Phase Transition in 2D q-state Clock Model
Kingshuk Sarkar
Indian Institute of Science, Bangalore 560012
November 5, 2012
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
1 / 29
References
True and quasi-long-range order in the generalized
q-state clock model
S. K. Baek, P. Minnhagen, B. J. Kim, Phy. Rev. E
80,060101 (2009)
Non-Kosterlitz-Thouless transitions for the q-state
clock model
S. K. Baek, P. Minnhagen, Phys. Rev. E 82, 031102
(2010)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
2 / 29
Clock Model or Planer Potts Model
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
3 / 29
Clock Model or Planer Potts Model
Ising Model:
P Two component spin model.
Si Sj where Si = +1 or -1
H = −J
<ij>
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
3 / 29
Clock Model or Planer Potts Model
Ising Model:
P Two component spin model.
Si Sj where Si = +1 or -1
H = −J
<ij>
Clock Model: A system of spins confined in a plane with each spin
pointing to one of the q equally spaced direction.
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
3 / 29
Clock Model or Planer Potts Model
Ising Model:
P Two component spin model.
Si Sj where Si = +1 or -1
H = −J
<ij>
Clock Model: A system of spins confined in a plane with each spin
pointing to one of the q equally spaced direction.
θn = 2π
q n n=0,1,...,(q-1)
(q-state clock model) Discrete Zq symmetry
Si = (cosθi , sinθi )
.
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
3 / 29
Clock Model or Planer Potts Model
Ising Model:
P Two component spin model.
Si Sj where Si = +1 or -1
H = −J
<ij>
Clock Model: A system of spins confined in a plane with each spin
pointing to one of the q equally spaced direction.
θn = 2π
q n n=0,1,...,(q-1)
(q-state clock model) Discrete Zq symmetry
Si = (cosθi , sinθi )
.
XY Model: q → ∞ continuous U(1) symmetry
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
3 / 29
Clock Model or Planer Potts Model
Ising Model:
P Two component spin model.
Si Sj where Si = +1 or -1
H = −J
<ij>
Clock Model: A system of spins confined in a plane with each spin
pointing to one of the q equally spaced direction.
θn = 2π
q n n=0,1,...,(q-1)
(q-state clock model) Discrete Zq symmetry
Si = (cosθi , sinθi )
.
XY Model: q → ∞ continuous U(1) symmetry
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
3 / 29
Potts Model
P
H = −J
[qδsi ,sj − 1]
<ij>
F. Y. Wu. Rev. Mod. Phys. 54,235(1982)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
4 / 29
True Long range order(LRO):
x→∞
< s(x).s(0) > = constant
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
5 / 29
True Long range order(LRO):
x→∞
< s(x).s(0) > = constant
Disorder:
x→∞
< s(x).s(0) > ∼ exp(−|x|)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
5 / 29
Mermin-Wagner Theorem: No LRO for d ≤ 2 in systems having
continuous symmetry at non-zero T
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
6 / 29
Mermin-Wagner Theorem: No LRO for d ≤ 2 in systems having
continuous symmetry at non-zero T
For discrete symmetry LRO possible in d = 2 (e.g. Ising symmetry)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
6 / 29
Mermin-Wagner Theorem: No LRO for d ≤ 2 in systems having
continuous symmetry at non-zero T
For discrete symmetry LRO possible in d = 2 (e.g. Ising symmetry)
For 2d xy spins
x→∞
< s(x).s(0) > ∼
1
|x|η(T )
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
6 / 29
Mermin-Wagner Theorem: No LRO for d ≤ 2 in systems having
continuous symmetry at non-zero T
For discrete symmetry LRO possible in d = 2 (e.g. Ising symmetry)
For 2d xy spins
x→∞
< s(x).s(0) > ∼
Quasi-LRO
1
|x|η(T )
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
6 / 29
Mermin-Wagner Theorem: No LRO for d ≤ 2 in systems having
continuous symmetry at non-zero T
For discrete symmetry LRO possible in d = 2 (e.g. Ising symmetry)
For 2d xy spins
x→∞
< s(x).s(0) > ∼
Quasi-LRO
1
|x|η(T )
QLRO → Disorder phase transition is known as KosterlitzThouless(KT) transition
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
6 / 29
KT transition: Doesn’t involve any broken symmetry.
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
7 / 29
KT transition: Doesn’t involve any broken symmetry.
