Cluster DMFT and Nonlocal-Correlation Validity

Cluster DMFT restores correlations within a finite cluster while embedding that cluster in a dynamical bath. Cellular DMFT (CDMFT) works in real space and breaks translations inside the approximation; the dynamical cluster approximation (DCA) uses periodic momentum patches. A nonlocal conclusion is credible only if it persists across cluster size, geometry, boundary convention, and reconstruction of lattice quantities.

Required background. Use the single-site DMFT mapping. Helpful background. Convergence and extrapolation supplies general certification principles.

Cellular and momentum-space clusters

CDMFT maps the lattice to an $N_c$-site impurity with a matrix Weiss field. Its self-consistency has the form

$$\mathbf G_c(i\omega_n)=\frac{N_c}{N}\sum_{\widetilde{\mathbf k}} \left[(i\omega_n+\mu)\mathbf1 -\mathbf t(\widetilde{\mathbf k})-\boldsymbol\Sigma_c(i\omega_n) \right]^{-1}.$$

The cluster self-energy is exact inside the chosen impurity problem, but its use between clusters is approximate. DCA instead assumes the self-energy is constant within momentum cells:

$$\overline G_{\mathbf K}(i\omega_n)=\frac{N_c}{N} \sum_{\widetilde{\mathbf k}} \frac{1}{i\omega_n+\mu-\epsilon_{\mathbf K+\widetilde{\mathbf k}} -\Sigma_{\mathbf K}(i\omega_n)}.$$

DCA preserves cluster translations but coarse-grains momentum; CDMFT resolves real-space inhomogeneity but has boundary and translation artifacts. Their errors need not approach the limit monotonically. The constructions and their scaling properties are reviewed in Maier et al. 2005, §§ II–IV.

Periodization is part of the result

To visualize a lattice spectrum from CDMFT, one may periodize $\Sigma$, the cumulant $M=(i\omega_n+\mu-\Sigma)^{-1}$, or $G$. Near a self-energy pole these choices can disagree qualitatively. Report the directly computed cluster quantities first, then show whether a momentum-selective gap survives at least two justified reconstructions.

A convergence protocol

Hold the Hamiltonian, temperature, solver tolerances, and continuation prior fixed while varying:

1. $N_c$ and clusters with different point-group bias;
2. CDMFT versus DCA when both are appropriate;
3. bath resolution and impurity-solver error;
4. self-energy, cumulant, and Green-function periodization; and
5. imaginary-axis observables before analytic continuation.

A four-site plaquette can establish short-range momentum differentiation within that approximation. It cannot establish a thermodynamic phase boundary without drift data. Long correlation lengths, competing stripes, or a cluster commensurate with one order and not another are characteristic failure cases.

Exercises

Why can self-energy and cumulant periodization disagree most strongly near a Mott gap?

Solution

A Mott gap is associated with a self-energy pole. Interpolating a divergent $\Sigma$ is ill conditioned, whereas the cumulant can remain smoother; away from that regime the reverse may be true. Their disagreement measures reconstruction sensitivity rather than an uncertainty that can be hidden.

References

• Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler, “Quantum Cluster Theories,” Reviews of Modern Physics 77 (2005) 1027–1080, §§ II–IV, doi:10.1103/RevModPhys.77.1027.