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Thank you : Teachers, Collaborators, Students and Postdocs

Apologies for a few omissions!. Thank you : Teachers, Collaborators, Students and Postdocs. Thank you:. Colleagues. Tata Institute of Fundamental Research. Bell Laboratories. IISc, UC SC.

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Thank you : Teachers, Collaborators, Students and Postdocs

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  1. Apologies for a few omissions! Thank you : Teachers, Collaborators, Students and Postdocs

  2. Thank you: Colleagues Tata Institute of Fundamental Research Bell Laboratories IISc, UC SC

  3. An Exactly solvable model for Strontium Copper Borate: Mott Hubbard Physics on an Archimedean Lattices APS March Meeting 18th March 2009 Session T-8 Sriram Shastry, UCSC, Santa Cruz, CA Lars Onsager Prize Talk Exact analytical theory: Experimentally relevant theory: Why choose- have them both!!

  4. 1981: Quantum Heisenberg Antiferromagnetic Model for spin half particles in 2-dimensions on a frustrated lattice. Exact Solution by Shastry and Sutherland. • 1999: Experimental realization of the model in SrCu2(BO3)2. Topologically equivalent lattice where condition for solvability is easily satisfied. • 2000-2007: Magnetization plateaux found, many new experiments at high fields and their theory. • 2007-2009: Several new materials found with same lattice structure, with Ising and XY symmetries. New experiments with interesting magnetic structures. • 2000- 2009: Theoretical Proposal for doping these systems to test Mott Hubbard Anderson ideas of correlated superconductivity. Nature follows Models: (Life follows fiction) Hence “A Natural Model”!! Many physicists and chemists are involved in the new systems, dozens of papers. A sample here.

  5. 1981 Original Motivation Bill Sutherland and SS • Understanding Chanchal Majumdar’s (with D Ghosh) 1968 model in 1-d more closely. Majumdar broke out of the Bethe Ansatz (1932) mould of nearest nbr models and put in a second nbr interaction. He found a surprising instance of an exactly solvable model !! • Understanding Anderson’s RVB (1973) ideas more closely. Anderson had targeted the triangular lattice, since it contains frustration. His work (with P Fazekas) was highly stimulating, without being fully understandable!! Majumdar found twofold degenerate groundstate

  6. Key Question:: How does one prove that a given state is the ground state? Need 1) an eigenstate 2) an argument for GS Bosons: (Well known) A Nodeless eigenstate is THE ground state Fermions: 1-d Lieb Mattis (node counting) but higher dimensions???? Generically: Rayleigh Ritz (RR) + Divide and Conquer Given a H, and a wave fn RR give us an upper bound to the GSE . This is not good enough since we need a lower bound on the energy. Let us divide the Hamiltonian into two pieces as reuse RR GS Found!! UB and LB coincide!! If miraculously

  7. Triangles Rule 2 4 6 5 1 3 Also Anderson RVB intended for triangular lattice Therefore search for a triangle decomposable lattice: Need something in between Triangular lattice and Honeycomb

  8. Exactly Solvable spin ½ for J1> 2 J2 • Remarkable that one can find an exact solution, or even an exact eigenstate of such a difficult problem!! • Condition for solvability seemed too unnatural to achieve!! J2 J1

  9. Jumping ahead to 1999, we need not have been so pessimistic. Nature finds a way. • The lattice turns out to be related to a little known classic going back to Archimedes- • Topological equivalence to another lattice makes the solvability condition natural. Change the angle q to obtain version (c) from (a). Also note that when the angle is 2p/3, we have an equilateral triangle and squares: Archimedes enters

  10. Archimedes was born c. 287 BC in the seaport city of Syracuse (Sicily) Archimedes is considered to be one of the greatest mathematicians of all time. He thought about tiling the 2-D plane with symmetric polygons:

  11. We learn about Archimedes from Plutarch • Story of the splashing water in the bath tub and eureka!! • Plutarch “Oftimes Archimedes' servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, …..so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.” We begin to draw a conclusion: Maybe he didn’t like taking a bath …!!

