Reduced Density Matrices (RDMs) for Neutron Drops using Semi-Definite Programming (SDP)

A program to output an SDP data format file for neutrons in a harmonic trap interacting via the Minnesota potential. This can then be solved using a SDP solver to find the ground state energy (along with the associated one- and two-body RDMs). Only the m scheme harmonic oscillator basis is currently supported.

To render equations on this page in Chrome, enable the Github with mathjax extension

Usage Instructions

There are three steps that must be followed before running the RDM program (which creates the SDP file).

Create one-body matrix elements using the Mathematica notebook in the src directory. Simply open the notebook and run it. Note however that the output filename, basis $\hbar \omega$, $n{\text{max}}$, and $l{\text{max}}$ currently all needs to be input by hand. Choices of $n{\text{max}}$ and $l{\text{max}}$ should match some choice of total $N$ for $N_{\text{max}}$ truncation, $N = 2n + l$.
Create two-body matrix elements using Morten’s ME code and then put the TBME output file and single-particle output file in the correct directories (me_files/tbme/ and me_files/ref_files/ respectively). Again, the filenames, basis $\hbar \omega$, $n{\text{max}}$, and $l{\text{max}}$ all need to be put in by hand. These are all specified in the renorm.ini file. Then run the program using ./vrenorm.exe.
Create the single-particle orbital states list, two-body basis states list, and data files with SDP constraint information using the python file nmax_count.py. The user needs to specify the nmax variable in the file (where the nmax variable is our $N_{\text{max}}$ truncation) and then run the python file.

After this is all done, the user specifies the $N_{\text{max}}$ truncation, basis $\hbar \omega$, the number of particles, and the chosen constraint flags in the inputs.inp file. Then make the program with make and run with ./run_rdm.

After the SDP file is generated (currently only with filename sdp_files/test_sdp.dat-s), the SDP solver can then be called on it using for example:

csdp test_sdp.dat-s
sdpa test_sdp.dat-s solution_out.dat

File Hierarchy and Description

Program source files are kept in the src directory, input m scheme matrix elements and single-particle files and two-particle basis files in the me_files directory, input flag files from the python program in the flag_files directory, and output data files in the sdp_files directory.

SDP Data File Format

The SDP data file is outputted in sparse data format as indicated by the


at the end of the file. Note that the sparse format (and solvers that we use in general) assumes that all constraint matrices are symmetric and sub-block formats between matrices are identical. Output format is given as:
* The total number of constraints
* The number of sub-blocks 
* The relevant sizes of each sub-block in order (pure consecutive diagonal entires are given with a - prefix)
* The constant term for each constraint matrix relation
* A line for each non-zero entry in the upper triangle part of a given constraint matrix
    * which constraint matrix is being specified starting from zero for the matrix in the objective function
    * which matrix block is specified (starting from 1)
    * the relevant row (starting from 1)
    * the relevant column (starting from 1)
    * the value of the matrix 
Note that rows and columns reset for each block. For example, supposing the matrix

[[6 1 0 0 0]
[1 8 0 0 0]
[0 0 4 0 0]
[0 0 0 3 0]
[0 0 0 0 7]]


was our first and only constraint matrix term with the constant value 4.5 in the matrix relation. Our SDP file would be given by,

1
2
2 -3
4.5
0 1 1 1 6
0 1 1 2 1
0 1 2 2 8
0 2 1 1 4
0 2 2 2 3
0 2 3 3 7
```

Benchmarks

Calculations for benchmarks assume the following parameter values:

$\hbar c = 197.326 \ \text{MeV-fm}$
$m = 938.92 \ \text{MeV}$
$\hbar \omega = 10 \ \text{MeV}$
$b = \frac{\hbar c}{\sqrt{\hbar \omega m}} = 2.0364 \ \text{fm}$