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