# A command line tool to compute OTOC for N=4 supersymmetric Yang–Mills theory

This is a command line tool to numerically compute Out-of-time-ordered correlator (OTOC) for the N=4 SYM and beta deformed N=4 SYM, for both inifnite N and finite N cases.

All computations are limited to SU(2) sector where trace states are built out of from X and Z. The number of fields in a trace state is called trace length, denoted by integer number L. The X field are also called magnon, its number is denoted by an integer M <= L.

For more details of the background, read the paper One-Loop Non-Planar Anomalous Dimensions in Super Yang-Mills Theory and Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?

## Features

- Generate trace states of SU(2) sector and Hamiltonian matrix in trace states.
- Generate norm matrix, whose entry at
`(i,j)`

represents the overlap between trace states`i`

and`j`

. - Compute energy spectrum for Hamiltonian matrix
- Compute OTOC

## Build & Run

- Install the BOOST C++ library
- Install CMake
- Change
`BOOST_ROOT`

in CMakeLists.txt in the project root:

```
SET(BOOST_ROOT ${MYDEV}/boost_1_70_0) --> SET(BOOST_ROOT path/toboost/library)
```

- Create a
`build`

directory under project root - Run command line
`cmake ../`

from the`build`

directory - Build the project:
- In Windows/Visual Studio, double click the .sln file in the
`build`

directory, then build the solution - In Linux, run command line
`make`

in the`build`

directory

- In Windows/Visual Studio, double click the .sln file in the
- Run
`otoc.exe`

from command line to see help & examples

## Examples

Examples are displayed with helps when `otoc.exe`

is executed without option.

- Generate Hamiltonian matrix and trace states for L=4, M=2

`otoc ham -L 4 -M 2`

- Generate all Hamiltonian matrix and trace states for L=4

```
otoc ham -L 4
```

- Generate norm matrix for L=4, M=2

```
otoc norm -L 4 -M 2
```

- Generate all norm matrix for L=4

```
otoc norm -L 4
```

- Compute energy spectrum for L=4, M=2, N=17, beta=0.9

```
otoc spec -L 4 -M 2 -N 17 -b 0.9
```

- Generate matrix for operator counting traces containing X for L=4

```
otoc operator -o NTrX -L 4
```

- Compute OTOC for W=Tr(Xd/dZ), V=Tr(Zd/dX), L=4, N=5, beta=0.9, beta of temperature=0.5, and time run from 0.0 to 5.0 with step size 0.1. Note that to run the example one needs to run
`otoc norm - L 4`

first to generate all norm matrices for L = 4 and then copy the generated files to otocdata folder

```
otoc otoc -W XZ -V ZX -L 4 -N 5 -b 0.9 -bt 0.5 -tmin 0.0 -tmax 5.0 -tstep 50
```

- Compute OTOC for W=Tr(XZ d/dX d/dZ), V=Tr(ZX d/dX d/dZ), L=4, M=2, beta=0.9, beta of temperature=0.5, and N=Infinity. As -n option is specified, it will compute norm matrix rather than load it from local files.

```
otoc otoc -W XZXZ -V XZXZ -L 4 -M 2 -b 0.9 -bt 0.5 -tmin 0.0 -tmax 5.0 -tstep 50 -n
```

- Compute C(t) for W=Tr(XX d/dX d/dZ), V=Tr(ZZ d/dZ d/dX), L=4, N=5. Zero energy states are excluded.

```
otoc otoc -W XXXZ -V ZZZX -L 4 -N 5 -b 0.9 -bt 0.5 -tmin 0.0 -tmax 5.0 -tstep 50 -nozero
```

- Compute C(t) for W=Tr(XX d/dX d/dZ), V=Tr(ZZ d/dZ d/dX), L=4, N=5. With regularization parameters alpha=0.5 and sigma=0.25.

```
otoc otoc -W XXXZ -V ZZZX -alpha 0.5 -sigma 0.25 -L 4 -N 5 -b 0.9 -bt 0.5 -tmin 0.0 -tmax 5.0 -tstep 50
```

- Compute OTOC for composite operators. Currently, addition (+), substraction(-), multiplication(*), and exponentiate (exp), and normal ordered (~) are supported. No division (/) and space is allowd in the expression. Complex number should write in the form (real,imag).

```
otoc otoc -W -0.2*exp(0.2*XXXZ+0.3*XZ)+(0.3,-0.5) -V ~(-XZXZ+0.5*ZZZX-0.2*ZX) -L 4 -N 5 -b 0.9 -bt 0.5 -tmin 0.0 -tmax 5.0 -tstep 50
```