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General information

The Schwinger model code for two degenerate flavors has two classes: GaugeConf and HMC, which contain the relevant functions/properties of the gauge configurations and the Hybrid Monte Carlo algorithm. Some other modules are added to perform the matrix operations, statistical analysis, global variables and operator overloading. All the $\psi$ fields have the following format:

std:vector<std::vector<std::complex<double>>> Psi[Nx*Nt][2]

The first entry corresponds to the volume coordinate and the second one to the spin. Essentially it is just a rearrangment of the vector $\psi(n)_\alpha$.

We briefly describe the content of the .h files in the include directory

variables.h

• Declares the lattice dimensions (the values for $N_x$ and $N_t$ are extracted from the CMakeLists.txt file)
• Declares a 2d vector which contains the vectorized version of the 2d coordinates: Coords[x][t] = $x\cdot N_t + t$. Coords is defined with the void Coordinates() function located in gauge_conf.h
• x_1_t1 is a 2d array that has the vectorized coordinates of the neighbors $(t-1,x+1)$ (with periodic boundary).
• x1_t_1 is a 2d array that has the vectorized coordinates of the neighbors $(t+1,x-1)$ (with periodic boundary). Both arrays are initialized in matrix_operations.h, in the function periodic_boundary().
• RightPB stores the neighbor coordinates Coords[x, t + 1], Coords[x + 1,t].
• LeftPB stores the neighbor coordinates Coords[x - 1,t], Coords[x, t + 1]. Again, the arrays are initialized in the function periodic_boundary().

statistics.h

• Has some functions to obtain the mean of a set of measurements and the jackknife error.

operator_overloads.h

• Overloads the operators *,+,- for matrix-vector multiplication, scalar-matrix multiplication, matrix addition and substraction, vector addition and substraction and scalar-vector multiplication.
• Defines the alias

typedef std::complex<double> c_double;
typedef std::vector<c_double> c_vector;
typedef std::vector<c_vector> c_matrix; 

• Defines a complex dot product between complex fields arranged as psi[Nx*Nt][2]
• Defines a complex dot product between complex vectors arranged as psi[no_of_entries]

gauge_conf.h

class GaugeConf

• Intializes and stores the gauge configuration.
• Computes and stores the plaquettes.
• Computes and stores the staples.
• Measures the average plaquette value.

hmc.h

class HMC

• Intializes random momenta.
• Computes the force for the gauge term.
• Computes the force for the fermionic term.
• Performs the leapfrog algorithm.
• Computes the action (needed for the accept/reject step).
• Accept/reject step.
• Updates the configuration with leapfrog and the accept/reject step.

matrix_operations.h

• Defines the $\sigma$ matrices.
• Implements the periodic boundary and anti-periodic boundary conditions.
• Implements the operations $D\phi$, $D^\dagger \phi$, $DD^\dagger \phi$ and $\psi^\dagger \frac{\partial D}{\partial \omega_\mu (z)}\chi$, for arbitrary fields. These operations don't assemble the Dirac matrix.

conjugate_gradient.h

• Conjugate gradient to invert $(DD^\dagger)^{-1} \phi$ for arbitrary $\phi$.

config.h and config.h.in are just files needed for cmake.