Stabilization Parameters

Stabilization Parameters

Numerical stabilization settings in ALF. These apply to every simulation regardless of the update scheme.

Reference: The stabilization algorithm is described in the documentation (PDF), Section on numerical stabilization.

Parameters

Nwrap

Set in the &VAR_QMC namelist in the parameters file.

Parameter	Default	Description
Nwrap	10 (in pyALF/examples; 0 internally if not set)	Number of imaginary-time slices between QR stabilizations

The Green's function is recomputed from scratch after every Nwrap × Dtau interval of imaginary time. The total number of stabilization checkpoints is Ltrot / Nwrap (rounded up if Ltrot is not evenly divisible).

Stabilization Scheme

Set at compile time via configure.sh:

source configure.sh GNU noMPI STAB3    # or STAB1, STAB2, LOG

Scheme	Flag	Description
Default	(none)	Standard stabilization — generally works well
STAB1	-DSTAB1	Older UDV decomposition
STAB2	-DSTAB2	UDV with additional normalizations
STAB3	-DSTAB3	Newest: separates large and small scales
LOG	-DSTABLOG	Logarithmic storage of internal scales — extends accessible parameter ranges

Guidelines

Choosing Nwrap

• Smaller Nwrap = more stable but slower. Each stabilization costs a full Green's function recomputation.
• Larger Nwrap = faster but risks numerical errors. The propagated Green's function drifts further from the exact value between stabilizations.
• Nwrap = 10 is a good default for moderate interaction strengths and typical Dtau values.
• Reduce Nwrap (e.g., to 5 or even 1) when:
  • Working at strong coupling (large ham_U)
  • Using large Dtau
  • The info file reports poor precision
  • You see the error message "Try with smaller Nwrap or dtau"
• Nwrap must be ≥ 1. Setting it to 0 is not valid.

Choosing a Stabilization Scheme

• Start with the default (no STAB flag). It is fast and works well for most problems.
• Use LOG when you need to reach large $\beta$ (low temperature) or strong interactions. It solves NaN issues by storing internal scales in logarithmic form, extending the accessible parameter range at some computational overhead.
• STAB1 and STAB2 are older schemes kept for backward compatibility. STAB1 uses UDV decompositions; STAB2 adds additional normalizations. They are generally not recommended for new simulations.
• STAB3 separates large and small scales explicitly. It can be useful for specific problems but is not the default.

Interaction with Dtau

Nwrap and Dtau together determine the imaginary-time interval between stabilizations: $\Delta\tau_\text{stab} = \texttt{Nwrap} \times \texttt{Dtau}$. If you reduce Dtau (for Trotter error reasons), you can often afford a proportionally larger Nwrap while maintaining the same stability.

Diagnostics

The info file reports the precision of the Green's function — the maximum deviation between the propagated and freshly computed Green's function at stabilization checkpoints. Values should be small (e.g., $< 10^{-6}$). If the precision is poor:

1. Reduce Nwrap
2. Reduce Dtau
3. Switch to LOG stabilization

If the deviation exceeds 10, ALF prints the error: 'Try with smaller Nwrap or dtau.'

Known Pitfalls

• Nwrap too large for strong coupling. At large interaction strengths, scales separate rapidly in imaginary time. The default Nwrap = 10 may be too aggressive.
• Changing stabilization scheme requires recompilation. The scheme is set via preprocessor flags, not at runtime.
• Ltrot not divisible by Nwrap. This is handled (the last interval is shorter), but can lead to slightly uneven stabilization. Choose Dtau and Beta such that Ltrot = Beta/Dtau is a nice multiple of Nwrap when possible.