Input files for Standard mode

An example of input file for the standard mode is shown below:

W = 2
L = 4
model = "spin"

lattice = "triangular lattice"
//mu = 1.0
// t = -1.0
// t' = -0.5
// U = 8.0
//V = 4.0
//V'=2.0
J = -1.0
J'=-0.5
// ncond = 8

Basic rules for input files

  • In each line, there is a set of a keyword (before an “ =”) and a parameter(after an “ =”); they are separated by “ =”.

  • You can describe keywords in a random order.

  • Empty lines and lines beginning in a “//”(comment outs) are skipped.

  • Upper- and lowercase are not distinguished. Double quotes and blanks are ignored.

  • There are three kinds of parameters.

    1. Parameters that must be specified (if not, vmcdry.out will stop with error messages),

    2. Parameters that is not necessary be specified (if not, default values are used),

    3. Parameters that must not be specified (if specified, vmcdry.out will stop with error messages).

    An example of 3 is transfer \(t\) for the Heisenberg spin system. If you choose “model=spin”, you should not specify “\(t\)”.

We explain each keywords as follows:

Parameters about the kind of a calculation

  • model

    Type : String (Choose from "Fermion Hubbard", "Spin", "Kondo Lattice" , "Fermion HubbardGC", "SpinGC", "Kondo LatticeGC" ) [1]

    Description : The target model is specified with this parameter; "Fermion Hubbard" denotes the canonical ensemble of the Fermion in the Hubbard model

    (1)\[\begin{aligned} H = -\mu \sum_{i \sigma} c^\dagger_{i \sigma} c_{i \sigma} - \sum_{i \neq j \sigma} t_{i j} c^\dagger_{i \sigma} c_{j \sigma} + \sum_{i} U n_{i \uparrow} n_{i \downarrow} + \sum_{i \neq j} V_{i j} n_{i} n_{j}, \end{aligned}\]

    "Spin" denotes canonical ensemble in the Spin model( \(\{\sigma_1, \sigma_2\}={x, y, z}\))

    (2)\[\begin{split}\begin{aligned} H &= -h \sum_{i} S_{i z} - \Gamma \sum_{i} S_{i x} + D \sum_{i} S_{i z} S_{i z} \nonumber \\ &+ \sum_{i j, \sigma_1}J_{i j \sigma_1} S_{i \sigma_1} S_{j \sigma_1}+ \sum_{i j, \sigma_1 \neq \sigma_2} J_{i j \sigma_1 \sigma_2} S_{i \sigma_1} S_{j \sigma_2} , \end{aligned}\end{split}\]

    "Kondo Lattice" denotes canonical ensemble in the Kondo lattice model

    (3)\[\begin{aligned} H = - \mu \sum_{i \sigma} c^\dagger_{i \sigma} c_{i \sigma} - t \sum_{\langle i j \rangle \sigma} c^\dagger_{i \sigma} c_{j \sigma} + \frac{J}{2} \sum_{i} \left\{ S_{i}^{+} c_{i \downarrow}^\dagger c_{i \uparrow} + S_{i}^{-} c_{i \uparrow}^\dagger c_{i \downarrow} + S_{i z} (n_{i \uparrow} - n_{i \downarrow}) \right\}. \end{aligned}\]

    "Fermion HubbardGC", "SpinGC", and "Kondo LatticeGC" indicate the \(S_z\)-unconserved Fermion in the Hubbard model [Eqn. (1) ], the \(S_z\)-unconserved Spin model [Eqn. (2) ], and the \(S_z\)-unconserved Kondo lattice model [Eqn. (3) ], respectively. Note: Although these flags has a word “GC”(=grandcanonical), the number of electrons are conserved in these system.

  • lattice

    Type : String (Choose from "Chain Lattice", "Square Lattice", "Triangular Lattice", "Honeycomb Lattice", "Kagome", "Ladder")

    Description : The lattice shape is specified with this parameter; above words denote the one dimensional chain lattice (Fig. 2 (a)), the two dimensional square lattice (Fig. 2 (b)), the two dimensional triangular lattice (Fig. 2 (c)), the two dimensional anisotropic honeycomb lattice (Fig. 3 ), the Kagome Lattice(Fig. 4 ), and the ladder lattice (Fig. 5 ), respectively.

