Transcript APS_Onsager_2009_mac..

Apologies
for a few
omissions!
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Teachers, Collaborators, Students and Postdocs
Abrahams E
Kennedy T
Agrawal GS
Krishnamurthy HR
Altshuler B
Kumar B
Anderson PW
Lieb EH
Barma M
Majumdar CK
Mattis D
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
Millis A
Mucciolo E
Mukherjee S
Ramirez A
Rigol M
Sen D
Shenoy SR
Shraiman B
Siddharthan R
Simons B
Singh RRP
Singh V
Narayan O
Sutherland B
Taniguchi, N
Peterson M
Walstedt R
Rajagopal AK
Young AP
Ramakrishnan TV
Yuzbashyan E
Z Zhou
Thank you:
Tata Institute of
Fundamental Research
Bell Laboratories
IISc, UC SC
Colleagues
An Exactly solvable model for Strontium Copper Borate:
Mott Hubbard Physics on an Archimedean Lattices
APS March
Meeting
18th March 2009
Sriram Shastry,
UCSC, Santa Cruz,
CA
Lars Onsager Prize Talk
Session T-8
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.
Nature follows Models:
(Life follows fiction)
Many physicists and chemists are involved in
the new systems, dozens of papers. A
sample here.
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
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
If miraculously
GS Found!!
UB and LB
coincide!!
Triangles Rule
2
1
4
3
6
5
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
J1
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!!
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
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 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 …!!
Archimedes had found 11 special lattices in 2-d in 250 BC
Grunbaum and Shepard
Tilings and patterns
Suding Ziff, PRE 60, 275(1999)
1995 Kageyama et al
Discovery of spin gapped
nature of Sr Cu_2 (BO_3)_2
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
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
Magnetization plateaux argued for at m=
1/N for all N, and also 2/9
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
Challenging question:
why does the MFT
with fermions, the CS
theory work so well!!
“Wigner crystallization” picture of the plateaux phases.
Other realizations of SSL



Rare earth tetraborides (2006-2008) and
R2T2M
List 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:
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)
TBA: Shastry Kumar 2000
LDA Liu, Trivedi Lee, Harmon,
Schmalian 2007
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
Spin liquid breaks no symmetry ( spatial)
1-d HAFM Bethe state or 1/r^2 Gutzwiller state
y
T
SC
AFM
AFM state is a “nuisance”
Get rid using a quantum
disorder parameter “y” to
get a spin liquid.
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
Shastry Kumar 2001
MFT Exact at x=0 !!
Mean field theory gives s+id superconductivity
for either hole or electrondoping. Tc~10K
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.
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 )