Competing phases of XXZ spin
chain with frustration
Akira Furusaki (RIKEN)
July 10, 2011
Symposium on Theoretical and
Mathematical Physics, The Euler
International Mathematical Institute
Collaborators:
Shunsuke Furukawa (U. Tokyo)
Toshiya Hikihara (Gunma U.)
Shigeki Onoda (RIKEN)
Masahiro Sato (Aoyama Gakuin U.)
Thanks: Sergei Lukyanov (Rutgers)
Outline
1. Introduction: frustrated spin-1/2 J1-J2 XXZ chain
2. XXZ chain (J2=0): review of bosonization approach
3. Phase diagram of J1-J2 XXZ spin chain
a. J1>0 (antiferromagnetic)
b. J2<0 (ferromagnetic)
frustrated spin-1/2 J1-J2 chain
H

x x
y y
z z
J
S
S

S
S


S
  n j j n j j n j S j n
n1,2
j
J1
J2> 0(AF)
If J2 is antiferromagnetic,
spins are frustrated regardless of the sign of J1.
J1-J2 spin chain is the simplest spin model
with frustration.

Materials: Quasi-1D cuprates (multi-ferroics)
Cu
(3dx2-y2)
Cu2+ spin S=1/2
J
O(2px) 1
LiCuVO4
LiCu2O2
J2
NaCu2O2
PbCuSO4(OH)2
Rb2Cu2Mo3O12
a
b
c
Quasi-1D spin-1/2 frustrated magnets with ferro J1
-4
-3
-2
Li2ZrCuO4
Drechsler et al.
PRL,2007
-1
LiCu2O2
Masuda et al.
PRL,2004; PRB,2005
0
LiCuVO4
Enderle et al.
Europhys.Lett.,2005
 Multiferroicity
 CuO2 chain: edge-sharing network
ferromagnetic J1
(Kanamori-Goodenough rule)
Observation of chiral ordering
through electric polarization P
J2 >0
a
c
J1 <0
b
Cu
O
Seki et al.,
PRL, 2008
(LiCu2O2)
Model
Frustrated spin-1/2 J1-J2 XXZ chain
y
easy-plane anisotropy
Frustration occurs when J2>0,
irrespective of the sign of J1.
J2 (>0, antiferro)
J1
Classical ground state
-4
0
Spin spiral
Ferro
pitch angle
finite chirality
4
Antiferro
The phase diagram
is symmetric.
J1>0 and J1<0 are
equivalent under
(applies only in
the classical case)
x
Quantum case S=1/2
 Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D.
 Antiferromagnetic case (J1>0,J2>0) is well understood.
- Singlet dimer order is stabilized (J2/J1>0.24).
Haldane, PRB1982
Nomura & Okamoto, J.Phys.A 1994
White & Affleck, PRB 1996
Eggert, PRB 1996
- Vector chiral ordered phase (quantum remnant of the spiral phase)
is found for small J1 J 2  0.8,   0.2 Nersesyan,Gogolin,& Essler, PRL 1998
Hikihara,Kaburagi,& Kawamura, PRB 2001
Previous study of spin-1/2 J1-J2 chain
Ground-state phase diagram for AF-J1 case
K. Okamoto and K. Nomura, Phys. Lett. A (1992).
T. Hikihara, M. Kaburagi, and H. Kawamura, PRB (2001), etc.
J1 chain
Two decoupled J2 chains
Majumdar-Ghosh line
Quantum case S=1/2
 Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D.
 Antiferromagnetic case (J1>0,J2>0) is well understood.
- Singlet dimer order is stabilized (J2/J1>0.24).
Haldane, PRB1982
Nomura & Okamoto, J.Phys.A 1994
White & Affleck, PRB 1996
Eggert, PRB 1996
- Vector chiral ordered phase (quantum remnant of the spiral phase)
is found for small J1 J 2  0.8,   0.2 Nersesyan,Gogolin,& Essler, PRL 1998
Hikihara,Kaburagi,& Kawamura, PRB 2001
 The ferromagnetic-J1 case (J1<0,J2>0) is less understood.
Goal: to determine the ground-state phase diagram
Our strategy
• perturbative RG analysis around J1=0 or J2=0.
XXZ spin chain: exactly solvable
low-energy effective theory (bosonization)
• numerical methods
density matrix renormalization group (DMRG)
time evolving block decimation for infinite system (iTEBD)
XXZ spin chain: brief review
mostly standard textbook material,
plus some relatively new developments
XXZ spin chain

H   S jx S jx1  S jy S jy1  S jz S jz1

• Exactly solvable: Bethe ansatz
gapless phase 1    1
ferromagnetic
Ising order
LRO
energy gap
-1