    _images/chap04_1_lattice.png

    Figure 2: Schematic illustration of (a) one dimensional chain lattice, (b) two dimensional square lattice, and (c) two dimensional triangular lattice. They have \(t\), \(V\), and \(J\) as a nearest neighbor hopping, an offsite Coulomb integral, and a spin-coupling constant, respectively (magenta solid lines); They also have \(t'\), \(V'\), and \(J'\) as a next nearest neighbor hopping, offsite Coulomb integral, and spin-coupling constant, respectively (green dashed line).

    _images/chap04_1_honeycomb.png

    Figure 3: Schematic illustration of the anisotropic honeycomb lattice. The first/second/third nearest neighbor hopping integral, spin coupling, and offsite Coulomb integral depend on the bond direction.

    _images/kagome.png

    Figure 4: Schematic illustration of the Kagome lattice.

    _images/ladder.png

    Figure 5: Schematic illustration of the ladder lattice.

Parameters for the lattice

Chain lattice

Fig. 2 (a)

  • L

    Type : Integer

    Description : The length of the chain is specified with this parameter.

Ladder lattice

Fig. 5

  • L

    Type : Integer

    Description : The length of the ladder is specified with this parameter.

  • W

    Type : Integer

    Description : The number of the ladder is specified with this parameter.

    _images/chap04_1_unitlattice.png

    Figure 6: The shape of the numerical cell when \({\vec a}_0 = (6, 2), {\vec a}_1 = (2, 4)\) in the triangular lattice. The region surrounded by \({\vec a}_0\) (Magenta dashed arrow) and \({\vec a}_1\) (Green dashed arrow) becomes the cell to be calculated (20 sites).

Square lattice , Triangular lattice, Honeycomb lattice, Kagome lattice

Square lattice [Fig. 2 (b)], Triangular lattice[Fig. 2 (c)], Honeycomb lattice(Fig. 3 ), Kagome lattice(Fig. 4 )

In these lattices, we can specify the shape of the numerical cell by using the following two methods.

  • W, L

    Type : Integer

    Description : The alignment of original unit cells (dashed black lines in Figs. 2 - 4 ) is specified with these parameter.

  • a0W, a0L, a1W, a1L

    Type : Integer

    Description : We can specify two vectors (\({\vec a}_0, {\vec a}_1\)) that surrounds the numerical cell (Fig. 6 ). These vectors should be specified in the Fractional coordinate.

If we use both of these method, vmcdry.out stops.

We can check the shape of the numerical cell by using a file lattice.gp (only for square, trianguler, honeycomb, and kagome lattice) which is written in the Standard mode. This file can be read by gnuplot as follows:

$ gnuplot lattice.gp

Sublattice

By using the following parameters, we can force the pair-orbital symmetrical to the translation of the sublattice.

  • a0Wsub, a0Lsub, a1Wsub, a1Lsub, Wsub, Lsub

    Type : Positive integer. In the default setting, a0Wsub=a0W, a0Lsub=a0L, a1Wsub=a1W, a1Lsub=a1L, Wsub=W, and Lsub=L. Namely, there is no sublattice.

    Description : We can specify these parameter as we specify a0W, a0L, a1W, a1L, W, L. If the sublattice is incommensurate with the original lattice, vmcdry.out stops.

Parameters for the Hamiltonian

A default value is set as \(0\) unless a specific value is not defined in a description. Table [table_interactions] shows the parameters for each models. In the case of a complex type, a file format is “ a real part, an imaginary part “ while in the case of a real type, only “ a real part “.

Local terms

  • mu

    Type : Real

    Description : (Hubbard and Kondo lattice model) The chemical potential \(\mu\) (including the site potential) is specified with this parameter.

  • U

    Type : Real

    Description : (Hubbard and Kondo lattice model) The onsite Coulomb integral \(U\) is specified with this parameter.

  • Jx, Jy, Jz, Jxy, Jyx, Jxz, Jzx, Jyz, Jzy

    Type : Real

    Description : (Kondo lattice model) The spin-coupling constant between the valence and the local electrons is specified with this parameter. If the exchange coupling J is specified in the input file, instead of Jx, Jy, Jz, the diagonal exchange couplings, Jx, Jy, Jz, are set as Jx = Jy = Jz = J. When both the set of exchange couplings (Jx, Jy, Jz) and the exchange coupling J are specified in the input file, vmcdry.out will stop.

  • h, Gamma, D

    Type : Real

    Description : (Spin model) The longitudinal magnetic field, transverse magnetic field, and the single-site anisotropy parameter are specified with these parameters. The single-site anisotropy parameter is not available for model=SpinGCBoost.

