Imaginary time and diffusion Monte Carlo
Prerequisites: VMC; gradients; operator evolution
Prerequisite lessons: Variational Monte Carlo and local energy
1. Project instead of oscillate
Use atomic units and imaginary time \(\tau\), not real-time trajectory time. Higher-energy components decay faster relative to the lowest state with initial overlap, provided suitable gap and convergence conditions. \(E_T\) controls normalization or population; it is not necessarily the final estimator.
2. Derive importance-sampled evolution
For \(H=-\nabla^2/2+V\), introduce a real positive guide inside a nodal pocket and \(f=\Psi_T\Phi\). Apply the product rule and define drift \(v_D=\nabla\ln|\Psi_T|\). Diffusion and divergence drift terms move walkers; local energy minus reference controls growth or decay.
3. A short-step schematic, not a production solver
With independent standard-normal components \(\xi\), the simple proposal has Gaussian displacement plus drift. Weights approximately represent branching; symmetric endpoint treatments improve the short-time approximation. Nodes, singular drifts, acceptance correction and population control need care. Finite timestep, population and cell sizes are separate errors.
4. Fermions require a sign or nodal treatment
Positive walkers cannot freely represent changing sign. Fixed-node DMC uses trial nodes as boundaries. For real antisymmetric states and a local Hamiltonian, ideal fixed-node energy bounds the fermionic ground energy from above; exact nodes remove nodal bias. More samples reduce noise, not nodal error. Nonlocal pseudopotential approximations can alter the simple upper-bound claim.
Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.
Worked oscillator guide
For \(\Psi_T=e^{-x^2/2}\), drift is \(-x\) and \(E_L=1/2\). With \(E_T=1/2\), branching is constant. The limiting process \(dx=-x\,d\tau+dW\) has stationary variance \(1/2\), matching \(|\Psi_T|^2\). The simple Euler proposal alone is biased at finite \(\Delta\tau\) and need not reproduce that variance exactly.
Mixed estimates and check yourself
The usual stationary distribution is \(\Psi_T\Phi_{FN}\), not \(|\Phi_{FN}|^2\). Mixed energy is special because \(H\) acts on its eigenstate; general noncommuting observables retain trial-state bias without pure estimators or suitable corrections. Will halving an error bar eliminate incorrect nodes?
Answer and reasoning
No.
Is imaginary time a physical dynamical clock?
Answer and reasoning
No.
References and further reading
Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.