oleg starykh, andreas schnyder and leon balents- spatially anisotropic s=1/2 heisenberg kagome...

8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet

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Spatially anisotropic S=1/2

Heisenberg Kagomeantiferromagnet

Oleg Starykh, University of Utah

Andreas Schnyder, KITP

Leon Balents, KITP and UCSB

Thanks to J.-S. Caux for numerical data

PRB 78, 174420 (2008)


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Kagome spin-1/2 antiferromagnet:

from model to experiments

herbersmithite

volborthite

vesignieite

Hiroi et al. 2001, Shores et al. 2005, Okamoto et al. 2009

no order for T


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Outline

Motivation

- strong spatial anisotropy offers well controlled analysis

- relevant experimentally

Description of the J


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Volborthite Cu3V2O7(OH)2 2H2O


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Volborthite Cu3V2O7(OH)2 2H2O


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k,

jk

,j

k,

j"

JJJ!!!

J

J'

2i 2i+1 2i+2

2y

2y+1

2y-1

J'

Spatially anisotropic geometry


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The model

Chain spins S (black dots), exchange interactionJ (blue lines)

Interstitial (interchain) spins (red dots), exchange J

Quasi-one-dimensional limit: J


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How small should J/J be ? Frustration greatly enhances region of small interchain J

Numerics: no interchain correlations for J/J < 0.6 - 0.7Weng et al 2006, Hayashi Ogata 2007, Heidarian Sorella Becca 2009

Example: spatially anisotropic triangularAFM

Collinear AFM state,generated by quantum fluctuations,coupling between NN chains (J/J)4

Pardini Singh 2008 - no; Bishop et al 2008 - yes

(J)4/J3

Starykh, Balents 2007

interchain spin correlations,

J=0.6 J;

exponential decay

Weng et al 2006


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The decoupled limit, J 0, is very singular

collection ofindependentspin chains and interstitial spins

JS S


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Separation of scales

local Kondo coupling J (marginal, TK ~ e-c J/J )

RKKY interaction between interstitial spins (via chains)

Tinterstitial ~ (J)2/J

The biggest non-frustrated interaction energy

Tinterstitial >> Tchains >> TK

Interstitial-mediated coupling between spin chainsTchains ~ (J)4/J3

=>

Ny+1

Ny

y+1

y


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Frustrated vs Non-Frustrated geometry

Imagine non-frustrated rectangulargeometry:then Tinterstitial ~ Tchain ~ (J)2/J

=>no clear separation of energy scales,

difficult to analyze.e.g. Kondo necklace, Essler Kuzmenko Zaliznyak 2007

=>

Frustrated Kagome geometry:

Ny+1

Ny

y+1

y

(S2n + S2n+1)y My + x Ny

V(1)ch =

y

dx Nx Ny x Ny+1 + M My My+1

(J)2/J order: marginal coupling

(J)4/J order: relevant coupling

V(2)ch =

y dx N Ny Ny+1 + yy+1

M xN

=>

uniform magnetization

staggered magnetization


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RKKY interaction between interstitial spins

a polarizes spin chain, which in turn couples to b,c: interactionbetween s connected to the same spin chain.

JS

J

S

c

b

a


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Spatially anisotropic triangular lattice formed byK1 and K2 bonds, withfurther-neighbor interactions K3,4 etc.


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K1: nearest-neighbor interaction between s


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K2: next nearest-neighbor interaction


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K3: further-neighbor interactions


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Inter- interactions are determined by dynamical structure factor of critical spin-1/2 chain



K1 = 2(J)2A(1)

K2 = 4(J)2A(2)

A(r) =8

0

d

0

dqS(q, )

cos2(q

2) cos(qr)

< 0 ferro> 0 antiferro

K1

K2K2

K1 0.7

because (J/J)4


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: Spatially anisotropic triangular lattice

K1: nearest-neighbor interaction


K1 = 2(J)2A(1)

K2 = 4(J)2A(2)

< 0 ferro

> 0 antiferro

K1

K2

H =

q

K(q)q q

K(q) = 2K1 cos qx cos qy + K2 cos(2qx) + 2K3 cos(3qx)cos qy + K4 cos(4qx)

2-spinon approximation for A(r) : rotating spiral ground state, q=2(0.08,0)

ABACUS database (N=500 site chain): ferromagnetic ground state, q=0

J.S.Caux, U Amsterdam

= s0[x cos(qx) + y sin(qx)]

2d magnetic orderamongst

interstitial spins!