The non-local terms described below should be specified in different ways depending on the lattice structure: For lattice=Ladder, the non-local terms are specified in the different way from lattice=Chain Lattice, Square Lattice, Triangular Lattice, Honeycomb Lattice, Kagome. Below, the available parameters for each lattice are shown in Table [table_interactions].

Interactions

1D chain

2D square

2D triangular

Honeycomb

Kagome

Ladder

J, t, V (simplified)

OK

OK

OK

OK

OK

NG

J0, t0, V0

OK

OK

OK

OK

OK

OK

J1, t1, V1

NG

OK

OK

OK

OK

OK

J2, t2, V2

NG

NG

OK

OK

OK

OK

J', t', V' (simplified)

OK

OK

OK

OK

OK

NG

J0', t0', V0'

OK

OK

OK

OK

OK

NG

J1', t1', V1'

NG

OK

OK

OK

OK

OK

J2', t2', V2'

NG

NG

OK

OK

OK

OK

J'', t'', V'' (simplified)

OK

OK

OK

OK

NG

NG

J0'', t0'', V0''

OK

OK

OK

OK

NG

NG

J1'', t1'', V1''

NG

OK

OK

OK

NG

NG

J2'', t2'', V2''

NG

NG

OK

OK

NG

NG

Table: Interactions for each models defined in an input file. We can define spin couplings as matrix format.

Non-local terms for Ladder lattice

Fig. 5

  • t0, t1, t1', t2, t2'

    Type : Complex

    Description : (Hubbard and Kondo lattice model) Hopping integrals in the ladder lattice (See Fig. 5 ) is specified with this parameter.

  • V0, V1, V1', V2, V2'

    Type : Real

    Description : (Hubbard and Kondo lattice model) Offsite Coulomb integrals on the ladder lattice (Fig. 3 are specified with these parameters.

  • J0x, J0y, J0z, J0xy, J0yx, J0xz, J0zx, J0yz, J0zy

  • J1x, J1y, J1z, J1xy, J1yx, J1xz, J1zx, J1yz, J1zy

  • J1'x, J1'y, J1'z, J1'xy, J1'yx, J1'xz, J1'zx, J1'yz, J1'zy

  • J2x, J2y, J2z, J2xy, J2yx, J2xz, J2zx, J2yz, J2zy

  • J2'x, J2'y, J2'z, J2'xy, J2'yx, J2'xz, J2'zx, J2'yz, J2'zy

    Type : Real

    Description : (Spin model) Spin-coupling constants in the ladder lattice (See Fig. 5 ) are specified with these parameter. If the simplified parameter J0 is specified in the input file instead of the diagonal couplings, J0x, J0y, J0z, these diagonal couplings are set as J0x = J0y = J0z = J0. If both J0 and the set of the couplings (J0x, J0y, J0z) are specified, vmcdry.out will stop. The above rules are also valid for the simplified parameters, J1, J1', J2, and J2'.

Non-local terms for other than Ladder

Figs. 2 , 3 , 4

  • t, t0, t1, t2

    Type : Complex

    Description : (Hubbard and Kondo lattice model) The nearest neighbor hoppings for each direction (see fig_chap04_1_latticefig_chap04_1_lattice - fig_kagomefig_kagome ) are specified with these parameters. If there is no bond dependence of the hoppings, the simplified parameter t is available to specify t0, t1, and t2 as t0 = t1 = t2 = t. If both t and the set of the hoppings (t0, t1, t2) are specified, the program will stop.

  • t', t0', t1', t2'

    Type : Complex

    Description : (Hubbard and Kondo lattice model) The second nearest neighbor hoppings for each direction (see fig_chap04_1_latticefig_chap04_1_lattice - fig_kagomefig_kagome ) are specified with these parameter. If there is no bond dependence of the hoppings, the simplified parameter t' is available to specify t0', t1', and t2' as t0' = t1' = t2' = t'. If both t' and the set of the hoppings (t0', t1', t2') are specified, the program will stop.

  • t'', t0'', t1'', t2''

    Type : Complex

    Description : (Hubbard and Kondo lattice model) The third nearest neighbor hoppings for each direction (see fig_chap04_1_latticefig_chap04_1_lattice - fig_kagomefig_kagome ) are specified with these parameter. If there is no bond dependence of the hoppings, the simplified parameter t'' is available to specify t0'', t1'', and t2'' as t0'' = t1'' = t2'' = t''. If both t'' and the set of the hoppings (t0'', t1'', t2'') are specified, the program will stop.

