1) Could you please outline how one should determine a file to be used as an initial state? Can one set up and initial state without the need to creat such files?
2) My second question reagrds creating an specific form of hamiltonian:
![spinless fermions coupled via spins] (https://image.ibb.co/bKi835/Selection_044.png)
How can one spcefiy such terms?
3) I am also intrested in calculating such dynamics for fermionic number operator:
![ staggered fermion density] (https://image.ibb.co/eGbd35/Selection_045.png)
4) And the last question, does "QT" keyword works in the latest version? That is, averaging over different realization. How coud I use this?
Thank you very much for helping.
Last edit: Moein Najafi Ivaki 2017-05-04
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1) The default initial state is the ground state of a system. So you could design a Hamiltonian with different couplings and/or additional terms, which has your target state as ground state. Afterwards, you switch to your actual Hamiltonian for the time-evolution. This approach works for number conserving systems and in the grand-canonical ensemble.
Now as you pointed out, you can provide a file with the initial state. This solutions targets simulations in the grand-canonical ensemble. For example, product states are covered in a python function and you would not write down the file yourself, but call the function. Number conserving simulations write initial guesses for the ground state search, but they are not accessible as initial state right now.
Let me know if you want to use number conservation and how your intial states look like, and then we can work on more specific directions.
2) I think I need your feedback here to give you convenient advice. How are the sigma(i,j) defined? They are defined on two sites? As first input, the "hopping" term could be a bond term and you may use the decomposition request. Right now I am not sure if the Hamiltonian fulfills the bond term with a phase operator. Second term coupled with h could be a MBString term, although you would need to decompose it into products of local operators acting on i-1, i, and i+1 (O_1 x O_2 x O_3).
3) Your notation is in the Heisenberg picture? Assuming |0> = |psi(t=0)> is your initial state and you want to measure <psi(t) |="" n_j="" -="" n_(j+1)="" psi(t)="">, you could simply add the number operator as site observable. The difference in the density can be calculated during the postprocessing.</psi(t)>
4) If you are using the MPS methods, the QT (quantum trajectory) is a flag for the future use of the Lindblad master equation, but currently not available. For the exact diagonalization package inside openMPS, local Lindblad operators are available. Either as density matrix, or via quantum trajectories.
So far for the initial feedback. Once you helped me to understand the operators and the type of initial state, I can certainly give you a more detailed feedback.
Kind regards,
Daniel
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I really appreciate the swift response.
Let's just start with the basic questions first.
1) Consider for example a state like what we call density-wave, |101010...>, and some other like domain wall, |11...11000...0>. I would like to use U(1) symmetry for fermions (Just in case it might be of use). I would appreciate very much if you write down a minimal piece of code, just to know how to implement it in a right manner. I also would like to make sure the dynamics is using the specified initial state correctly, so I can handle it myself in case of future difficulties.
2) Right away I am not sure. It might be good for the start to know how to creat such terms:.
In addition, I have no idea of the decompoition process with package!
3) Yes, in the Heisenberg picture. Could you please drop me a piece of code? It is not clear to me how the dyanmics will do the diffrences in site observables. I undrestand the examples and various correlation within the package etc, however, definig this in the package, although it is highly trivial, does not seem straightforward to me. Sorry. I am also intrested in such connected correlators <0|n_{j}(t)+n_{j+l}(t)|0>.
Thank you very much.
Added notes, May 6:
More details on the structure of the given Hamiltonian:
Actually fermions are supposed to be placed on sites, and spins on bonds. There could several approaches be suggested, the efficeincy of which should be examined regarding the TEBD and tensor network, like placing fermions on odd sites and spins on even ones, or pattern like {spin1, spin2, fermion3, fermion4, spin5, spin6,...}. But these have heavy numrical cost. Thus, one should think about somthing like ladder geometry (Just schematically). Is it possible to implement such propoition with the right tensor and dimension, taknig into account both Z_2 and U(1) for related particles separately? We can forget about Z_2 however. I have prepared some notes and handy graphic (handwritten), let me know if I need to show you.
Last edit: Moein Najafi Ivaki 2017-05-06
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sorry, I had to catch up with some other things after a presentation on Monday.
1) I hope to implement the initial state wih quantum numbers tomorrow. Right now that is not possible.
2) I maybe did not phrase my question very well, but I would still like to see the matrix representation of these operators sigma_{i,j}^{z} if I did not make the right guess below. I'll explain the decomposition along with the initial state with quantum numbers once they are ready.
3) Simulations are in the Schroedinger picture, but that should not cause to much trouble.
