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research27 min studyLesson 765 in Physics

Build reversible proposals with an acceptance correction

Separate fictitious dynamics, pseudofermion measures, and exactness assumptions.

Before you begin
On this page

The question for today

What you’ll understand

  • Explain the auxiliary Hamiltonian and pseudofermion representation.
  • Check leapfrog reversibility on a finite example.
  • State which errors a Metropolis correction removes and which errors remain.

Hybrid Monte Carlo uses auxiliary momenta and reversible numerical trajectories to propose global moves through configuration space. Its trajectory time is a sampling device rather than physical evolution. An acceptance step corrects the energy error of a suitable integrator, allowing finite integration steps without the corresponding equilibrium bias in ideal arithmetic. With dynamical fermions, auxiliary bosonic fields can represent a positive determinant. The method’s correctness depends on that representation, reversibility, volume preservation, and sufficiently accurate numerical solves, in addition to the usual exploration checks.

Enlarge the measure without changing its marginal

For two degenerate fermion flavors at zero chemical potential, a common representation uses the positive matrix M=D†D, assuming invertibility on the sampled configurations. The determinant det M arises from integrating a complex bosonic pseudofermion with action φ†M⁻¹φ. Refreshing that field with the correct Gaussian construction includes the fermion determinant in the gauge measure; omitting it would define a quenched theory, not merely a cheaper estimator of the same ensemble.

Introduce Gaussian auxiliary momenta, represented by Hermitian traceless matrices for each gauge link, and add their quadratic kinetic energy to the action. Gauge-link updates remain on the compact group through exponential maps, while momenta live in an unconstrained tangent algebra. Marginalizing the Gaussian momenta returns the desired gauge/pseudofermion measure. Other flavor numbers can require rational approximations or additional determinant treatments, whose approximation and solver errors must be controlled explicitly.

det⁡M∝∫dϕ†dϕ  e−ϕ†M−1ϕ,H=Sg+ϕ†M−1ϕ+12∑ℓtr⁡Πℓ2\det M\propto\int d\phi^\dagger d\phi\;e^{-\phi^\dagger M^{-1}\phi},\qquad \mathcal H=S_g+\phi^\dagger M^{-1}\phi+\frac12\sum_\ell\operatorname{tr}\Pi_\ell^2
M is a finite positive Hermitian fermion matrix such as D†D. The action and fictitious Hamiltonian are dimensionless; momentum normalization is absorbed consistently into Π and its Gaussian measure.↔ Scroll wide equations horizontally.

Correct a reversible finite step

A leapfrog-like integrator alternates momentum and coordinate updates. Its step sequence is symmetric under time reversal and preserves phase-space volume in exact arithmetic. Combined with the appropriate momentum reversal, it defines a reversible proposal. Its Hamiltonian need not be exactly conserved: the Metropolis decision accepts according to the total auxiliary energy difference. Rejection keeps the old configuration in the Markov record, followed by the algorithm’s prescribed refresh operations.

The correction removes the integration-step bias under those assumptions, rather than demanding every proposed trajectory be almost exact. An excessively large step can nevertheless make acceptance poor or cause numerical instability. In gauge simulations, inaccurate linear solves and floating-point reversibility violations require monitoring and tolerance studies. High acceptance alone does not establish fast exploration: slowly changing topology or long-lived modes may remain despite apparently gentle trajectories.

Pacc=min⁡(1,e−ΔH),ΔH=Hproposed−HoldP_{\rm acc}=\min(1,e^{-\Delta\mathcal H}),\qquad \Delta\mathcal H=\mathcal H_{\rm proposed}-\mathcal H_{\rm old}
This acceptance rule assumes a reversible, volume-preserving proposal and a correct target Hamiltonian. For a more general proposal the density/Jacobian ratio must be included.↔ Scroll wide equations horizontally.

