Topological phase transitions
Beyond the Landau paradigm Beyond the Landau paradigm
Topological phase transitions ” for topological phase transitions and topological phases of matter ”
Nobel Prize in Physics in 2016 Nobel Prize in Physics in 2016
” for topological phase transitions and topological phases of matter ” David Thouless Michael Kosterlitz Duncan Haldane U. of Washington Brown U. Princeton U. Berezinskii - Kosterlitz - Thouless transition
BKT transition
Vadim Berezinskii Michael Kosterlitz David Thouless
topological phase transitions BKT transition
Vadim Berezinskii Michael Kosterlitz David Thouless
topological phase transitions
Berezinskii - Kosterlitz - Thouless transition topological phase transitions
Berezinskii - Kosterlitz - Thouless transition
BKT transition
Vadim Berezinskii Michael Kosterlitz David Thouless Ising model, si = ±1,
x y XY model, ~si = (si , si ) = (cos θi , sin θi )
P HIsing = −J si sj discrete sym.
P P HXY = −J ~si · ~sj = −J cos θi − θj continuous sym.
Ising modeld ` = 1
XY modeld ` = 2
symmetry Ising modeld ` = 1
XY modeld ` = 2
symmetry
Ising model, si = ±1,
x y XY model, ~si = (si , si ) = (cos θi , sin θi )
P HIsing = −J si sj discrete sym.
P P HXY = −J ~si · ~sj = −J cos θi − θj continuous sym.
Ising model, si = ±1,
x y XY model, ~si = (si , si ) = (cos θi , sin θi )
P HIsing = −J si sj discrete sym.
P P HXY = −J ~si · ~sj = −J cos θi − θj continuous sym.
Ising modeld ` = 1
XY modeld ` = 2 discrete symmetry
d = 1: m(T , 0) = 0 (T > 0)
0, T > Tc d ≥ 2: m(T , 0) = 6= 0, T < Tc Mermin - Wagner theorem (’66)
d = 1, 2, m~ (T , 0) = 0 (T > 0)
0, T > Tc d ≥ 3, m~ (T , 0) = 6= 0, T < Tc
continuous symmetry continuous symmetry Mermin - Wagner theorem (’66)
d = 1, 2, m~ (T , 0) = 0 (T > 0)
0, T > Tc d ≥ 3, m~ (T , 0) = 6= 0, T < Tc no long-range order in two-dimensional XY- model
no spontaneous symmetry breaking in two-dimensional XY- model no spontaneous symmetry breaking in two-dimensional XY- model
no long-range order in two-dimensional XY- model discrete symmetry Ising model d > 1
Γ(r, T , h = 0) = lim [ < s~r 0 s~r 0+~r > − < s~r 0 > < s~r 0+~r > ] ∞
−r/ξ(T ) e , T → T r d−2+η c Γ(r, T , h = 0) ∼ 1 , T = T r d−2+η c
correlation function Ising model d > 1
Γ(r, T , h = 0) = lim [ < s~r 0 s~r 0+~r > − < s~r 0 > < s~r 0+~r > ] ∞
−r/ξ(T ) e , T → T r d−2+η c Γ(r, T , h = 0) ∼ 1 , T = T r d−2+η c
correlation function discrete symmetry Γ(r, T , h = 0) = lim [ < s~r 0 s~r 0+~r > − < s~r 0 > < s~r 0+~r > ] ∞
−r/ξ(T ) e , T → T r d−2+η c Γ(r, T , h = 0) ∼ 1 , T = T r d−2+η c
correlation function discrete symmetry Ising model d > 1 correlation function discrete symmetry Ising model d > 1
Γ(r, T , h = 0) = lim [ < s~r 0 s~r 0+~r > − < s~r 0 > < s~r 0+~r > ] ∞
−r/ξ(T ) e , T → T r d−2+η c Γ(r, T , h = 0) ∼ 1 , T = T r d−2+η c continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2 =⇒ m~ 0 = (m1,0, 0) = A e~1
I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ] I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2 K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt
correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) > correlation functions continuous symmetry d > 2
