Pure estimators and extrapolation technique

As a result of the VMC calculation one obtains a variational esimator for a quantity (let it be described by an operator $\hat A$), which corresponds to an average over the trial wave function $\psi _T$:

$$\langle \hat A \rangle_{var.} = \frac{\langle\psi_T\vert\hat A\vert\psi_T\rangle}{\langle\psi_T\vert\psi_T\rangle}$$ (2.157)

Instead, the DMC method asymptotically provides a more precise mixed estimator, which we denote as

$$\langle \hat A \rangle_{mix.} = \frac{\langle\phi_0\vert\hat A\vert\psi_T\rangle}{\langle\phi_0\vert\psi_T\rangle}$$ (2.158)

Still, this type of average can differ from the ``pure'' ground state average, which corresponds to the true quantum-mechanical equilibrium value at a zero temperature

$$\langle \hat A \rangle_{pure} = \frac{\langle\phi_0\vert\hat A\vert\phi_0\rangle}{\langle\phi_0\vert\phi_0\rangle}$$ (2.159)

The DMC method gives an exact result for the energy, as the mixed average of the local energy $E_{loc} = \psi_T^{-1}\hat H\psi_T$ coincides with the pure estimator. This can be easily seen by noticing that when $\langle \phi_0$ acts on $\hat H$, it gives exactly the ground state energy.

We will show that averages of local operators can be calculated in a ``pure'' way. This means that the pair distribution function, radial distribution, size of the condensate can be found essentially exactly. We assume that $\langle {\bf R}\vert\hat A \vert{\bf R}'\rangle = A({\bf R}) \langle {\bf R}\vert{\bf R}'\rangle$. The ``pure'' average can be related to the mixed one in the following way

$$\langle \hat A \rangle_{pure} = \frac{\langle\phi_0\vert A({\bf R... ...\langle A({\bf R}) P({\bf R}) \rangle_{mix}}{\langle P({\bf R}) \rangle_{mix}},$$ (2.160)

where $P({\bf R})$ is defined as

$$P({\bf R}) = \frac{\phi_0({\bf R})}{\psi_T({\bf R})} \langle \phi_0\vert\psi_T\rangle$$ (2.161)

and gives the number of descendants of a walker ${\bf R}$ for large times $\tau\to\infty$. Practically it is enough to wait a sufficiently large, but a finite time $T$. One makes measurements of a local quantity for all of the walkers, but calculates the average after the time $T$, so that each walker was replicated according to the weight $P({\bf R})$.

One of important examples of a non-local quantity is the non-diagonal element of the one-body density matrix (see (2.141)). This quantity deserves a special attention, so we will explain an extrapolation technique, which can be applied for finding averages of a non-local operators.

Let us denote the difference between the trial wave function and ground-state wave function as $\delta\psi$

$$\phi_0 = \psi_T + \delta \psi$$ (2.162)

Then the ground-state average can be written as

$$\langle\hat A\rangle_{pure} = \langle\phi_0\vert\hat A\vert\phi_0... ...at A\vert\delta\psi\rangle + \langle\delta\psi\vert\hat A\vert\delta\psi\rangle$$ (2.163)

If $\delta\psi$ is small the second order term $\langle\delta\psi\vert\hat A\vert\delta\psi\rangle$ can be neglected. After substitution $\langle\phi_0\vert\hat A\vert\delta\psi\rangle = \langle\psi_T\vert\hat A\vert\phi_0\rangle - \langle\psi_T\vert\hat A\vert\psi_T\rangle$ the extrapolation formula becomes

$$\langle\hat A\rangle_{pure} \approx 2 \langle\hat A\rangle_{mix.} - \langle\hat A\rangle_{var.}$$ (2.164)

It is possible to write another extrapolation formula of the same order of accuracy:

$$\langle\hat A\rangle_{pure} \approx \frac{\langle\hat A\rangle^2_{mix.}}{\langle\hat A\rangle_{var.}}$$ (2.165)

Of course, if the extrapolation technique is applicable, formulae (2.164) and (2.165) give the same result. The second formula is preferable for extrapolation of a non-negative quantity (e.g. the OBDM), if the function can be very close to zero, as (2.165) preserves the sign of the function.

In the end of this Section we will mention that it can be proven that the measurement of the superfluid density in DMC method is a pure estimator and to a large extent is not biased by the chose of a trial wave function [Ast01,AG02].