Do not read fictitious paths as gauge physics

A force along an HMC trajectory is a derivative of the target action used to generate proposals. It is not the Minkowski color force experienced by a quark, and its fictitious energy is not a measured hadron energy. Trajectory length, step size, preconditioning, and solver strategy affect sampling efficiency. Physical observables still require ensemble averages, autocorrelation analysis, and lattice-spacing/volume studies after the algorithm has sampled the intended measure.

The oscillator lab contrasts conservative and dissipative motion in a separate exact model. Reset and set damping ratio to 0 and frequency to 1, then Step twice to time 1; predict x=cos1 and velocity −sin1. Reset and set damping to 1 and frequency to 1, then Step twice; predict x=2/e and velocity −1/e. The latter loses mechanical energy. This analytic unit-mass oscillator has no leapfrog proposal, gauge links, fermion solve, or accept/reject step; dissipation cannot substitute for reversible HMC sampling.

p1/2=p0−h2q0,q1=q0+hp1/2,p1=p1/2−h2q1p_{1/2}=p_0-\frac h2q_0,\qquad q_1=q_0+hp_{1/2},\qquad p_1=p_{1/2}-\frac h2q_1
This finite example is leapfrog for H=(q²+p²)/2 in dimensionless canonical coordinates. It illustrates proposal algebra only; the gauge-group implementation needs its corresponding group-valued updates.↔ Scroll wide equations horizontally.

Work it through

Reverse one leapfrog step

For H=(q²+p²)/2, start at q=1,p=0 and use one leapfrog step h=1. Find the proposal, its energy change and acceptance probability, then reverse its momentum and check the return step.

  1. The half momentum is −1/2, the new coordinate is 1/2, and the final momentum is −1/2−(1/2)(1/2)=−3/4. The proposal is therefore (1/2,−3/4).

  2. The initial energy is 1/2 and proposed energy is [(1/2)²+(3/4)²]/2=13/32. Thus ΔH=−3/32 and acceptance is one, despite the finite energy error.

  3. Reverse momentum to +3/4. A second h=1 leapfrog step gives half momentum 1/2, coordinate 1, and final momentum 0. It returns to the initial point, demonstrating reversibility rather than exact energy conservation.

The result

The proposal is (1/2,−3/4), ΔH=−3/32, and acceptance is one. Momentum reversal followed by the same step returns to (1,0).

The living notebook

Explore the model

Interactive lab

Interactive lab · oscillator

Contrast reversible and damped motion

Reset and set ζ=0,ω₀=1, then Step twice to t=1; predict x=cos1 and v=−sin1. Reset and set ζ=1,ω₀=1, then Step twice; predict x=2/e and v=−1/e.

Contrast reversible and damped motionUnit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Current readings: Displacement x(t): 1; Velocity v(t): 0; Regime: underdamped. All parameters are adjustable with the labeled controls below.02.557.510−1.15−0.5700.571.15Time · sDisplacementunderdamped
Displacement x(t)Current time
Displacement x(t)
1
Velocity v(t)
0
Regime
underdamped
t = 0 s

Unit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThis analytic oscillator starts at x=1,v=0. It illustrates energy-conserving versus dissipative motion, not HMC integration, proposals, or gauge dynamics.

Pause & reason

Check your understanding

What does a valid HMC Metropolis step correct?

Your turn

Try it yourself

A valid HMC trajectory proposes ΔH=ln4. Find its acceptance probability and explain what to record after rejection. Would keeping accepted configurations only preserve the same chain?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Apply min(1,exp[−ΔH]) and remember the self-transition probability.

Compare with the solution

Acceptance is 1/4. A rejection records the previous configuration again, with any subsequent momentum refresh performed according to the algorithm. Removing rejected repetitions generally biases the configuration distribution because acceptance depends on state. A smaller integrator step can improve accuracy but must be assessed together with computational cost and autocorrelation.

Keep the idea

Take it with you

  • HMC momenta and time are auxiliary sampling variables.
  • Reversibility and volume preservation enable the finite-step acceptance correction.
  • Algorithmic exactness and efficient, well-diagnosed exploration are distinct requirements.

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