Transcript Competing phases of XXZ spin chain with frustration

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)