Notes: yes, any reference or notes would be great. If you don't want to have your notes publicly accessible, you can send them along per mail (djaschke@mymail.mines.edu) and I will treat them confidential. But I think I got it now ... the 1d systems looks like
where F is a fermion and S a spin. So sigma_{i,j}^{z} is actual the Pauli operator acting on the spin between the fermion i and j. The index is only running over the fermions. That will help me to think about the best implementation, but I must say that looks tricky.
Thank you for your patience and kind regards,
Daniel
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I truly apperiate your concern. You got the right idea in its best, and I should stress that one should think about the most efficient way.
For the first stage I really like to know the initial state problem and that \delta\rho(t) computation.
I will get in touch with email.
Best.
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with a combined Hilbert space containing one spin and one fermion
(S1,F1) --- (S2,F2) --- (S3,F3) --- (S3,F4)
where each parenthesis is one site in our MPS. To represent the Hamiltonian in your first post, we have to consider the following points:
1) MPS does not have a combined Hilbert space spin-fermion, but Fermi-Bose operators. We define the spin operators on a hard-core boson local Hilbert space.
2) The xx-spin interaction in the second term should not act on MPS-sites 1 and 2. We can resolve that with a space-dependent coupling.
3) The operator s(i,i+1) f(i+1) act solely on one MPS-site. We can simply multiply them.
There is a reason for not doing this as alternating spin fermion chain: I do not see any good option to overcome the problem with the phase operator of the Jordan-Wigner transformation mapping the fermions to the chain, which would act only on every second site containing the fermions.
I attached a toy model which feeds in the input state with and without symmetry. I tested it against each other and they deliver the same results. If you run into any error, please let me know. You need to download the most recent version via svn from SourceForge. The initial product state with quantum numbers were not supported before.
IMPORTANT: Do not use the Krylov time evolution! It fails and has known issues if one does not start with ground state. The TDVP works, TEBD works as well but TDVP seems to be faster.
I hope that helps you to get started and let me know if you have any other questions once you played around with the sketched script (and check it carefully).
I want to know if i can calculate the mixed state using OSMPS
(djaschke: I moved this question to a new topic as it is not directly related to the initial state. Thus, I'll delete the question here at some point.)
Last edit: Daniel Jaschke 2018-12-18
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Hello everybody,
1) Could you please outline how one should determine a file to be used as an initial state? Can one set up and initial state without the need to creat such files?
2) My second question reagrds creating an specific form of hamiltonian:
![spinless fermions coupled via spins] (https://image.ibb.co/bKi835/Selection_044.png)
How can one spcefiy such terms?
3) I am also intrested in calculating such dynamics for fermionic number operator:
![ staggered fermion density] (https://image.ibb.co/eGbd35/Selection_045.png)
4) And the last question, does "QT" keyword works in the latest version? That is, averaging over different realization. How coud I use this?
Thank you very much for helping.
Last edit: Moein Najafi Ivaki 2017-05-04
Hallo Moein,
1) The default initial state is the ground state of a system. So you could design a Hamiltonian with different couplings and/or additional terms, which has your target state as ground state. Afterwards, you switch to your actual Hamiltonian for the time-evolution. This approach works for number conserving systems and in the grand-canonical ensemble.
Now as you pointed out, you can provide a file with the initial state. This solutions targets simulations in the grand-canonical ensemble. For example, product states are covered in a python function and you would not write down the file yourself, but call the function. Number conserving simulations write initial guesses for the ground state search, but they are not accessible as initial state right now.
Let me know if you want to use number conservation and how your intial states look like, and then we can work on more specific directions.
2) I think I need your feedback here to give you convenient advice. How are the sigma(i,j) defined? They are defined on two sites? As first input, the "hopping" term could be a bond term and you may use the decomposition request. Right now I am not sure if the Hamiltonian fulfills the bond term with a phase operator. Second term coupled with h could be a MBString term, although you would need to decompose it into products of local operators acting on i-1, i, and i+1 (O_1 x O_2 x O_3).
3) Your notation is in the Heisenberg picture? Assuming |0> = |psi(t=0)> is your initial state and you want to measure <psi(t) |="" n_j="" -="" n_(j+1)="" psi(t)="">, you could simply add the number operator as site observable. The difference in the density can be calculated during the postprocessing.</psi(t)>
4) If you are using the MPS methods, the QT (quantum trajectory) is a flag for the future use of the Lindblad master equation, but currently not available. For the exact diagonalization package inside openMPS, local Lindblad operators are available. Either as density matrix, or via quantum trajectories.
So far for the initial feedback. Once you helped me to understand the operators and the type of initial state, I can certainly give you a more detailed feedback.
Kind regards,
Daniel
I really appreciate the swift response.
Let's just start with the basic questions first.
