Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization
Abstract
:1. Introduction
2. Model
3. The Real Space Quantum Cluster Extension of TMT
3.1. Typical Medium Theory: TMT
3.2. Real Space Cluster-TMT
4. Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
- Anderson, P.W. Absence of Diffusion in Certain Random Lattices. Phys. Rev. 1958, 109, 1492–1505. [Google Scholar] [CrossRef]
- Abrahams, E. (Ed.) 50 Years of Anderson Localization; World Scientific: Singapore, 2010. [Google Scholar]
- Vollhardt, D.; Wölfle, P. Anderson Localization in d<∼2 Dimensions: A Self-Consistent Diagrammatic Theory. Phys. Rev. Lett. 1980, 45, 842–846. [Google Scholar] [CrossRef]
- Vollhardt, D.; Wölfle, P. Diagrammatic, self-consistent treatment of the Anderson localization problem in d≤2 dimensions. Phys. Rev. B 1980, 22, 4666–4679. [Google Scholar] [CrossRef]
- Kramer, B.; MacKinnon, A. Localization: Theory and experiment. Rep. Prog. Phys. 1993, 56, 1469. [Google Scholar] [CrossRef]
- John, S. Electromagnetic Absorption in a Disordered Medium near a Photon Mobility Edge. Phys. Rev. Lett. 1984, 53, 2169–2172. [Google Scholar] [CrossRef]
- John, S. Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 1987, 58, 2486–2489. [Google Scholar] [CrossRef] [Green Version]
- Wolf, P.E.; Maret, G. Weak Localization and Coherent Backscattering of Photons in Disordered Media. Phys. Rev. Lett. 1985, 55, 2696–2699. [Google Scholar] [CrossRef] [PubMed]
- Albada, M.P.V.; Lagendijk, A. Observation of Weak Localization of Light in a Random Medium. Phys. Rev. Lett. 1985, 55, 2692–2695. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Tsang, L.; Ishimaru, A. Backscattering enhancement of random discrete scatterers. J. Opt. Soc. Am. A 1984, 1, 836–839. [Google Scholar] [CrossRef]
- Wiersma, D.S.; Bartolini, P.; Lagendijk, A.; Righini, R. Localization of light in a disordered medium. Nature 1997, 390, 671–673. [Google Scholar] [CrossRef]
- Störzer, M.; Gross, P.; Aegerter, C.M.; Maret, G. Observation of the critical regime near Anderson localization of light. Phys. Rev. Lett. 2006, 96, 063904. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Sperling, T.; Buehrer, W.; Aegerter, C.M.; Maret, G. Direct determination of the transition to localization of light in three dimensions. Nat. Photonics 2013, 7, 48–52. [Google Scholar] [CrossRef] [Green Version]
- Skipetrov, S.; Page, J.H. Red light for Anderson localization. New J. Phys. 2016, 18, 021001. [Google Scholar] [CrossRef]
- Skipetrov, S.E.; Sokolov, I.M. Absence of Anderson Localization of Light in a Random Ensemble of Point Scatterers. Phys. Rev. Lett. 2014, 112, 023905. [Google Scholar] [CrossRef] [Green Version]
- Sperling, T.; Schertel, L.; Ackermann, M.; Aubry, G.J.; Aegerter, C.M.; Maret, G. Can 3D light localization be reached in ‘white paint’? New J. Phys. 2016, 18, 013039. [Google Scholar] [CrossRef]
- Ángel, J.C.; Guzmán, J.T.; de Anda, A.D. Anderson localization of flexural waves in disordered elastic beams. Sci. Rep. 2019, 9, 3572. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Frank, R.; Lubatsch, A.; Kroha, J. Theory of strong localization effects of light in disordered loss or gain media. Phys. Rev. B 2006, 73, 245107. [Google Scholar] [CrossRef] [Green Version]
- Lubatsch, A.; Frank, R. Self-consistent quantum field theory for the characterization of complex random media by short laser pulses. Phys. Rev. Res. 2020, 2, 013324. [Google Scholar] [CrossRef] [Green Version]
