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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 materials|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 [[Electronic band structure|band theory]], as seen in [[density functional theory]], such as the independent particle approximation and material homogeneity.<ref>{{cite journal | last1=Ekuma | first1=C.E. | last2=Terletska | first2=H. | last3=Tam | first3=K.-M. | last4=Meng | first4=Z.-Y. | last5=Moreno | first5=J. | last6=Jarrell | first6=M. | title=Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions | journal=Physical Review B | volume=89 | issue=8 | pages=081107(R) | year=2014 | doi=10.1103/PhysRevB.89.081107|url= https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081107| arxiv=1402.4190 | bibcode=2014PhRvB..89h1107E }}</ref> <ref>{{cite journal | last1=Ekuma | first1=C.E. | last2=Dobrosavljević | first2=V. | last3=Gunlycke | first3=D. | title=First-Principles-Based Method for Electron Localization: Application to Monolayer Hexagonal Boron Nitride | journal=Physical Review Letters | volume=118 | pages=106404 | year=2017 | issue=10 | doi=10.1103/PhysRevLett.118.106404|url= https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.106404| pmid=28339229 | arxiv=1701.03842 | bibcode=2017PhRvL.118j6404E }}</ref><ref>{{cite journal |last1=Ekuma |first1=C. E. |last2=Moore |first2=C. |last3=Terletska |first3=H. |last4=Tam |first4=K.-M. |last5=Moreno |first5=J. |last6=Jarrell |first6=M. |last7=Vidhyadhiraja |first7=N. S. |title=Finite-cluster typical medium theory for disordered electronic systems |journal=Phys. Rev. B |volume=92 |issue= |pages=014209 |year=2015 |doi=10.1103/PhysRevB.92.014209|url= https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.014209 }}</ref><ref name=":0">{{cite thesis | last=Ekuma | first=Chinedu | title=Towards the Realization of Systematic, Self-Consistent Typical Medium Theory for Interacting Disordered Systems | year=2015 | publisher=Louisiana State University and Agricultural and Mechanical College | degree=PhD | department=Physics and Astronomy | url=https://repository.lsu.edu/gradschool_dissertations/2391/| accessdate=2024-06-04}}</ref>
The TMDCA is a variant of the [[dynamical mean field theory|dynamical mean field approximation]] (DMFA),<ref>{{cite journal |last1=Georges |first1=Antoine |last2=Kotliar |first2=Gabriel |last3=Krauth |first3=Werner |last4=Rozenberg |first4=Marcelo J. |title=Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and the Limit of Infinite Dimensions |journal=Rev. Mod. Phys. |volume=68 |issue=1 |pages=13–125 |year=1996|publisher=American Physical Society |doi=10.1103/RevModPhys.68.13 |url=https://link.aps.org/doi/10.1103/RevModPhys.68.13}}</ref><ref name="Vollhardt">{{cite journal | author = D. Vollhardt | title = Dynamical mean-field theory for correlated electrons | journal = [[Annalen der Physik]] | volume = 524 | issue = 1 | pages = 1–19 | year = 2012 | doi = 10.1002/andp.201100250 | bibcode = 2012AnP...524....1V | doi-access = free }}</ref> built on the dynamical cluster approximation (DCA).<ref>{{cite journal |last1=Maier |first1=Thomas |last2=Jarrell |first2=Mark |last3=Pruschke |first3=Thomas |last4=Hettler |first4=Matthias H. |title=Quantum cluster theories |journal=Rev. Mod. Phys. |volume=77 |pages=1027 |year=2005 |doi=10.1103/RevModPhys.77.1027 |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.77.1027 }}</ref> It is designed to more accurately handle the combined impacts of disorder and electron-electron interactions in strongly correlated systems. <ref>{{cite journal | last1=Ekuma | first1=C. E. | last2=Yang | first2=S.-X. | last3=Terletska | first3=H. | last4=Tam | first4=K.-M. | last5=Vidhyadhiraja | first5=N. S. | last6=Moreno | first6=J. | last7=Jarrell | first7=M. | year=2015 | title=Metal-insulator transition in a weakly interacting disordered electron system | journal=Phys. Rev. B | volume=92 | pages=201114(R) | doi=10.1103/PhysRevB.92.201114 | url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.201114}}</ref> 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 <math>N_c</math> primitive cells, resulting in the first [[Brillouin zone]] of the original lattice being divided into <math>N_c</math> non-overlapping cells. Each cell, centered at the wave vector <math>\mathbf{K}</math>, contains a set of wave vectors <math>\tilde{\mathbf{k}} \equiv \mathbf{k} - \mathbf{K}</math>, where <math>\tilde{\mathbf{k}}</math> and <math>\mathbf{k}</math> 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 <math>N_c</math>, it is possible to systematically incorporate longer-range spatial fluctuations. <ref>{{cite journal | last1=Ekuma | first1=C.E. | last2=Terletska | first2=H. | last3=Tam | first3=K.-M. | last4=Meng | first4=Z.-Y. | last5=Moreno | first5=J. | last6=Jarrell | first6=M. | title=Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions | journal=Physical Review B | volume=89 | issue=8 | pages=081107(R) | year=2014 | doi=10.1103/PhysRevB.89.081107|url= https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081107| arxiv=1402.4190 | bibcode=2014PhRvB..89h1107E }}</ref> <ref>{{cite journal | last1=Ekuma | first1=C.E. | last2=Dobrosavljević | first2=V. | last3=Gunlycke | first3=D. | title=First-Principles-Based Method for Electron Localization: Application to Monolayer Hexagonal Boron Nitride | journal=Physical Review Letters | volume=118 | pages=106404 | year=2017 | issue=10 | doi=10.1103/PhysRevLett.118.106404|url= https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.106404| pmid=28339229 | arxiv=1701.03842 | bibcode=2017PhRvL.118j6404E }}</ref><ref
