Path-Integral Discretization

Path-integral discretization replaces $e^{-\beta H}$ by a product of short-time kernels. It creates a systematic approximation before any Monte Carlo sampling occurs. This page derives that approximation and its continuum-time checks; path-integral Monte Carlo treats how the resulting paths are sampled.

Required background. Euclidean path integrals explains imaginary time and thermal traces. Here $\beta=1/(k_BT)$ has inverse-energy units.

Symmetric short-time kernels

For $H=T+V$ with $T=p^2/(2m)$, choose $M$ slices and $\epsilon=\beta/M$. A symmetric factorization is

$$e^{-\epsilon H} =e^{-\epsilon V/2}e^{-\epsilon T}e^{-\epsilon V/2} +O(\epsilon^3)$$

in an appropriate operator sense for sufficiently regular problems. Repeating $M$ factors gives a second-order global discretization at fixed $\beta$ when the required commutators and estimator limits are controlled. Singular interactions or domain issues require a more careful short-time density matrix.

The free kernel in one dimension is

$$\rho_0(x,x';\epsilon) =\left(\frac{m}{2\pi\hbar^2\epsilon}\right)^{1/2} \exp\left[-\frac{m(x-x')^2}{2\hbar^2\epsilon}\right].$$

Its normalization is essential for partition functions and thermodynamic energy estimators, even though it cancels in some move-acceptance ratios.

The closed-path action

In a thermal trace, $x_M=x_0$, and the endpoint half-potentials combine. The primitive ring-polymer approximation is

$$Z_M=C_M\int\prod_{j=0}^{M-1}dx_j\,e^{-S_M},$$ $$S_M=\sum_{j=0}^{M-1} \left[ \frac{m(x_{j+1}-x_j)^2}{2\hbar^2\epsilon} +\epsilon V(x_j) \right], \quad C_M=\left(\frac{m}{2\pi\hbar^2\epsilon}\right)^{M/2}.$$

$S_M$ is dimensionless. The bead spacing in imaginary physical time is $\hbar\epsilon$, so inserting another $\hbar$ into the potential term would change units incorrectly.

For identical particles, closure includes permutations. The displayed formula is for one distinguishable particle; exchange is a separate part of the configuration space.

A finite-slice oscillator result

Set $m=\hbar=\omega=1$. The action is quadratic, with Fourier-mode precision eigenvalues

$$a_n=\epsilon+\frac4\epsilon\sin^2\frac{\pi n}{M}, \qquad n=0,\ldots,M-1.$$

Averaging the inverse precision gives the bead variance,

$$\langle x_j^2\rangle_M=\frac1M\sum_n\frac1{a_n}.$$

At $M=1$, this yields $1/\beta$, the classical configurational result. As $M\to\infty$ at fixed $\beta$, it approaches $\frac12\coth(\beta/2)$. This closed finite-$M$ expression is a stronger sampler test than comparing every coarse discretization directly with the continuum answer.

Diagonalizing the quadratic action also suggests efficient independent normal-mode proposals. It separates a faulty sampler from a perfectly sampled but biased discretization.

Extrapolation without hiding statistics

Measure at several increasing $M$ while keeping $\beta$ fixed. In a verified second-order regime, fit $O_M=O_\infty+c/M^2+\cdots$, and check stability when the coarsest point is removed. Include the statistical covariance when runs share random numbers. A two-point extrapolation cannot demonstrate the assumed power law by itself.

Separately increase $\beta$ if a ground-state observable is desired. More slices at fixed $\beta$ converge a finite-temperature result; lower temperature with fixed $M$ makes each slice larger and can worsen discretization.

Exercises

At $\beta=1$ and $M=2$, compute the oscillator bead variance from the mode formula.

Solution

$\epsilon=1/2$, so $a_0=1/2$ and $a_1=17/2$. The variance is $(2+2/17)/2=18/17\approx1.05882$. The continuum value is $\frac12\coth(1/2)\approx1.08198$. Their difference is deterministic slice bias, not a statistical error.

References

• Ceperley, David M., “Path integrals in the theory of condensed helium,” Reviews of Modern Physics 67, 279–355 (1995), doi:10.1103/RevModPhys.67.279 — density matrices, path sampling, and estimators.