- Razo-López, L.; Fernández-Marín, A.; Méndez-Bermúdez, J.; Sánchez-Dehesa, J.; Gopar, V. Delay time of waves performing Lévy walks in 1D random media. Sci. Rep. 2020, 10, 20816. [Google Scholar] [CrossRef]
- Kostadinova, E.G.; Padgett, J.L.; Liaw, C.D.; Matthews, L.S.; Hyde, T.W. Numerical study of anomalous diffusion of light in semicrystalline polymer structures. Phys. Rev. Res. 2020, 2, 043375. [Google Scholar] [CrossRef]
- Ziegler, K. Ray Modes in Random Gap Systems. Ann. Phys. 2017, 529, 1600345. [Google Scholar] [CrossRef]
- Leseur, O.; Pierrat, R.; Sáenz, J.J.; Carminati, R. Probing two-dimensional Anderson localization without statistics. Phys. Rev. A 2014, 90, 053827. [Google Scholar] [CrossRef] [Green Version]
- Chabé, J.; Rouabah, M.T.; Bellando, L.; Bienaimé, T.; Piovella, N.; Bachelard, R.; Kaiser, R. Coherent and incoherent multiple scattering. Phys. Rev. A 2014, 89, 043833. [Google Scholar] [CrossRef] [Green Version]
- Mafi, A.; Karbasi, S.; Koch, K.W.; Hawkins, T.; Ballato, J. Transverse Anderson Localization in Disordered Glass Optical Fibers: A Review. Materials 2014, 7, 5520–5527. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- White, D.H.; Haase, T.A.; Brown, D.J.; Hoogerland, M.D.; Najafabadi, M.S.; Helm, J.L.; Gies, C.; Schumayer, D.; Hutchinson, D.A.W. Observation of two-dimensional Anderson localisation of ultracold atoms. Nat. Commun. 2020, 11, 4942. [Google Scholar] [CrossRef] [PubMed]
- Abou-Chacra, R.; Thouless, D.J.; Anderson, P.W. A Selfconsistent Theory of Localization. J. Phys. C Solid State Phys. 1973, 6, 1734–1752. [Google Scholar] [CrossRef]
- Soven, P. Coherent-Potential Model of Substitutional Disordered Alloys. Phys. Rev. 1967, 156, 809–813. [Google Scholar] [CrossRef]
- Shiba, H. A Reformulation of the Coherent Potential Approximation and Its Applications. Prog. Theor. Phys. 1971, 46, 77. [Google Scholar] [CrossRef] [Green Version]
- Velický, B.; Kirkpatrick, S.; Ehrenreich, H. Single-Site Approximations in the Electronic Theory of Simple Binary Alloys. Phys. Rev. 1968, 175, 747–766. [Google Scholar] [CrossRef]
- Kirkpatrick, S.; Velický, B.; Ehrenreich, H. Paramagnetic NiCu Alloys: Electronic Density of States in the Coherent-Potential Approximation. Phys. Rev. B 1970, 1, 3250–3263. [Google Scholar] [CrossRef]
- Onodera, Y.; Toyozawa, Y. Persistence and Amalgamation Types in the Electronic Structure of Mixed Crystals. J. Phys. Soc. Jpn. 1968, 24, 341–355. [Google Scholar] [CrossRef]
- Taylor, D. Vibrational Properties of Imperfect Crystals with Large Defect Concentrations. Phys. Rev. 1967, 156, 1017–1029. [Google Scholar] [CrossRef]
- Yonezawa, F. A Systematic Approach to the Problems of Random Lattices. I: A Self-Contained First-Order Approximation Taking into Account the Exclusion Effect. Prog. Theor. Phys. 1968, 40, 734–757. [Google Scholar] [CrossRef] [Green Version]
- Weh, A.; Zhang, Y.; Östlin, A.; Terletska, H.; Bauernfeind, D.; Tam, K.M.; Evertz, H.G.; Byczuk, K.; Vollhardt, D.; Chioncel, L. Dynamical mean-field theory of the Anderson-Hubbard model with local and nonlocal disorder in tensor formulation. Phys. Rev. B 2021, 104, 045127. [Google Scholar] [CrossRef]
