2007.4.19~202007.4.19~20
Monte Carlo Study of the Monte Carlo Study of the JJ11-J-J2 2 antiferromagnetic antiferromagnetic XYXY model model
on the triangular latticeon the triangular lattice
Department of PhysicsDepartment of Physics
Sungkyunkwan UniversitySungkyunkwan University
Jin-Hong Park and Jung Hoon HanJin-Hong Park and Jung Hoon Han
Two types of the transition found on triangular lattice Two types of the transition found on triangular lattice
Classical Classical XYXY model Hamiltonian model Hamiltonian
.
XYXY model on triangular model on triangular latticelattice
1.1.Kosterlitz-Thouless(KT) transitionKosterlitz-Thouless(KT) transition
2.2.Chirality transitionChirality transition
The separation of the phase temperatures is extremely small.The separation of the phase temperatures is extremely small.The chirality-ordered phase is not well-defined.The chirality-ordered phase is not well-defined.
Sooyeul Lee and Koo-Chul Lee,Sooyeul Lee and Koo-Chul Lee,Phys. Rev. B 57, 8472 (1998)Phys. Rev. B 57, 8472 (1998)
TTMagneticMagnetic ParamagnetiParamagneti
ccTTKT KT TT
XYXY model on triangular model on triangular latticelattice
--++++
++++++
++++
++
++
-- --
------
-- ----ChiralChiral
Biquadratic interaction Biquadratic interaction on triangular lattice ?on triangular lattice ?
If the spin-spin interaction is biquadratic, If the spin-spin interaction is biquadratic, a spin-nematic order is realized instead.a spin-nematic order is realized instead. .
Biquadratic interaction supports a spin nematic order.Biquadratic interaction supports a spin nematic order.
== oror
We want to study a variant of the We want to study a variant of the XYXY model in which the chirality order e model in which the chirality order exists over an extended region of the phase diagram by combining quadrxists over an extended region of the phase diagram by combining quadratic and bi-quadratic interactionsatic and bi-quadratic interactions
JJ11-J-J22 XYXY model model
JJ22/J/J11
TT
paramagneticparamagnetic
magneticmagnetic
chiral, chiral, non-magnetnon-magneticic
JJ22/J/J11=9=9
We focus on JWe focus on J22/J/J11 = 9. = 9.
A chiral phase is seen to exist A chiral phase is seen to exist over an extended temperature over an extended temperature region when Jregion when J22/J/J11 is large is large
0.1 0.2 0.3 0.4 0.5 0.6 0.70.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
S
peci
fic h
eat
T
L15 L30 L60
TT11 TT22
JJ22/J/J11= 9 = 9 (L = 15, 30, (L = 15, 30, 60).60).
Specific heatSpecific heat
Two phase transitions Two phase transitions clearly identifiedclearly identified
Magnetic order parameterMagnetic order parameter
Nematic order parameterNematic order parameter
Magnetic/nematic parametersMagnetic/nematic parameters
We study the nature of the phases with the magnetic and nematic order We study the nature of the phases with the magnetic and nematic order parametersparameters
Chiral order parameterChiral order parameter
Chiral order parameterChiral order parameter
++--
11 33
22 44
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.0
0.2
0.4
0.6
0.8
1.0
M1
T
L=15 L=30 L=45 L=60
Magnetic orderMagnetic order
0.1 0.2 0.3 0.4 0.5 0.6 0.70.0
0.2
0.4
0.6
0.8
1.0
N1
T
L15 L30 L45 L60
0.1 0.2 0.3 0.4 0.5 0.6 0.70.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
N2
T
L15 L30 L45 L60
0.42 0.44 0.46 0.48 0.500.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
Bin
der
cum
ulen
t
T
L30 L45 L60
Binder cumulentBinder cumulent
Nematic orderNematic order
TTKTKT = 0.460 = 0.460
Helicity Modulus
0.1 0.2 0.3 0.4 0.5 0.6 0.70.0
0.5
1.0
1.5
2.0
2.5
Y
T
L=15 L=30 L=45 L=60 fit
Helicity modulusHelicity modulus
.
TTKTKT = 0.459 = 0.459
This TThis TKTKT must agree with the one must agree with the one obtained from Binder cumulent obtained from Binder cumulent in the previous page.in the previous page.
criticalcritical disorderdisorder
TTKTKT
Critical phase for nematic order below TCritical phase for nematic order below TKTKT
We find critical dependence of NWe find critical dependence of N1 1 and Nand N22 on the lattice dimension L below Ton the lattice dimension L below TKT.KT.
Chiral orderChiral order
Chiral order undergoes two phase transitions. Chiral order undergoes two phase transitions. The first one at higher temperature obeys a The first one at higher temperature obeys a scaling plot. A scaling plot of chirality using the scaling plot. A scaling plot of chirality using the 0.15, 0.15, 0.69, and T 0.69, and T = 0.462. = 0.462. This TThis T is higher than T is higher than TKT KT of the nematic order. of the nematic order.
TTMagneticMagnetic ChiralChiral ParamagneticParamagnetic
TTMagneticMagnetic ParamagneticParamagnetic
Phase diagramPhase diagram
JJ22/J/J11 =0 =0
JJ22/J/J11 =9 =9
By introducing frustration in the form of JBy introducing frustration in the form of J22 we find an extended we find an extended region of chiral phaseregion of chiral phase
1.1. We find a clear separation of magnetic (TWe find a clear separation of magnetic (T11) and nematic (T) and nematic (T22) phase transition ) phase transition for Jfor J22/J/J11 = 9. = 9.
3.3. This is the first demonstration of the clear separation of the chiral phase tranThis is the first demonstration of the clear separation of the chiral phase transition and the magnetic phase transition in sition and the magnetic phase transition in XYXY-like models.-like models.
SummarySummary
2.2. Quite remarkably, the staggered chirality order sets in at T=TQuite remarkably, the staggered chirality order sets in at T=T22, where , where the nematic order occurs.the nematic order occurs.
AppendixAppendix
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