Unconventional phase transitions in frustrated systems (March, 2014)

Unconventional Phase Transitions
in Frustrated Systems
Shu Tanaka (The University of Tokyo)
Collaborators:
Ryo Tamura (NIMS)
Naoki Kawashima (ISSP)
2D case: PRB 87, 214401 (2013), 3D case: PRE 88, 052138 (2013).

Main results
To investigate unconventional phase transition
behavior in geometrically frustrated systems.
2D 3D
SO(3)xZ2 SO(3)xC3
- Z2 vortex dissociation
- 2nd-order PT w/ Z2 breaking
(2-dim. Ising universality)
at the same temperature.
- 1st-order PT w/ SO(3)xC3
breaking
- increases, decreases.J E

Conventional phase transitions
Ferromagnets Antiferromagnets In the ground state, all spin
pairs form stable spin
configurations.
Type
Order parameter
space
1D 2D 3D
Ising Z2 × √ √
XY U(1) × KT √
Heisenberg S2 × × √
Temperature
Ordered phase
Tc
Disordered phase
Phase transition occurs.

Frustration: random spin systems
E. Vincent, Lecture Notes in Physics 716 (2007),
Slow relaxation Novel order
We study the universality classes of phase transitions of
our model. In the phase diagram (see Fig. 5), there are two
types of phase boundaries. To make clear the universality
classes of each phase transition, x is set to 3/16 = 0.1875 such
that transition temperatures are separated sufficiently. For this
parameter, the intermediate phase is the (πππ) ordered phase
(see the dotted arrow in Fig. 5).
First, we investigate the higher-temperature phase transition
from the paramagnetic phase to the (πππ) ordered phase.
From the Harris criterion,36
we expect that the higher-
temperature phase transition belongs to the three-dimensional
Heisenberg universality class. This is because the critical
exponent α is negative in the three-dimensional Heisenberg
model, and thus the disorder should not affect the universality
class. To obtain the transition temperature and confirm the
critical exponents, we calculate the correlation function Gc
(rc),
a
b
c
(a) (b)
Random Fan-Out State
θ
θ
FIG. 9. (Color online) (a) “Average” spin directions in the spin
configuration of the random fan-out state. In each layer (ab plane),
R. Tamura, N. Kawashima, H. Kageyama et al.,
PRB 84, 214408 (2011)
Ferromagnetic
interaction
Antiferromagnetic
interaction
Even in the GS,
locally unstable spin state
appears due to frustration.
layered perovskite
SrFe1-xMnxO2
H. Takano and S. Miyashita, JPSJ 64, 423 (1995).

Frustration: geometrically frustrated systems
Ising model Heisenberg model
Residual entropy
(macroscopically degenerated states)
Single-q state
(120-degree structure, spiral spin texture)
Antiferromagnets on triangle-based
lattice structures
Geometrical frustration

