Typical medium dynamical cluster approximation

The Typical Medium Dynamical Cluster Approximation (TMDCA) is a non-perturbative approach designed to model and obtain the electronic ground state of strongly correlated many-body systems. It addresses critical aspects of mean-field treatments of strongly correlated systems, such as the lack of an intrinsic order parameter to characterize quantum phase transitions and the description of spatial (or momentum) dependent features. Additionally, the TMDCA tackles the challenge of accurately modeling strongly correlated systems when imperfections disrupt the fundamental assumptions of band theory, as seen in density functional theory, such as the independent particle approximation and material homogeneity.

The TMDCA is a variant of the dynamical mean field approximation (DMFA), built on the dynamical cluster approximation (DCA). It is designed to more accurately handle the combined impacts of disorder and electron-electron interactions in strongly correlated systems. Through a set of self-consistent equations, the TMDCA maps a lattice onto a finite cluster embedded in a typical medium. This cluster is a periodically repeated cell containing $N_{c}$ primitive cells, resulting in the first Brillouin zone of the original lattice being divided into $N_{c}$ non-overlapping cells. Each cell, centered at the wave vector $\mathbf {K}$, contains a set of wave vectors ${\tilde {\mathbf {k} }}\equiv \mathbf {k} -\mathbf {K}$, where ${\tilde {\mathbf {k} }}$ and $\mathbf {k}$ are wave vectors generated by the translational symmetry of the cluster and the original lattice, respectively. These clusters allow for resonance effects, and by increasing $N_{c}$, it is possible to systematically incorporate longer-range spatial fluctuations. This approach bridges the gap between the single-site approximation of DMFA and the realities of spatial correlations and randomly distributed disorder, providing a more nuanced understanding of phenomena such as Anderson localization, the Mott transition, and the metal-insulator transition in disordered systems.

TMDCA has notably elucidated Anderson localization, offering a mean-field model that precisely captures the re-entrance of the mobility edge in the three-dimensional Anderson model. TMDCA clearly delineates the metal-insulator transition in weakly interacting disordered electron systems, highlighting that interactions stabilize the metallic phase and induce a soft pseudogap near the critical disorder strength. Furthermore, it confirms that the mobility edge remains stable as long as the chemical potential exceeds or meets the mobility edge energy. TMDCA also sheds light on the cause of photoluminescent quenching in two-dimensional $MoS_{2}$ observed experimentally and defect-tolerant behavior in 2D monolayers PbSe and PbTe where impurity states forming shallow levels rather than localized deep levels. Its utility extends to characterizing real materials in conjunction with various functionals within density functional theory.

Background and Description of TMDCA

The Ising model laid the foundation of mean-field theories used today to model many-body systems. The Ising model maps the lattice problem onto an effective single-site problem, with a magnetization such as to reproduce the lattice magnetization through an effective "mean-field". This condition is called the self-consistency condition. It stipulates that the single-site observables should reproduce the lattice "local" observables through an effective field. However, unlike the Ising model and by extension the DMFT, the TMDCA maps the lattice problem onto a finite cluster that is embedded in an effective self-consistent typical medium characterized by a nonlocal hybridization function $\Gamma (\mathbf {K} ,\omega )$ or non-local self-energy $\Sigma (\mathbf {K} ,\omega )$. By mapping a $d$-dimensional lattice to a finite small cluster containing $N_{c}=L_{c}^{d}$ sites, where $L_{c}$ is the linear dimension of the cluster, we dramatically reduce the computation effort. Thus, the foundation of TMDCA is based on the combination of DMFA and the coherent potential approximation (CPA), extended to a finite cluster rather than single sites with the self-consistency characterized by an effective typical medium. Hence, the TMDCA ensures that nonlocal spatial fluctuations, neglected in single-site approaches, are systematically incorporated as the cluster size $N_{c}$ increases. The short length scale correlations are treated exactly inside the cluster, while the long length scale correlations are treated within the effective typical medium.

Self-consistency equations and TMDCA loop

To model the interplay of Anderson physics – disorder and Mott physics – electron interactions in strongly correlated systems, the TMDCA utilizes the Anderson – Hubbard model, expressed as:

$H=H_{0}+\sum _{i\alpha \sigma }V_{i\sigma }^{\alpha }n_{i\sigma }^{\alpha }+U\sum _{i\alpha }n_{i\uparrow }^{\alpha }n_{i\downarrow }^{\alpha }$

Here, $H_{0}$ represents the single-particle Hamiltonian, primarily comprising the hopping terms $t_{ij}$. The term $V_{i\sigma }^{\alpha }$ denotes a disorder potential, and $n_{i\sigma }^{\alpha }$ is the number operator, with indices $i$, $\alpha$, and $\sigma$ representing site, orbital, and spin, respectively. These terms describe single-particle motion, disorder (typically modeled as random variables with a specified probability distribution), and electron interactions. A crucial focus is the single-particle Green's function and the associated density of states, determined by the hybridization function ensuring that the corresponding coarse-grained Green’s function ${\bar {G}}_{c}(\mathbf {K} ,\omega )$ coincides with the local lattice Green's function $G_{c}^{t}(\mathbf {K} ,\omega )$.