  • V, V0, V1, V2

    Type : Real

    Description : (Hubbard and Kondo lattice model) The nearest neighbor offsite Coulomb integrals \(V\) for each direction (see fig_chap04_1_latticefig_chap04_1_lattice - fig_kagomefig_kagome ) are specified with these parameters. If there is no bond dependence of the offsite Coulomb integrals, the simplified parameter V is available to specify V0, V1, and V2 as V0 = V1 = V2 = V. If both V and the set of the Coulomb integrals (V0, V1, V2) are specified, the program will stop.

  • V', V0', V1', V2'

    Type : Real

    Description : (Hubbard and Kondo lattice model) The second nearest neighbor-offsite Coulomb integrals \(V'\) for each direction (see fig_chap04_1_latticefig_chap04_1_lattice - fig_kagomefig_kagome ) are specified with this parameter. If there is no bond dependence of the offsite Coulomb integrals, the simplified parameter V' is available to specify V0', V1', and V2' as V0' = V1' = V2' = V'. If both V' and the set of the Coulomb integrals (V0', V1', V2') are specified, the program will stop.

  • V'', V0'', V1'', V2''

    Type : Real

    Description : (Hubbard and Kondo lattice model) The third nearest neighbor-offsite Coulomb integrals \(V'\) for each direction (see fig_chap04_1_latticefig_chap04_1_lattice - fig_kagomefig_kagome ) are specified with this parameter. If there is no bond dependence of the offsite Coulomb integrals, the simplified parameter V'' is available to specify V0'', V1'', and V2'' as V0'' = V1'' = V2'' = V''. If both V'' and the set of the Coulomb integrals (V0'', V1'', V2'') are specified, the program will stop.

  • J0x, J0y, J0z, J0xy, J0yx, J0xz, J0zx, J0yz, J0zy

  • J1x, J1y, J1z, J1xy, J1yx, J1xz, J1zx, J1yz, J1zy

  • J2x, J2y, J2z, J2xy, J2yx, J2xz, J2zx, J2yz, J2zy

    Type : Real

    Description : (Spin model) Nearest-neighbor exchange couplings for each direction are specified with thees parameters. If the simplified parameter J0 is specified, instead of J0x, J0y, J0z, the exchange couplings, J0x, J0y, J0z, are set as J0x = J0y = J0z = J0. If both J0 and the set of the exchange couplings (J0x, J0y, J0z) are specified, vmcdry.out will stop. The above rules are valid for J1 and J2.

    If there is no bond dependence of the nearest-neighbor exchange couplings, the simplified parameters, Jx, Jy, Jz, Jxy, Jyx, Jxz, Jzx, Jyz, Jzy, are available to specify the exchange couplings for every bond as J0x = J1x = J2x = Jx. If any simplified parameter (Jx \(\sim\) Jzy) is specified in addition to its counter parts (J0x \(\sim\) J2zy), vmcdry.out will stop. Below, examples of parameter sets for nearest-neighbor exchange couplings are shown.

    • If there are no bond-dependent, no anisotropic and offdiagonal exchange couplings (such as \(J_{x y}\)), please specify J in the input file.

    • If there are no bond-dependent and offdiagonal exchange couplings but are anisotropic couplings, please specify the non-zero couplings in the diagonal parameters, Jx, Jy, Jz.

    • If there are no bond-dependent exchange couplings but are anisotropic and offdiagonal exchange couplings, please specify the non-zero couplings in the nine parameters, Jx, Jy, Jz, Jxy, Jyz, Jxz, Jyx, Jzy, Jzx.

    • If there are no anisotropic and offdiagonal exchange couplings, but are bond-dependent couplings, please specify the non-zero couplings in the three parameters, J0, J1, J2.

    • If there are no anisotropic exchange couplings, but are bond-dependent and offdiagonal couplings, please specify the non-zero couplings in the nine parameters, J0x, J0y, J0z, J1x, J1y, J1z, J2x, J2y, J2z.

    • If there are bond-dependent, anisotropic and offdiagonal exchange couplings, please specify the non-zero couplings in the twenty-seven parameters from J0x to J2zy.

