Typical medium dynamical cluster approximation: Difference between revisions

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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 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>{{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> xxxxxxxxxxx

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>{{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> 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.