- Dobrosavljević, V.; Pastor, A.A.; Nikolić, B.K. Typical medium theory of Anderson localization: A local order parameter approach to strong-disorder effects. EPL Europhys. Lett. 2003, 62, 76. [Google Scholar] [CrossRef] [Green Version]
- Schubert, G.; Schleede, J.; Byczuk, K.; Fehske, H.; Vollhardt, D. Distribution of the local density of states as a criterion for Anderson localization: Numerically exact results for various lattices in two and three dimensions. Phys. Rev. B 2010, 81, 155106. [Google Scholar] [CrossRef] [Green Version]
- Byczuk, K.; Hofstetter, W.; Vollhardt, D. Mott-Hubbard Transition versus Anderson Localization in Correlated Electron Systems with Disorder. Phys. Rev. Lett. 2005, 94, 056404. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Semmler, D.; Byczuk, K.; Hofstetter, W. Mott-Hubbard and Anderson metal-insulator transitions in correlated lattice fermions with binary disorder. Phys. Rev. B 2010, 81, 115111. [Google Scholar] [CrossRef] [Green Version]
- Murphy, N.C.; Wortis, R.; Atkinson, W.A. Generalized inverse participation ratio as a possible measure of localization for interacting systems. Phys. Rev. B 2011, 83, 184206. [Google Scholar] [CrossRef] [Green Version]
- Aguiar, M.C.O.; Dobrosavljević, V.; Abrahams, E.; Kotliar, G. Critical Behavior at the Mott-Anderson Transition: A Typical-Medium Theory Perspective. Phys. Rev. Lett. 2009, 102, 156402. [Google Scholar] [CrossRef] [Green Version]
- Aguiar, M.C.O.; Dobrosavljević, V. Universal Quantum Criticality at the Mott-Anderson Transition. Phys. Rev. Lett. 2013, 110, 066401. [Google Scholar] [CrossRef] [Green Version]
- Oliveira, W.S.; Aguiar, M.C.O.; Dobrosavljević, V. Mott-Anderson transition in disordered charge-transfer model: Insights from typical medium theory. Phys. Rev. B 2014, 89, 165138. [Google Scholar] [CrossRef] [Green Version]
- Bragança, H.; Aguiar, M.C.O.; Vučičević, J.; Tanasković, D.; Dobrosavljević, V. Anderson localization effects near the Mott metal-insulator transition. Phys. Rev. B 2015, 92, 125143. [Google Scholar] [CrossRef] [Green Version]
- Dobrosavljević, V. Typical-Medium Theory of Mott–Anderson Localization. Int. J. Mod. Phys. B 2010, 24, 1680–1726. [Google Scholar] [CrossRef] [Green Version]
- Byczuk, K.; Hofsletter, W.; Yu, U.; Vollhardt, D. Correlated electrons in the presence of disoder. Eur. Phys. J. Spec. Top. 2009, 180, 135–151. [Google Scholar] [CrossRef] [Green Version]
- Byczuk, K.; Hofstetter, W.; Vollhardt, D. Anderson Localization VS. Mott-Hubbard Metal-Insulator Transition in Disordered, Interacting Lattice Fermion Systems. Int. J. Mod. Phys. B 2010, 24, 1727–1755. [Google Scholar] [CrossRef] [Green Version]
- Alvermann, A.; Schubert, G.; Weiße, A.; Bronold, F.; Fehske, H. Characterisation of Anderson localisation using distribution. Phys. B Condens. Matter 2005, 359–361, 789–791. [Google Scholar] [CrossRef] [Green Version]
- Janssen, M. Mutifractal Analysis of Broadly Distributed Observables at Criticality. Int. J. Mod. Phys. B 1994, 8, 943. [Google Scholar] [CrossRef] [Green Version]
- Janssen, M. Statistics and scaling in disordered mesoscopic electronic systems. Phys. Rep. 1998, 295, 1–91. [Google Scholar] [CrossRef] [Green Version]
- Logan, D.E.; Wolynes, P.G. Dephasing and Anderson localization in topologically disordered systems. Phys. Rev. B 1987, 36, 4135–4147. [Google Scholar] [CrossRef]
- Ekuma, C.E.; Terletska, H.; Tam, K.M.; Meng, Z.Y.; Moreno, J.; Jarrell, M. Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions. Phys. Rev. B 2014, 89, 081107. [Google Scholar] [CrossRef] [Green Version]
