Green Functions

Overview

class

cntr::herm_matrix<T>

This class contains the data structures for representing two-time contour functions \(C(t,t')\) on the equidistantly discretized time-contour \(\mathcal{C}\) with points \(\{i\Delta t: i=0,1,2,...,{\tt nt}\}\) along the real time branch, and \(\{i\Delta\tau,i=0,1,...,{\tt ntau}\}\) along the imaginary branch. The Green’s functions are defined by the following parameters:

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

    #define GREEN cntr::herm_matrix<double>

  • nt and ntau (integer): number of discretization points on the real and imaginary time axis.

  • size1 (integer): orbital dimension. Each element \(C(t,t')\) is a square matrix of dimension size1 \(\times\) size1.

  • sig (integer): Takes the values FERMION (defined as -1) or BOSON (+1).

herm_matrix stores the following Keldysh components on the equidistantly discretized Keldysh contour with time discretization \(\Delta t\) and \(\Delta \tau\) on the real and imaginary time, respectively (The real-time domain of a herm_matrix object is illustrated in the Figure):

Storage diagram for herm_matrix

  • Matsubara component \(C^\mathrm{M}(i\Delta \tau)\) for i=0,...,ntau,

  • retarded component \(C^\mathrm{R}(i\Delta t,j\Delta t)\) for i=0,...,nt, j=0,...,i,

  • lesser component \(C^<(i\Delta t,j\Delta t)\) for j=0,...,nt, i=0,...,j,

  • left-mixing component \(C^\rceil(i\Delta t,j\Delta \tau)\) for i=0,...,nt, j=0,...,ntau.

If nt = -1, only the Matsubara component is stored.

Size arguments of herm_matrix are returned by the following read-only member functions:

GREEN G;
cout << "The object G of type herm_matrix has the following dimensions:" << endl;
cout << "nt= " << G.nt() << endl;
cout << "ntau= " << G.ntau() << endl;
cout << "size1= " << G.size1() << endl;
cout << "sig= " << G.sig() << endl;

Hermitian symmetry and conjugate:

In the following, we call the components mentioned above the hermitian domain of a two-time contour function.

The Hermitian conjugate \(C^\ddagger\) of a Green’s function \(C\) is defined (\(\xi=\) +1 and -1 for BOSON and FERMION, respectively):

  • \([C^\ddagger]^\mathrm{R}(t,t^\prime) = \left( C^\mathrm{A}(t^\prime,t)\right)^\dagger \ ,\)

  • \([C^\ddagger]^\gtrless(t,t^\prime) = -\left( C^\gtrless(t^\prime,t) \right)^\dagger \ ,\)

  • \([C^\ddagger]^{\rceil}(t,\tau) = - \xi \left( C^\lceil(\beta-\tau,t) \right)^\dagger \ ,\)

  • \([C^\ddagger]^{\mathrm{M}}(\tau) =\left( C^{\mathrm{M}}(\tau) \right)^\dagger \ .\)

Apparently, \(C=(C^\ddagger)^\ddagger\).

Hence, if a Green’s function has hermitian symmetry (\(C=C^\ddagger\)), it is sufficient to use one object of type herm_matrix and store the hermitian domain. If a Green’s function is not hermitian, one must store the hermitian domain of \(C\) and of \(C^\ddagger\) in two separate herm_matrix variables.

For a summary of all member functions, see Summary: Member Functions of herm_matrix and herm_matrix_timestep[_view]. The following paragraphs give detailed explanations of some functionalities.

Constructors

herm_matrix<T>()

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

herm_matrix<T>(int nt,int ntau,int size1,int sig)

Allocate memory, and sets all entries to 0. It requires nt>=-1, ntau>0, size1>0, and sig \(\in\{\) FERMION, BOSON \(\}\).

herm_matrix<T>(int nt,int ntau)

Equivalent to herm_matrix(int nt,int ntau,int size1,int sig) with size1=1 and sig=FERMION.

Accessing individual elements

The following routines allow to read/write the elements of a contour function \(C(t,t')\) stored as herm_matrix 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).

Member functions of herm_matrix:

Member functions of herm_matrix allow to read/write elements of \(C\) in the hermitian domain. To read elements of \(C\) outside the hermitian domain (which are reconstructed from \(C\) and \(C^\ddagger\)), one can use the non-member element access functions defined later.

The following member functions set components of a contour function \(C\) in the hermitian domain from M:

C.set_les(i,j,M)

\(C^<(i\Delta t,j\Delta t)\) is set to M

required: 0 <= j <= C.nt(), 0<=i<=j

C.set_ret(i,j,M)

\(C^R(j\Delta t,i\Delta t)\) is set to M

required: 0 <= i <= C.nt(), 0<=j<=i

C.set_tv(i,j,M)

\(C^{\rceil}(i\Delta t,j\Delta \tau)\) is set to M

required: 0 <= i<= C.nt(), 0 <= j<= C.ntau()

C.set_mat(i,M)

\(C^M(i\Delta \tau)\) is set to M

required: 0 <= i <= C.ntau()

  • If C.size1()>1, M must be a complex eigen matrix (cdmatrix for double precision)

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

  • If M is a Matrix, it must be a square matrix of dimension size1

The following member functions read components of a contour function \(C\) in the hermitian domain to M:

C.get_les(i,j,M)

M is set to \(C^<(i\Delta t,j\Delta t)\)

required: 0 <= j <= C.nt(), 0<=i<=j

C.get_ret(i,j,M)

M is set to \(C^R(i\Delta t,j\Delta t)\)

required: 0 <= i <= C.nt(), 0<=j<=i

C.get_tv(i,j,M)

M is set to \(C^{\rceil}(i\Delta t,j\Delta \tau)\)

required: 0 <= i<= C.nt(), 0 <= j<= C.ntau()

C.get_mat(i,M)

M is set to \(C^M(i\Delta \tau)\)

required: 0 <= i <= C.ntau()

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

  • If M is a scalar, only the (0,0) entry of \(C(t,t')\) is read.