  • J'x, J'y, J'z, J'xy, J'yx, J'xz, J'zx, J'yz, J'zy

  • J0'x, J0'y, J0'z, J0'xy, J0'yx, J0'xz, J0'zx, J0'yz, J0'zy

  • J1'x, J1'y, J1'z, J1'xy, J1'yx, J1'xz, J1'zx, J1'yz, J1'zy

  • J2'x, J2'y, J2'z, J2'xy, J2'yx, J2'xz, J2'zx, J2'yz, J2'zy

    Type : Real

    Description : (Spin model) The second nearest neighbor exchange couplings are specified. However, for lattice = Honeycomb Lattice and lattice = Kagome with model=SpinGCCMA, the second nearest neighbor exchange couplings are not available in the \(Standard\) mode. If the simplified parameter J' is specified, instead of J'x, J'y, J'z, the exchange couplings are set as J'x = J'y = J'z = J'. If both J' and the set of the couplings (J'x, J'y, J'z) are specified, the program will stop.

  • J''x, J''y, J''z, J''xy, J''yx, J''xz, J''zx, J''yz, J''zy

  • J0''x, J0''y, J0''z, J0''xy, J0''yx, J0''xz, J0''zx, J0''yz, J0''zy

  • J1''x, J1''y, J1''z, J1''xy, J1''yx, J1''xz, J1''zx, J1''yz, J1''zy

  • J2''x, J2''y, J2''z, J2''xy, J2''yx, J2''xz, J2''zx, J2''yz, J2''zy

    Type : Real

    Description : (Spin model) The third nearest neighbor exchange couplings are specified. However, for lattice = Honeycomb Lattice and lattice = Kagome with model=SpinGCCMA, the third nearest neighbor exchange couplings are not available in the \(Standard\) mode. If the simplified parameter J'' is specified, instead of J''x, J''y, J''z, the exchange couplings are set as J''x = J''y = J''z = J''. If both J'' and the set of the couplings (J''x, J''y, J''z) are specified, the program will stop.

  • phase0, phase1

    Type : Double (0.0 as defaults)

    Description : We can specify the phase for the hopping through the cell boundary with these parameter (unit: degree). These factors for the \(\vec{a}_0\) direction and the \(\vec{a}_1\) direction can be specified independently. For the one-dimensional system, only phase0 can be used. For example, a fopping from \(i\)-th site to \(j\)-th site through the cell boundary with the positive direction becomes as

    \[\begin{aligned} \exp(i \times {\rm phase0}\times\pi/180) \times t {\hat c}_{j \sigma}^\dagger {\hat c}_{i \sigma} + \exp(-i \times {\rm phase0}\times\pi/180) \times t^* {\hat c}_{i \sigma}^\dagger {\hat c}_{j \sigma} \end{aligned}\]

Parameters for the numerical condition

  • ncond

    Type : int-type (must be specified)

    Description : The number of itinerant electrons. It is the sum of the \(\uparrow\) and \(\downarrow\) electrons.

  • NVMCCalMode

    Type : int-type (default value: 0)

    Description : [0] Optimization of variational parameters, [1] Calculation of one body and two body Green’s functions.

  • NDataIdxStart

    Type : int-type (default value: 1)

    Description : An integer for numbering of output files. For NVMCCalMode = 0 , NDataIdxStart is added at the end of the output files. For NVMCCalMode = 1, the files are outputted with the number from NDataIdxStart to NDataIdxStart + NDataQtySmp-1.

  • NDataQtySmp

    Type : int-type (default value: 1)

    Description : The set number for outputted files (only used for NVMCCalMode = 1).

  • NSPGaussLeg

    Type : int-type (Positive integer, default value is 8 for 2Sz=0)

    Description : The mesh number for the Gauss-legendre quadrature about \(\beta\) integration (\(S_y\) rotation) for the spin quantum-number projection in actual numerical calculation.

  • NSPStot

    Type : int-type ( greater equal 0, default value is 0 for 2Sz=0)

    Description : The total spin quantum-number.

  • 2Sz

    Type : int-type ( greater equal 0, default value is 0)

    Description : The spin quantum-number \(S_z\).

  • NMPTrans

    Type : int-type (Positive integer. Default 1)

    Description : The number of the momentum and lattice translational quantum-number projection. In the case of not to apply the projection, this value must be set as 1.

  • NSROptItrStep

    Type : int-type (Positive integer, default value: 1000)

    Description : The whole step number to optimize variational parameters by SR method. Only used for NVMCCalMode =0.

  • NSROptItrSmp

    Type : int-type (Positive integer, default value: NSROptItrStep/10)

    Description : In the NSROptItrStep step, the average values of the each variational parameters at the NSROptItrStep step are adopted as the optimized values. Only used for NVMCCalMode =0.

