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.

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Usage Instructions

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

1. 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, $N = 2n + l$.

2. 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.

3. 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

-s

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:

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