## RTNI_light_web   Tweet

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",
J. Phys. A: Math. Theor. 52 425303 (2019); BibTeX.

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:

• 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.
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:
RTNI_light at GitHub

Besides, one can get the Weingarten functions for upto partitions of 20 as well.

I hope that there is no mistake in the program. :)

Note: 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).

• [ ] 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.
Some examples (you do not need "" for the input.):
• "Trace[U^(**)AU] xxTrace[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-)".
Note that those top-right symbols in AsciiMath: T, - and ** are used in the output format, to represent transpose, complex conjugate and adjoint, respectively.

### Getting Weingarten functions.

An input is a set of numbers separated by commas. For example,

• "4,2,5" will give you the Weingarten function of the type (5,4,2).
where the input numbers are ordered automatically.