NRG and CT-QMC Impurity Solvers: Validity and Error

Numerical renormalization group (NRG) and continuous-time quantum Monte Carlo (CT-QMC) resolve complementary impurity observables. NRG follows exponentially separated real-frequency scales but introduces logarithmic discretization, state truncation, and spectral broadening. CT-QMC samples an exact continuous-time expansion up to statistical and implementation error, but produces imaginary-time data and may face a sign problem and ill-conditioned analytic continuation. Agreement is meaningful only after both methods compute the same normalized observable and their distinct errors are varied independently.

Required background. The Anderson model supplies a shared benchmark. Convergence certification and covariance and resampling supply numerical error discipline.

Helpful background. Spectral moments supplies exact high-frequency checks.

NRG error budget

NRG logarithmically discretizes a bath using $\Lambda>1$, maps it to a Wilson chain with asymptotic hopping

$$t_n\propto D\Lambda^{-n/2},$$

and diagonalizes iteratively while retaining a finite set of low-energy states. Scale separation makes low-temperature thermodynamics and impurity fixed points exceptionally accessible. The controlled sequence is not “increase kept states once,” but:

1. vary $\Lambda\to1^+$ and use several interleaved $z$ shifts;
2. increase kept multiplets or reduce an energy truncation threshold;
3. exploit exact symmetries without accidentally removing perturbations under study;
4. verify thermodynamic sum rules and known fixed-point spectra; and
5. vary spectral broadening independently of discretization.

A broadened NRG spectrum is not raw resolution. Kernel width, log-Gaussian-to-linear crossover, $z$ averaging, and complete-basis construction all affect peaks. Integrated weights and moments should converge even when pointwise line shapes do not. Bulla, Costi, and Pruschke 2008, §§II–IV reviews the method and complete-basis advances.

CT-QMC error budget

CT-HYB expands the partition function in impurity–bath hybridization; CT-INT expands in interactions. A measured imaginary-time Green function has Monte Carlo covariance,

$$G(\tau)=-\int_{-\infty}^{\infty}\mathrm d\omega\, \frac{e^{-\tau\omega}}{1+e^{-\beta\omega}}\mathcal A(\omega).$$

Here $\mathcal A=-\pi^{-1}\operatorname{Im}G^R$ is the unit-normalized local density of states used throughout this impurity chapter; equivalently, the volume-wide spectral convention is $A=2\pi\mathcal A$. The forward relation is smoothing, so recovering $\mathcal A(\omega)$ is ill-conditioned. A continuation output inherits the prior, regularizer, frequency grid, covariance treatment, and default model. It must be tested on synthetic spectra with comparable noise and on held-out imaginary-time or Matsubara data.

The Monte Carlo report includes expansion-order distribution, warmup, chain or bin length, integrated autocorrelation, independent replicas, covariance, sign or phase, and numerical stabilization. Increasing samples reduces statistical error but cannot repair an incorrect local trace, bath convention, or symmetry block.

Gull et al. 2011, §§II, V, and IX supplies the algorithmic families and sign/measurement considerations.

Cross-solver benchmark

For one Anderson-impurity parameter set, compare quantities that require no continuation first:

Observable	NRG check	CT-QMC check	Shared convention
Occupancy $n_d$	density-matrix trace and discarded-weight stability	equal-time Green function and direct estimator	per orbital and spin sum
Double occupancy	operator expectation under truncation	local estimator with covariance	same $U$, temperature, and action sign
Susceptibility	conserved total-spin response and field derivative	integrated correlation with tail control	impurity contribution versus local response
$G(i\omega_n)$	spectral transform of complete-basis output	direct Matsubara estimator	Fourier sign and high-frequency moments
Spectral weight	broadened integral and moment	continuation posterior predictive check	$\int \mathcal A\,\mathrm d\omega=1$ per spin

Only after these pass should one compare a real-frequency Kondo width. Report a difference as the combination of NRG discretization/truncation/broadening and CT-QMC sampling/continuation errors, not as a single unexplained solver spread.

The structure diagram places both solvers after the model and before observable inference.

NRG and CT-QMC occupy the controlled-solution stage but have different error sources. Cross-validation begins with convention-matched observables that do not require broadening or analytic continuation. Original schematic, not to scale.

The impurity claim test matrix states the solver-specific stopping rules.

Exercise

Diagnose a false agreement. NRG and CT-QMC give the same occupancy, but their real-frequency peak widths disagree by 30%. List the next variations.

Solution

For NRG, vary $\Lambda$, $z$ averaging, kept-state threshold, and broadening kernel separately, checking moments after each change. For CT-QMC, inspect covariance and autocorrelation, vary statistics, continuation method, default model, frequency grid, and synthetic-recovery tests. Compare $G(i\omega_n)$ directly before continuing. Occupancy agreement validates one integral but cannot determine a narrow spectral width.

References

• Bulla, R., Costi, T. A., and Pruschke, T. (2008). “Numerical renormalization group method for quantum impurity systems.” Reviews of Modern Physics 80, 395–450. doi:10.1103/RevModPhys.80.395.
• Gull, E., Millis, A. J., Lichtenstein, A. I., Rubtsov, A. N., Troyer, M., and Werner, P. (2011). “Continuous-time Monte Carlo methods for quantum impurity models.” Reviews of Modern Physics 83, 349–404. doi:10.1103/RevModPhys.83.349.