  • DSROptRedCut

    Type : double-type (default value: 0.001)

    Description : The stabilized factor for the SR method by truncation of redundant directions corresponding to \(\varepsilon_{\rm wf}\) in the ref. [Tahara2008 ].

  • DSROptStaDel

    Type : double-type (default value: 0.02)

    Description : The stabilized factor for the SR method by modifying diagonal elements in the overwrap matrix corresponding to \(\varepsilon\) in the ref. [Tahara2008 ].

  • DSROptStepDt

    Type : double-type (default value: 0.02)

    Description : The time step using in the SR method.

  • NVMCWarmUp

    Type : int-type (Positive integer, default value: 10)

    Description : Idling number for the Malkov chain Montecarlo Methods.

  • NVMCInterval

    Type : int-type (Positive integer, default value: 1)

    Description : The interval step between samples. The local update will be performed Nsite × NVMCInterval times.

  • NVMCSample

    Type : int-type (Positive integer, default value: 1000)

    Description : The sample numbers to calculate the expected values.

  • NExUpdatePath

    Type : int-type (Positive integer)

    Description : The option for local update about exchange terms. 0: not update, 1: update. The default value is set as 1 when the local spin exists, otherwise 0.

  • RndSeed

    Type : int-type (default value: 123456789)

    Description : The initial seed of generating random number. For MPI parallelization, the initial seeds are given by RndSeed +my rank+1 at each ranks.

  • NSplitSize

    Type : int-type (Positive integer, default value: 1)

    Description : The number of processes of MPI parallelization.

  • NStore

    Type : int-type (0 or 1, default value: 1)

    Description : The option of applying matrix-matrix product to calculate expected values \(\langle O_k O_l \rangle\) (0: off, 1: on). This speeds up calculation but increases the amount of memory usage from \(O(N_\text{p}^2)\) to \(O(N_\text{p}^2) + O(N_\text{p}N_\text{MCS})\), where \(N_\text{p}\) is the number of the variational parameters and \(N_\text{MCS}\) is the number of Monte Carlo sampling.

  • NSRCG

    Type : int-type (0 or 1, default value: 0)

    Description : The option of solving \(Sx=g\) in the SR method without constructing \(S\) matrix [NeuscammanUmrigarChan ]. (0: off, 1: on). This reduces the amount of memory usage from \(O(N_\text{p}^2) + O(N_\text{p}N_\text{MCS})\) to \(O(N_\text{p}) + O(N_\text{p}N_\text{MCS})\) when \(N_\text{p} > N_\text{MCS}\).

  • ComplexType

    Type : int-type (0 or 1. Default value is 0 for the \(S_z\)-conserved system and 1 for the \(S_z\)-unconserved system.)

    Description : If it is 0, only the real part of the variational parameters are optimized. And the real and the imaginary part of them are optimized if this parameter is 1.

  • OutputMode

    Type : Choose from "none", "correlation", and "full" (correlation as a default)

    Description : Indices of correlation functions are specified with this keyword. "none" indicates correlation functions will not calculated. When outputmode="correlation", the correlation function supported by the utility fourier is computed. For more details, see the document in doc/fourier/. If "full" is selected, \(\langle c_{i \sigma}^{\dagger}c_{j \sigma'} \rangle\) is computed at all \(i, j, \sigma, \sigma'\), and \(\langle c_{i_1 \sigma}^{\dagger}c_{i_2 \sigma} c_{i_3 \sigma'}^{\dagger}c_{i_4 \sigma'} \rangle\) is computed at all \(i_1, i_2, i_3, i_4, \sigma, \sigma'\).

    In spin system, indices are specified as those on the Bogoliubov representation (See [sec_bogoliubov_rep]).

  • CDataFileHead

    Type : string-type (default : "zvo")

    Description : A header for output files. For example, the output filename for one body Green’s function becomes “ xxx_cisajs_yyy.dat” (xxx are characters set by CDataFileHead and yyy are numbers given by numbers from NDataIdxStart to NDataIdxStart + NDataQtySmp).

  • CParaFileHead

    Type : string-type (default : "zqp")

    Description : A header for output files of the optimized variational parameters. For example, the optimized variational parameters are outputted as zzz_opt_yyy.dat (zzz are characters set by CParaFileHead and yyy are numbers given by numbers from NDataIdxStart to NDataIdxStart + NDataQtySmp -1).