2007.4.19~20 Monte Carlo Study of the J 1 -J 2 antiferromagnetic XY model on the triangular lattice Department of Physics Sungkyunkwan University Jin-Hong.

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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