Unconventional behaviors in GFMs
Chirality and Z2 vortex Reentrant phase transition
ParaAntiferroParaFerro
Temperature
Slow relaxation
T ! 0þ, we expect that nloop must be the maximum value
and the spin structure becomes the so-called
ffiffiffi
3
p
Â
ffiffiffi
3
p
structure.
Next, we study the relaxation of magnetization and nloop.
We ready the three types of initial configurations, i.e., (a) theffiffiffi
3
p
Â
ffiffiffi
3
p
structure, (b) the q ¼ 0 structure, and (c) a random
structure. The configurations (a) and (b) are typical ground
states of the present model, and the configuration (c)
corresponds to a state just after quench the temperature
from a high temperature.
In Fig. 4, the relaxation processes at T ¼ 0:05J are
plotted. In the cases (a) and (b), the magnetization is
maximum at t ¼ 0, and it relaxes very fast to uniformly
magnetized ordered state. The relaxation of magnetization to
the equilibrium is depicted in the inset. In contrast, in the
case (c), i.e., from a random state, it takes some time to
realize the uniformly magnetized state. Thus we regard
the relaxation time in the case (c) as the intrinsic relaxation
time of the magnetization mag.
0 [×10
+7
]
0
0.04
0.08
0.12
0.16
0 5000 10000
0
0.04
0.08
0.12
0.16
Monte Carlo Step (MCS)
Magnetization
Magnetization
0 [×10
+7
]
0
0.5
1
NumberofWeathervaneLoops
(a)
(c)
(b)
54321 54321
Fig. 4. (Color online) Relaxation of the magnetization and nloop at T ¼ 0:05J from (a)
ffiffiffi
3
p
Â
ffiffiffi
3
p
configuration, (b) q ¼ 0 configuration, and (c) random
configuration.
10
0
10
2
10
4
10
6
10
8
10
10
0
0.5
1
NumberofWeathervaneloops
T=0.0425J
T=0.04J
T=0.045J
T=0.0475J
T=0.05J
T=0.055J
T=0.06J
T=0.065J
T=0.07J
T=0.1J
T=0.09J
T=0.08J
Fig. 5. (Color online) Relaxation of nloop from
ffiffiffi
3
p
Â
ffiffiffi
3
p
structure at
several temperatures. Dashed lines denote the fittling curves estimated
by eq. (2).
J. Phys. Soc. Jpn., Vol. 76, No. 10 LETTERS S. TANAKA and S. MIYASHITA
A. Kuroda and S. Miyashita, JPSJ 64, 4509 (1995).
S. Tanaka and S. Miyashita, JPSJ 76, 103001 (2007).
S. Miyashita and H. Shiba, JPSJ 53, 1145 (1984).
H. Kawamura and S. Miyashita, JPSJ 53, 4138 (1984).
X. Hu, S. Miyashita, and M. Tachiki, PRL 79, 3498 (1997).
R. Tamura, S. Tanaka, and N. Kawashima, PRB 87, 214401 (2013).
H. Kitatani, S. Miyashita, and M. Suzuki, JPSJ 55, 865 (1986).
S. Miyashita, S. Tanaka, and M. Hirano, JPSJ 76, 083001 (2007).
Successive phase transitions
MoO4)3 featuring (a) MnO5 polyhedra, (b) equilateral triangular lattices
rlayer distances between Mn2+
ions are given by a = 6.099 ˚Aand c/2 =
S. Miyashita and H. Kawamura, JPSJ 54, 3385 (1985).
S. Miyashita, JPSJ 55, 3605 (1986).
R. Ishii, S. Tanaka, S. Nakatsuji et al. EPL 94, 17001 (2011).

Phase transition in 2D GFMs
H = J1
i,j
si · sj J3
i,j 3
si · sj
The 1st n.n. interaction The 3rd n.n. interaction
J3/J10-1/4
Ferromagnetic (S2) Spiral-spin structure (SO(3)xC3)
J1: Ferro
J3/J10-1/9
Degenerated GSs 120-degree structure (SO(3))
Order by disorder
J1: Antiferro
2D triangular lattice
NiGa2S4
S. Nakatsuji, Y. Nambu, Y. Maeno et al.,
Science 309, 1697 (2005).
1st-order PT w/ 3-fold symmetry breaking and
Z2 vortex dissociation occur.
R. Tamura and N. Kawashima,
JPSJ 77, 103002 (2008), JPSJ 80, 074008 (2011).
Z2 vortex dissociation

Model
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
The 1st n.n. interaction
along axes 2 and 3
The 3rd n.n. interactionThe 1st n.n. interaction
along axis 1

Model
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
along axes 2 and 3
along axis 1
4 types of ground states for ferromagnetic J1
1. Ferromagnetic state (S2)
2. Single-q spiral state (SO(3))
3. double-q spiral state (SO(3)xZ2)
4. triple-q spiral state (SO(3)xC3)
No phase transition occurs at finite T (Mermin-Wagner theorem).
Z2 vortex dissociation occurs at finite T.
1st-order PT and Z2 vortex dissociation occur at the same T.
N. D. Mermin and H. Wagner, PRL 17, 1133 (1966).
H. Kawamura and S. Miyashita, JPSJ 53, 4138 (1984).
R. Tamura and N. Kawashima, JPSJ 77, 103002 (2008).
R. Tamura and N. Kawashima, JPSJ 80, 074008 (2011).

