.. _CF_states: *********************** Program: CF_states *********************** This program generates configuration files for excited composite fermion states, for different configurations of quasiholes and quasielectrons and different values of the quantum numbers :math:`M` and :math:`L` of the operators :math:`L_z` and :math:`L^2` (z-component and total relative angular momentum respectively). These files can be used with :ref:`Run_MC_sphere`. =============== Input options =============== ====================================== ============================================================================================================================================================================== Input Description ====================================== ============================================================================================================================================================================== :code:`-n` Number of occupied CF Landau levels in the ground state. :code:`--twice-flux` or :code:`-2q` Twice the effective CF flux. The lowest CF Landau level has :math:`2q+1` states. :code:`--twice-L` or :code:`-2L` Twice the quantum number :math:`L` of total angular momentum. :code:`--twice-Lz` or :code:`-2Lz` Twice the quantum number :math:`M` of the z-component of angular momentum. :code:`--qh-LLs` or :code:`-qh` The CF Landau levels in which to insert quasiholes (counting from 0). :code:`--qe-LLs` or :code:`-qe` The CF Landau levels in which to insert quasielectrons (counting from 0). :code:`--out-prefix` or :code:`-op` Prefix for the files containing the result states (compatible with "Run_MC_sphere"). :code:`--Lz-prefix` or :code:`-Lzp` When activated all the :math:`L_z`-states are written to files compatible with "Run_MC_sphere" using this prefix. :code:`--det-file-out` or :code:`-do` When activated all the :math:`L_z`-states are written to this file; compatible with this program ("CF_states"). :code:`--det-file-in` or :code:`-di` When activated all the :math:`L_z`-states are read from this file. :code:`--kernel-method` Elects what matrix decomposition is used to find the kernel giving the :math:`L`-states, choosing from sparse QR, QR, SVD and LU. *Note*: LU can be numerically unstable. :code:`--test-kernel` When activated a check is performed to test that the kernel states are actually zero. :code:`--Lz-only` Programs stops running after finding the :math:`L_z`-states. :code:`--verbose` or :code:`-v` Print more information than usual to the screen. :code:`--quiet` or :code:`-q` Print less information than usual to the screen. ====================================== ============================================================================================================================================================================== =============== Example usage =============== """""""""""""""""""""""""""""""""""""""""""""""""" Example 1: :math:`\nu=2/5` with a single exciton """""""""""""""""""""""""""""""""""""""""""""""""" Our starting point is :math:`\nu=2/5`, i.e. two CF Landau levels filled: :math:`n=2`. Note that :math:`n=2` also corresponds to other states; increasing the number of attached flux on the electrons from :math:`2p=2` to :math:`2p=4,6,\ldots` gives :math:`\nu=3/7,4/9,\ldots`, while reverse flux composite fermions at :math:`n=2` has a corresponding series :math:`\nu=2/3,3/5,4/7,\ldots`. If we want e.g. eight electrons there will be three composite fermions in the lowest CF Landau level and five in the first, which corresponds to effective flux :math:`2q=2`. This can also be seen from the formula :math:`2q=N_e/n-n=8/2-2=2`. Then we excite a single composite fermion, which is modelled using a quasihole and a quasielectron. If the quasihole is in the first Landau level (counting from zero) and the quasielectron is in the second, we use :code:`--qh-LLs 1 --qe-LLs 2`. An example :math:`L_z`-state from this configuration can be diagrammatically represented as: :: _ _ _ _ x _ _ x x x _ x x x x Finally lets assume we want to have a total relative angular momentum quantum number :math:`L=2` with z-component :math:`L_z=0` and save the results in files starting with "nu_2_5_exciton." We can then run: :: CF_states -n 2 -2q 2 -2L 4 -2Lz 0 -qh 1 -qe 2 -op nu_2_5_exciton It turns out there is only one state with these criteria, which is then saved in "nu_2_5_exciton.dat" and looks like this: :: 0.37796447300922736 0 -1 0 0 0 1 1 -2 1 0 1 1 1 2 2 -1 -0.59761430466719667 0 -1 0 0 0 1 1 -1 1 0 1 1 1 2 2 -2 -0.37796447300922753 0 -1 0 0 0 1 1 -2 1 -1 1 0 1 2 2 1 0.59761430466719678 0 -1 0 0 0 1 1 -2 1 -1 1 0 1 1 2 2 Every line corresponds to an :math:`L_z`-state, where the first number gives the coefficient in the superposition of the :math:`L`-state. After that the CF LL and momentum for the individual composite fermions are given alternatingly. Let's check that the first state (first line) has the correct total z-component, which should be zero. We skip the superposition coefficient and then count every other integer: :math:`L_z=-1+0+1-2+0+1+2-1=0`.