- Ekuma, C.E.; Moore, C.; Terletska, H.; Tam, K.M.; Moreno, J.; Jarrell, M.; Vidhyadhiraja, N.S. Finite-cluster typical medium theory for disordered electronic systems. Phys. Rev. B 2015, 92, 014209. [Google Scholar] [CrossRef] [Green Version]
- Terletska, H.; Zhang, Y.; Tam, K.M.; Berlijn, T.; Chioncel, L.; Vidhyadhiraja, N.; Jarrell, M. Systematic quantum cluster typical medium method for the study of localization in strongly disordered electronic systems. Appl. Sci. 2018, 8, 2401. [Google Scholar] [CrossRef] [Green Version]
- Jarrell, M.; Krishnamurthy, H.R. Systematic and causal corrections to the coherent potential approximation. Phys. Rev. B 2001, 63, 125102. [Google Scholar] [CrossRef] [Green Version]
- Jarrell, M.; Maier, T.; Huscroft, C.; Moukouri, S. Quantum Monte Carlo algorithm for nonlocal corrections to the dynamical mean-field approximation. Phys. Rev. B 2001, 64, 195130. [Google Scholar] [CrossRef] [Green Version]
- Sen, S.; Terletska, H.; Moreno, J.; Vidhyadhiraja, N.S.; Jarrell, M. Local theory for Mott-Anderson localization. Phys. Rev. B 2016, 94, 235104. [Google Scholar] [CrossRef] [Green Version]
- Terletska, H.; Zhang, Y.; Chioncel, L.; Vollhardt, D.; Jarrell, M. Typical-medium multiple-scattering theory for disordered systems with Anderson localization. Phys. Rev. B 2017, 95, 134204. [Google Scholar] [CrossRef] [Green Version]
- Terletska, H.; Ekuma, C.E.; Moore, C.; Tam, K.M.; Moreno, J.; Jarrell, M. Study of off-diagonal disorder using the typical medium dynamical cluster approximation. Phys. Rev. B 2014, 90, 094208. [Google Scholar] [CrossRef] [Green Version]
- Mondal, W.R.; Berlijn, T.; Jarrell, M.; Vidhyadhiraja, N.S. Phonon localization in binary alloys with diagonal and off-diagonal disorder: A cluster Green’s function approach. Phys. Rev. B 2019, 99, 134203. [Google Scholar] [CrossRef] [Green Version]
- Mondal, W.R.; Vidhyadhiraja, N.S. Effect of short-ranged spatial correlations on the Anderson localization of phonons in mass-disordered systems. Bull. Mater. Sci. 2020, 43, 314. [Google Scholar] [CrossRef]
- Zhang, Y.; Terletska, H.; Moore, C.; Ekuma, C.; Tam, K.M.; Berlijn, T.; Ku, W.; Moreno, J.; Jarrell, M. Study of multiband disordered systems using the typical medium dynamical cluster approximation. Phys. Rev. B 2015, 92, 205111. [Google Scholar] [CrossRef] [Green Version]
- Zhang, Y.; Nelson, R.; Siddiqui, E.; Tam, K.M.; Yu, U.; Berlijn, T.; Ku, W.; Vidhyadhiraja, N.S.; Moreno, J.; Jarrell, M. Generalized multiband typical medium dynamical cluster approximation: Application to (Ga,Mn)N. Phys. Rev. B 2016, 94, 224208. [Google Scholar] [CrossRef] [Green Version]
- Zhang, Y.; Nelson, R.; Tam, K.M.; Ku, W.; Yu, U.; Vidhyadhiraja, N.S.; Terletska, H.; Moreno, J.; Jarrell, M.; Berlijn, T. Origin of localization in Ti-doped Si. Phys. Rev. B 2018, 98, 174204. [Google Scholar] [CrossRef] [Green Version]
- Östlin, A.; Zhang, Y.; Terletska, H.; Beiuşeanu, F.; Popescu, V.; Byczuk, K.; Vitos, L.; Jarrell, M.; Vollhardt, D.; Chioncel, L. Ab initio typical medium theory of substitutional disorder. Phys. Rev. B 2020, 101, 014210. [Google Scholar] [CrossRef] [Green Version]
- Zhang, Y.; Terletska, H.; Tam, K.M.; Wang, Y.; Eisenbach, M.; Chioncel, L.; Jarrell, M. Locally self-consistent embedding approach for disordered electronic systems. Phys. Rev. B 2019, 100, 054205. [Google Scholar] [CrossRef] [Green Version]