１
Tomonaga-Luttinger
liquid
gapless excitations
power-law correlations
antiferromagnetic
Ising order
LRO
energy gap
• Effective field theory: bosonization
H eff
v
1
2
2

  dx  K   x     x    cos
2
K

 ( x),  ( x) : bosonic field



16 

  x  ,  y  y    i  x  y 
relevant for   1
cos

16

is
irrelevant for   1
marginally irrelevant for   1
For
 1
1
  
   cos 
 : K  1 at   0, K  at   1
2
 2K 
v
sin  
2 1   

1
1
 1  cos 1 
2K

Luther, Peschel
In the critical phase 1    1
The cosine term is irrelevant in the low-energy limit
H
v
1
2
2

dx
K





 x 
 x  : Gaussian model


2
K


 ( x),  ( y)  i ( x  y)
 ( x),  ( x) : bosonic field

Spin operators

1
  (l )  a (1) sin 

Sl  ei
Slz 
 ( l )
 b0 (1)l  b1 sin

l
x
1
v, 
z
1
x
0

4 (l )  
4 (l ) 
y
1
2K


x
0
S S
x
r
S0z S rz
(1) r
1
x
A

A

1
r
r 1/
r
1
z ( 1)
  2 2  A1   
4  r
r
x
0
: exactly determined by Bethe ansatz
x
0
A ,A ,A ,
 a1, b0 , b1,... 
not directly obtained
from Bethe ansatz
x
0
S S
x
r
S0z S rz
(1) r
1
x
A

A

1
r
r 1/
r
1
z ( 1)
  2 2  A1   
4  r
r
x
0
Lukyanov &
Zamolodchikov (1997)

Ax 
1
8 1  
2
  dt 
    2  2   

sinh t 
2t
e 

 exp  


 0 t  sinh  t  cosh 1   t 
 4  1  2  2   

1
  dt 
sinh  2  1 t 
2     2  2   
2  1 2t   Lukyanov (1998)
Az  2 

e 
 exp   


  4  1  2  2   

 0 t  sinh t  cosh 1    t 
  more recently,

Maillet et al.
T. Hikihara & AF (1998)
Az
exact
numerics

from S. Lukyanov, arXiv:cond-mat/9809254
dimer correlation
(staggered) dimer correlation: as important as the spin correlations
scaling dimension = 1/2 at AF Heisenberg point
NN bond (energy) operators


1  
O (l )  Sl Sl 1  Sl Sl1 ,
4

e
Oez (l )  Slz Slz1
Oe (l )  c0  c1  1 cos  4 ( xl ) 



l


  x ( xl ) 
c
2
c
(   , z, xl  l  1 2 )





c0 , c , c ,...

c1


  x ( xl ) 
2

[cf. Eggert-Affleck (1992)]
known (can be obtained from energy density etc.)
unknown: we have determined numerically using DMRG
Analytic results for uniform components
Uniform part of dimer operators = energy density in uniform chain
We can evaluate the coefficients of the uniform comp.
from the exact results of the energy density
1
cos 
c 
I1 
I2
2
2 sin  
2
1
cos 
1
c0z  
I1  2 I 2
4  sin  


0
sin 2   2 1   
2
16 2 1    sin  
sin     1    cos 
z
c2 
2
4 2 1    sin  
c2 
sinht 
0
sinht  cosh1   t 

t cosht 
I 2   dt
0
sinht  cosh2 1   t 

I1   dt
cos 
8 2 1   
1
c2z  2
4  1   
c2 
Dimer operators in finite open chain
Dimer order induced at open boundaries
penetrates into bulk decaying algebraically
Open boundary condition
 (0)   ( L  1)  0
Dirichlet b.c. for boson field :
mode expansion
Oe (l ) ( xc) 0  c1 x
 1l



sin qn x f ( x)  2( L  1) sin  x 
n  n 
 2( L  1) 

2





f
(
2
l

1
)
 ( L  1)
 n 1  n
0 
1
1
2

cos

 qn x 1
 n   n2  
 ( x) c2  0  i 2 c2

12( L  1) n 1
nf (2l  1)
DMRG results

Calculate the local dimer operator Oe (l ) for a finite open chain using DMRG
fit the data to the form obtained by bosonization to determine
c1
excellent agreement between DMRG data and bosonization forms
Numerics (DMRG)
Oe (l ) 
Staggered part of the dimer operators

Oe (l )

