Heisenberg model

One of the most basic models that describes the spin degrees of freedom in solid, which is a reasonable description of many materials in low-energy (low-temperature) scale. There are many variants according to the types and ranges of the interactions, and lattice structures. In particular, the antiferromagnetic Heisenberg model can be obtained from the half-filled Hubbard model through the perturbation with respect to the hopping constant, which makes the model relevant in the study of high-temperature cuprate superconductivity. The case where frustration exists, as in the case of the antiferromagnetic model on a kagome lattice, is a target of active research with the expectation of novel quantum states. The cases without frustration can be studied by quantum Monte Carlo and software packages such as ALPS and DSQSS are available whereas more general cases can be dealt with by exact diagonalization using for exacmple TITPACK, SpinPack, KobePack, and Hphi. Variational Monte Carlo is also an option in the latter case, for which mVMC is available.

Hubbard model

A theoretical model proposed by Hubbard, Kanaori and Gutzwiller for describing the electronic states of transition metal oxides. It consists of the transfer term and on-site Coulomb interaction. Despite its simplicity, its mathematical treatment is rather difficult, allowing exact solution only for very restricted cases, e.g., the one-dimensional system. The model is extensively studied since it exhibits many phases of interest: ferromagnetic, antiferromagnetic, Mott insulator, high-temperature superconducting, etc. Exact diagonalization of small clusters of the model can be done with ALPS, SpinPack, Hphi, etc. By Hphi, calculation of dynamic properties, and finite-temperature calculation is also possible. Calculation by dynamical mean-field theory can be done by ALPS and pyDMFT. Variational Monte Carlo calculation can be done with mVMC.

Ising Model

One of the most fundamental models in statistical physics that is first introduced as an effective model of the ferromagnetic-paramagnetic phase transition. This model consists of “spins” that take value of +/- 1 on a lattice. When each spin tends to take the same value as that of neighbor spins (ferromagnetic interaction), all spins take the same value at absolute zero (ferromagnetic phase or ordered phase). At the high temperature limit, on the other hand, all spins takes independent values from each other (paramagnetic phase or disordered phase). Finite-temperature phase transition is a phenomena that the system changes from one phase to the other phase at some temperature. When randomness of interactions is introduced, this model becomes difficult to solve even numerically. For example, the ground state cannot be found in polynomial time and the thermal-relaxation time becomes very long (spin glass). Ising model is a classical model originally, but one can obtain quantum version of Ising model by introducing transverse magnetic field flipping a spin as a source of quantum fluctuation. In this model (transverse field Ising model,) ferromagnetic-paramagnetic phase transition manifests oneself even at absolute zero by controlling the strength of a field (quantum phase transition). Since Ising model itself is very simple but supports many phenomena, it has been well studied and used in not only classical / quantum statistical physics but also classical / quantum information theory and machine learning.

Related software

Job management

For use of a massively large computer system, an user requires a machine resource (the number of CPU cores and wall-time) and runs one’s calculation on assigned resources from job manager of the supercomputer. Some tools and libraries implement useful functions that automatically require resources, effectively use assigned resources (load balancing), save a snapshot of the simulation and restart from the snapshot (checkpoint), and store pairs of input and output of each simulation. MateriApps project refers these functions as “job management.”

Related software

KKR method

One of the first-principles calculation methods. The name of the method comes from the capital letters of the developers, Korringa, Kohn, and Rostoker. In this method, the Green’s function which describes multiple electron scattering due to effective potentials is numerically solved. By combining with density functional theory (DFT), electronic structure calculation can be performed efficiently. Furthermore, by combining with CPA (coherent potential approximation), this method can be applied to electronic state calculation for impurity systems and random alloys. Although various methods related to DFT can, in principle, be used, open packages usually employs pseudo-potential methods and muffin-tin potentials.

Related software

Kohn-Sham method

The most majorly-used calculation methodology based on density functional theory (DFT). The Kohn-Sham method was developed to provide a good approximation for the kinetic energy of many-electron systems, which was found to be very difficult when trying to parametrize as a pure (orbital-free) density functional. The kinetic energy, which holds the largest portion of the total energy, is approximated by the kinetic energy of a reference noninteracting system with the same electron density. The rest of the total energy was assigned to the exchange-correlation energy functional, and as a result, the many-body problem was reformulated as a system of single-electron Schroedinger-like equations (Kohn-Sham equations) in an effective potential.

Lanczos method

Although there are several packages for the full diagonalization such as lapack, it is almost impossible to perform the full diagonalization for large-scale matrices whose dimension is more than one million. In the condensed matter physics, we want to obtain the lowest (ground-state) eigenvalue and the eigenvector for characterizing the nature of the target quantum many-body systems. For that purpose, the Lanczos method is commonly used for obtaining the eigenvalues and the eigenvector of ground state.

In the Lanczos method, we successively multiply the Hamiltonian to the initial vector (typically we take the random vector as the initial vector). Then, we can obtain the lowest eigenvectors. Only two vectors are necessary for performing the Lanczos method, we can obtain the ground state of larger matrices whose dimension is up to tens of billion.

The Lanczos method is implemented in several exact diagonalization packages such as
TITPACK,KobePACK,SpinPACK,ALPS and HΦ. Especially, in HΦ, recently developed modern algorithm for obtaining several low-energy excited states (the LOBPCG method) is implemented. By using the LOPBCG method, we can obtain the several excited states at one calculations.