'''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]]. <ref>{{cite journal | last1=Ekuma | first1=C.E. | last2=Terletska | first2=H. | last3=Tam | first3=K.-M. | last4=Meng | first4=Z.-Y. | last5=Moreno | first5=J. | last6=Jarrell | first6=M. | title=Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions | journal=Physical Review B | volume=89 | issue=8 | pages=081107(R) | year=2014 | doi=10.1103/PhysRevB.89.081107|url= https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081107| arxiv=1402.4190 | bibcode=2014PhRvB..89h1107E }}</ref> 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. <ref>{{cite journal | last1=Ekuma | first1=C. E. | last2=Yang | first2=S.-X. | last3=Terletska | first3=H. | last4=Tam | first4=K.-M. | last5=Vidhyadhiraja | first5=N. S. | last6=Moreno | first6=J. | last7=Jarrell | first7=M. | year=2015 | title=Metal-insulator transition in a weakly interacting disordered electron system | journal=Phys. Rev. B | volume=92 | pages=201114(R) | doi=10.1103/PhysRevB.92.201114 | url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.201114}}</ref> 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 <math>MoS_2</math> observed experimentally and defect-tolerant behavior in 2D monolayers PbSe and PbTe where impurity states forming shallow levels rather than localized deep levels. <ref>{{cite journal | last1=Ekuma | first1=C. E. | last2=Gunlycke | first2=D. | year=2018 | title=Optical absorption in disordered monolayer molybdenum disulfide | journal=Phys. Rev. B | volume=97 | pages=201414(R) | doi=10.1103/PhysRevB.97.201414 | url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.201414}}</ref><ref name="Ekuma2019">{{cite journal |last1=Ekuma |first1=Chinedu E. |title=Fingerprints of native defects in monolayer PbTe |journal=Nanoscale Adv. |year=2019 |volume=1 |issue=2 |pages=513-521 |publisher=RSC |doi=10.1039/C8NA00125A|url= https://pubs.rsc.org/en/content/articlelanding/2019/na/c8na00125a }}</ref><ref name="Ekuma2018">{{cite journal |last1=Ekuma |first1=Chinedu E. |title=Effects of vacancy defects on the electronic and optical properties of monolayer PbSe |journal=The Journal of Physical Chemistry Letters |year=2018 |volume=9 |number=13 |pages=3680-3685 |publisher=American Chemical Society |doi=10.1021/acs.jpclett.8b01585|url= https://pubs.acs.org/doi/10.1021/acs.jpclett.8b01585 }}</ref> 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 <math>\Gamma(\mathbf{K}, \omega)</math> or non-local self-energy <math>\Sigma(\mathbf{K}, \omega)</math>. By mapping a <math>d</math>-dimensional lattice to a finite small cluster containing <math>N_c = L^d_c</math> sites, where <math>L_c</math> is the linear dimension of the cluster, we dramatically reduce the computation effort.<ref>{{cite journal | last1=Ekuma | first1=C.E. | last2=Terletska | first2=H. | last3=Tam | first3=K.-M. | last4=Meng | first4=Z.-Y. | last5=Moreno | first5=J. | last6=Jarrell | first6=M. | title=Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions | journal=Physical Review B | volume=89 | issue=8 | pages=081107(R) | year=2014 | doi=10.1103/PhysRevB.89.081107|url= https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081107| arxiv=1402.4190 | bibcode=2014PhRvB..89h1107E }}</ref><ref>{{cite journal | last1=Jarrell | first1=M. | last2=Maier | first2=Th. | last3=Huscroft | first3=C. | last4=Moukouri | first4=S. | year=2001 | title=Quantum Monte Carlo algorithm for nonlocal corrections to the dynamical mean-field approximation | journal=Phys. Rev. B | volume=64 | pages=195130 | doi=10.1103/PhysRevB.64.195130 | url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.64.195130 }}</ref><ref
===Self-consistency equations and TMDCA loop===
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<math>H = H_0 + \sum_{i\alpha\sigma} V_{i\sigma}^\alpha n_{i\sigma}^\alpha + U\sum_{i\alpha} n^\alpha_{i \uparrow} n^\alpha_{i \downarrow}</math>
Here, <math>H_0</math> represents the single-particle Hamiltonian, primarily comprising the hopping terms <math>t_{ij}</math>. The term <math>V_{i\sigma}^\alpha</math> denotes a disorder potential, and <math>n_{i\sigma}^\alpha</math> is the number operator, with indices <math>i</math>, <math>\alpha</math>, and <math>\sigma</math> 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 <math>\bar{G}_c(\mathbf{K}, \omega)</math> coincides with the local lattice Green's function <math>G_c^t(\mathbf{K}, \omega)</math>.
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 (mathematics)|critical point]].<ref
# Start with an initial guess for the hybridization function <math>\Gamma(\mathbf{K}, \omega)</math>, which describes the coupling between the cluster and the effective typical medium.
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