Thermally generated vortex-pair unbinds at TKT
Elastic rigidity (spin stiffness) goes to zero with a universal jump 2TπKT
Figure: vorticity n=1 and n=-1
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
7 / 29
Overall phase transitions of q-state Clock Models
(q ≤ 4) single phase transition.
q=2 Ising transition
q=3 Three-state Potts transition
q=4 Ising-like transition M. Suzuki, Prog. Theor. Phys. 37, 770
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
8 / 29
Overall phase transitions of q-state Clock Models
(q ≤ 4) single phase transition.
q=2 Ising transition
q=3 Three-state Potts transition
q=4 Ising-like transition M. Suzuki, Prog. Theor. Phys. 37, 770
(q > 4) two phase transitions.
LRO→QLRO→Disorder
Jose et. al., Phys. Rev. B16,1217 (1977)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
8 / 29
Overall phase transitions of q-state Clock Models
(q ≤ 4) single phase transition.
q=2 Ising transition
q=3 Three-state Potts transition
q=4 Ising-like transition M. Suzuki, Prog. Theor. Phys. 37, 770
(q > 4) two phase transitions.
LRO→QLRO→Disorder
Jose et. al., Phys. Rev. B16,1217 (1977)
When q → ∞ we get QLRO→Disorder (Kosterlitz-Thouless
transition)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
8 / 29
Overall phase transitions of q-state Clock Models
(q ≤ 4) single phase transition.
q=2 Ising transition
q=3 Three-state Potts transition
q=4 Ising-like transition M. Suzuki, Prog. Theor. Phys. 37, 770
(q > 4) two phase transitions.
LRO→QLRO→Disorder
Jose et. al., Phys. Rev. B16,1217 (1977)
When q → ∞ we get QLRO→Disorder (Kosterlitz-Thouless
transition)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
8 / 29
Numerical study for eight-state clock model
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
9 / 29
Numerical study for eight-state clock model
P
cos(θi − θj )
Clock Model: H = −J
<ij>
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
9 / 29
Numerical study for eight-state clock model
P
cos(θi − θj )
Clock Model: H = −J
<ij>
Monte Carlo simulation with single-cluster algorithm (Wolff
Algorithm)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
9 / 29
Numerical study for eight-state clock model
P
cos(θi − θj )
Clock Model: H = −J
<ij>
Monte Carlo simulation with single-cluster algorithm (Wolff
Algorithm)
Wolff algorithm very effecient for large system size near critical temp
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
9 / 29
Numerical study for eight-state clock model
P
cos(θi − θj )
Clock Model: H = −J
<ij>
Monte Carlo simulation with single-cluster algorithm (Wolff
Algorithm)
Wolff algorithm very effecient for large system size near critical temp
A square lattice of size L × L
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
9 / 29
Numerical study for eight-state clock model
P
cos(θi − θj )
Clock Model: H = −J
<ij>
Monte Carlo simulation with single-cluster algorithm (Wolff
Algorithm)
Wolff algorithm very effecient for large system size near critical temp
A square lattice of size L × L
P
Order parameter m = N −1 e i θj = |m|e i φ
j
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
9 / 29
q=8 (system size 8 × 8)
Figure: T=1.50 J/kB Disordered Phase
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
10 / 29
q=8 (system size 8 × 8)
Figure: T=1.50 J/kB Disordered Phase
Figure: T=0.70 J/kB QLRO phase
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
10 / 29
q=8 (system size 8 × 8)
Figure: T=0.36 J/kB LRO phase
QLRO phase exists between LRO and Disorder.
How to detect the transition temperatures?
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
11 / 29
The appearence of QLRO phase can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
12 / 29
The appearence of QLRO phase can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
<|m|4 >
2<|m|2 >
|m|e i φ so Um
Um = 1 −
m=
direction)
can’t detect Order → QLRO (differ only angular
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
12 / 29
The appearence of QLRO phase can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
<|m|4 >
2<|m|2 >
|m|e i φ so Um
Um = 1 −
m=
direction)
can’t detect Order → QLRO (differ only angular
m has been modified to mφ = cos(qφ)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
12 / 29
The appearence of QLRO phase can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
<|m|4 >
2<|m|2 >
|m|e i φ so Um
Um = 1 −
m=
direction)
can’t detect Order → QLRO (differ only angular
m has been modified to mφ = cos(qφ)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
12 / 29
q=8
QLRO→Disorder
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
13 / 29
q=8
QLRO→Disorder
Order→QLRO
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
13 / 29
Position of the crossing point (T∗ ) corresponds to the transition
temperature in thermodynamic limit.
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
14 / 29
Position of the crossing point (T∗ ) corresponds to the transition
temperature in thermodynamic limit.
T∗ for both transitions obtained from lnL ∼ |T∗ − Tc |−1/2
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
14 / 29
Position of the crossing point (T∗ ) corresponds to the transition
temperature in thermodynamic limit.
