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Mixed Exciton-Boson models

Technical
lello
2019-09-24
2019-09-25
  • lello

    lello - 2019-09-24

    Hi,

    I just started to use the OMPS package but I cannot understant how to write an exciton-boson Hamiltonian, or if it is possible at all.
    The Hamiltonian is technically comprised of a first site with and arbitrary (Hermitian) Hamiltonian H_e and additional boson sites interacting with the first site. The formal structure of the Hamiltonian would be

    H = ∑{n,m} H |n>< m| + ∑k ω_k n_k − ∑ |n><n|g_{nk} (b_k + b_k)

    In the above expression the parameters H_{nm} define the exction part and should be arbitrary, and moreover each bosonic site should allow for a different size of the basis (i.e. a different maximum excitation number); n_k is the k-th occupation number; ω_k the frequency of the k-th boson; the constants g_k defines the coupling between the state |n> of the first site and the k-th boson.

    Any help is appreciated,
    Raffaele

     
  • lello

    lello - 2019-09-25

    Hi,

    the main problem I am facing at the moment is how to define a copuling operator between the first site and all the other sites. Due to lack of symmetries of such a model I think it should be possible to use the rule "ProductTerm" to construct very general exciton-boson MPOs and then add all of them toghether.
    However, I don't understand how OMPS performs the summation of MPO operators. Is it possible to perform addition and truncation of operators using SVD to keep bond dimensions at a reasonable level?

    Thanks
    R

     
    • Daniel Jaschke

      Daniel Jaschke - 2019-09-25

      Hi,

      the ProductTerm is an n-body interaction, not a two-site term. From your
      question, I would have thought that is not what you want.

      But I would look into (and the weights for the coupling are crucial):
      - ExpTerm with site-dependent coupling only activating the first site. You
      could also get a decay depending on the distance. With the proper decay
      parameter, it does not decay.
      - InfiniteFunction with site-dependent coupling activating the first site.
      It uses ExpRule to fit other decay functions and is more expensive.
      - TTerm is an interaction (equal strength) of sites 1 .. (kk - 1) with site
      k, where you could probably which kk as the last site instead of the first
      as you had in mind. I hope there is no check that kk must be smaller than
      the system. (This one is less used)

      You can find the details in the documentation.
      https://openmps.sourceforge.io/pyautodoc/mpoterm.html?highlight=expterm#mpoterm.ExpTerm
      https://openmps.sourceforge.io/pyautodoc/mpoterm.html?highlight=infinite#mpoterm.InfiniteFunc
      https://openmps.sourceforge.io/pyautodoc/mpoterm.html?highlight=tterm#mpoterm.TTerm

      There are no SVDs on the representation of the MPO, but for the
      infiniteFunction you have cut-off criteria how many functions you want to
      use. If look a bit into our OSMPS paper in CPC, you can estimate the bond
      dimension of the MPO to get an estimate of how expensive the simulations
      will be.

      Let me know if you have further questions once you had a chance to look
      into the terms.

      Best regards,

      Daniel

      On Wed, Sep 25, 2019 at 2:25 PM lello lelloborrelli@users.sourceforge.net
      wrote:

      Hi,

      the main problem I am facing at the moment is how to define a copuling
      operator between the first site and all the other sites. Due to lack of
      symmetries of such a model I think it should be possible to use the rule
      "ProductTerm" to construct very general exciton-boson MPOs and then add all
      of them toghether.
      However, I don't understand how OMPS performs the summation of MPO
      operators. Is it possible to perform addition and truncation of operators
      using SVD to keep bond dimensions at a reasonable level?

      Thanks
      R


      Mixed Exciton-Boson models
      https://sourceforge.net/p/openmps/discussion/tech/thread/9a4064774b/?limit=25#2427


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