Ground state phase diagram
SO(3)xC3
SO(3)xZ2
(i) ferromagnetic
(ii) single-k spiral
(iii) double-k spiral
(iv)triple-kspiral
(ii) single-k spiral
4 independent
sublattices
structure
structure
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
along axes 2 and 3
along axis 1

Model
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
axis 1
axis2
axis3
along axes 2 and 3
along axis 1
Order parameter space: SO(3)xZ2
J1/J3 = 0.4926 · · · , = 1.308 · · ·

Physical quantities
SECOND-ORDER PHASE TRANSITION IN THE . . .
1
2
3
0.49 0.495 0.5
U4
T/J3
(c)
0
0.05
0.1
m
2

(b)
0
5
10
15
20
C
(a)
L=144
L=216
L=288
0
0.2
0.4
0.6
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
χLη-2
(T-Tc)L1/ν
/J3
(f)
1
2
3
U4
(e)
-2.6
-2.4
-2.2
-2.0
2.00 2.02 2.04 2.06 2.08
ln(nv)
J3/T
Arrhenius law
(d)
0
0.2
0.4
0.6
1
Tc/J3
(
FIG.
model fo
open squ
ﬁrst-orde
solid circ
specific heat
order parameter
Binder ratio
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
axis 1
axis2
axis3
SECOND-ORDER PHASE TRANSITION IN THE . . .
3
4
0
0.05
0.1
m
2

(b)
0
5
10
15
20
C
(a)
L=144
L=216
L=288
0.6 (f)
1
2
3U4
(e)
-2.6
-2.4
-2.2
-2.0
2.00 2.02 2.04 2.06 2
ln(nv)
J3/T
Arrhenius law
(d)
(t)
:= s
(t)
1 · s
(t)
2 s
(t)
3 , m :=
t
(t)
/N
J1/J3 = 0.4926 · · · , = 1.308 · · ·
Order parameter detecting Z2 breaking
U4 :=
m4
m2 2
Binder ratio
Crossing point

-2.6
-2.4
-2.2
-2.0
2.00 2.02 2.04 2.06 2.08
ln(nv)
J3/T
Arrhenius law
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
along axes 2 and 3
along axis 1
J1/J3 = 0.4926 · · · , = 1.308 · · ·
No phase transition w/ SO(3)
breaking occurs at finite T.
(Mermin-Wagner theorem)
Point defect: 1(SO(3)) = Z2
Z2 vortex dissociation can occur
at finite T.
Z2 vortex density
Z2 vortex dissociation occurs
at the 2nd-order PT point (Tc).

Finite size scaling
0
0.2
0.4
0.6
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
L
-2
(T-Tc)L
1/
/J3
1
2
3
U4
= 1, = 1/4
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
along axes 2 and 3
along axis 1
J1/J3 = 0.4926 · · · , = 1.308 · · ·
Binder ratio
Susceptibility
Finite size scaling relations
U4 f (T Tc)L1/
L2
g (T Tc)L1/
2D Ising universality class
= 1, = 1/4
Z2 vortex dissociation does not
aﬀect the phase transition nature.

Phase diagram
E . . . PHYSICAL REVIEW B 87, 214401
(f)
(e)
2.04 2.06 2.08
J3/T
rhenius law
(d)
0
0.2
0.4
0.6
1 1.5 2 2.5 3
Tc/J3
λ
(a)
0.48
0.5
0.52
1 1.1 1.2
1.0
2.0
3.0
U4
(b)
L=108
L=144
L=180
L=216
0.0
0.2
0.4
-4 -2 0 2
χL
η-2
(T-Tc)L1/ν
/J3
(c)
FIG. 3. (Color online) (a) Phase diagram of the distorted
H = J1
i,j axis 1
si · sj J1
i,j axis 2,3
si · sj J3
i,j 3
si · sj
along axes 2 and 3
along axis 1
J1/J3 = 0.7342 · · ·
SO(3)xC3
1st-order PT w/ C3
breaking
occur at the same T.
SO(3)xZ2 SO(3)
R. Tamura and N. Kawashima,
JPSJ 77, 103002 (2008).
JPSJ 80, 074008 (2011).
SO(3)xZ2
2nd-order PT w/ Z2
breaking
occur at the same T.
2D Ising universality
SO(3)
occur at finite T.
H. Kawamura and S. Miyashita,
JPSJ 53, 4138 (1984).