S


 Oe (l )  c0  c2

 c1
2
12( L  1) 2
 c2


1  
Sl Sl 1  Sl Sl1 ,
4
1
 f (2l  1)2
 1l
 f (2l  1)
1
2
Oez (l )  Slz Slz1
f ( x) 
2( L  1)

 x 

sin

2
(
L

1
)



coefficient c1
Exact formulas for
c1
are not known.
Hikihara, AF & Lukyanov, unpublished
• Effective field theory
H eff
v
1
2
2

  dx  K   x     x    cos
2
K

2

     1      1    2  2   
  4sin   

 
       4  1  1  2  2   
2



16 

2
Lukyanov & Zamolodchikov
NPB (1997)
1st order perturbation in  gives the leading boundary contribution to
free energy of semi-infinite (or finite) spin chains for 1 2    1  2 3    1
 cos

16

 0 for Dirichlet b.c.,   x  0  0
2
1 
   T 
Boundary specific heat: Cb     1  1     3 2  1   

v v 
 
2
 1     3  2    2  2 ' 1    2 T  3
Boundary susceptibility: b 


4v 2  2     2  1  
 v 
2
• Boundary energy of open XXZ chain
L spins
E (m, L) : lowest energy of a finite open chain with
 ( m)
E (m, L)  L 0 (m)   1 (m)  2
L 1
2
1
(2m) 2

1 (m)  1 (0)  h1 (2m) 
2 2

h1 
2
 1

 3   sin    1
      
  1
 2  1 
2  1  1
    
1

arccos  
2
 1
3
AF & T. Hikihara, PRB 69, 094429 (2004)
z
Stot
 mL
J1-J2 spin chain with
antiferromagnetic J1
Ground-state phase diagram for AF-J1 case
J1 chain
Two decoupled J2 chains
Majumdar-Ghosh line
S j  S j 1  S j 1  S j  2  0
dimer phase
Haldane ‘82
White & Affleck ’96
……..
S j  S j 1  c0  c1  1 cos  4 ( xl )  
v
1
2
2


H eff   dx  K   x     x    cos 16 
2
K




l

J2>0 changes  and scaling dimension of cos
If cos


16 is relevant and   0 ,
then   x  is pinned at
  0.or  4.



16 .
cos

4

0
dimer LRO


16 is relevant and   0 ,
then   x  is pinned at    .  4 or   4.
If cos
sin

4

Neel LRO
0
Ground-state phase diagram for AF-J1 case
J1 chain
Two decoupled J2 chains
Majumdar-Ghosh line
Two decoupled J2 chains
Perturbation around J1=0
H eff
2
2
v
1

   dx  K   x      x    cos
K

 1,2 2
S ,l  e
S z ,l 


1
  (l )  a (1) sin 

i   ( l )
 b0 (1)l  b1 sin

x 

l
1
J1 S1, j  S1, j 1 S2, j  h.c.
 
g1 cos
1
1
1  2  ,   1  2 
2
2


4 (l )  
4 (l ) 



16 




 
8 cos
dimer order

8   g 2 x  sin

8 
vector chiral order

Vector chiral phase
When
g2
d 
sin
dx

8 

p-type nematic Andreev-Grishchuk (1984)
relevant → sin
 Vector chiral order


 sl  sl 1 
z
~  sin

8
d 
dx
Opposite sign   (1) ,  (2)    ,  


 0,
d 
0
dx
Nersesyan-Gogolin-Essler (1998)
Characteristics of the vector chiral state
(1)
l

8 
Vector chiral order


l( 2)  sl  sl  2 z ~ 

SxSx
x x
0 r
s s
&
SxSy
~r
1
or  ,  
no net spin current flow
J1l(1)  2J 2 l( 2)  0
spin correlation
4 K
cosqr
x y
0 r
s s
~ r
1
4 K
sin qr
power-law decay, incommensurate
A quantum counterpart of the classical helical state
J1-J2 spin chain with
ferromagnetic J1
Phase diagram & chiral order parameter
ferromagnetic J1
antiferromagnetic J1
The vector chiral order phase is large in the ferromagnetic J1 case
and extends up to the vicinity of the isotropic case   1.
Two decoupled J2 chains
Perturbation around J1=0
H eff
2
2
v
1

   dx  K   x      x    cos
K

 1,2 2
S ,l  e
S z ,l 


1
  (l )  a (1) sin 

i   ( l )
 b0 (1)l  b1 sin

l
x 
1

J1 S1, j  S1, j 1 S2, j  h.c.
g1 cos
1
1
1  2  ,   1  2 
2
2
 


2
at   1




 
8 cos

8   g 2 x  sin
dimer order
2
2
J1   x     x  


J
K  K K 2 1
v
1
J1 S1,z j  S1,z j 1 S2,z j
K 0  1 at   0, K 0 


4 (l )  
4 (l ) 


