Spin-1/2 Chains in Uniform and Staggered Fields

Collin Broholm * Johns Hopkins University and NIST Center for Neutron Research

Y. Chen M. Kenzelmann C. P. Landee K. Lefmann Y. Qiu D. H. Reich C. Rische M. B. Stone M. M. Turnbull LANL JHU & NIST Clarke University Risø National Lab NIST & Univ. Maryland JHU Univ. of Copenhagen Penn State University Clarke University

* Supported by the National Science Foundation

Spin-1/2 chain preliminaries

H

 

n



J

S

n



S

n

 1 

g



B HS z

  Simple Hamiltonian; complex properties  Good model materials and experimental tools  Integrable through Bethe Ansatz: – – Ground state energy 1

N H

 Equal time correlation function 2

J

 1  4

S S

0 

n

ln 2   Quantum critical  Exact results for dynamic spin correlations

n n

 1 SCTMC 8/5/03

Copper pyrazine dinitrate

Hammar et al. (1999)

T 2 (K 2 ) //a SCTMC 8/5/03

Cu(C 4 H 4 N 2 )(NO 3 ) 2

Magnetic Neutron Scattering

k



i

2  

Q k



f



Q



k



i



k



f

  

E i



E f

The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function

S

  (

Q

,  )  1 2   

dt e



i



t

1

N

  

R R

'

e



i Q

 ( 

R

 

R

' )  S  R (

t

) S 

R

' ( 0 )  SCTMC 8/5/03

NIST Center for Neutron Research

Neutron Scattering from Spin-1/2 chain

Stone et al., PRL (2003)

Fermions in spin ½ chain Uniform spin-1/2 chain (XY case for simplicity) H 0   H 

J

 1 2  H

Z



i



S i



S i

  1 

S i



S i

  1  

g



B H



i S i z

Jordan-Wigner transformation

S S i z i

  

a i



a

 

a

exp     1

i

 

j



i a j



a j



i i

Diagonalizes H || H 0 

k



k

2

a k



a k



g

2 

B HN

Non interacting fermionic lattice gas  

J

 cos

k



g



B H

SCTMC 8/5/03 

a i

,

a



j

  

ij

q (  )

From band-structure to bounded continuum SCTMC 8/5/03

q (



) Q (



) S

 

Q

,    2   

G S Q

  2       

Neutron Scattering

Stone et al. (2003).

Exact two-spinon cross-section

Karbach et al. 2000

Neutron Data & Two-Spinon Cross section

q

  1.1 meV 0.6 meV

q

 0.7



q

 0.5

 0.3 meV 1.0

Stone et al., PRL (2003)

Spinons in magnetized spin- ½ chain   2 

m Broholm et al. (2002)

SCTMC 8/5/03

SCTMC 8/5/03 0.0 T Uniform Spin ½ chain

Stone et al. (2003)

SCTMC 8/5/03 8.7 T Uniform Spin ½ chain  ||

Stone et al. (2003)

Diagonalization of spin-½ chain in a field +   1 4

S zz



S

  

S

  

Stone et al. (2003)

Neutron Scattering Pentium Scattering

Stone et al. (2003)

Spin-½ chain with two spins per chain unit

g

1

g

1

g

2

g

2

Landee et al. (1986) CuCl 2 .

2(dimethylsulfoxide)

H    

n n

 

J

S

J

S

n n

 

S S

n n

 1  1  

D S

n n HS n z

 

h s

S

n

 1   

n



S B n x

Hg S

n



n



Oshikawa and Affleck (1997)

The staggered field is given by

h

s

SCTMC 8/5/03

g

1 

g

2 2

H

 1 2

J

D



g

1 

g

2 2

H

3 H=0 T 2 1 0

q

 

Kenzelmann et al. (2003)

3 H=11 T 2 1 0

q

 

Kenzelmann et al. (2003)

Bound states from 2-spinon continuum

q

 

q

 0.77



Kenzelmann et al. (2003)

SCTMC 8/5/03

Why staggered field yields bound states Zero field state quasi-long range AFM order Without staggered field distant spinons don’t interact With staggered field solitons separate “good” from “bad” domains, which leads to interactions and bound states SCTMC 8/5/03

Sine-Gordon mapping of spin-1/2 chain Effective staggered + uniform field spin hamiltonian H eff  

n



J

S S

n



n

 1 

HS n z



h s n S



n x

   to incommensurate quasi-long-range order with Lagrangian density L  1 2  

   

t x

 2   

Ch s



 



This is sine-Gordon model with interaction term proportional to

h s

Spectrum consists of • Solitons, anti-solitons 2   1   2 • Breather bound states

M n M

 

J A

2

M

  sin

H



J n

 / 2 

Oshikawa and Affleck (1997)

SCTMC 8/5/03

Bound states from 2-spinon continuum Breathers n=1,2 and possibly 3

q

  Soliton, M

q

 0.77



Kenzelmann et al. (2003)

SCTMC 8/5/03

SCTMC 8/5/03 Testing sine-Gordon predictions

Theory by Essler-Tsvelik (1998) Cu-Benz Dender et al. (1997).

0

Kenzelmann et al. (2003)

Conclusions: S=½ Chain in Uniform Field  H=0: data well described by exact two-spinon continuum scattering  H>0: – – Incommensurate correlations from shifted fermi points Gapless excitation at

q=

 and

q=



-2



m

– Neutron scattering data in excellent agreement with finite chain calculations

Publications and viewgraphs at http://www.pha.jhu.edu/~broholm/homepage/

SCTMC 8/5/03

Conclusions: S=½ Chain in Staggered Field  Staggered g-tensor and DM interaction inherent to multi atom cell and produce effective staggered field  Staggered field yields bound states  Features described by sine-Gordon model: – Relative energies of bound states at

q=

 and 

-2



m

– – Relative intensities of breather excitations Field dependent incommensurability  Excellent experimental realization of quantum sine-Gordon model

Publications and viewgraphs at http: // www.pha.jhu.edu

/ ~broholm / homepage /

SCTMC 8/5/03