Post on 25-Jul-2020
Introduction to Path Integral Monte Carlo
Xinyu SongDepartment of Physics, Peking University
Dec 17, 2019
Importance Sampling
Simple Sampling • numerical integration of given functions, random walks
Importance sampling • average value of thermodynamic quantities, such as energy, heat
capacity, magnetization etc• configuration in phase space or space of microscopic states
Configuration, probability distribution and average value
Choose a set of M configurations {𝑐𝑐𝑖𝑖} from 𝛺𝛺 , according to the distribution 𝑝𝑝 𝑐𝑐𝑖𝑖 .The average is then approximated by the sample mean
Statistical Error𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑐𝑐𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑡𝑡𝑎𝑎 𝜏𝜏𝐴𝐴 describes the time it takes for two measurements to be decorrelated.
The advantage of Monte Carlo MethodFast“ it allows phase space integrals for many-particle problems to be evaluatedin a time that scales only polynomially with the particle number N, althoughthe configuration space grows exponentially with N.”
Precise“ Using the Monte Carlo approach the same average can be estimated to anydesired accuracy in polynomial time, as long as the autocorrelation time 𝜏𝜏𝐴𝐴does not increase faster than polynomially with N.”
Reference: Matthias Troyer and Uwe-Jens Wiese, Phys. Rev. Lett. 94, 170201
Example: Simple Ising Model
Markov Chain
• A Markov chain is a sequence of random variables x1, x2, x3, ... with the property that the future state depends only on the past via the present state.
• Variables updates by state transformation matrix
Detailed balance
Conservation of probability
Result for 2D Ising
Path Integral formulation
evaluate thermal averages in the framework of quantum statistical mechanics
example
Problem: no explicit diagonalization scheme(eigenstates)Find a way to solve Schrodinger equation and deal with off diagonal term(non-commuting operators ).
Trotter product formula:
Ref: David P. Landau, Kurt Binder-A Guide to Monte Carlo Simulations in Statistical Physics-Cambridge University Press (2005)
From one particle to many particles
Quantum Monte Carlo Most general definition : A stochastic method to solve the Schrödinger equation
Path Integral Monte Carlo:• Mapping D dimension Quantum system to D+1 dimension Classical
system one particle -> closed particle chainsconfiguration in real space -> configuration in real space plus imaginary time • Do classical Monte Carlo on the equivalent problem
• bosonic Hubbard model
Bosonic Systems
Ref: https://arxiv.org/pdf/0910.1393.pdfNikolay Prokof’ev and Boris Svistunov
Bosonic Systemsequilibrium properties of many-body 4𝐻𝐻𝑎𝑎 at all temperatures
Ref: https://www.cond-mat.de/events/correl13/manuscripts/ceperley.pdf David Ceperley
PIMC for fermions : sign problem
s represents the sign of each configuration
“For fermionic or frustrated models this mapping may yield configurationswith negative Boltzmann weights, resulting in an exponential growth ofthe statistical error and hence the simulation time with the number ofparticles, defeating the advantage of the Monte Carlo method.”Reference: Matthias Troyer and Uwe-Jens Wiese, Phys. Rev. Lett. 94, 170201
Application on uniform electron gas at finite temperature
Ref: V. S. Filinov, V. E. Fortov, M. Bonitz, and Zh. Moldabekov Phys. Rev. E 91, 033108
Calculate Total energy per particle for polarized ideal and interacting electron gas under different temperature