m~ = m1 e~1 + m2 e~2 = A cos θ e~1 + A sin θ e~2
I state with spontaneously broken continuous symmetry: q 6a|τ| A = A = u , θ = θ = 0 =⇒ m~ 0 = (m1,0, 0) = A e~1 I consider a small fluctuation around this state: m~ 0 −→ m~ = A + Φ`(~r) e~1 + Φt (~r) e~2
I Φ`(~r) - longitudinal fluctuation, Φt (~r) - transverse fluctuation I Gaussian Heff [m~ ]
Z ( 1 2 + K 2 aτ + 2 A u 2 H [Φ , Φt ] = G + d~r (∇Φ ) + (Φ ) + eff ` 0 2 ` 2 `
1 2 ) K 2 aτ + 6 A u 2 + (∇Φt ) + (Φt ) 2 2
I correlation functions:
0 00 0 00 Γ`,`(~r , ~r ) = < Φ`(~r )Φ`(~r ) >
0 00 0 00 Γt,t (~r , ~r ) = < Φt (~r )Φt (~r ) >
K 1 2 I 2 = aτ + 2 A u ξ` K 1 2 2 = aτ + 6 A u ξt K aτ , τ > 0 = 2 0 , τ < 0 ξt
ξt = ∞ for τ < 0
η Γt,t (r; T < Tc ) ∼ 1/r power law decay
K aτ , τ > 0 = 2 2a|τ| , τ < 0 ξ` ξt = ∞ for τ < 0
η Γt,t (r; T < Tc ) ∼ 1/r power law decay
K aτ , τ > 0 = 2 2a|τ| , τ < 0 ξ`
K aτ , τ > 0 = 2 0 , τ < 0 ξt K aτ , τ > 0 = 2 2a|τ| , τ < 0 ξ`
K aτ , τ > 0 = 2 0 , τ < 0 ξt
ξt = ∞ for τ < 0
η Γt,t (r; T < Tc ) ∼ 1/r power law decay Yes, there is a topological phase transition.
XY model, d=2
V. Berezinskii (’72), D. Thouless and J. Kosterlitz (’72, ’73)
Is there anything interesting going on in two-dimensional XY model ? XY model, d=2
V. Berezinskii (’72), D. Thouless and J. Kosterlitz (’72, ’73)
Is there anything interesting going on in two-dimensional XY model ?
Yes, there is a topological phase transition. Vortices in two-dimensional XY-model Vortices in two-dimensional XY-model Vortices in two-dimensional XY-model Vortices in two-dimensional XY-model spin configurations that cannot be smoothly transformed into ground state configuration
vortices m topological defects vortices m
topological defects
spin configurations that cannot be smoothly transformed into ground state configuration spin wave vortex
ground state configurations spin wave vortex
ground state configurations spin wave vortex
ground state configurations vortex
ground state configurations
spin wave ground state configurations
spin wave vortex ground state configurations
spin wave vortex ground state configurations
spin wave vortex continuum limit ∂θ(~r) θ(~r + ae~i ) − θ(~r) −→ a , i = 1, 2 ∂xi
X Z a2 −→ d~r ~r
X X HXY = −J ~si · ~sj = −J cos θi − θj ≈
NzJ J X 2 J X 2 ≈ − + (θi − θj ) + ··· = E0 + (θ(~r + ~a) − θ(~r)) + ... 2 2 2
X Z a2 −→ d~r ~r
X X HXY = −J ~si · ~sj = −J cos θi − θj ≈
NzJ J X 2 J X 2 ≈ − + (θi − θj ) + ··· = E0 + (θ(~r + ~a) − θ(~r)) + ... 2 2 2
continuum limit X Z a2 −→ d~r ~r
X X HXY = −J ~si · ~sj = −J cos θi − θj ≈
NzJ J X 2 J X 2 ≈ − + (θi − θj ) + ··· = E0 + (θ(~r + ~a) − θ(~r)) + ... 2 2 2
continuum limit ∂θ(~r) θ(~r + ae~i ) − θ(~r) −→ a , i = 1, 2 ∂xi X X HXY = −J ~si · ~sj = −J cos θi − θj ≈
NzJ J X 2 J X 2 ≈ − + (θi − θj ) + ··· = E0 + (θ(~r + ~a) − θ(~r)) + ... 2 2 2
continuum limit ∂θ(~r) θ(~r + ae~i ) − θ(~r) −→ a , i = 1, 2 ∂xi
X Z a2 −→ d~r ~r I minimizing configuration : ∆θ(~r) = 0 I
I lim < ~s(~r) >h = 0 for T > 0 h→0 I I correlation function : h i Γ(~r) = < ~s(~r) · ~s(~0) > = < cos θ(~r) − θ(~0) > = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2
Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q I
I lim < ~s(~r) >h = 0 for T > 0 h→0 I I correlation function : h i Γ(~r) = < ~s(~r) · ~s(~0) > = < cos θ(~r) − θ(~0) > = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2
Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q
I minimizing configuration : ∆θ(~r) = 0 I I correlation function : h i Γ(~r) = < ~s(~r) · ~s(~0) > = < cos θ(~r) − θ(~0) > = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2
Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q
I minimizing configuration : ∆θ(~r) = 0 I
I lim < ~s(~r) >h = 0 for T > 0 h→0 h i = < cos θ(~r) − θ(~0) > = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2
Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q
I minimizing configuration : ∆θ(~r) = 0 I
I lim < ~s(~r) >h = 0 for T > 0 h→0 I I correlation function : Γ(~r) = < ~s(~r) · ~s(~0) > h i = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2
Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q
I minimizing configuration : ∆θ(~r) = 0 I
I lim < ~s(~r) >h = 0 for T > 0 h→0 I I correlation function : Γ(~r) = < ~s(~r) · ~s(~0) > = < cos θ(~r) − θ(~0) > I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2
Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q
I minimizing configuration : ∆θ(~r) = 0 I
I lim < ~s(~r) >h = 0 for T > 0 h→0 I I correlation function : h i Γ(~r) = < ~s(~r) · ~s(~0) > = < cos θ(~r) − θ(~0) > = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 Z J 2 J X 2 2 H ≈ E0 + d~r (∇θ) = E0 + q |θ˜ | XY 2 2V ~q ~q
I minimizing configuration : ∆θ(~r) = 0 I
I lim < ~s(~r) >h = 0 for T > 0 h→0 I I correlation function : h i Γ(~r) = < ~s(~r) · ~s(~0) > = < cos θ(~r) − θ(~0) > = < Re ei(θ(~r)−θ(~0)) > − 1 <(θ(~r)−θ(~0))2> − 1 g(r) = e 2 = e 2 I x2 2 2 1 R − ix 1 R − 1 ( x −iσ)2 − σ − σ I √ dxe 2σ2 e = √ dxe 2 σ e 2 = e 2 2πσ2 2πσ2 K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 > V → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | P[θ] ∼ e ~q 0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q 1 0 00 = − C2(~r − ~r ) K
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 0 00 0 00 1 X ei(~q·(~r −~r ) 1 Z dq~ ei~q·(~r −~r ) = → VK q2 K (2π)2 q2 ~q 2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q 2 ˜ → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
∆Cd (~r) = δ(~r) Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1 Z Z ~ → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
∆Cd (~r) = δ(~r) 2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V 2−d d−1 dCd r → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
Cd (~r) = Cd (r) r 2−d → Cd (r) = + c Ωd−1(2 − d)
< (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
dC C (~r) = C (r) → Ω r d−1 d = 1 d d d−1 dr < (θ(~r) − θ(~0))2 >
K P 2 ˜ 2 − 2V q |θ~q | ~q V P[θ] ∼ e → < θ˜ 0 θ˜ 00 > = δ 0 00 ~q ~q K q02 ~q ,−~q
0 00 1 X X i(~q 0·~r 0+~q 00·~r 00) < θ(~r )θ(~r ) > = e < θ˜ 0 θ˜ 00 > = V 2 ~q ~q ~q 0 ~q 00 i(~q·(~r 0−~r 00) Z i~q·(~r 0−~r 00) 1 X e 1 dq~ e 1 0 00 = → = − C2(~r − ~r ) VK q2 K (2π)2 q2 K ~q
2 ˜ ∆Cd (~r) = δ(~r) → −q Cd (q~) = 1
Z Z ~ ∆Cd (~r) = δ(~r) → d~r∆Cd (~r) = dS · ∇Cd (~r) = 1 V ∂V
