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Abrahams E
Agrawal GS
Altshuler B
Anderson PW
Barma M
Blumberg GCava RCooper SLDoucot B
Dhar AEdwards D MGiamarchi T
Haas S
Haerter J
Hansen D
Huse D
Jha SS
Ramirez A
Rigol M
Sen D
Shenoy SR
Shraiman B
Siddharthan RSimons BSingh RRPSingh V
Sutherland BTaniguchi, N
Walstedt R
Young AP
Yuzbashyan EZ Zhou
Kennedy T
Krishnamurthy HR
Kumar B
Lieb EH
Majumdar CKMattis D
Millis A
Mucciolo E
Mukherjee S
Narayan O
Peterson M
Rajagopal AK
Ramakrishnan TV
Thank you :
Teachers, Collaborators, Students and Postdocs
Apologies for a few omissions!
APS March
Meeting
18th March 2009
Session T-8
Sriram Shastry,UCSC, Santa Cruz,
CA
Lars Onsager Prize Talk
An Exactly solvable model for Strontium Copper Borate: Mott Hubbard Physics on an Archimedean Lattices
Exact analytical theory:Experimentally relevant theory:
Why choose- have them both!!
•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.
Many physicists and chemists are involved in the new systems, dozens of papers. A
sample here.
Nature follows Models:
(Life follows fiction)Hence “A Natural
Model”!!
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
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
This is not good enough since we need a lower bound on the energy.
GS Found!!
UB and LB coincide!!
Given a H, and a wave fn RR give us an upper bound to the GSE .
Let us divide the Hamiltonian into two pieces as
reuse RR
If miraculously
Triangles Rule
1 3 5
2 4 6
Also Anderson RVB intended for triangular lattice
Therefore search for a triangle decomposable lattice:
Need something in between
Triangular lattice and Honeycomb
Exactly Solvable spin ½ for J1> 2 J2
J1J2
•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!!
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 to obtain version (c) from (a).
Also note that when the angle is 2/3, we have an equilateral triangle and squares: Archimedes enters
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:
We learn about Archimedes from PlutarchWe learn about Archimedes from Plutarch
Story of the splashing water in the bath tub and eureka!!Story of the splashing water in the bath tub and eureka!! Plutarch “Oftimes Archimedes' servants got him against his will to the 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 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 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 himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.” the study of geometry.”
Maybe he didn’t like taking a bath …!!
We begin to draw a conclusion:
Archimedes had found 11 special lattices in 2-d in 250 BC
Suding Ziff, PRE 60, 275(1999)
Grunbaum and Shepard Tilings and patterns
VOLUME 84, PHYSICAL REVIEW LETTERS 8 MAY 2000Quantum Phase Transitions in the Shastry-Sutherland
Model for SrCu2BO32Akihisa Koga and Norio Kawakami
Meanwhile, transition from Neel order to QSL state has more complexity. Many works…one sample
Plaquette ordered states occur in between.
SrCu2(BO3)2 is very close to a QCP
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
What about excited states?
VOLUME 84 PHYSICAL REVIEW LETTERS 19 JUNE 2000Direct Evidence for the Localized Single-Triplet Excitations
and the Dispersive Multitriplet Excitations in SrCu2BO32H. 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
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.
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
“Wigner crystallization” picture of the plateaux phases.
Challenging question: why does the MFT
with fermions, the CS theory work so well!!
Other realizations of SSLOther realizations of SSL Rare earth tetraborides (2006-2008) andRare earth tetraborides (2006-2008) and R2T2M R2T2M List includes very good metals + local momentsList includes very good metals + local moments
Kim Bennett Aronson, BNL
Ising limit with weak long ranged RKKY interactions. (Metallic system).Very convenient energy scale for magnetic field experiments.
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::::?
SrCu2(BO3)2 is a Mott Insulator:
TBA: Shastry Kumar 2000
LDA Liu, Trivedi Lee, Harmon, Schmalian 2007
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!!
Analogy to bilayer graphene, but strongly correlated
Towards Superconductivity via Mott Phases
T
x
y
AFM state is a “nuisance” Get rid using a quantum disorder parameter “y” to
get a spin liquid.SCAFM
Spin liquid breaks no symmetry ( spatial)
1-d HAFM Bethe state or 1/r^2 Gutzwiller state
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
Mean field theory gives s+id superconductivity for either hole or electrondoping. Tc~10K
Shastry Kumar 2001
MFT Exact at x=0 !!
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!!
What are we learning from these studies?
1. Standard approximations are as yet non-standardized!! Different answers emerge for e.g. symmetry of Order, depending on methods used.
2. Mott Hubbard models are at the frontier of research since 1987!! Little theoretical progress.
3. 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.
4. 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.
5. Exactly solvable models are very useful, especially if they are realizable in nature.