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
• Run otoc.exe from command line to see help & examples

Examples

Examples are displayed with helps when otoc.exe is executed without option.

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

otoc ham -L 4 -M 2

2. Generate all Hamiltonian matrix and trace states for L=4

otoc ham -L 4

3. Generate norm matrix for L=4, M=2

otoc norm -L 4 -M 2

4. Generate all norm matrix for L=4

otoc norm -L 4

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

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

6. Generate matrix for operator counting traces containing X for L=4

otoc operator -o NTrX -L 4

7. 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

8. 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

9. 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

10. 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

11. 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