Kagome Spin LiquidAssa AuerbachRanny Budnik

Erez Berg

Classical Heisenberg AFM

Macroscopic degeneracy

Kagome

O(3)xO(2)/O(2) -> O(4) critical pt

Three sublattice N’eel stateHuse, Singh

Triangular

ac b

ab

c

a b

b

Experiments

Strong quantum spin fluctuations (spin gap?)

S=3/2 layered Kagome

‘90

‘90

However: Large low T specific heat

2TC

S=1/2 Kagome: Numerical Results

1 .Short range spin correlations : Zheng & Elser ’90; Chalker & Eastmond ‘92

Spin gap

0.06J

2 .Finite spin gap

E(Smin+1)-E(Smin)=

Lots of Low Energy Singlets

Mambrini & Mila

Finite T=0 entropy?

energy

Log

(# s

tate

s)

Number of sites

Misguich&Lhuillier

Log

(# s

tate

s)

2k1k

Massless nonmagnetic modes?

S=0 S=1

E

RVB on the Kagome Mambrini & Mila, EPJB 2000

Weak bonds

strong bonds

6-site singlet “dimer”

Perturbation theory in weak/strong bonds .

1 .Number of dimer coverings is N

15361.2 .Dimers (10-5 of all singlets N=36) exhaust low energy spectrum.

Contractor Renormalization (CORE)C.Morningstar, M.Weinstein, PRD 54, 4131 (1996).

E. Altman and A. A, PRB 65, 104508, (2002).

Details: Ehud Altman's Ph.D. Thesis.

Truncate small longer range interactions

2. Interactions range N

subclus

iiii

renNN

NN

NhHh

'

',...

,...,,),...(1

111

From exact diagonalization of clusters

2. Effective Hamiltonian (exact)

..

...ijk

ijkijk

ijki ij

ijieff hhhhH

Kagome CORE step 1 Triangles on a triangular superlattice

s l

States of

2/

z zx ˆˆ 23

21

S

zx ˆˆ 23

21 ze

2

el ˆ

e

Dominant range 2 interactions

2 triangles

cb

21212112 SSJhclhblSSJH se

)ˆ)(ˆ(

HeisenbergDimerization field

TEST Supertriangle has 4-fold degeneracyFor Heisenberg, and CORE range 2

supertriangle

Range 3 corrections

Effective Bond Interactions

Large Dimerization fields. Contributions will cancel

for uniform <SS!<

cb

0.953

0.2111

0.053

0.1079

0.2805

0.0598

0.038

21 SS

2121

yy llSS

clblSS ˆˆ 2121

2121 llSS

clblSS ˆˆ 1221

clbl ˆˆ 21

21yy ll

)( corrh12

01 zcbl ˆˆˆ

Variational theory

Columnar dimers win!

Barrier between ground states is 0.66/site

Spin OrderE = -0.134/site

2021 .SS

Columnar Dimers. E=-0.2035/site

1243

21 SS

Energies of dimer configurationsDefect in Columnar state: 02720.E

Flipping dimers using yy

yy llJ 21

0.038

Quantum Dimer Model (Rokhsar, Kivelson)

H = -t +V

0.038 -0.0272

Quantum Dimer Model

Quantum Dimer Model (Rokhsar, Kivelson)

H = -t +V

0.038 -0.0272

Moessner& Sondhi:For t/V=1: an exponentially disordereddimer liquid phase!

Here t/V<0.

Long Wavelength GL Theory

/expexp

)cos(

3636

6

020

0222

21

mmm

mxddS low

2+1 dimensional N=6 Clock model ,

Exponentially suppressed mas gap.Extremely close to the 2+1 D O(2) modelCv ~ T2

The triangular Heisenberg Antiferromagnet

2121212112 SSllJSSJhclhclSSJH yyyyse

)ˆ)(ˆ(

Comparison to the Kagome:1. Je, and h are smaller.2. Jyy is negative!3. Variationally: Triangular Heisenberg also prefers Columnar Dimers.

