HMC Parameters

Tuning the Hybrid Monte Carlo (HMC) updating scheme in ALF. HMC is available for models with continuous auxiliary fields (OP_V%type=3).

Reference: The algorithm is derived in the documentation (PDF), Section on Hybrid Monte Carlo.

Parameters¶

All HMC parameters are set in the &VAR_QMC namelist in the parameters file.

The trajectory length is $T_m$ = Delta_t_Langevin_HMC × Leapfrog_Steps.

Example¶

&VAR_QMC
HMC                    = .true.
Sequential             = .true.
Delta_t_Langevin_HMC   = 0.1d0
Leapfrog_Steps         = 10
N_HMC_sweeps           = 5
NSweep                 = 100
NBin                   = 20
Nwrap                  = 10
/

Guidelines¶

Step Size and Acceptance Rate¶

The leap-frog integrator introduces an $\mathcal{O}(\delta t_m^2)$ discretization error in the Hamiltonian. The Metropolis accept/reject step corrects for this, but large errors lead to low acceptance.

• Target acceptance rate: ~70–80%.

• Check the acceptance rate in the info file after a run (Acceptance_HMC).

• If acceptance is too low: reduce Delta_t_Langevin_HMC.

• If acceptance is very high (>95%): you’re being too conservative — increase Delta_t_Langevin_HMC or Leapfrog_Steps to explore phase space faster.

Trajectory Length¶

• The trajectory length $T_m$ controls how far each HMC proposal moves in configuration space.

• Too short: proposals are close to the current state (inefficient, like random walk).

• Too long: computational cost per trajectory is high without proportional gain.

• A reasonable starting point is $T_m \approx 1$.

Combining with Sequential Updates¶

Always keep Sequential = .true. when using HMC. The documentation explicitly warns:

“In order to address potential ergodicity issues it is crucial to integrate both sequential and hybrid molecular dynamics moves, rather than relying solely on hybrid molecular dynamics.”

N_HMC_sweeps controls how many HMC trajectories are performed between sequential sweeps. Increasing this value shifts the balance toward HMC moves.

Mass Matrix (Apply_B_HMC)¶

The HMC Hamiltonian is:

where $M$ is a positive-definite mass matrix. It is factored as $M^{-1} = B^T B$, with the constraint $\text{Ker}(B) = \{0\}$.

By default, $M = \mathbf{1}$ (identity) — the base implementation of Apply_B_HMC in the code is a no-op.

A non-trivial mass matrix can precondition the dynamics:

• Assign larger mass to fast modes → they evolve slowly in HMC time.

• Assign smaller mass to slow modes → they evolve faster.

• This can significantly reduce autocorrelation times.

To use a custom mass matrix, override the Apply_B_HMC subroutine in your Hamiltonian submodule. It receives a force/momentum array and a logical flag ltrans:

• ltrans = .false.: apply $B$ (used in momentum updates)

• ltrans = .true.: apply $B^T$ (used in position updates)

Note: No shipped Hamiltonians currently override Apply_B_HMC. This is an advanced feature for users who understand the mode structure of their model.

Validation¶

The documentation provides a validation benchmark: 6-site Hubbard chain at $\beta t = 4$:

Use small exact-diagonalization benchmarks like this to validate your HMC setup before production runs.

Known Pitfalls¶

• HMC requires continuous fields. If any interaction vertex has OP_V%type ≠ 3, the code will terminate with an error. Set Continuous = .true. in your Hamiltonian namelist (e.g. &VAR_Hubbard).

• Forgetting to enable HMC. Setting Delta_t_Langevin_HMC and Leapfrog_Steps without HMC = .true. has no effect.

• Zero defaults. Both Delta_t_Langevin_HMC and Leapfrog_Steps default to 0 — you must set both to nonzero values.

• Langevin vs. HMC. The Delta_t_Langevin_HMC parameter is shared between Langevin and HMC dynamics. Do not enable both simultaneously unless you understand the interplay.