Model
H = J1
i,j
si · sj J3
i,j 3
si · sj J
i,j
si · sj
The 3rd n.n. interaction
intralayer
interlayer
intralayer

Ground state
H = J1
i,j
si · sj J3
i,j 3
si · sj J
i,j
si · sj
/2 /2 /2
/2 /2
/2
Order parameter space: SO(3)xC3
intralayer
interlayer
intralayer

Internal energy and specific heat
H = J1
i,j
si · sj J3
i,j 3
si · sj J
i,j
si · sj
J3/J1 = 0.85355 · · · , J /J1 = 2INTERLAYER-INTERACTION DEPENDENCE OF LATENT . .
20
30
40
-2.3
-2.2
-2.1
(a)
(d)
(b)
0
0.05
0.1
0 15 30 45
0
5
10
15
20
25
30
INTERLAYER-INTERACTION D
0.02
0
10
20
30
40
-2.3
-2.2
-2.1
(a)
(b)
(c)
INTERLAYER-INTERACTION DEPENDENCE OF LATENT . .
0
10
20
30
40
-2.3
-2.2
-2.1
1.53
1.54
1.55
0 0.00004 0.000
(a)
(d)
(e)
(b)
(c)
0
0.05
0.1
0 15 30 45
0
5
10
15
20
25
30
-2.3 -2.2 -2.1
Internal energy Specific heat
Phase transition occurs
at finite T.
0
0.01
0.02
1.52 1.53 1.54 1.55
0
10
20
30
40
-2.3
1.53
1.54
1.55
0 0.00004 0.0000
0
20
40
60
0 20000 40000 60000
(a)
(e)
(f)
(b)
(c)
0
5
10
15
-2.3 -2.2 -2.1
FIG. 4. (Color online) Temperature dependence of (a) intern
0
0.01
0.02
1.52 1.53 1.54 1.55
0
10
20
30
40
-2.3
-2.2
(a)
(b)
(c)
intralayer
interlayer
intralayer

