### Path-integral Monte Carlo

A Markov-chain Monte Carlo method for quantum many body systems. Simulation by this method is based on a Markov process in the space of d+1 dimensional configurations obtained via path-integral representation of d dimensional quantum many-body systems. The method is also called the world-line Monte Carlo. While the weight of a given state can be easily computed in classical systems, its computational cost for quantum systems is exponentially high. Therefore, the mapping into the d+1 dimensional classical problem through Suzuki-Trotter transformation or high-temperature series expansion is necessary for reducing the cost to a manageable level. The method is applied to various strongly-correlated lattice models such as transverse Ising model, Heisenberg model, and Hubbard model, as well as bosonic systems such as He4. However, in application to frustrated Heisenberg models and fermionic systems, the weight can be negative in general, making the method unpractical. This problem is called the negative sign problem, the most severe limitation of the method in studying quantum many-body systems.

### Plane wave basis

In electronic structure calculations, the wave function is often expanded as a linear combination of plane waves. Plane waves comprise an orthonormal basis set, so that increasing the basis size (using many plane waves with different wavelengths) leads to a monotonic improvement in the reproduction of the wave function. However, it is unsuitable for describing steeply-varying wave functions near the core since disproportionately many plane waves are necessary for expanding steep functions. The (L)APW and pseudopotential methods were developed to circumvent this difficulty.

### Pseudopotential method

Core electrons play a very small part in chemical bond formation, so that computational load can be decreased with small loss in accuracy by replacing core electrons by a pseudopotential that act on the valence electrons. In this manner, only the relatively slowly-varying valence wave functions need to be considered explicitly, and this allows for decreasing the basis set size when using a plane wave basi

### Quantum Monte Carlo method (QMC)

A Monte Carlo method is referred to as a quantum Monte Carlo method when it is applied to quantum many-body systems. There is a variety of the methods including: Path-integral Monte Carlo, which is a Markov-chain Monte Carlo in the d+1 dimensional path-integral representation of d dimensional quantum system, diffusion Monte Carlo, in which ground state wave functions are sampled from the random walk of the diffusion process in the imaginary time evolution, and variational Monte Carlo, in which the expectation value of the energy of the trial wave function is evaluated by stochastic sampling.

### Recursive Green’s function method

An efficient calculation method for treating scattering of electrons in a potential. Transmission and reflection amplitudes are evaluated by directly calculating the Green’s function of the electrons. It is possible to evaluate electron transport properties of nanoscale devices by combining with Laundauer’s formula. In the recursive Green’s function method, fast computation of the scattering states of electrons are realized by sequentially calculating the Green’s function along the transport direction. The method can also treat magnetic materials and superconducting materials. The method is implemented in Kwant.