Langevin updates

ALF’s Langevin mode updates continuous real auxiliary fields. It is a global update: during one Langevin step every field is changed and there is no Metropolis acceptance/rejection step.

When it can be used¶

• Enable the mode with Langevin = .T. in &VAR_QMC.

• All auxiliary fields must be of ALF type 3 (continuous); ALF stops during setup if any field has another type.

• The Hamiltonian must provide the derivative of its bosonic action, Ham_Langevin_HMC_S0. The base implementation assumes $S_0=0$ and prints a warning because this is usually incorrect. Existing model implementations may already supply this routine; a new Hamiltonian needs a correct implementation of $\partial S_0/\partial s_{n,\tau}$.

• Langevin mode turns off sequential, HMC, global, MALA, and tempering updates. Do not combine them in the same run.

Langevin = .T. is the current input switch. Do not add Global_update_scheme = Langevin: it is not a VAR_QMC input variable in the current ALF source.

Update equation¶

For a field vector $\mathbf{s}$, the Langevin equation documented by ALF is

where $\boldsymbol{\eta}$ consists of independent unit Gaussian random variables. The current implementation uses the identity form $Q=I$; Fourier acceleration is not implemented.

In the code, the drift force contains the bosonic contribution $F_0$ and the phase-reweighted fermionic contribution $F_\mathrm{F}$:

Each continuous field is then updated as

Minimal input¶

Place the following in the &VAR_QMC namelist of the parameters file. The numerical values are examples, not universal defaults.

&VAR_QMC
        Langevin             = .T.
        Delta_t_Langevin_HMC = 0.01d0
        Max_Force            = 5.0d0
        NSweep               = 100
        NBin                 = 100
/

Delta_t_Langevin_HMC is the nominal, dimensionless Langevin-time step. Max_Force is the threshold for adaptive stepping. Explicitly set both to positive values: their current runtime defaults are zero and the code does not perform a positivity check.

ALF performs one Langevin update for each NSweep iteration, so a run makes NBin * NSweep updates. A bin contains the measurements accumulated during its NSweep updates.

What Max_Force does¶

Max_Force does not clip the force. For the largest force estimate $X_\mathrm{max}$ in a configuration, ALF uses

Thus, a large force makes the current step shorter while leaving the force unchanged. ALF weights measurements with this running time step when adaptive stepping is active. The info file reports mean and maximum forces at the end of the run; use it to identify frequent or very large excursions. An adaptive step can prevent a crash, but it does not make a simulation with singular or unbounded forces reliable.

Choosing parameters and checking results¶

A nonzero Delta_t_Langevin_HMC introduces a systematic integration error. Run several simulations with decreasing nominal time steps, keep all other physical parameters fixed, and verify convergence (or extrapolate) as

Choose Max_Force to protect against rare force spikes, then inspect the force statistics and the amount of adaptive stepping. Do not use a small Max_Force as a substitute for a time-step study: it only slows the Langevin-time advance near a large force.

Measurements begin immediately. Discard equilibration bins during analysis with n_skip; it is not a Langevin warm-up parameter. To restart an orderly completed run, use out_to_in.sh to turn the written confout_* fields into input configurations before running ALF again. ALF also writes the latest running time step to Langevin_time_steps for continuation.

A worked pyALF example is available in A Jupyter notebook for: Metropolis, MALA, HMC and Langevin updates..