Order parameter (C3 and SO(3))
H = J1
i,j
si · sj J3
i,j 3
si · sj J
i,j
si · sj
J3/J1 = 0.85355 · · · , J /J1 = 2
0
0.01
0.02
1.52 1.53 1.54 1.55
0
10
20
30
1.53
1.54
1.55
0 0.00004 0.0
0
20
40
60
0 20000 40000 60000
(e)
(f)
(b)
(c)
0
5
10
-2.3 -2.2 -2
FIG. 4. (Color online) Temperature dependence of (a) inte
energy per site E/J1, (b) specific heat C, and (c) order par
2
INTERLAYER-INTERACTION DEPENDENCE OF LATENT . .
0
10
20
30
40
-2.3
-2.2
-2.1
1.54
1.55
(a)
(d)
(e)
(b)
0
0.05
0.1
0 15 30 45
0
5
10
15
20
25
30
-2.3 -2.2 -2.1
C3 symmetry breaks at Tc.
Order parameter (C3)
peaks E(L)/J1. (e) Plot of Tc(L)/J1 as a function of L−3
. (f) Plot
of Cmax(L) as a function of L3
. Lines are just visual guides and error
bars in all figures are omitted for clarity since their sizes are smaller
than the symbol size.
of analysis. One is the finite-size scaling and the other is
a naive analysis of the probability distribution P(E; Tc(L)).
The scaling relations for the first-order phase transition in
d-dimensional systems [82] are given by
Tc(L) = aL−d
+ Tc, (9)
Cmax(L) ∝
( E)2
Ld
4T 2
c
, (10)
where Tc and E are, respectively, the transition temperature
and the latent heat in the thermodynamic limit. The coefficient
of the first term in Eq. (9), a, is a constant. Figures 4(e)
and 4(f) show the scaling plots for Tc(L)/J1 and Cmax(L),
respectively. Figure 4(e) indicates that Tc is a nonzero value
in the thermodynamic limit. Figure 4(f) shows an almost
linear dependence of Cmax(L) as a function of L3
. However,
using the finite-size scaling, we cannot obtain the transition
temperature and latent heat in the thermodynamic limit with
high accuracy because of the strong finite-size effect. Next we
directly calculate the size dependence of the width between
bimodal peaks of the energy distribution shown in Fig. 4(d).
The width for the system size L is represented by E(L) =
E+(L) − E−(L), where E+(L) and E−(L) are the averages of
the Gaussian function in the high-temperature phase and that in
the low-temperature phase, respectively. In the thermodynamic
limit, each Gaussian function becomes the δ function and then
E(L) converges to E [82]. The inset of Fig. 4(d) shows the
size dependence of the width E(L)/J1. The width enlarges as
value in the thermodynamic limit. Figure 5(a) shows the
temperature dependence of the largest value of structure factors
S(k∗
) calculated by six wave vectors in Eq. (4). Here S(k∗
)
becomes zero in the thermodynamic limit above the first-
order phase transition temperature. The structure factor S(k∗
)
becomes a nonzero value at the first-order phase transition
temperature. Moreover, as temperature decreases, the structure
factor S(k∗
) increases. The structure factors at kz = 0 in the
first Brillouin zone at several temperatures for L = 40 are also
shown in Fig. 5(b). As mentioned in Sec. II, the spiral-spin
structure represented by k is the same as that represented by
−k in the Heisenberg models. Figure 5(b) confirms that one
distinct wave vector is chosen from three types of ordered
(a)
(b)
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2
10-5
10-4
10-3
10-2
10-1
FIG. 5. (Color online) (a) Temperature dependence of the largest
value of structure factors S(k∗
) calculated by six wave vectors in
Eq. (4) for J3/J1 = −0.853 55 . . . and J⊥/J1 = 2. Error bars are
Order parameter (SO(3))
SO(3) symmetry breaks at Tc.
intralayer
interlayer
intralayer

Energy histogram
H = J1
i,j
si · sj J3
i,j 3
si · sj J
i,j
si · sj
J3/J1 = 0.85355 · · · , J /J1 = 2
N DEPENDENCE OF LATENT . . .
(d)
0
0.05
0.1
0 15 30 45
0
5
10
15
20
25
30
-2.3 -2.2 -2.1
first-order phase tr
finite temperature.
We further inv
mentioned above,
SO(3) × C3. It was
the first-order phas
Heisenberg model
a nearest-neighbor
space is SO(3), a s0.01
0.02
0
10
20
30
40
-2.3
-2.2
-2.1
1.53
1.54
1.55
0
40
60
(a)
(e)
(f)
(b)
(c)
0
0.05
0.1
0 1
0
5
10
15
20
25
30
-2.3
P(E; T) = D(E)e E/kBT
D(E) : density of states
E(L) : width between two peaks
Bimodal distribution
1st-order PT w/ SO(3)xC3
breaking occurs.
intralayer
interlayer
intralayer

