Thank you :

1
Thank you Teachers, Collaborators, Students
and Postdocs
Apologies for a few omissions!
Ramirez A
Rigol M
Sen D
Shenoy SR
Shraiman B
Siddharthan R Simons B Singh RRP Singh V
Sutherland B Taniguchi, N
Walstedt R
Young AP
Yuzbashyan E Z Zhou
Kennedy T
Krishnamurthy HR
Kumar B
Lieb EH
Majumdar CK Mattis D
Millis A Mucciolo E
Mukherjee S
Narayan O

Peterson M
Rajagopal AK
Ramakrishnan TV
Abrahams E
Agrawal GS
Altshuler B
Anderson PW
Barma M
Blumberg G Cava R Cooper SL Doucot B
Dhar A Edwards D M Giamarchi T
Haas S
Haerter J
Hansen D
Huse D
Jha SS
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Colleagues
Thank you
Tata Institute of Fundamental Research
Bell Laboratories
IISc, UC SC
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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!!
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• 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.
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1981 Original Motivation Bill Sutherland and SS

• Understanding Chanchal Majumdars (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 Andersons 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
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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
GS Found!! UB and LB coincide!!
reuse RR
If miraculously
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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
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Exactly Solvable spin ½ for J1gt 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
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• 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
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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
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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 didnt like taking a bath !!
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Archimedes had found 11 special lattices in 2-d
in 250 BC
Grunbaum and Shepard Tilings and patterns
Suding Ziff, PRE 60, 275(1999)
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1995 Kageyama et al Discovery of spin gapped
nature of Sr Cu_2 (BO_3)_2
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Meanwhile, transition from Neel order to QSL
state has more complexity. Many worksone 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
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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
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Magnetization plateaux argued for at m 1/N for
all N, and also 2/9
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• 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.

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Chern Simon Transmutation in 2-space dimensions.
Similar to Jordan Wigner in 1-d
fs fermions bs Bosons E. Fradkin
Hardcore bosons

Mean Field Appx

Spinless Electrons in a fictitious (orbital)
magnetic field
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Challenging question why does the MFT with
fermions, the CS theory work so well!!
Wigner crystallization picture of the plateaux
phases.
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Other realizations of SSL

• Rare earth tetraborides (2006-2008) and
• R2T2M
• List includes very good metals local moments

Kim Bennett Aronson, BNL
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Ising limit with weak long ranged RKKY
interactions. (Metallic system).Very convenient
energy scale for magnetic field experiments.
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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?
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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
es/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
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Towards Superconductivity via Mott Phases
Spin liquid breaks no symmetry ( spatial) 1-d
HAFM Bethe state or 1/r2 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
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Shastry Kumar 2001 MFT Exact at x0 !!
Mean field theory gives sid superconductivity
for either hole or electrondoping. Tc10K
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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!!
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• 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.

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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!!!
E4/3 (I1I2 )