2−d d−1 dCd r Cd (~r) = Cd (r) → Ωd−1 r = 1 → Cd (r) = + c dr Ωd−1(2 − d) d = 2 log(r/a) C2(r) = Kπ
1 − 1 <(θ(~r)−θ(~0))2> − log(r/a) a 2Kπ Γ(r; T , h = 0) = e 2 = e K2π = r
1 1 k T Γ(r; T , h = 0)| ∼ → η = = B criticalpoint r η 2Kπ 2πJ
2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d) log(r/a) C2(r) = Kπ
1 − 1 <(θ(~r)−θ(~0))2> − log(r/a) a 2Kπ Γ(r; T , h = 0) = e 2 = e K2π = r
1 1 k T Γ(r; T , h = 0)| ∼ → η = = B criticalpoint r η 2Kπ 2πJ
2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d)
d = 2 1 − 1 <(θ(~r)−θ(~0))2> − log(r/a) a 2Kπ Γ(r; T , h = 0) = e 2 = e K2π = r
1 1 k T Γ(r; T , h = 0)| ∼ → η = = B criticalpoint r η 2Kπ 2πJ
2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d)
d = 2 log(r/a) C2(r) = Kπ a 1 = 2Kπ r
1 1 k T Γ(r; T , h = 0)| ∼ → η = = B criticalpoint r η 2Kπ 2πJ
2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d)
d = 2 log(r/a) C2(r) = Kπ
− 1 <(θ(~r)−θ(~0))2> − log(r/a) Γ(r; T , h = 0) = e 2 = e K2π 1 1 k T Γ(r; T , h = 0)| ∼ → η = = B criticalpoint r η 2Kπ 2πJ
2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d)
d = 2 log(r/a) C2(r) = Kπ
1 − 1 <(θ(~r)−θ(~0))2> − log(r/a) a 2Kπ Γ(r; T , h = 0) = e 2 = e K2π = r 1 k T → η = = B 2Kπ 2πJ
2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d)
d = 2 log(r/a) C2(r) = Kπ
1 − 1 <(θ(~r)−θ(~0))2> − log(r/a) a 2Kπ Γ(r; T , h = 0) = e 2 = e K2π = r
1 Γ(r; T , h = 0)| ∼ criticalpoint r η 2 r 2−d − a2−d < (θ(~r) − θ(~0))2 > = K Ωd−1 (2 − d)
d = 2 log(r/a) C2(r) = Kπ
1 − 1 <(θ(~r)−θ(~0))2> − log(r/a) a 2Kπ Γ(r; T , h = 0) = e 2 = e K2π = r
1 1 k T Γ(r; T , h = 0)| ∼ → η = = B criticalpoint r η 2Kπ 2πJ ⇓
low - temperature phase with pairs of bound vortices (T < TBKT )
high - temperature phase with free vortices (T > TBKT ) high - temperature phase with free vortices (T > TBKT )
⇓
low - temperature phase with pairs of bound vortices (T < TBKT ) I at low temperatures (T < TBKT ) there are pairs of bound vortices and Γ(r; T ) ∼ 1/r η(T )
quasi-long-range order I √ const 1 ξ(T ) ∝ exp , η(TBKT ) = 4 T −TBKT
π J TBKT = 2kB
I at high temperatures (T > TBKT ) there are free vortices in the system and Γ(r; T ) ∼ e−r/ξ(T ) I √ const 1 ξ(T ) ∝ exp , η(TBKT ) = 4 T −TBKT
π J TBKT = 2kB
I at high temperatures (T > TBKT ) there are free vortices in the system and Γ(r; T ) ∼ e−r/ξ(T )
I at low temperatures (T < TBKT ) there are pairs of bound vortices and Γ(r; T ) ∼ 1/r η(T )
quasi-long-range order I at high temperatures (T > TBKT ) there are free vortices in the system and Γ(r; T ) ∼ e−r/ξ(T )
I at low temperatures (T < TBKT ) there are pairs of bound vortices and Γ(r; T ) ∼ 1/r η(T )
quasi-long-range order I √ const 1 ξ(T ) ∝ exp , η(TBKT ) = 4 T −TBKT
π J TBKT = 2kB vortices (topological defects) ! y 1 1 1 ∇θ(~r) = q ∇ϕ(~r) = q − , = x2 y 2 x y 2 1 + x 1 + x y x q q = q − , = (− sin ϕ, cos ϕ) = e~ϕ x2 + y 2 x2 + y 2 r r
I I I q dθ(~r) = d~` · ∇θ(~r) = Rdϕe~ϕ · e~ϕ = 2πq R
Z 2 Z Z J 2 Jq 1 2 L Evortex = d~r |∇θ(~r)| = dϕ dr = πJq log 2 2 r a
single vortex y θ(~r) = q ϕ(~r) + ϕ0 , ϕ(~r) = arctan , q = ±1, ±2,... x q = e~ϕ r
I I I q dθ(~r) = d~` · ∇θ(~r) = Rdϕe~ϕ · e~ϕ = 2πq R