The main challenge of using mean-field approximations to study disordered electronic systems is the lack of a single-particle order parameter to distinguish between localized and extended states — a quantum phase transition — that is generally observed in many physical systems. In the typical medium dynamical cluster approximation, the self-consistency field (SCF) is defined by a typical medium, where the typical density of states (TDoS) is used instead of the arithmetically averaged local density of states. The TDoS, which is approximated using geometrical averaging over disorder configurations, provides a more accurate representation of the most probable value of the local density of states, particularly near a critical point. The self-consistency field procedure of the TMDCA involves:

1. Start with an initial guess for the hybridization function $\Gamma (\mathbf {K} ,\omega )$, which describes the coupling between the cluster and the effective typical medium.
2. Calculate the fully dressed cluster Green’s function: $G^{c}(\mathbf {K} ,\omega )=({\mathcal {G}}^{-1}(\mathbf {K} ,\omega )-V-\Sigma ^{\mathrm {Int} })^{-1}$, where ${\mathcal {G}}^{-1}(\mathbf {K} ,\omega )$ is the cluster-excluded Green’s function, $V$ is the disorder potential, and $\Sigma ^{\mathrm {Int} }(\mathbf {K} ,\omega )$ is the full self-energy, obtained within a secondary SCF loop. Currently, the full self-energy due to interactions is computed up to the second-order expansion of the interactions, represented as $\Sigma _{c}^{Int}(i,j,\omega )=\Sigma _{c}^{H}[{\mathcal {\tilde {G}}}]+\Sigma _{c}^{(SOPT)}[{\mathcal {\tilde {G}}}]$, Where the first term, $\Sigma _{c}^{H}[{\mathcal {\tilde {G}}}]$, is the static Hartree correction, and the second term, $\Sigma _{c}^{(SOPT)}[{\mathcal {\tilde {G}}}]$, is the non-local second-order perturbation theory (SOPT) contribution. It is important to note that the computational cost grows exponentially with each order of the perturbation series, making it numerically prohibitive to include more diagrams in the calculation.
3. Compute the cluster density of states $\rho ^{c}(\mathbf {K} ,\omega )=-{\frac {1}{\pi }}\Im G^{c}(\mathbf {K} ,\omega )$, and averaging over a large number of configurations to obtain the wave-vector-resolved, non-self-averaged typical density of states: $\rho _{t}^{c}(\mathbf {K} ,\omega )=\left\langle \rho _{i}^{c}\right\rangle _{\mathrm {geom} }\left\langle {\frac {\rho ^{c}(\mathbf {K} )}{{\frac {1}{N_{c}}}\sum _{i}\rho _{i}^{c}}}\right\rangle _{\mathrm {arit} }$, where $\left\langle \rho _{i}^{c}\right\rangle _{\mathrm {geom} }=\exp \left(\left\langle \ln \rho _{i}\right\rangle _{\mathrm {arit} }\right)$ is the geometric mean of the diagonal elements of the density of states, capturing non-local fluctuations.
4. Calculate the cluster typical Green’s function $G_{t}^{c}(\mathbf {K} )$ from the Kramers Kronig transform of the density of states, used to compute the coarse-grained Green’s function: ${\bar {G}}(\mathbf {K} )={\frac {N_{c}}{N}}\sum _{\tilde {k}}\left[\left(G_{t}^{c}(\mathbf {K} )^{-1}+\Gamma (\mathbf {K} )-H_{0}(k)+{\bar {H}}_{0}(\mathbf {K} )+\mu \right)^{-1}\right]$, where $\mu$ is the Fermi level obtained in the secondary SCF loop.
5. Obtain a new hybridization function based on a mixture of old and updated functions, with the linear mixing parameter $\zeta$: $\Gamma _{n}(\mathbf {K} ,\omega )=(1-\zeta )\Gamma _{o}(\mathbf {K} ,\omega )+\zeta \left[(G^{c})^{-1}-{\bar {G}}^{-1}\right]$.
6. Go back to step 2 until convergence is achieved, which is $G_{t}^{c}\approx {\bar {G}}$.

TMDCA Outcomes and Implications

The TMDCA preserves causality as all the Green’s functions are causal; consequently, both the “algebraically averaged density of states (ADoS) and the typical density of states calculated from them are positive definite. As $N_{c}$ increases, the TMDCA systematically interpolates between the local typical medium theory and numerically exact methods such as exact diagonalization, transfer matrix method, and the Kernel polynomial method, achieving these results at a fraction of the cost — over ten times less expensive than these numerically exact methods. A key output of the TMDCA SCF is the typical density of states (TDoS), which is calculated from the imaginary part of the typical Green's function $\rho _{\text{typ}}(\omega )=\exp \left(\langle \log(-{\text{Im}}G(\omega ))\rangle \right)$, where the average is taken over disorder configurations and the cluster sites. The TDoS provides insights into the density of electronic states that are most representative (“typical”) of the system under study.

Applications

TMDCA has been applied to a wide range of model systems and real materials, exploring the effects of disorder in novel two-dimensional materials, magnetic materials, and strongly correlated systems. Its capacity to account for cluster-level spatial correlations makes it especially valuable for predicting new phenomena in these complex materials.

The nonlocal lattice Green's function and other cluster observables enable the calculation of various physical quantities:

• Optical properties: Analysis of absorption spectra and optical properties using the Kubo Greenwood formalism and exciton absorption spectra in nanostructured materials like two-dimensional materials via two-particle Green’s function within the Bethe-Salpeter theory.
• Electronic properties: Investigation of photoemission spectra in strongly correlated systems, including the analysis of spectral function.

See Also

• Dynamical Mean Field Approximation
• Coherent Potential Approximation
• Strongly correlated material
• Anderson Localization
• Mott Localization