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Resonating valence bond theory
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<h2 id="description">Description</h2> <p>The physics of Mott insulators is described by the repulsive <a href="/facts/Hubbard_model/5pphLNde">Hubbard model</a> <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian</a>: </p> H = − t ∑ ⟨ i j ⟩ ( c i σ † c j σ + h.c. ) + U ∑ i n i ↑ n i ↓ {\displaystyle H=-t\sum _{\langle ij\rangle }(c_{i\sigma }^{\dagger }c_{j\sigma }+{\text{h.c.}})+U\sum _{i}n_{i\uparrow }n_{i\downarrow }} <p>In 1971, Anderson first suggested that this Hamiltonian can have a non-degenerate ground state that is composed of disordered spin states. Shortly after the high-temperature superconductors were discovered, Anderson and Kivelson et al. proposed a <i>resonating valence bond</i> ground state for these materials, written as </p> | RVB ⟩ = ∑ C | C ⟩ {\displaystyle |{\text{RVB}}\rangle =\sum _{C}|C\rangle } <p>where C {\displaystyle C} represented a covering of a lattice by nearest neighbor dimers. Each such covering is weighted equally. In a <a href="/facts/Mean_field_approximation/tJrwSURE">mean field approximation</a>, the RVB state can be written in terms of a Gutzwiller projection, and displays a superconducting phase transition per the <a href="/facts/Berezinskii%E2%80%93Kosterlitz%E2%80%93Thouless_transition/Fzav8LFh">Kosterlitz–Thouless</a> mechanism.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> However, a rigorous proof for the existence of a superconducting ground state in either the Hubbard or the t-J Hamiltonian is not yet known.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Further the stability of the RVB ground state has not yet been confirmed.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mann, Adam (2011). "High-temperature superconductivity at 25: Still in suspense". Nature. 475 (7356): 280–282. Bibcode:2011Natur.475..280M. doi:10.1038/475280a. PMID 21776057. <a href="https://doi.org/10.1038%2F475280a" target="_blank">https://doi.org/10.1038%2F475280a</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Cho, Adrian (30 March 2020). "Philip Anderson, legendary theorist whose ideas shaped modern physics, dies". Science. AAAS. doi:10.1126/science.abb9809. Retrieved 25 May 2020. <a href="https://www.science.org/content/article/philip-anderson-legendary-theorist-whose-ideas-shaped-modern-physics-dies" target="_blank">https://www.science.org/content/article/philip-anderson-legendary-theorist-whose-ideas-shaped-modern-physics-dies</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Zaanen, Jan (2010). "A modern, but way too short history of the theory of superconductivity at a high temperature". arXiv:1012.5461 [cond-mat.supr-con]. <a href="/wiki/Jan_Zaanen" target="_blank">/wiki/Jan_Zaanen</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Weber, Cédric (2007). Variational Study of Strongly Correlated Electron Models (PDF). École Polytechnique Fédérale de Lausanne. Archived from the original (PDF) on 2018-10-01. Retrieved 2012-04-08. <a href="https://web.archive.org/web/20181001234920/http://www.tcm.phy.cam.ac.uk/~cw489/Cedric_homepage/Welcome_files/these.pdf" target="_blank">https://web.archive.org/web/20181001234920/http://www.tcm.phy.cam.ac.uk/~cw489/Cedric_homepage/Welcome_files/these.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Piazza, B. Dalla (2015). "Fractional excitations in the square-lattice quantum antiferromagnet". Nature Physics. 11 (1): 62–68. arXiv:1501.01767. Bibcode:2015NatPh..11...62D. doi:10.1038/nphys3172. PMC 4340518. PMID 25729400. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4340518" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4340518</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"How electrons split: New evidence of exotic behaviors". Nanowerk. École Polytechnique Fédérale de Lausanne. Dec 23, 2014. Retrieved Dec 23, 2014. <a href="http://www.nanowerk.com/nanotechnology-news/newsid=38557.php" target="_blank">http://www.nanowerk.com/nanotechnology-news/newsid=38557.php</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Baskaran, Ganapathy (2009). "Five-fold way to new high Tc superconductors" (PDF). Pramana. 73 (1): 61–112. Bibcode:2009Prama..73...61B. doi:10.1007/s12043-009-0094-8. S2CID 73670216. Retrieved 8 April 2012. <a href="http://www.ias.ac.in/pubs/splpubs/pjubileebook/279.pdf" target="_blank">http://www.ias.ac.in/pubs/splpubs/pjubileebook/279.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Baskaran, Ganapathy (2009). "Five-fold way to new high Tc superconductors" (PDF). Pramana. 73 (1): 61–112. Bibcode:2009Prama..73...61B. doi:10.1007/s12043-009-0094-8. S2CID 73670216. Retrieved 8 April 2012. <a href="http://www.ias.ac.in/pubs/splpubs/pjubileebook/279.pdf" target="_blank">http://www.ias.ac.in/pubs/splpubs/pjubileebook/279.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Dombre, Thierry; Gabriel Kotliar (1989). "Instability of the long-range resonating valence bond state in the mean-field approach" (PDF). Physical Review B. 39 (1): 855–857. Bibcode:1989PhRvB..39..855D. doi:10.1103/PhysRevB.39.855. PMID 9947250. Retrieved 8 April 2012. <a href="http://www.physics.rutgers.edu/~kotliar/papers/PRB_39_855.pdf" target="_blank">http://www.physics.rutgers.edu/~kotliar/papers/PRB_39_855.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>