1) Consider for example a state like what we call density-wave, |101010...>, and some other like domain wall, |11...11000...0>. I would like to use U(1) symmetry for fermions (Just in case it might be of use). I would appreciate very much if you write down a minimal piece of code, just to know how to implement it in a right manner. I also would like to make sure the dynamics is using the specified initial state correctly, so I can handle it myself in case of future difficulties.
2) Right away I am not sure. It might be good for the start to know how to creat such terms:
.
In addition, I have no idea of the decompoition process with package!
3) Yes, in the Heisenberg picture. Could you please drop me a piece of code? It is not clear to me how the dyanmics will do the diffrences in site observables. I undrestand the examples and various correlation within the package etc, however, definig this in the package, although it is highly trivial, does not seem straightforward to me. Sorry. I am also intrested in such connected correlators <0|n_{j}(t)+n_{j+l}(t)|0>.
Thank you very much.
Added notes, May 6:
More details on the structure of the given Hamiltonian:
Actually fermions are supposed to be placed on sites, and spins on bonds. There could several approaches be suggested, the efficeincy of which should be examined regarding the TEBD and tensor network, like placing fermions on odd sites and spins on even ones, or pattern like {spin1, spin2, fermion3, fermion4, spin5, spin6,...}. But these have heavy numrical cost. Thus, one should think about somthing like ladder geometry (Just schematically). Is it possible to implement such propoition with the right tensor and dimension, taknig into account both Z_2 and U(1) for related particles separately? We can forget about Z_2 however. I have prepared some notes and handy graphic (handwritten), let me know if I need to show you.
Last edit: Moein Najafi Ivaki 2017-05-06
No answer?
Hallo Moein,
sorry, I had to catch up with some other things after a presentation on Monday.
1) I hope to implement the initial state wih quantum numbers tomorrow. Right now that is not possible.
2) I maybe did not phrase my question very well, but I would still like to see the matrix representation of these operators
sigma_{i,j}^{z}if I did not make the right guess below. I'll explain the decomposition along with the initial state with quantum numbers once they are ready.3) Simulations are in the Schroedinger picture, but that should not cause to much trouble.
Notes: yes, any reference or notes would be great. If you don't want to have your notes publicly accessible, you can send them along per mail (djaschke@mymail.mines.edu) and I will treat them confidential. But I think I got it now ... the 1d systems looks like
F1 --- S1,2 --- F2 --- S2,3 --- F3 --- S3,4 --- F4
where F is a fermion and S a spin. So
sigma_{i,j}^{z}is actual the Pauli operator acting on the spin between the fermion i and j. The index is only running over the fermions. That will help me to think about the best implementation, but I must say that looks tricky.Thank you for your patience and kind regards,
Daniel
I truly apperiate your concern. You got the right idea in its best, and I should stress that one should think about the most efficient way.
For the first stage I really like to know the initial state problem and that \delta\rho(t) computation.
I will get in touch with email.
Best.
Hallo Moein,
so the solution is to rewrite the original model
F1 --- S1,2 --- F2 --- S2,3 --- F3 --- S3,4 --- F4
with a combined Hilbert space containing one spin and one fermion
(S1,F1) --- (S2,F2) --- (S3,F3) --- (S3,F4)
where each parenthesis is one site in our MPS. To represent the Hamiltonian in your first post, we have to consider the following points:
1) MPS does not have a combined Hilbert space spin-fermion, but Fermi-Bose operators. We define the spin operators on a hard-core boson local Hilbert space.
2) The xx-spin interaction in the second term should not act on MPS-sites 1 and 2. We can resolve that with a space-dependent coupling.
3) The operator s(i,i+1) f(i+1) act solely on one MPS-site. We can simply multiply them.
There is a reason for not doing this as alternating spin fermion chain: I do not see any good option to overcome the problem with the phase operator of the Jordan-Wigner transformation mapping the fermions to the chain, which would act only on every second site containing the fermions.
I attached a toy model which feeds in the input state with and without symmetry. I tested it against each other and they deliver the same results. If you run into any error, please let me know. You need to download the most recent version via svn from SourceForge. The initial product state with quantum numbers were not supported before.
IMPORTANT: Do not use the Krylov time evolution! It fails and has known issues if one does not start with ground state. The TDVP works, TEBD works as well but TDVP seems to be faster.
I hope that helps you to get started and let me know if you have any other questions once you played around with the sketched script (and check it carefully).
Kind regards,
Daniel
Very well done Daniel. I will be testing it soon to see if it gives the right output for various quantities.
Kind regards.
Last edit: Moein Najafi Ivaki 2017-05-12
I want to know if i can calculate the mixed state using OSMPS
(djaschke: I moved this question to a new topic as it is not directly related to the initial state. Thus, I'll delete the question here at some point.)
Last edit: Daniel Jaschke 2018-12-18