Finite size scaling
H = J1
i,j
si · sj J3
i,j 3
si · sj J
i,j
si · sj
J3/J1 = 0.85355 · · · , J /J1 = 2
5
1.53
1.54
1.55
0 0.00004 0.00008
0
20
40
60
0 20000 40000 60000
(d)
(e)
(f)
0
0.05
0 15 30 45
0
5
10
15
20
25
-2.3 -2.2 -2.1
mperature dependence of (a) internal
finite temperature.
We further investigate th
mentioned above, the order pa
SO(3) × C3. It was confirmed
the first-order phase transition
Heisenberg model on a stack
a nearest-neighbor interactio
space is SO(3), a single peak i
dependence of the specific heat
finite-temperature phase transi
state and magnetic ordered sta
is broken. Then, in our mode
break at the first-order phase tr
heat has a single peak corresp
transition. To confirm this w
dependence of the structure fac
1
0
0.01
0.02
1.52 1.53 1.54 1.55
0
10
20
30
40
1.53
1.54
1.55
0 0.00004 0.00008
0
20
40
60
0 20000 40000 60000
(e)
(f)
(b)
(c)
0
5
10
-2.3 -2.2 -2.1
FIG. 4. (Color online) Temperature dependence of (a) internal
energy per site E/J1, (b) specific heat C, and (c) order param-
eter |µ|2
, which can detect the C3 symmetry breaking of the
model with J3/J1 = −0.853 55 . . . and J⊥/J1 = 2 for L = 24,32,40.
(d) Probability distribution of the internal energy P(E; T (L)). The
Max of specific heatTc(L)
Tc(L) = aL d
+ Tc Cmax(L)
( E)2
Ld
4T2
c
M. S. S. Challa, D. P. Landau, and K. Binder, PRB 34, 1841 (1986).
1st-order PT w/ SO(3)xC3 breaking occurs.
intralayer
interlayer
intralayer

Interlayer interaction dependence fixing J3/J1PHYSICAL REVIEW E 88, 052138 (2013)
0
0.1
0.2
0
0.05
0.1
0.15
0.7
0.75
0.8
0.85
0
20
40
-3
-2.5
-2
-1.5
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
(a)
(b)
(c)
(d)
(e)
0.25 0.50 0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
0.25
0.50
0.75
1.00
1.25 1.50 1.75 2.00 2.25 2.50
0.25
0.25
0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
0.50
0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
2.50
FIG. 6. (Color online) Interlayer-interaction J⊥/J1 dependence
of (a) internal energy per site E/J1, (b) specific heat C, (c) uniform
magnetic susceptibility χ, (d) order parameter |µ|2
, which can
detect the C3 symmetry breaking, and (e) largest value of structure
factors S(k∗
) calculated by six wave vectors in Eq. (4) for L = 24.
Error bars in all figures are omitted for clarity since their sizes are
J /J1 increases
INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .
0
10
20
-2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4
0
10
20
0
10
20
0
10
0
10
0
10
0
10
0
10
0
10
0
10
0
0.04
0.08
0 1 2
0.5
1
1.5
(a) (b)
(c)
FIG. 7. (Color online) (a) Interlayer-interaction J⊥/J1 depen-
dence of the probability distribution of internal energy P(E; Tc(L))
when the specific heat becomes the maximum value for L = 24.
on
fir
J3
no
fo
In
ity
be
de
la
tri
or
bu
ob
th
Fo
in
m
tra
fie
in
an
in
ju
th
INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .
0
10
20
-2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4
0
10
20
0
10
20
0
10
0
10
0
10
0
10
0
10
0
10
0
10
0
0.04
0.08
0 1 2
0.5
1
1.5
(a) (b)
(c)
FIG. 7. (Color online) (a) Interlayer-interaction J⊥/J1 depen-
dence of the probability distribution of internal energy P(E; Tc(L))
when the specific heat becomes the maximum value for L = 24.
o
fi
J
n
fo
In
it
b
d
la
tr
o
b
o
th
F
in
m
tr
fi
in
an
in
ju
th
Transition
temperature
Latent heat
As the interlayer interaction increases,
transition temperature increases but latent heat decreases.

Conclusion
We investigated unconventional phase transition
behavior in geometrically frustrated systems.
2D 3D
SO(3)xZ2 SO(3)xC3
- Z2 vortex dissociation
- 2nd-order PT w/ Z2 breaking
(2-dim. Ising universality)
at the same temperature.
- 1st-order PT w/ SO(3)xC3
breaking
- increases, decreases.J E

Thank you for your attention!!
2D case: PRB 87, 214401 (2013), 3D case: PRE 88, 052138 (2013).

Unconventional phase transitions in frustrated systems (March, 2014)

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