Quantum Monte Carlo methods - a large family of methods for the study of complex quantum systems . One of the main tasks is to provide a reliable solution (or a fairly accurate approximation) of the . Different versions of this method have a common feature: they use the Monte Carlo method to calculate the multidimensional integrals that arise in various formulations of the many-body problem. The quantum Monte Carlo methods make it possible to describe the complex effects of many particles encrypted in the wave function , going beyond the mean field theory and in some cases offering exact solutions to the many-body problem. In particular, there is a numerically accurate and polynomial scalable algorithm for the exact study of the static properties of a system of bosons without geometric frustration . For fermions , such algorithms are not known, but there are separately algorithms that give very good approximations of their static properties, and separately quantum Monte Carlo algorithms that are numerically accurate but exponentially scalable.
Content
Introduction
In principle, any physical system is described by the Schrödinger equation for many particles, unless the particles move too fast (that is, so that their speed remains low compared to the speed of light , and relativistic effects can be neglected). This requirement is fulfilled for a wide range of electronic problems in condensed matter physics, in the Bose-Einstein condensate, and in superfluid liquids like liquid helium. The ability to solve the Schrödinger equations for a given system allows us to predict its behavior and has important applications in many fields of science, from materials science to complex biological systems. The difficulty is that solving the Schrödinger equation requires knowledge of the many-particle wave function in a multi-dimensional Hilbert space , the size of which, as a rule, grows exponentially with increasing number of particles.
A solution for a large number of particles is basically impossible in a reasonable amount of time, even for modern parallel computing . The approximations of many-particle antisymmetric functions composed of single-particle molecular orbitals [1] are traditionally used, which reduces the problem of solving the Schrödinger equation to a form with which one can work. Formulations of this kind have several drawbacks. They are either limited by taking into account quantum correlations, for example, the Hartree-Fock method , or converge very slowly, as in the case of the application of configuration interactions in quantum chemistry .
Quantum Monte Carlo methods open the way to the direct study of multiparticle problems and multiparticle wave functions without these restrictions. The most advanced quantum Monte Carlo methods give exact solutions to the many-particle problems of a boson system without frustration, simultaneously with an approximate, but usually correct description of fermion systems with interaction. Most methods have the goal of finding the wave function of the ground state of the system, with the exception of the Monte Carlo methods for path integrals and the Monte Carlo method for finite temperatures, which are used to calculate the density matrix. In addition to stationary problems, it is also possible to solve the time-dependent Schrödinger equation, although only approximately, limiting the functional form of the time-dependent wave function. For this, a time-dependent Monte Carlo variational method has been developed. From the point of view of probability theory, the calculation of the leading eigenvalues and the corresponding ground state wave functions relies on a numerical solution of the integral problem along the Feynman-Kaka trajectories [2] [3] . The mathematical basis of the Feynman-Kaka particle absorption model, the Monte Carlo sequencing method, and mean-field interpretations were laid in [4] [5] [6] [7] [8] .
There are several quantum Monte Carlo methods, in each of them Monte Carlo is used to solve the problem of many bodies in various ways.
Methods
Zero temperature (ground state only)
- Monte Carlo variational method : a good starting point; used in solving a wide range of different quantum problems.
- Monte Carlo diffuse method : the most popular high-precision method for an electron system (i.e., for chemical calculations), since it converges relatively effectively to the exact value of the ground state energy. It is also used to reproduce the quantum behavior of atoms and the like.
- The Monte Carlo Reputation Method : a modern method of computing at zero temperature associated with path integrals, the scope is the same as the Monte Carlo diffusion method, but the assumptions are different, so the advantages and disadvantages differ. Reputation is a term from the physics of polymers that describes the creeping of long chains by a snake.
- Gaussian quantum Monte Carlo method
- Finding the ground state through integrals along the trajectories : mainly used for a system of bosons; for those where the physical observable quantities can be calculated accurately, that is, with an arbitrarily small error.
Nonzero temperatures (thermodynamics)
- Monte Carlo auxiliary field method : mainly used for problems defined on the lattice, although there are new works that apply this method to electrons in chemical systems.
- Quantum Monte Carlo method with continuous time .
- Determinant Monte Carlo Quantum Method or Hirsch-Faye Quantum Monte Carlo Method
- Monte Carlo Hybrid Quantum Method
- Quantum Monte Carlo method through integrals along trajectories : a calculation procedure at nonzero temperatures, which is mainly used for systems where temperature effects are of great importance, in particular for superfluid helium.
- Stochastic algorithm for the Green's function [9] : an algorithm designed for bosons models a Hamiltonian defined on a lattice of any complexity, unless it has a sign problem.
- Quantum Monte Carlo method of world lines.
Real-time dynamics (closed quantum systems)
- Time-dependent variational quantum Monte Carlo method : extension of the variational Monte Carlo method to the dynamics of pure quantum states.
Projects and software products
Links
- ↑ Functional form of the wave function Archived July 18, 2009 on Wayback Machine
- ↑ Caffarel, Michel; Claverie, Pierre. Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman – Kac formula. I. Formalism (English) // Journal of Chemical Physics : journal. - 1988. - Vol. 88 , no. 2 . - P. 1088-1099 . - ISSN 0021-9606 . - DOI : 10.1063 / 1.454227 . - . Archived June 12, 2015. Archived June 12, 2015 on Wayback Machine
- ↑ Korzeniowski, A .; Fry, JL; Orr, DE; Fazleev, NG Feynman-Kac path-integral calculation of the ground-state energies of atoms ( Physical ) // Physical Review Letters : journal. - 1992 .-- 10 August ( vol. 69 , no. 6 ). - P. 893-896 . - DOI : 10.1103 / PhysRevLett . 69.893 . - .
- ↑ EUDML | Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman – Kac semigroups - P. Del Moral, L. Miclo. . eudml.org . Date of treatment June 11, 2015.
- ↑ Del Moral, Pierre; Doucet, Arnaud. Particle Motions in Absorbing Medium with Hard and Soft Obstacles (English) // Stochastic Analysis and Applications: journal. - 2004 .-- 1 January ( vol. 22 , no. 5 ). - P. 1175-1207 . - ISSN 0736-2994 . - DOI : 10.1081 / SAP-200026444 .
- ↑ Del Moral, Pierre. Mean field simulation for Monte Carlo integration . - Chapman & Hall / CRC Press, 2013. - P. 626. - "Monographs on Statistics & Applied Probability".
- ↑ Del Moral, Pierre. Feynman-Kac formulae. Genealogical and interacting particle approximations . - Springer, 2004. - P. 575. - "Series: Probability and Applications".
- ↑ Del Moral, Pierre. Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering. / Pierre Del Moral, Laurent Miclo. - 2000. - Vol. 1729. - P. 1–145. - DOI : 10.1007 / bfb0103798 .
- ↑ Rousseau, VG Stochastic Green function algorithm (Eng.) // Physical Review E : journal. - 2008 .-- 20 May ( vol. 77 ). - P. 056705 . - DOI : 10.1103 / physreve.77.056705 . - . - arXiv : 0711.3839 . (inaccessible link)