  12. Archimedes had found 11 special lattices in 2-d in 250 BC Grunbaum and Shepard Tilings and patterns Suding Ziff, PRE 60, 275(1999)

  13. 1995 Kageyama et al Discovery of spin gapped nature of Sr Cu_2 (BO_3)_2

  14. Meanwhile, transition from Neel order to QSL state has more complexity. Many works…one sample SrCu2(BO3)2 is very close to a QCP Plaquette ordered states occur in between. VOLUME 84, PHYSICAL REVIEW LETTERS 8 MAY 2000 Quantum Phase Transitions in the Shastry-Sutherland Model for SrCu2BO32 Akihisa Koga and Norio Kawakami

  15. What about excited states? VOLUME 84 PHYSICAL REVIEW LETTERS 19 JUNE 2000 Direct Evidence for the Localized Single-Triplet Excitations and the Dispersive Multitriplet Excitations in SrCu2BO32 H. Kageyama,1,* M. Nishi,2 N. Aso,2 K. Onizuka,1 T. Yosihama,2 K. Nukui,2 K. Kodama,3 K. Kakurai,2 and Y. Ueda1 Triplons on bonds do not propagate well, only pairs do. Massive interacting boson representation is feasible, they Wigner crystallize hence give a variety of insulating states Plateaus: numerical and theoretical + experiments. K Ueda and S Miyahara

  16. Magnetization plateaux argued for at m= 1/N for all N, and also 2/9

  17. A few competing theories. • Inspite of knowing the GS, it is a hard problem • Analogy to Quantum Hall Effect by Fermionizing the Bosons, followed by MFT of Fermi system. • Non MFT numerical techniques using numerical RG. • Important Work of J Dorier, K Schmidt and F Mila PRL 2008 and A Abendschien and S Capponi. PRL 2008 agrees with Fermionization results but some fractions are not found. • Some fractions are more strong than others….work in progress.

  18. Chern Simon Transmutation in 2-space dimensions. Similar to Jordan Wigner in 1-d f’s fermions b’s Bosons E. Fradkin Hardcore bosons Mean Field Appx Spinless Electrons in a fictitious (orbital) magnetic field

  19. Challenging question: why does the MFT with fermions, the CS theory work so well!! “Wigner crystallization” picture of the plateaux phases.

  20. Other realizations of SSL • Rare earth tetraborides (2006-2008) and • R2T2M • List includes very good metals + local moments Kim Bennett Aronson, BNL

  21. Ising limit with weak long ranged RKKY interactions. (Metallic system).Very convenient energy scale for magnetic field experiments.

  22. Futuristic: INSERTING CHARGE i.e. DOPING the MOTT INSULATOR A few theories including mine Valiant experimental effort by David C. Johnston and coworkers at Ames Laboratory ::::Why is this an important problem::::?

  23. SrCu2(BO3)2 is a Mott Insulator: Experimentally it is an insulator with a large gap ~ 1eV, whereas it should have been a semimetal with quadratic touching of bands (due to non symmorphic space group symmetry). (4 e’s/unitcell) Breaks no spatial symmetry in order to avoid the (semi)metallic state hence a Mott Insulator!! TBA: Shastry Kumar 2000 LDA Liu, Trivedi Lee, Harmon, Schmalian 2007 Analogy to bilayer graphene, but strongly correlated

  24. Towards Superconductivity via Mott Phases Spin liquid breaks no symmetry ( spatial) 1-d HAFM Bethe state or 1/r^2 Gutzwiller state y T AFM state is a “nuisance” Get rid using a quantum disorder parameter “y” to get a spin liquid. SC AFM x Study t_J model on this lattice allowing for possibility of Superconductivity. SC is obtained by “unleashing” preexisting singlets in the insulating state- Anderson 1986

  25. Shastry Kumar 2001 MFT Exact at x=0 !! Mean field theory gives s+id superconductivity for either hole or electrondoping. Tc~10K

  26. VMC calculations with projected BCS wave functions. Large scale numerics involved with many variational parameters and also allowed inhomogenous solutions D-wave solution with inhomogeneous charges wins!!

  27. What are we learning from these studies? • Standard approximations are as yet non-standardized!! Different answers emerge for e.g. symmetry of Order, depending on methods used. • Mott Hubbard models are at the frontier of research since 1987!! Little theoretical progress. • A model such the present, becomes a testing and a proving ground for techniques and ideas in correlated electron physics. Charge and spin gap at half filling makes many perturbations “irrelevant” as in QHE. Scope for very clean theory. • Known ground state is reproduced by the approximations, so that at half filling they are already doing well. This is in contrast to say High Tc. • Exactly solvable models are very useful, especially if they are realizable in nature.

  28. Another Onsager Story: A silent slide (no words needed)! Onsager loved to play with multiple integrals, as we can figure out, while reading one of his lesser famous papers!! NOTE the independent variables in the integrals!!! E=4/3 (I1+I2 )

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