T∗ for both transitions obtained from lnL ∼ |T∗ − Tc |−1/2
Figure: Tc1 (lower ) = 0.417(3) Tc2 (upper ) = 0.894(1)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
14 / 29
Position of the crossing point (T∗ ) corresponds to the transition
temperature in thermodynamic limit.
T∗ for both transitions obtained from lnL ∼ |T∗ − Tc |−1/2
Figure: Tc1 (lower ) = 0.417(3) Tc2 (upper ) = 0.894(1)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
14 / 29
Scaling function f (mφ ) =< |m| > Lη/2
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
15 / 29
Scaling function f (mφ ) =< |m| > Lη/2
By appropriate choice of η all the data for different system sizes will
fall on a single curve
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
15 / 29
Scaling function f (mφ ) =< |m| > Lη/2
By appropriate choice of η all the data for different system sizes will
fall on a single curve
Figure: η/2 = 0.031
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
15 / 29
Generalized
P q-state clock model
Clock Model: H = −J
cos(θi − θj )
<ij>
H(θ) = H(−θ) = H(θ + 2π)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
16 / 29
Generalized
P q-state clock model
Clock Model: H = −J
cos(θi − θj )
<ij>
H(θ) = H(−θ) = H(θ + 2π)
P
Generalized Clock Model:H =
<ij>
2J
[1
p2
2
− cos 2p (
θi −θj
)]
2
Domany et. al., Phys. Rev. Lett., 52, 1535 (1984)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
16 / 29
Generalized
P q-state clock model
Clock Model: H = −J
cos(θi − θj )
<ij>
H(θ) = H(−θ) = H(θ + 2π)
P
Generalized Clock Model:H =
<ij>
2J
[1
p2
2
− cos 2p (
θi −θj
)]
2
Domany et. al., Phys. Rev. Lett., 52, 1535 (1984)
For p=1 q-state clock model
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
16 / 29
Generalized
P q-state clock model
Clock Model: H = −J
cos(θi − θj )
<ij>
H(θ) = H(−θ) = H(θ + 2π)
P
Generalized Clock Model:H =
<ij>
2J
[1
p2
2
− cos 2p (
θi −θj
)]
2
Domany et. al., Phys. Rev. Lett., 52, 1535 (1984)
For p=1 q-state clock model
Two phase separation line mearges as p approaches 3
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
16 / 29
Generalized
P q-state clock model
Clock Model: H = −J
cos(θi − θj )
<ij>
H(θ) = H(−θ) = H(θ + 2π)
P
Generalized Clock Model:H =
<ij>
2J
[1
p2
2
− cos 2p (
θi −θj
)]
2
Domany et. al., Phys. Rev. Lett., 52, 1535 (1984)
For p=1 q-state clock model
Two phase separation line mearges as p approaches 3
Figure: Phase Diagram for q=8
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
16 / 29
Susceptibility χ = N(kB T )−1 (< |m|2 > −< |m| >2 ) corresponds to
sum of correlations
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
17 / 29
Susceptibility χ = N(kB T )−1 (< |m|2 > −< |m| >2 ) corresponds to
sum of correlations
Divergence of χ implies long ranged correlations.
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
17 / 29
Susceptibility χ = N(kB T )−1 (< |m|2 > −< |m| >2 ) corresponds to
sum of correlations
Divergence of χ implies long ranged correlations.
For small p in QLRO phase χ diverges over a finite temperature range
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
17 / 29
Susceptibility χ = N(kB T )−1 (< |m|2 > −< |m| >2 ) corresponds to
sum of correlations
Divergence of χ implies long ranged correlations.
For small p in QLRO phase χ diverges over a finite temperature range
For p ≃ 2.8 χ ∼ |T − Tc |−1.2 (one singular point at T = Tc )
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
17 / 29
Susceptibility χ = N(kB T )−1 (< |m|2 > −< |m| >2 ) corresponds to
sum of correlations
Divergence of χ implies long ranged correlations.
For small p in QLRO phase χ diverges over a finite temperature range
For p ≃ 2.8 χ ∼ |T − Tc |−1.2 (one singular point at T = Tc )
QLRO phase shrinks to a single point at p ≃ 2.8
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
17 / 29
Systematic study for different values of q
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
18 / 29
Systematic study for different values of q
The appearence of QLRO(quasiliquid) can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
18 / 29
Systematic study for different values of q
The appearence of QLRO(quasiliquid) can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
Helicity modulus(Υ) is the response of the system to a twist in one
direction.
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
18 / 29
Systematic study for different values of q
The appearence of QLRO(quasiliquid) can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
Helicity modulus(Υ) is the response of the system to a twist in one
direction.
