The early years of quantum Monte Carlo (2): finite-temperature simulations

Abstract

In this article, we present the second part of our historical survey on quantum Monte Carlo methods. We focus on the simulations performed at a finite temperature and based on Feynman’s path-integral formulation of quantum mechanics. We introduce the method and insist on the central role played by the description of the transition to superfluidity for Helium 4.

Appendix 1: Derivation of equation (1)

We consider the quantum canonical ensemble of N atoms enclosed in a volume V and maintained at temperature T. It is being modelled by N point particles with mass m and interaction potential energy $V(r_{ij}$), with $r_{ij}$ the absolute value of the relative distance. The Hamiltonian is a sum of kinetic, K, and potential energy, V; it reads

$$\begin{aligned} H=\mathop {\sum }\limits _{i=1}^N {\frac{p_{i}^{2} }{2m}} +\mathop {\sum }\limits _{i\ne j} {V\left( r_{ij} \right) =K+V} \end{aligned}$$

(A1)

The Helmholtz free energy, $F_{N}$(V,T), is related to the density operator $\rho$ and the partition function Z through the usual relation

(A2)

In the main text, the 3N-dimensional element of integration is represented by the symbol dR. As usual, $\beta$ is 1/k$_{B}T$

The starting point of the analysis is a rather trivial identity for the equilibrium density operator:

$$\begin{aligned} \rho =e^{-\beta H}=\left( {e^{-\tau H}} \right) ^{M}with\,\,\,\,\tau =\frac{\beta }{M}\,\,\, \end{aligned}$$

(A2)

the quantity $\tau$ is called the timestep and it will be used as a smallness parameter. By analogy with the usual time propagator, $\exp \left( {-\frac{iHt}{\hslash }} \right)$, we can imagine following an imaginary time¹³ between 0 and $\beta$, and $\tau =\beta /M$ is then called the timestep.

Quantum effects are known to appear at low temperatures. More precisely, the usual transition between classical and quantum behaviour occurs when the average distance between particles becomes of the order of the so-called de Broglie wavelength

$$\begin{aligned} \Lambda =\frac{h}{\sqrt{2\pi mk_{B} T} } \end{aligned}$$

(A3)

Or, for a system of number density $n=N/V$, quantum effects will appear while lowering the temperature so that

$$\begin{aligned} n\Lambda ^{3}\simeq o(1) \end{aligned}$$

(A4)

Since K and V are non-commuting operators,

$$\begin{aligned} e^{-\tau H}=e^{-\tau K}e^{-\tau V}+o(\tau ^{2}) \end{aligned}$$

(A5)

or

$$\begin{aligned} e^{-\beta H}=\lim \limits _{M\rightarrow \infty } \left( {e^{-\tau K}e^{-\tau V}} \right) ^{M} \end{aligned}$$

(A6)

The approximation in Eq. (A5) is called the primitive approximation and is essential in the present analysis. It allows an explicit writing of Eq. (A6) in a position representation. Equation (A6) is only exact when M goes to infinity, which is known as the Trotter formula, but one expects the expression given in Eq. (A5) to be useful for M sufficiently large.

The eigenfunctions of the kinetic energy operator are plane waves

(A7)

where k is a three-dimensional integer vector times 2$\pi /L$. The density matrix for a free particle in a box $V=L^{3}$ is explicitly obtained using Eq. (A7):

(A8)

Using Eqs. (A6) and (A8), one obtains the density matrix elements which can be written explicitly as the following expression:

$$\begin{aligned} \left\langle {R_{0} } \left| \rho R_{M} \right| \right\rangle= & {} \int \cdots \int d \mathrm{R}_{1}\ldots d \mathrm{R}_{M-1} \frac{1}{(4\pi \lambda \tau )^{3NM/2}}\nonumber \\&\mathrm{exp}\left\{ -\mathop {\sum }\limits _{k=1}^{M}\left[ \frac{(R_{k-1}-R_{k})^{2}}{4\lambda \tau }\right] \right. \nonumber \\&\left. +\frac{\tau }{2}\left( V(R_{k-1})+V(R_{k})\right) \right\} \end{aligned}$$

(A9)

In order to go from Eqs. (A9) to (1) of the present article, one has to take into account the symmetry of the wave functions under permutation of any two particle indices. This step can be found in several textbooks, see, for example, [47]. Note that the kinetic energy has been transformed into the energy of a harmonic string between intermediate configurations. It can also be looked as the square of a velocity in imaginary time (difference of positions divided by the timestep).