1. Second-Order Phase Transition in Heisenberg Model
on Triangular Lattice with Competing Interactions
Ryo Tamura, Shu Tanaka, and Naoki Kawashima
Physical Review B 87, 214401 (2013)

2. Main Results
SECOND-ORDER PHASE TRANSITION IN THE . . .
We studied the phase transition nature of a frustrated Heisenberg
model on a distorted triangular lattice.
20
L=144
L=216
L=288
-2.2
ln(nv)
5
(a)
2
<m >
0
0.1
(b)
Arr
hen
ius
-2.6
2.00
2.02
0.05
s2
U4
U4
axi
s3
2.06
2.08
0
2
(e)
1
2
(c)
1
A second-order phase transition occurs.
2.04
J3/T
law
3
0
3
axi
axis 1
0
-2.4
Tc/J3
10
0.49
0.495
T/J3
0.5
χLη-2
C
15
0
(d)
-2.0
(f)
0.6
0.4
0.2
0
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
(T-T c)L1/ν/J3
FIG.
(Color
- At the second-order phase transition physical 2.quantitiesonline) Temperature Jdependence offor J /J =
point, Z2 of the distorted -J model equilibrium
−0.4926 . . and λ = 1.308 .
(a) Specific
(b) Square of
symmetry (lattice reflection symmetry)order.broken.. (c). ..Binder ratio Uheat C.Log of number
is parameter m
. (d)
the
1
2
3
1
3
4
density of Z2 vortex nv versus J3 /T . The dotted vertical line indicates
the transition temperature Tc /J3 = 0.4950(5). (e) and (f) Finite-size
scaling of the Binder ratio U4 and that of the susceptibility χ using
the critical exponents of the 2D Ising model (ν = 1 and η = 1/4)
and the transition temperature. Error bars are omitted for clarity since
their sizes are smaller than the symbol sizes.
- The universality class of the phase transition is
the same as that of the 2D Ising model.
- Dissociation of Z2 vortices occurs at the secondIn antiferromagnetic Heisenberg models on a triangular
lattice, the dissociation of the Z vortices occurs at finite
order phase transition point.
temperature.
13,27
2
In order to confirm the dissociation of the
mod
ope
first
soli
pha
size
λ=
the
are
of t
vie
figu
smo
and
wh
sca
λ=

3. Background
Unfrustrated systems (ferromagnet, bipartite antiferromagnet)
Ferromagnet
Bipartite antiferromagnet
Model
Ising
Order parameter space
Z2
XY
Heisenberg
U(1)
S2

4. Background
Frustrated systems
Antiferromagnetic Ising model
on triangle
Antiferromagnetic XY/Heisenberg model
on triangle
?
triangular lattice
kagome lattice
pyrochlore lattice

5. Background
Order parameter space in antiferromagnet on triangular lattice.
Model
Ising
XY
Order parameter space
--U(1)
Phase transition
--KT transition
Heisenberg
SO(3)
Z2 vortex dissociation

6. Motivation
To investigate the finite-temperature properties in two-dimensional
systems whose order parameter space is SO(3)xZ2.
- Phase transition occurs?
- Z2 vortex dissociation?

7. Model
H = J1
i,j
axis 1
si · sj + J1
1st nearestneighbor
axis 1
i,j
axis 2,3
si · sj + J3
1st nearestneighbor
axis 2, 3
i,j
si · sj
> 0, J3 > 0
3rd nearestneighbor
s2
axi
axi
s3
si : Heisenberg spin
(three components)
axis 1

8. Ground State
Spiral-spin configuration
si = R cos(k · ri )
I sin(k · ri )
R, I are two arbitrary orthogonal unit vectors.
Fourier transform of interactions
J(k)
J1
2J1
kx
=
cos kx +
cos
cos
N J3
J3
J3
2
3ky
+ cos 2kx + 2 cos kx cos
2
3ky
Find k that minimizes the Fourier transform of interactions!
4 < J1 /J3 < 0

9. SO(3) x C3 & SO(3) x Z2
(ii) single-k spiral
(a)
(c)
structure
sp
ira
l
4 independent
sublattices
)t
rip
lek
structure
(iv
axis 3 axis 2
axis 1
(b)
(iii) double-k spiral
(ii) single-k spiral
(i) ferromagnetic
R. dotted hexagonal area in Soc. The
Fig. 1. (a) Triangular lattice with L x × Ly sites. (b) Enlarged view of theTamura and N. Kawashima, J. Phys. (a). Jpn., 77, 103002 (2008).
R. Tamura and N. Kawashima, J. i-th site are
thick and thin lines indicate λJ1 and J1 , respectively. The third nearest-neighbor interactions at thePhys. Soc. Jpn., 80, 074008 (2011).
R. Tamura and S. categorized into
depicted. (c) Ground-state phase diagram of the model given by Eq. (1). Ground states can be Tanaka, Phys. Rev. E, 88, 052138 (2013).
R. Tamura, S. Tanaka, and N. Kawashima, to appear in Proceedings of APPC12.
five types. More details in each ground state are given in the main text.
J1-J3 model on triangular lattice
discussed the connection between frustrated continuous spin systems and a fundamental discrete spin
Order parameter space
Order of phase transition
system by using a locally defined parameter. The most famous example is the chiral phase transition
in the antiferromagnetic XY model on a triangular lattice. The relation between the phase transition
SO(3)xC3 that of the Ising model has been established [24, 25]. In this paper,
1st order
of the continuous spin system and
we study finite-temperature properties in the J1 -J3 model on a distorted triangular lattice depicted in
SO(3)xZ2
2nd order (Ising universality)
Figs. 1(a) and (b) from a viewpoint of the Potts model with invisible states.