Remark: Member functions set_les and set_ret can return also values outside the hermitian domain (assuming that \(C\) has hermitian symmetry), but this functionality is no longer supported. Similarly, there are non-supported member functions like get_gtr(...) which return other Keldysh components. Use instead the general access functions described in the next paragraph.

General element access:

The following functions read components of a contour function \(C\) to M:

cntr::get_les(i,j,M,C,Ccc)

M is set to \(C^<(i\Delta t,j\Delta t)\)

required: 0 <= i,j <= C.nt()

cntr::get_gtr(i,j,M,C,Ccc)

M is set to \(C^>(i\Delta t,j\Delta t)\)

required: 0 <= i,j <= C.nt()

cntr::get_ret(i,j,M,C,Ccc)

M is set to \(C^>(i\Delta t,j\Delta t)-C^<(i\Delta t,j\Delta t)\)

required: 0 <= i,j <= C.nt()

cntr::get_tv(i,j,M,C,Ccc)

M is set to \(C^{\rceil}(i\Delta t,j\Delta \tau)\)

required: 0 <= i<= C.nt(), 0 <= j<= C.ntau()

cntr::get_vt(i,j,M,C,Ccc)

M is set to \(C^{\lceil}(i\Delta \tau,j\Delta t)\)

required: 0 <= i<= C.ntau(), 0 <= j<= C.nt()

cntr::get_mat(i,M,C,Ccc)

M is set to \(C^M(i\Delta \tau)\)

required: 0 <= i <= C.ntau()

  • C and Ccc are (size-matched) arguments of type herm_matrix which store the hermitian domain of \(C\) and its hermitian conjugate \(C^\ddagger\).

  • If the argument Ccc is omitted, hermitian symmetry \(C=C^\ddagger\) is assumed.

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

  • M may always be a scalar; in this case only the (0,0) entry of \(C(t,t')\) is read.

Note

All set_XXX and get_XXX element access functions are not optimised for massive read/write operations. (In particular, set_XXX and get_XXX are not used inside the numerically costly tasks like the solution of integral equations). Mostly, data are anyway read and written timestep-wise (see Timeslices).

Density matrix

The equal-time elements of Green’s functions contain expectation values of physical observables. If \(C_{ab}(t,t') = -i \langle T_\mathcal{C} \hat A_a(t)\hat B_b(t')\rangle\) is the contour ordered expectation value of two operators \(\hat A\) and \(\hat B\) (such as creation and annihilation operators with orbital indices a and b) then the time-dependent expectation value \(\rho_{ab}(t) = \langle B_{b} A_{a} \rangle_t\) is given by (\(\xi\) is the BOSON/FERMION sign):

\[\begin{split}\rho_{ab}(t) &= i \xi C^<_{a,b}(t,t) \quad \text{for general real times} \\ \rho_{ab} &= - C^M_{a,b}(\beta) \quad \text{for the equilibrium initial state}\end{split}\]

We therefore define the Density matrix of a two-time Green’s function on the discretized contour at timestep tstp as:

\[\begin{split}\text{DensityMatrix}[C]({\tt tstp}) &= - C^M({\tt ntau}\,\Delta\tau) \quad \text{for the equilibrium initial state}\,({\tt tstp}=-1) \\ &= i \xi C^<_{a,b}({\tt tstp}\Delta t,{\tt tstp}\Delta t) \quad \text{for}\,{\tt tstp}>=0.\end{split}\]

The following member function of a cntr::herm_matrix<T> object C writes the Density matrix to an object M:

C.density_matrix(int tstp,M)

M is set to \(\text{DensityMatrix}[C]({\tt tstp})\)

  • tstp <=  C.nt() required.

  • If C.size1()>1, M must be a complex eigen matrix (cdmatrix for double precision)

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

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

File I/O

Reading/writing to text files

The member functions print_to_file and read_to_file of a herm_matrix implement text-file access, as apparent by example:

int nt=10;
int ntau=10;
int size1=2;
int sig=FERMION;
GREEN A(nt,ntau,size1,sig);
// ... set elements of  A ...

//WRITE:
// create a file filename.txt and store the data of A:
A.print_to_file("filename.txt");

// READ
GREEN B;
// if the file "filename.txt" has been written previously with print_to_file ,
// the parameters (nt,ntau,size1,sig) and the data of B are modified
// according to the information in the file:
B.read_from_file("filename.txt");
// now B is resized to B.nt()=10, B.ntau()=10,B.size1()=2,B.sig()=FERMION, and the data of B match the data of A.

The file format is rather explicit (and storage intensive):

  • First line: # nt ntau size1 sig

  • The following lines list the Matsubara component in the format: mat: i Re_C^mat(i)_{0,0} Im_C^mat(i)_{0,0} ... Re_C^mat(i)_{size1,size1} Im_C^mat(i)_{size1,size1}

  • If nt>-1, the following lines list the other components in the format: XXX: i j Re_C^XXX(i,j)_{0,0} Im_C^XXX(i,j)_{0,0} ... Re_C^XXX(i,j)_{size1,size1} Im_C^XXX(i,j)_{size1,size1} where XXX is ret for \(C^R\), les for \(C^<\), tv for \(C^{\rceil}\), and i,j loop through the corresponding time arguments of the hermitian domain.

Note

Writing to text files is supposed as a quick way to generate human-readible data. For compressed storage, one should use the HDF5 format below.

Reading/writing to hdf5 files

The HDF5 format, as well as some scripts to interpret the hdf5 files are discussed in a separate section HDF5 Usage below.