Z 2 Z Z J 2 Jq 1 2 L Evortex = d~r |∇θ(~r)| = dϕ dr = πJq log 2 2 r a
single vortex y θ(~r) = q ϕ(~r) + ϕ0 , ϕ(~r) = arctan , q = ±1, ±2,... x
! y 1 1 1 ∇θ(~r) = q ∇ϕ(~r) = q − , = x2 y 2 x y 2 1 + x 1 + x y x q = q − , = (− sin ϕ, cos ϕ) x2 + y 2 x2 + y 2 r I I I q dθ(~r) = d~` · ∇θ(~r) = Rdϕe~ϕ · e~ϕ = 2πq R
Z 2 Z Z J 2 Jq 1 2 L Evortex = d~r |∇θ(~r)| = dϕ dr = πJq log 2 2 r a
single vortex y θ(~r) = q ϕ(~r) + ϕ0 , ϕ(~r) = arctan , q = ±1, ±2,... x
! y 1 1 1 ∇θ(~r) = q ∇ϕ(~r) = q − , = x2 y 2 x y 2 1 + x 1 + x y x q q = q − , = (− sin ϕ, cos ϕ) = e~ϕ x2 + y 2 x2 + y 2 r r Z 2 Z Z J 2 Jq 1 2 L Evortex = d~r |∇θ(~r)| = dϕ dr = πJq log 2 2 r a
single vortex y θ(~r) = q ϕ(~r) + ϕ0 , ϕ(~r) = arctan , q = ±1, ±2,... x
! y 1 1 1 ∇θ(~r) = q ∇ϕ(~r) = q − , = x2 y 2 x y 2 1 + x 1 + x y x q q = q − , = (− sin ϕ, cos ϕ) = e~ϕ x2 + y 2 x2 + y 2 r r
I I I q dθ(~r) = d~` · ∇θ(~r) = Rdϕe~ϕ · e~ϕ = 2πq R Jq2 Z Z 1 L = dϕ dr = πJq2 log 2 r a
single vortex y θ(~r) = q ϕ(~r) + ϕ0 , ϕ(~r) = arctan , q = ±1, ±2,... x
! y 1 1 1 ∇θ(~r) = q ∇ϕ(~r) = q − , = x2 y 2 x y 2 1 + x 1 + x y x q q = q − , = (− sin ϕ, cos ϕ) = e~ϕ x2 + y 2 x2 + y 2 r r
I I I q dθ(~r) = d~` · ∇θ(~r) = Rdϕe~ϕ · e~ϕ = 2πq R
Z J 2 Evortex = d~r |∇θ(~r)| 2 single vortex y θ(~r) = q ϕ(~r) + ϕ0 , ϕ(~r) = arctan , q = ±1, ±2,... x
! y 1 1 1 ∇θ(~r) = q ∇ϕ(~r) = q − , = x2 y 2 x y 2 1 + x 1 + x y x q q = q − , = (− sin ϕ, cos ϕ) = e~ϕ x2 + y 2 x2 + y 2 r r
I I I q dθ(~r) = d~` · ∇θ(~r) = Rdϕe~ϕ · e~ϕ = 2πq R
Z 2 Z Z J 2 Jq 1 2 L Evortex = d~r |∇θ(~r)| = dϕ dr = πJq log 2 2 r a −→ ∆F < 0 −→ free vortices
2kB T < πJ −→ ∆F > 0 −→ pairs of bound vortices
πJ unbinding transition TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ free vortices
2kB T < πJ −→ ∆F > 0 −→ pairs of bound vortices
πJ unbinding transition TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 2kB T < πJ −→ ∆F > 0 −→ pairs of bound vortices
πJ unbinding transition TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 −→ free vortices −→ ∆F > 0 −→ pairs of bound vortices
πJ unbinding transition TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 −→ free vortices
2kB T < πJ −→ pairs of bound vortices
πJ unbinding transition TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 −→ free vortices
2kB T < πJ −→ ∆F > 0 πJ unbinding transition TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 −→ free vortices
2kB T < πJ −→ ∆F > 0 −→ pairs of bound vortices πJ TBKT = 2kB
L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 −→ free vortices
2kB T < πJ −→ ∆F > 0 −→ pairs of bound vortices
unbinding transition L ∆F = ∆E − T ∆S = (πJ − 2k T ) ln B a
2kB T > πJ −→ ∆F < 0 −→ free vortices
2kB T < πJ −→ ∆F > 0 −→ pairs of bound vortices
πJ unbinding transition TBKT = 2kB T < TBKT T > TBKT
Γ(r; T ) ∼ r −η(T ) Γ(r; T ) ∼ e−r/ξ(T )
vortex - antivortex pair
|~r1 − ~r2| Ev,av ∼ −q1 q2 ln a vortex - antivortex pair
|~r1 − ~r2| Ev,av ∼ −q1 q2 ln a
T < TBKT T > TBKT
Γ(r; T ) ∼ r −η(T ) Γ(r; T ) ∼ e−r/ξ(T ) (Kosterlitz, Thouless, Halperin, Nelson, Young ’78, ’79)
How does a (two-dimensional) crystals turn into a (two-dimensional) liquid ?
