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 \(M\) and \(L\) of the operators \(L_z\) and \(L^2\) (z-component and total relative angular momentum respectively). These files can be used with Program: Run_MC_sphere.

Input options

Input	Description
-n	Number of occupied CF Landau levels in the ground state.
--twice-flux or -2q	Twice the effective CF flux. The lowest CF Landau level has \(2q+1\) states.
--twice-L or -2L	Twice the quantum number \(L\) of total angular momentum.
--twice-Lz or -2Lz	Twice the quantum number \(M\) of the z-component of angular momentum.
--qh-LLs or -qh	The CF Landau levels in which to insert quasiholes (counting from 0).
--qe-LLs or -qe	The CF Landau levels in which to insert quasielectrons (counting from 0).
--out-prefix or -op	Prefix for the files containing the result states (compatible with “Run_MC_sphere”).
--Lz-prefix or -Lzp	When activated all the \(L_z\)-states are written to files compatible with “Run_MC_sphere” using this prefix.
--det-file-out or -do	When activated all the \(L_z\)-states are written to this file; compatible with this program (“CF_states”).
--det-file-in or -di	When activated all the \(L_z\)-states are read from this file.
--kernel-method	Elects what matrix decomposition is used to find the kernel giving the \(L\)-states, choosing from sparse QR, QR, SVD and LU. Note: LU can be numerically unstable.
--test-kernel	When activated a check is performed to test that the kernel states are actually zero.
--Lz-only	Programs stops running after finding the \(L_z\)-states.
--verbose or -v	Print more information than usual to the screen.
--quiet or -q	Print less information than usual to the screen.

Example usage

Example 1: \(\nu=2/5\) with a single exciton

Our starting point is \(\nu=2/5\), i.e. two CF Landau levels filled: \(n=2\). Note that \(n=2\) also corresponds to other states; increasing the number of attached flux on the electrons from \(2p=2\) to \(2p=4,6,\ldots\) gives \(\nu=3/7,4/9,\ldots\), while reverse flux composite fermions at \(n=2\) has a corresponding series \(\nu=2/3,3/5,4/7,\ldots\).

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 \(2q=2\). This can also be seen from the formula \(2q=N_e/n-n=8/2-2=2\).

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 --qh-LLs 1 --qe-LLs 2. An example \(L_z\)-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 \(L=2\) with z-component \(L_z=0\) 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 \(L_z\)-state, where the first number gives the coefficient in the superposition of the \(L\)-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: \(L_z=-1+0+1-2+0+1+2-1=0\).