after writing the files and before calling the fortran core running your simulation. But honestly, it is not a solution I would pursue myself. The initial guess for iMPS is starting from both ends starting with site pairs (1, L), then (2, L - 1), (3, L - 2) until one reaches the center site(s).
Passing an initial guess by filename is since a long time on our wishlist, so I will try to implement this feature in the next days and provide you with an example.
If you are not exploring the symmetry-broken state itself, you might want to go to odd system sizes and unit filling + 1 which should have a unique ground state.
Kind regards,
Daniel
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Thank you for the response, and I will try the odd system size to see the CDW phase.
And it would be really nice that we can give an initial gues. This my be also useful for sweeping in a phase space; we can use the ground state of similar parameters as an initial guess to acheive faster convergence (especially near the phase boundary).
Best,
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I have opened the possibility to give an initial guess for the ground state search. A quick comparison for our statics examples did actually see both an increase and a decrease in CPU time when passing the state with the closest parameters as the initial guess. Of course, examples are pretty small, I run it only in a virtual machine, and it might depend on how big the step size in the parameter is, etc. I assume it cannot be used blindly while expecting a speed-up. The modified Ising example is attached.
Back to the CDW problem. In the limit of nearest-neighbor interaction n_{i} n_{i+1} having an infinite coupling, the states you mentioned first are no local minima. The ground state has a degeneracy growing with the system size, e.g., for eight sites the ground states are |01010101>, |10101010>, |10010101>, |10100101>, and |10101001>. In between, it is hard to tell for me right now if they are local minima or not. You could try to get some excited states and see if their energy is below the ground state energy, which is challenging as the degeneracy grows with system size.
I have attached you an example of the ground state search with an initial guess for the CDW using the |20202020...> state as an initial guess. From your question with the symmetry-broken state it was not clear to me if you wanted to give (|2020...> + |0202...>) / sqrt(2) as the initial guess? This entangled superposition would be more cumbersome to write down with the U(1) symmetry and is not within the methods we provide right now.
Let me know if there are any open questions you might have.
I'm sorry for the late reply. I have been traveling last week.
Thank you so much for the exmaple codes! I want to try an initial guess like |20202020...>, so the implementation is good enough for me.
I agree that the naive speed-up may not be achieved with initial guess, since it depends on specifics of the simulations.
I also agree that when next-nearest neighbor repultion is infinite, there are many degenerate ground states. I was imagining the case when we can map the model to a Heisenberg model, where an antiferromagnetic state is the true ground state. However, the mapping is limited to the large S, so the details of systems may be importnat in this case.
I will try the example codes, and post another comment if I find something.
Best,
If you would like to refer to this comment somewhere else in this project, copy and paste the following link:
Hallo,
I'm looking into the extened Bose Hubbard model with filling n =1. My problem is that I cannot get the correct symmetry-broken ground state.
When the neaest-neighbot interaction V is strong, I would expect that the ground state looks like {202020....2020}, while the one I got is {2020...20020202.....0202}; there is always a kink in the middle.
I believe that this is due to the construction of the warm-up process. Can we give some initial guess to the ground state serach, so that we are not trapped in this local minimum?
Many thanks,
Hello Tomio,
Right now, you have the possibility to overwrite the initial guess for the quantum numbers after writing the files and before calling the fortran core running your simulation. But honestly, it is not a solution I would pursue myself. The initial guess for iMPS is starting from both ends starting with site pairs (1, L), then (2, L - 1), (3, L - 2) until one reaches the center site(s).
Passing an initial guess by filename is since a long time on our wishlist, so I will try to implement this feature in the next days and provide you with an example.
If you are not exploring the symmetry-broken state itself, you might want to go to odd system sizes and unit filling + 1 which should have a unique ground state.
Kind regards,
Daniel
Hi Daniel,
Thank you for the response, and I will try the odd system size to see the CDW phase.
And it would be really nice that we can give an initial gues. This my be also useful for sweeping in a phase space; we can use the ground state of similar parameters as an initial guess to acheive faster convergence (especially near the phase boundary).
Best,
Hello Tomio,
I have opened the possibility to give an initial guess for the ground state search. A quick comparison for our statics examples did actually see both an increase and a decrease in CPU time when passing the state with the closest parameters as the initial guess. Of course, examples are pretty small, I run it only in a virtual machine, and it might depend on how big the step size in the parameter is, etc. I assume it cannot be used blindly while expecting a speed-up. The modified Ising example is attached.
Back to the CDW problem. In the limit of nearest-neighbor interaction n_{i} n_{i+1} having an infinite coupling, the states you mentioned first are no local minima. The ground state has a degeneracy growing with the system size, e.g., for eight sites the ground states are |01010101>, |10101010>, |10010101>, |10100101>, and |10101001>. In between, it is hard to tell for me right now if they are local minima or not. You could try to get some excited states and see if their energy is below the ground state energy, which is challenging as the degeneracy grows with system size.
I have attached you an example of the ground state search with an initial guess for the CDW using the |20202020...> state as an initial guess. From your question with the symmetry-broken state it was not clear to me if you wanted to give (|2020...> + |0202...>) / sqrt(2) as the initial guess? This entangled superposition would be more cumbersome to write down with the U(1) symmetry and is not within the methods we provide right now.
Let me know if there are any open questions you might have.
Best regards,
Daniel
Hi Daniel,
I'm sorry for the late reply. I have been traveling last week.
Thank you so much for the exmaple codes! I want to try an initial guess like |20202020...>, so the implementation is good enough for me.
I agree that the naive speed-up may not be achieved with initial guess, since it depends on specifics of the simulations.
I also agree that when next-nearest neighbor repultion is infinite, there are many degenerate ground states. I was imagining the case when we can map the model to a Heisenberg model, where an antiferromagnetic state is the true ground state. However, the mapping is limited to the large S, so the details of systems may be importnat in this case.
I will try the example codes, and post another comment if I find something.
Best,