- Terletska, H.; Moilanen, A.; Tam, K.M.; Zhang, Y.; Wang, Y.; Eisenbach, M.; Vidhyadhiraja, N.; Chioncel, L.; Moreno, J. Non-local corrections to the typical medium theory of Anderson localization. Ann. Phys. 2021, 168454. [Google Scholar] [CrossRef]
- Tam, K.M.; Zhang, Y.; Terletska, H.; Wang, Y.; Eisenbach, M.; Chioncel, L.; Moreno, J. Application of the locally selfconsistent embedding approach to the Anderson model with non-uniform random distributions. Ann. Phys. 2021, 168480. [Google Scholar] [CrossRef]
- Georges, A.; Kotliar, G.; Krauth, W. Superconductivity in the Two-Band Hubbard Model in Infinite Dimensions. Z. Phys. B Condens. Matter 1993, 92, 313–321. [Google Scholar] [CrossRef] [Green Version]
- Biroli, G.; Kotliar, G. Cluster methods for strongly correlated electron systems. Phys. Rev. B 2002, 65, 155112. [Google Scholar] [CrossRef] [Green Version]
- Biroli, G.; Parcollet, O.; Kotliar, G. Cluster dynamical mean-field theories: Causality and classical limit. Phys. Rev. B 2004, 69, 205108. [Google Scholar] [CrossRef] [Green Version]
- Kotliar, G.; Savrasov, S.; Palsson, G.; Biroli, G. Cellular Dynamical Mean Field Approach to Strongly Correlated Systems. Phys. Rev. Lett. 2001, 87, 186401. [Google Scholar] [CrossRef] [Green Version]
- Lichtenstein, A.I.; Katsnelson, M.I. Antiferromagnetism and d-wave superconductivity in cuprates: A cluster dynamical mean-field theory. Phys. Rev. B 2000, 62, R9283–R9286. [Google Scholar] [CrossRef] [Green Version]
- Bulka, B.; Kramer, B.; MacKinnon, A. Mobility edge in the three-dimensional Anderson model. Z. Phys. B Condens. Matter 1985, 60, 13–17. [Google Scholar] [CrossRef]
- Li, J.; Chu, R.L.; Jain, J.K.; Shen, S.Q. Topological Anderson Insulator. Phys. Rev. Lett. 2009, 102, 136806. [Google Scholar] [CrossRef]
- Yonezawa, F.; Morigaki, K. Coherent Potential Approximation. Basic concepts and applications. Prog. Theor. Phys. Supp. 1973, 53, 1–76. [Google Scholar] [CrossRef] [Green Version]
- Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 1996, 68, 13–125. [Google Scholar] [CrossRef] [Green Version]
- Maier, T.; Jarrell, M.; Pruschke, T.; Hettler, M.H. Quantum cluster theories. Rev. Mod. Phys. 2005, 77, 1027–1080. [Google Scholar] [CrossRef] [Green Version]
- Lee, P.A.; Ramakrishnan, T.V. Disordered electronic systems. Rev. Mod. Phys. 1985, 57, 287–337. [Google Scholar] [CrossRef]
- Selvan, R.; Genish, I.; Perelshtein, I.; Moreno, J.; Gedanken, A. Single step, low temperature synthesis of submicron-sized rare earth hexaborides. J. Phys. Chem. C 2008, 112, 1795. [Google Scholar] [CrossRef]
- Bulka, B.; Schreibe, M.; Kramer, B. Localization, Quantum Interference, and the Metal-Insulator Transition. Z. Phys. B 1987, 66, 21–30. [Google Scholar] [CrossRef]
- Kramer, B.; Schreiber, M. Localization, quantum interference and transport in disordered solids. In Fluctuations and Stochastic Phenomena in Condensed Matter; Lecture Notes in Physics; Garrido, L., Ed.; Springer: Berlin/Heidelberg, Germany, 1987; Volume 268, pp. 351–375. [Google Scholar] [CrossRef]
- Kramer, B.; MacKinnon, A.; Ohtsuki, T.; Slevin, K. Finite Size Scaling Analysis of the Anderson Transition. Int. J. Mod. Phys. B 2010, 24, 1841–1854. [Google Scholar] [CrossRef] [Green Version]
