4.1.1. Parameters for the type of calculation

  • model

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

    Description : The target model is specified with this parameter; the expressions above denote the canonical ensemble of the Fermion in the Hubbard model

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

    the canonical ensemble in the Spin model(\(\{\alpha, \beta\}=\{x, y, z\}\))

    (4.2)\[\begin{split}\mathcal H &= -h \sum_{i} S^z_{i} - \Gamma \sum_{i} S^x_{i} + D \sum_{i} S^z_{i} S^z_{i} \nonumber \\ &+ \sum_{i j, \alpha}J_{i j \alpha} S^\alpha_{i} S^\alpha_{j}+ \sum_{i j, \alpha \neq \beta} J_{i j \alpha \beta} S_{i}^\alpha S_{j}^\beta,\end{split}\]

    the canonical ensemble in the Kondo lattice model

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

    the grand canonical ensemble of the Fermion in the Hubbard model [Eqn. (4.1) ], the grand canonical ensemble in the Spin model [Eqn. (4.2) ], and the grand canonical ensemble in the Kondo lattice model [Eqn. (4.3) ], respectively.

    When model="SpinGCCMA", by using a more efficient algorithm [2], \({\mathcal H}\Phi\) calculates a system that is the same as "SpinGC". However, supported models and MPI processes are highly limited. See "Lattice" section.

  • method

    Type : String (choose from "Lanczos", "TPQ", "Full Diag", "CG", "Time Evolution")

    Description : The calculation type is specified with this parameter; the above expressions above denote the single eigenstate calculation by using the Lanczos method, at the finite-temperature by using the thermally pure quantum state, the full diagonalization method, the multiple eigenstates calculation by using the LOBCG method [3] [4] , and the simulation of real-time evolution, respectively.

    The scheme employed for the spectrum calculation is also specified with this parameter. If "CG" is chosen, the shifted bi-conjugate gradient method [5] together with the seed-switch technique [6] is employed with the help of the \(K\omega\) library [7] .

  • lattice

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

    Description : The lattice shape is specified with this parameter; the expressions above denote the one-dimensional chain lattice ( Fig. 4.1 (a)), the two-dimensional square lattice ( Fig. 4.1 (b)), the two-dimensional triangular lattice ( Fig. 4.1 (c)), the two-dimensional anisotropic honeycomb lattice ( Fig. 4.2 ), the Kagome Lattice( Fig. 4.3 ), and the ladder lattice ( Fig. 4.4 ) respectively.

    In method="SpinGCCMA", only "Chain Lattice", "Honeycomb Lattice", "Kagome", and "Ladder" are supported. The limits of \(L\), \(W\), and the number of MPI processes (\(N_{\rm proc}\)) are as follows:

    • "Chain Lattice"

      \(L = 8n\) (where \(n\) is an integer number under the condition \(n\geq1\)), \(N_{\rm proc} \leq 2(L=8)\), \(N_{\rm proc} \leq 2^{L/2-2}(L>8)\).

    • "Honeycomb Lattice"

      \(W=3, L \geq 2\), \(N_{\rm proc} \leq 2(L=2)\), \(N_{\rm proc} \leq 64(L>2)\).

    • "Kagome"

      \(W=3, L \geq 2\), \(N_{\rm proc} \leq 1(L=2)\), \(N_{\rm proc} \leq 512(L>2)\).

    • "Ladder"

      \(W=2, L = 2n\) (where \(n\) is an integer number under the condition \(n\geq4\)), \(N_{\rm proc} \leq 2^{L-4}\).