Worldline Monte Carlo

Worldline Monte Carlo samples sequences of occupation or spin states in imaginary time. Hopping events connect those states, while the thermal trace closes the trajectories. The representation makes conservation constraints and topological sectors explicit; an update scheme must explore every sector included in the chosen ensemble.

Required background. Imaginary-time discretization supplies the short-time expansion and MCMC supplies the sampling requirements.

Local propagators create worldline events

For a nearest-neighbor hopping term on two sites,

$$h=-t(b_1^\dagger b_2+b_2^\dagger b_1),$$

the one-particle subspace $\{|10\rangle,|01\rangle\}$ has $h=-t\sigma_x$. Its exact bond propagator is

$$e^{-\epsilon h}= \begin{pmatrix} \cosh(\epsilon t)&\sinh(\epsilon t)\\ \sinh(\epsilon t)&\cosh(\epsilon t) \end{pmatrix}.$$

For $t\geq0$, all these local weights are nonnegative. Diagonal elements correspond to remaining on the same site; off-diagonal elements correspond to hopping. Checkerboard decompositions combine nonoverlapping bond propagators with a time-slice error. Continuous-time formulations instead expand in the number and times of events and remove that particular slice approximation.

Positivity depends on the Hamiltonian and chosen basis. Frustrated spin couplings or fermion exchange can produce negative weights. A worldline drawing does not by itself establish a sign-free algorithm.

Ergodicity is more than moving a local segment

Local plaquette changes can preserve particle number and often preserve the winding sector. If the desired ensemble permits different particle number or winding, those updates alone are insufficient. A worm update temporarily opens a worldline by inserting a source and sink, moves the endpoints, then closes the path; loop and directed-loop updates reorganize connected segments.

The extended configuration weights and endpoint proposal factors must be included in detailed or global balance. One cannot insert an open path and accept it with the closed-sector ratio by analogy. Measure partition-function observables in the appropriate closed sector and off-diagonal quantities with their defined extended-ensemble normalization.

Winding and superfluid response

In a periodic box, count the net unwrapped spatial displacement of all paths around each direction. For a continuum isotropic system of $N$ identical bosons of mass $m$ in a cubic box of side $L$, the winding estimator is

$$\frac{\rho_s}{\rho} =\frac{mL^2}{d\hbar^2\beta N} \langle \mathbf W^2\rangle.$$

Here $\mathbf W$ contains integer winding numbers and $\beta$ has inverse-energy units. This is a response estimator, not a condensate-fraction estimator. Lattice stiffness formulas use hopping and lattice-length conventions and should be derived for that Hamiltonian rather than copied with an undefined “mass.”

A run with identically zero winding can describe an insulating regime or an update that never changes winding. Inspect actual sector transitions and compare an algorithm capable of changing them.

A two-site thermal benchmark

For one particle on the displayed bond, eigenenergies are $-t$ and $t$. Thus

$$Z=2\cosh(\beta t), \qquad E=-t\tanh(\beta t).$$

The expansion of the trace contains an even number of hops because the path must return to its starting site. A sampler that permits an odd number in the closed one-particle sector violates the trace constraint. This example tests event parity, thermal weights, and energy before any large-lattice extrapolation.

Convergence and limitations

Converge time slices when using a discrete-time representation, mixing across slow sectors, system size, and inverse temperature separately. Record update units and observable-dependent autocorrelation times. Occupation cutoffs for soft-core bosons are another systematic error; increase them and check the probability near the cap.

Exercises

Why does the two-site partition function contain only even powers of $\beta t$?

Solution

Every hopping event exchanges $|10\rangle$ and $|01\rangle$. The trace requires the final occupation to equal the initial one, so an even number of exchanges is required. Algebraically, odd powers of $\sigma_x$ have zero trace while even powers equal the identity.

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.
• Pollock, E. L., and D. M. Ceperley, “Path-integral computation of superfluid densities,” Physical Review B 36, 8343–8352 (1987), doi:10.1103/PhysRevB.36.8343 — winding-number response.
• Troyer, Matthias, Fabien Alet, Simon Trebst, and Stefan Wessel, “Non-local updates for quantum Monte Carlo simulations,” AIP Conference Proceedings 690, 156 (2003), arXiv:physics/0306128 — loop, worm, and continuous-time updates.