- Rodriguez, A.; Vasquez, L.J.; Slevin, K.; Römer, R.A. Critical Parameters from a Generalized Multifractal Analysis at the Anderson Transition. Phys. Rev. Lett. 2010, 105, 046403. [Google Scholar] [CrossRef] [Green Version]
- Rodriguez, A.; Vasquez, L.J.; Slevin, K.; Römer, R.A. Multifractal finite-size scaling and universality at the Anderson transition. Phys. Rev. B 2011, 84, 134209. [Google Scholar] [CrossRef] [Green Version]
- Slevin, K.; Ohtsuki, T. Numerical verification of universality for the Anderson transition. Phys. Rev. B 2001, 63, 045108. [Google Scholar] [CrossRef] [Green Version]
- Slevin, K.; Ohtsuki, T. Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class. New J. Phys. 2014, 16, 015012. [Google Scholar] [CrossRef]
- Slevin, K.; Ohtsuki, T. Corrections to Scaling at the Anderson Transition. Phys. Rev. Lett. 1999, 82, 382–385. [Google Scholar] [CrossRef] [Green Version]
- Chang, T.; Bauer, J.D.; Skinner, J.L. Critical exponents for Anderson localization. J. Chem. Phys. 1990, 93, 8973–8982. [Google Scholar] [CrossRef]
- MacKinnon, A.; Kramer, B. The scaling theory of electrons in disordered solids: Additional numerical results. Z. Phys. B Condens. Matter 1983, 53, 1–13. [Google Scholar] [CrossRef]
- de Queiroz, S.L.A. Reentrant behavior and universality in the Anderson transition. Phys. Rev. B 2001, 63, 214202. [Google Scholar] [CrossRef] [Green Version]
- Grussbach, H.; Schreiber, M. Determination of the mobility edge in the Anderson model of localization in three dimensions by multifractal analysis. Phys. Rev. B 1995, 51, 663–666. [Google Scholar] [CrossRef]
- Sénéchal, D. An introduction to quantum cluster methods. arXiv 2010, arXiv:0806.2690. [Google Scholar]
- Kraberger, G.J.; Triebl, R.; Zingl, M.; Aichhorn, M. Maximum entropy formalism for the analytic continuation of matrix-valued Green’s functions. Phys. Rev. B 2017, 96, 155128. [Google Scholar] [CrossRef] [Green Version]
- Rowlands, D.A. Investigation of the nonlocal coherent-potential approximation. J. Phys. Condens. Matter 2006, 18, 3179–3195. [Google Scholar] [CrossRef]
- Zhang, Y.; Zhang, Y.F.; Yang, S.X.; Tam, K.M.; Vidhyadhiraja, N.S.; Jarrell, M. Calculation of two-particle quantities in the typical medium dynamical cluster approximation. Phys. Rev. B 2017, 95, 144208. [Google Scholar] [CrossRef] [Green Version]
- Roy, B.; Slager, R.J.; Juričić, V. Global Phase Diagram of a Dirty Weyl Liquid and Emergent Superuniversality. Phys. Rev. X 2018, 8, 031076. [Google Scholar] [CrossRef] [Green Version]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Tam, K.-M.; Terletska, H.; Berlijn, T.; Chioncel, L.; Moreno, J. Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization. Crystals 2021, 11, 1282. https://doi.org/10.3390/cryst11111282
Tam K-M, Terletska H, Berlijn T, Chioncel L, Moreno J. Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization. Crystals. 2021; 11(11):1282. https://doi.org/10.3390/cryst11111282
Chicago/Turabian StyleTam, Ka-Ming, Hanna Terletska, Tom Berlijn, Liviu Chioncel, and Juana Moreno. 2021. "Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization" Crystals 11, no. 11: 1282. https://doi.org/10.3390/cryst11111282
APA StyleTam, K. -M., Terletska, H., Berlijn, T., Chioncel, L., & Moreno, J. (2021). Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization. Crystals, 11(11), 1282. https://doi.org/10.3390/cryst11111282