Kagome Triangular

Iterated Core Transformations

Second Renormalization

0.081

0.005

0.039-

0.112

0.1

-0.018

0.004

21 SS

2121

yy llSS

clblSS ˆˆ 2121

2121 llSS

clblSS ˆˆ 1221

clbl ˆˆ 21

21yy ll

)( corrh12

0.039-

0.005

0.037-

0.038

0.05

-0.03

-0.05

21 SS

2121

yy llSS

clblSS ˆˆ 2121

2121 llSS

clblSS ˆˆ 1221

clbl ˆˆ 21

21yy ll

Kagome triangular)( corrh12

Dominant “ferromagnetic” interaction. Leads to <ly> > 0 in the ground state

Pseudospins align ferromagnetically in xz plane

Proposed RG flow

3 sublattice Neelspinwaves

O(2)-spin liquidMassless singlets

triangular Kagome

hJ yy

0yl 0bl ˆ

0

50.Spin gap, 6 sites

090.20.18 sites

54 sites

Conclusions• Using CORE, we derived effective low energy models for the

Kagome and Triangular AFM. • The Kagome model, describes local singlet formation, and a spin

gap.• We derive the Quantum Dimer Model parameters and find the

Kagome to reside in the columnar dimer phase. • Low excitations are described by a Quantum O(2) field theory, with

a 6-fold Clock model mass term. This leads to an exponentially small mass gap in the spinwaves.

• The triangular lattice flows to chiral symmetry breaking, probably the 3 sublattice Neel phase.

• Future: Investigations of the quantum phase transition in the effective Hamiltonian by following the RG flow.

Contractor Renormalization (CORE)C.Morningstar, M.Weinstein, PRD 54, 4131 (1996).

E. Altman and A. A, PRB 65, 104508, (2002).

Details: Ehud Altman's Ph.D. Thesis.

Step I: Divide lattice to disjoint blocks. Diagonalize H on each Block.

block excitations are the ''atoms'' (composite particles)

Truncate:

M lowest states per block

Niii 2 1 ,,

Reduced Hilbert space: ( dim= MN )

CORE Step II: The Effective Hamiltonian on a particular cluster

1. Diagonalize H on the connected cluster.

NN HHEPHH ',..'|..')('|,..

10

001

1

Old perturbative RG

n

M

nnn

renN

N

H ~~1

,,1

2. Project on reduced Hilbert space

nnn ~,

nnnnnnn P

'''

~~~nZ

1

3. Orthonormalize from ground state up. (Gramm-Schmidt)

CORE Step III: The Cluster Expansion

subclus

iiii

renNN

NN

NhHh

'

',...

,...,,),...(1

111 Effective Interactions:

2. CORE Exact Identity:

..

...ijk

ijkijk

ijki ij

ijieff hhhhH

+ + + + d>1: only rectangular shapes!

E. Altman's thesis.

3. If long range interactions are sufficiently small, truncate Heff at finite range.

coherence

4. is the size ("coherence length") of the renormalized degrees of freedom. Note:

Heff is not perturbative in hi j,and not a variational approximation.All the error is in the discarded longer range interactions.

pseudospin S=1/2

Tetrahedra Psedospins

2 JS=1 S=1 S=1

S=2

S=0 S=0

E

tetrahedron =

super-tetrahedron

pseudospin S=1/2

E. Berg, E. Altman and A.A,

cond-mat/0206384, PRL (03)

10-2

Cubic

16-site singlets

2 CORE Steps to Ground State

pyrochlore

1

E/J Heisenberg antiferromagnet

Fcc

10-1

CORE step 1

Anisotropic spin half model: frustrated

CORE step 2

Ising like model: not frustrated

spinJHhex

/.4150

Variational comparison (S=1/2)

Hexagons Versus Supertetrahedra

spinJHST

/.4440

What do experiments say?

Ground state

Moessner, Tshernyshyov, Sondhi

Domain wall singlet excitations

The Checkerboard )(.ˆˆ. SSSSJSSJH

i

ziijj

ijiji

effCORE

25050

Palmer and Chalker (2001)

Geometrical Frustration on Pyrochlores

2D Checkerboard3D Pyrochlore

constJJHtet

tetij

ji 2SSS

Non dispersive zero energy modes.

Spinwave theory is poorly controlled

Villain (79); Moessner and Chalker (98);

free hexagons Free plaquettes

Insufficient Renormalization!

Remaining Mean-Field zero energy modes

Perturbative Expansions+spinwave theory

Harris, Berlinsky,Bruder (92), Tsunetsugu (02)

Pseudospins defined on a FCC lattice

Range 3 CORE

+0.4 J (

0.1 J

Interactions between pseudospins

10-1

Fcc

pyrochlore

1No order!

Macroscopic degeneracy!

Spin-½ Pyrochlore AntiferromagnetE/J Mean Field OrderEffective model

4 sublattice “order”:Harris, Berlinsky,Bruder (92)

Pseudospins

Macroscopic degeneracy!

10-2

Cubic

Ising-like AFM: not frustrated

),( mqS

),( qS

)( qS

)( qS

)( qS

CORE:

Correlations: Theory vs Experiment

Ansatz:

Theory:

S=3/2

S=1/2

E. Berg AA.,, to be published

Tchernyshyov et.al.

S.H. Lee et. al.

magnon gap

fixed q

1 meV