10. SECOND-ORDER PHASE TRANSITION IN THE . . .
Specific heat, order parameter
SECOND-ORDER PHASE TRANSITION IN THE . . .
axi
U4
0.495
2
T/J3
(t)
0.5
3
(d)
0.4
(t)
2.02
(t)
s3
2.04
, m :=
J3/T
(e)
(f)
0.6
2
0.4
0.2
1
0
2
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
0.6
(T-T c)L1/ν/J3
0.4
0.2
2.06 0 2
(t)
/N 1
t
U4
Phase transition with FIG.
model fo
(e)
Z symmtery breaking open squ
(f)
first-orde
occurs.
solid circ
χL
(c)
(t)
:= s1 · s2
(
Tc/J3
ln(nv)
C
2
<m >
0.49
U4
2
U4
s2
1
2.00
si0.6j
·s
= 1.308 · · ·
rrhe
2.06 A2.08
niu
s la
w
2.04
J
-2.6 3/T
3 (b)
0.4926 · · · ,
i,j
law
2.02
1
0
3
ius
-2.4
axis 1
0.05
Binder ratio
2
(a)
axis 2,3
J1 /J3 =
hen
-2.2
2.00
2
Arr
si · sj + J3
(d)
i,j
-2.0
-2.6
(b)
0
0.1
0.05
0
3
(a)
specific heat
10
order 5
parameter
axi
<m >
0
0.1
-2.2
L=144
L=216
L=288
-2.4
15
5
axis 1
η-2
10
i,j
si · sj + J1
ln(nv)
20
-2.0
η-2
C
15
H = J1
L=144
L=216
L=288
s3
20

11. Number density of Z2 vortex
H = J1
i,j
axis 1
si · sj + J1
axis 2,3
J1 /J3 =
-2.0
0.4926 · · · ,
i,j
si · sj
= 1.308 · · ·
No phase transition with SO(3)
symmetry breaking occurs at
finite temperatures.
(Mermin-Wagner theorem)
ln(nv)
-2.2
-2.4
Arr
-2.6
2.00
i,j
si · sj + J3
2.02
hen
ius
2.04
J3/T
law
2.06
2.08
Dissociation of Z2 vortices occurs at the
second-order phase transition temperature.

12. Finite size scaling
H = J1
i,j
axis 1
si · sj + J1
U4
3
i,j
axis 2,3
si · sj + J3
J1 /J3 =
2
1
L
-2
= 1, = 1/4
0.6
0.4
0.2
0
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
1/
(T-Tc)L /J3
2D Ising universality class !!
0.4926 · · · ,
i,j
si · sj
= 1.308 · · ·

13. Phase diagram
0.6
0.4
0.52
Tc/J3
1st order phase
transition
R. Tamura and N. Kawashima,
J. Phys. Soc. Jpn., 77, 103002 (2008).
R. Tamura and N. Kawashima,
J. Phys. Soc. Jpn., 80, 074008 (2011).
R. Tamura, S. Tanaka, and N. Kawashima,
to appear in Proceedings of APPC12.
0.5
0.2
0.48
1
0
1
1.5
1.1
1.2
2
2.5
3
no phase transition
2nd order phase
transition

14. Conclusion
SECOND-ORDER PHASE TRANSITION IN THE . . .
We studied the phase transition nature of a frustrated Heisenberg
model on a distorted triangular lattice.
20
L=144
L=216
L=288
-2.2
ln(nv)
5
(a)
2
<m >
0
0.1
(b)
Arr
hen
ius
-2.6
2.00
2.02
0.05
s2
U4
U4
axi
s3
2.06
2.08
0
2
(e)
1
2
(c)
1
A second-order phase transition occurs.
2.04
J3/T
law
3
0
3
axi
axis 1
0
-2.4
Tc/J3
10
0.49
0.495
T/J3
0.5
χLη-2
C
15
0
(d)
-2.0
(f)
0.6
0.4
0.2
0
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
(T-T c)L1/ν/J3
FIG.
(Color
- At the second-order phase transition physical 2.quantitiesonline) Temperature Jdependence offor J /J =
point, Z2 of the distorted -J model equilibrium
−0.4926 . . and λ = 1.308 .
(a) Specific
(b) Square of
symmetry (lattice reflection symmetry)order.broken.. (c). ..Binder ratio Uheat C.Log of number
is parameter m
. (d)
the
1
2
3
1
3
4
density of Z2 vortex nv versus J3 /T . The dotted vertical line indicates
the transition temperature Tc /J3 = 0.4950(5). (e) and (f) Finite-size
scaling of the Binder ratio U4 and that of the susceptibility χ using
the critical exponents of the 2D Ising model (ν = 1 and η = 1/4)
and the transition temperature. Error bars are omitted for clarity since
their sizes are smaller than the symbol sizes.
- The universality class of the phase transition is
the same as that of the 2D Ising model.
- Dissociation of Z2 vortices occurs at the secondIn antiferromagnetic Heisenberg models on a triangular
lattice, the dissociation of the Z vortices occurs at finite
order phase transition point.
temperature.
13,27
2
In order to confirm the dissociation of the
mod
ope
first
soli
pha
size
λ=
the
are
of t
vie
figu
smo
and
wh
sca
λ=

15. Thank you !
Ryo Tamura, Shu Tanaka, and Naoki Kawashima
Physical Review B 87, 214401 (2013)