Contour Functions

Overview

class

cntr::function<T>

A contour function is a matrix or scalar-valued function \(f(t)\) which depends on physical time only:

  • On the imaginary time branch, it takes one given constant value \(f_{-1}\)

  • On the real-time branches, its value is the same for the same time \(t\) on the upper and lower contour branch.

A contour function is typically used to store time-dependent parameters of a system, or a time-dependent Hamiltonian.

The class cntr::function<T> is the container to store these data. It is characterized by the following parameters:

  • T (template parameter): Precision, usually set to double; we use the definition

    #define CFUNC cntr::herm_function<double>

  • nt (integer): number of discretization points on the real time axis. For nt=-1, only the (constant) value on the imaginary time axis is stored.

  • size1 (integer): orbital dimension. Each element \(f(t)\) is a square matrix of dimension size1 \(\times\) size1.

Constructors

function<T>()

Default constructor, does not allocate memory and sets nt=-2.

function<T>(int nt,int size1)

Allocate memory, set all entries to 0. It requires tstp>=-1, size1>0

Accessing individual matrix elements

The following routines allow to read/write the elements of a contour function \(f(t)\) stored as cntr::function at individual time arguments from/to another variable M. The latter can be either a scalar of type std::complex<T>, or a complex square matrix defined in Eigen (see Scalar and matrix types).

The following member functions of cntr::function<T> set components of a contour function \(f(t)\) from M:

f.set_value(j,M)

For j=-1: \(f_{-1}\) is set to M; For 0<=j<=f.nt(): \(f(j\Delta t)\) is set to M

  • If f.size1()>1, M must be a square matrix (cdmatrix for double precision)

  • If f.size1()==1, M can be also a scalar (cdouble) or a square matrix

  • If M is a matrix, it must be a square matrix of dimension f.size1()

The following member functions of cntr::function<T> read components of a contour function \(f\) to M:

f.get_value(j,M)

For j=-1: M is set to \(f_{-1}\); For 0<=j<=f.nt(): M is set to \(f(j\Delta t)\)

  • If M is a matrix, it is resized to a square matrix of dimension f.size1()

  • If M is a scalar, only the (0,0) entry of \(f(t)\) is read.