P
U(θi − θj )
Clock Model: H = −J
<ij>
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
18 / 29
Systematic study for different values of q
The appearence of QLRO(quasiliquid) can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
Helicity modulus(Υ) is the response of the system to a twist in one
direction.
P
U(θi − θj )
Clock Model: H = −J
<ij>
Free energy F = −kB TlnZ
2
Υ = ∂∂φF2 |φ→0 φ twist variable in one direction
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
18 / 29
Systematic study for different values of q
The appearence of QLRO(quasiliquid) can be detected by Binder’s
fourth-order cumulant(Um ) or helicity modulus(Υ)
Helicity modulus(Υ) is the response of the system to a twist in one
direction.
P
U(θi − θj )
Clock Model: H = −J
<ij>
Free energy F = −kB TlnZ
2
Υ = ∂∂φF2 |φ→0 φ twist variable in one direction
∂Υ
Ξ = ∂T
displays a phase-transition singularity manifested in a size
dependence for all q
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
18 / 29
= π2
H. Weber and P. Minnhagen, Phys. Rev. B 37, 5986 (1988)
Υ
Tc
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
19 / 29
q=2 (Ising Model)
Figure: helicity modulus
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
20 / 29
q=2 (Ising Model)
Figure: helicity modulus
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
20 / 29
q → ∞ (xy Model)
Figure: Υ jumps from 2Tc /π to 0 at Tc
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
21 / 29
q → ∞ (xy Model)
Figure: Υ jumps from 2Tc /π to 0 at Tc
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
21 / 29
√
Υ(T ) = Υ(Tc )[1
+
b
T − Tc ]
√
′
lnξ ∼ b / T − Tc
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
22 / 29
√
Υ(T ) = Υ(Tc )[1
+
b
T − Tc ]
√
′
lnξ ∼ b / T − Tc
Ξ ∼ lnL
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
22 / 29
q=3,4,5,6
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
23 / 29
Villain Approximation
Vvillain (φ) = β1 ln
∞
P
n=−∞
exp[−β(φ − 2πn)2 /2]
J. Villain, J. Phys. (Paris) 36, 581 (1975)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
24 / 29
Villain Approximation
Vvillain (φ) = β1 ln
∞
P
n=−∞
exp[−β(φ − 2πn)2 /2]
J. Villain, J. Phys. (Paris) 36, 581 (1975)
Periodic cosine potential has been replaced by
∞
P
ln
exp[−β(φ − 2πn)2 /2]
n=−∞
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
24 / 29
Villain Approximation
Vvillain (φ) = β1 ln
∞
P
n=−∞
exp[−β(φ − 2πn)2 /2]
J. Villain, J. Phys. (Paris) 36, 581 (1975)
Periodic cosine potential has been replaced by
∞
P
ln
exp[−β(φ − 2πn)2 /2]
n=−∞
Its a lattice version of clock model or xy model
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
24 / 29
Villain Approximation
Vvillain (φ) = β1 ln
∞
P
n=−∞
exp[−β(φ − 2πn)2 /2]
J. Villain, J. Phys. (Paris) 36, 581 (1975)
Periodic cosine potential has been replaced by
∞
P
ln
exp[−β(φ − 2πn)2 /2]
n=−∞
Its a lattice version of clock model or xy model
Villain approximation predicts five-state clock model has two
tranistions one of them being KT transition.
J. V. Jose, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson,
Phys. Rev. B 16,1217 (1977)
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
24 / 29
MC simulation of Villain model
Figure: a) 5-state Villain Model b)six-state generalized clock model with p=2
c)4-state Villain model
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
25 / 29
Figure: 5-state clock model
Figure: 5-state villain Model
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
26 / 29
Conclusions
Numerical simulation suggests clock model has two transition for
q>4
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
27 / 29
Conclusions
Numerical simulation suggests clock model has two transition for
q>4
KT transition obtained from a small modification to Villain
model for q = 5
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
27 / 29
Conclusions
Numerical simulation suggests clock model has two transition for
q>4
KT transition obtained from a small modification to Villain
model for q = 5
KT transition is seen for q ≥ 6
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
27 / 29
Conclusions
Numerical simulation suggests clock model has two transition for
q>4
KT transition obtained from a small modification to Villain
model for q = 5
KT transition is seen for q ≥ 6
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
27 / 29
Future Direction
classical xy spin → classical Heisenberg spin
These solids can fit into a sphere
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
28 / 29
Thank You
Kingshuk Sarkar (Indian Institute of Science,Phase
Bangalore
Transition
560012)
in 2D q-state Clock Model
November 5, 2012
29 / 29