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
// nelec = 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. 31 (a)), the two dimensional square lattice (Fig. 31 (b)), the two dimensional triangular lattice (Fig. 31 (c)), the two dimensional anisotropic honeycomb lattice (Fig. 32 ), the Kagome Lattice(Fig. 33 ), and the ladder lattice (Fig. 34 ), respectively.

Figure 31: 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).

Figure 32: Schematic illustration of the anisotropic honeycomb lattice. The nearest neighbor hopping integral, spin coupling, offsite Coulomb integral depend on the bond direction. Those between second nearest neighbor sites are not supported.

Figure 33: Schematic illustration of the Kagome lattice.

Figure 34: Schematic illustration of the ladder lattice.

Parameters for the lattice¶

Chain lattice¶

Fig. 31 (a)

L

Type : Integer

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

Ladder lattice¶

Fig. 34

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.

Figure 35: 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. 31 (b)], Triangular lattice[Fig. 31 (c)], Honeycomb lattice(Fig. 32 ), Kagome lattice(Fig. 33 )

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. 31 - 33 ) 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. 35 ). 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
`J'`, `t'`, `V'`	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
`J1'`, `t1'`, `V1'`	NG	NG	NG	NG	NG	OK
`J2'`, `t2'`, `V2'`	NG	NG	NG	NG	NG	OK

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

Non-local terms for Ladder lattice¶