With RTNI_light one can integrate matrices symbolically with respect to Haar unitary matrices.
RTNI_light is a light and user-friendly version of the program called RTNI, originally developed in the following paper:
M. Fukuda, R. Koenig, I. Nechita "RTNI - A symbolic integrator for Haar-random tensor networks",
The programs of the above paper can be found at the GitHub page:
RTNI at GitHub.
On this website, one can use the web version of RTNI_light, but
there are limitations:
The first condition is set for RTNI_light, and the second and third conditions are set to save computational resource of the server.
One can download and customize the program placed at:
- All matrices are `n xx n`.
- Only two i.i.d. Haar unitary matrices are allowed.
- The number of conjugate pairs of each Haar unitary matrix should not exceed two.
RTNI_light at GitHub
Besides, one can get the Weingarten functions for upto partitions of 20 as well.
Input data will not be stored or monitored on this webpage.
Integrating with respect to the Haar unitary.
An input is a sequence of two kinds of brackets and each bracket contains alphabets and some selected symbols (spaces will be ignored).
Some examples (you do not need "" for the input.):
- [ ] represents application of trace operation and ( ) otherwise.
- U and V (upper case) are reserved for i.i.d. Haar unitary matrices.
Any other upper and lower case alphabets can be used to represent matrices. Repetitions are allowed.
- +,-,* are used to represent transpose, complex conjugate and adjoint, respectively.
Note that those top-right symbols in AsciiMath: `T`, `-` and `**` are used in the output format,
to represent transpose, complex conjugate and adjoint, respectively.
- "Trace`[U^(**)AU] xx`Trace`[U^(T)BU^(-)]`" should be written as "[U*AU][U+BU-]".
- "Trace`[U^(**)AU] UCU^(**)`" should be written as "[U*AU](UCU*)".
- "`U^(**)AU^(T) ox UCDU^(**) ox VA^(T)DV^(-)`" should be written as "(U*AU+)(UCDU*)(VA+DV-)".
Getting Weingarten functions.
An input is a set of numbers separated by commas. For example,
where the input numbers are ordered automatically.
- "4,2,5" will give you the Weingarten function of the type (5,4,2).