Note: s0 ~ O(1)

consider both

cases!


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Ordering of chain spins I

Chain spins are subject to: external rotating field due to ordered s:

marginal backscattering term in every chain

(J/J)4 fluctuation-generated interchain interactions

Expect response to hx :static magnetization in x-yplane, of magnitude O(J/J)

hx = 2s0 cos(q/2)J[x cos(qx) + y sin(qx)]

V(2)ch =

y

dx N Ny Ny+1 + yy+1

Hbs = gbs

dx MR ML

q > 0


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Ordering of chain spins IV

Order in a rotated basis: Nx

y = (1)yM , M = c(J/J)2

M

a

R/L=

y= 0

Order in the original basis: S+x+1/2,y = h2 + d2

2vcos e

iqx

Szx+1/2,y = (1)x+y

M

yx

z

a

x

z

Non-coplanar order (q > 0)

x,y

O(1) Sx,y

O(J

/J) Sz

O(J

/J)2

~ J/J

~ (J/J)2

q > 0


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(b)xz + + + +

+ + + + +

+ + + + +

+ + + +

-yx

Ordering of chain spins: top viewinterstitial spins form spiral,

chain spins are locally anti-parallel to it,

with small staggered component normal to the spiral plane

basic physics: s=1/2 chain subject to magnetic field -components orthogonal to the field are most relevant


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Ordering of chain spins: q=0 case

Ferromagnetic order among interstitial spins,

predominantly ferromagnetic ordering among chain spins,

with weaker antiferromagnetic order along, and between, chains:

coplanar state, ferrimagnet.


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Comparison

Wang, Vishwanath, Kim 2007:q=0 state

Yavorskii, Apel, Everts 2007:extended region of incommensurate order q>0

among inter-chain spins, disordered chains

1d limit: J=1, J>>1

large-S semiclassical analysis large-N Sp(N)

very similar to our q=0 state


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Conclusions

Surprise: spatially anisotropic kagomeantiferromagnet is magnetically ordered

The order is non-coplanar (q > 0)- coplanar with q=0 is possible (ferrimagnetic state)

Interesting hierarchy of scales

- interstitial spins order at Tinterstitial ~ (J)2/J

- chain spins order at Tchain ~ (J)4/J3


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Heisenberg spin chain via free Dirac fermions

Spin-1/2 AFM chain = half-filled (1 electron per site, kF=!/2a ) fermion chain

Spin-charge separation

! 2kF(= !/a) fluctuations: charge density wave" , spin density wave N

Spin flip #S=1

#S=0

Staggered

Magnetization N

Staggered

Dimerization

"= (-1)x Sx Sx+a

Susceptibility

1/q

1/q

1/q

kF-kF

kF-kF

! q=0 fluctuations: right- and left- spin currents


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Quantumtriad: uniform magnetization M = JR

+ JL

,

staggered magnetization N and staggered dimerization != (-1)x Sx Sx+1!Components of Wess-Zumino-Witten-Novikov SU(2) matrix

Hamiltonian H ~ JRJR

+ JLJL+ "

bsJRJL

Operator product expansion

Scaling dimension 1/2 (relevant)

Scaling dimension 1 (marginal)

Low-energy degrees of freedom

(similar to commutation relations)

marginal perturbation


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S=1/2 AFM Chain in a Field

1

1/2

0 h/hsat1

M

1/2

XY AF correlations grow with hand remain commensurate Ising SDW correlations decrease with h and shift from !

Affleck and Oshikawa, 1999

Field-split Fermi momenta:

! Uniform magnetization

! Half-filled condition

Sz component ("S=0) peaked at

scaling dimension

increases

Sx,y components ("S=1) remain at !scaling dimension

decreases

Derived for free electrons but correct always - Luttinger Theorem

10 h/hsat

hsat=2J

oleg starykh, andreas schnyder and leon balents- spatially anisotropic s=1/2 heisenberg kagome...

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