16 


8 
vector chiral order
dimension
g1 : 2K    2 K  
g2 : 1   2K 
1
1

Phase diagram & chiral order parameter
ferromagnetic J1
d
J1  sin
dx

8 

antiferromagnetic J1
 sl  sl 1 
z
~  sin


8 ,
 sl  sl  2 
z
d 
~
dx
Ground-state phase diagram for Ferro-J1 case
PbCuSO4(OH)2
Li2ZrCuO4
Rb2Cu2Mo3O12
LiCu2O2
J1 chain
NaCu2O2
LiCuVO4
Two decoupled J2 chains
Sine-Gordon model for spin-1/2 J1-J2 XXZ chain
with ferromagnetic coupling J1
J1 chain
We begin with the J2=0 limit.
Ferromagnetic
Easy-plane
Ferromagnetic
SU(2) Heisenberg
Effective Hamiltonian (sine-Gordon model)
v
1
2
2

H eff   dx  K   x     x    cos
2
K

TL-liquid (free-boson) part
1
  cos   , 0   
2
1
TL-liquid parameter K 
2



16 

irrelevant perturbation
velocity v  J1
sin  
2 1  
Spin and dimer operators
1
j
z
Sj 
 x   1 sin 4 

j
S j  ei  b0   1 b1 sin 4  


j
S j  S j 1  S j  S j 1  c  1 cos 4 








If the cosine term  cos 16 becomes relevant, then
Neel order
dimer order
BKT-type RG equation
J1 chain
Exact coupling constant in the J1 chain (J2=0)
S. Lukyanov, Nucl. Phys. B (1998).
It vanishes and changes its sign at
i.e.,
Relation between
and excitation gaps
of finite-size systems from perturbation theory for cosine term
estimated by
numerical
diagonalization
“Dimer” gap
“Neel” gap
Exact value is
known in J1 chain
We can check the position of =0 from numerical-diagonalization result.
J2=0
This relation is stable against
perturbations conserving symmetries.
Generally the exact value of  is not known
in the presence of such perturbations (J2).
However, the position of =0 is determined by the equation
which can be numerically evaluated.
J2 perturbation makes the  term relevant .
Neel and dimer phases are expected to emerge.
Neel order
Gaussian phase transition point (c=1)
dimer order
Phase diagram and Neel/dimer order parameters
Ground-state phase diagram of easy-plane anisotropic J1-J2 chain
Neel
>0
dimer
<0
Furukawa, Sato & AF
PRB 81, 094410 (2010)
Neel
>0
Curves
of =0
dimer
<0
J1 chain
Irrelevant
relevant
Direct calculation of order parameters from iTEBD method
XY component of dimer
Z component of dimer
Neel operator (Z component of spin)
<0
>0
<0
Neel phase
The emergence of the Neel phase is against our intuition:
ferromagnetic J1  0 & easy-plane anisotropy   1.
Spin correlation functions in the Neel phase
Short-range behavior is different from
that of the standard Neel order.
Dimer phase
FM-J1 case
AF-J1 case
Neel
dimer phase in the AF-J1 region
dimer phase in the FM-J1 region
On the XY line (=0)
J1-J2 XY chain with FM J1
J1-J2 XY chain with AF J1
 rotation at every even site
“triplet” dimer
“singlet” dimer
Dimer order parameter
1
2
3
dimer
Different dimer order
S1  S2  0
dimer
Sato, Furukawa, Onoda & AF
Mod. Phys. Lett. 25, 901 (2011)
dimer order parameter
string order parameter
S

2 j 1

 S2 j

 k 1 
 
exp i  S 2l 1  S 2l  S 2k 1  S 2k
 l  j 1




 2 j 1,2 j  : dimerized bond
weak dimer order
zoom
long-range string order


The string op is short-ranged for S2j  S2j 1 .
1
2
3
Summary
ferromagnetic J1
antiferromagnetic J1
Furukawa, Sato & AF
PRB 81, 094410 (2010)
Sato, Furukawa, Onoda & AF
Mod. Phys. Lett. 25, 901 (2011)
J 2 J1
Construction of ground-state wave function of J1-J2 chain
a trimer state in every triangle
projection to single-spin space
Neel order!
Phase diagram in magnetic field (h>0, J1<0, J2>0,   1 )
Hikihara, Kecke, Momoi & AF
PRB 78, 144404 (2008);
Sudan et al. PRB 80, 140402 (2009)
Antiferro-triatic
Antiferro-nematic
Nematic
SDW2
SDW3
SDW2
SDW3
multi-magnon
instability
1
k (2)
k (1)
Nematic (IC)
Vector-chiral phase
k (2)
J1-J2 Heisenberg spin chain in magnetic field
J1<0
J1>0
J2>0
Okunishi & Tonegawa (2003);
McCulloch et al. (2008);
Okunishi (2008);
Hikihara, Momoi, AF, Kawamura (2010)