It goes via dissociation of topological defects
two-dimensional melting (KTHNY theory) How does a (two-dimensional) crystals turn into a (two-dimensional) liquid ?
It goes via dissociation of topological defects
two-dimensional melting (KTHNY theory)
(Kosterlitz, Thouless, Halperin, Nelson, Young ’78, ’79) It goes via dissociation of topological defects
two-dimensional melting (KTHNY theory)
(Kosterlitz, Thouless, Halperin, Nelson, Young ’78, ’79)
How does a (two-dimensional) crystals turn into a (two-dimensional) liquid ? two-dimensional melting (KTHNY theory)
(Kosterlitz, Thouless, Halperin, Nelson, Young ’78, ’79)
How does a (two-dimensional) crystals turn into a (two-dimensional) liquid ?
It goes via dissociation of topological defects dislocations
disclinations
dislocations and disclinations disclinations
dislocations and disclinations
dislocations disclinations
dislocations and disclinations
dislocations disclinations
dislocations and disclinations
dislocations dislocations and disclinations
dislocations
disclinations
T = T1 T = T2
crystal phase =⇒ hexatic phase =⇒ liquid phase translational order translational disorder translational disorder orientational order orientational order orientational disorder
pairs of dislocations unbound dislocations unbound disclinations pairs of disclinations
two-dimensional melting (KTHNY theory) T = T1 T = T2
crystal phase =⇒ hexatic phase =⇒ liquid phase translational order translational disorder translational disorder orientational order orientational order orientational disorder
pairs of dislocations unbound dislocations unbound disclinations pairs of disclinations
two-dimensional melting (KTHNY theory) two-dimensional melting (KTHNY theory)
T = T1 T = T2 crystal phase =⇒ hexatic phase =⇒ liquid phase translational order translational disorder translational disorder orientational order orientational order orientational disorder pairs of dislocations unbound dislocations unbound disclinations pairs of disclinations quantum antiferromagnetic spin chains (d=1) half-integer versus integer spin saga
s = 1/2 (Bethe ’31 , Lieb, Schultz, Mattis ’61, ... )
I no long-range order, gapless excitations (no gap) I power-law decay of correlations
s = 1 (Haldane ’83, ’85)
I no long-range order, non-zero gap I exponential decay of correlations
Duncan Haldane s = 1/2 (Bethe ’31 , Lieb, Schultz, Mattis ’61, ... )
I no long-range order, gapless excitations (no gap) I power-law decay of correlations
s = 1 (Haldane ’83, ’85)
I no long-range order, non-zero gap I exponential decay of correlations
Duncan Haldane quantum antiferromagnetic spin chains (d=1) half-integer versus integer spin saga (no gap) I power-law decay of correlations
s = 1 (Haldane ’83, ’85)
I no long-range order, non-zero gap I exponential decay of correlations
Duncan Haldane
quantum antiferromagnetic spin chains (d=1) half-integer versus integer spin saga
s = 1/2 (Bethe ’31 , Lieb, Schultz, Mattis ’61, ... )
I no long-range order, gapless excitations s = 1 (Haldane ’83, ’85)
I no long-range order, non-zero gap I exponential decay of correlations
Duncan Haldane
quantum antiferromagnetic spin chains (d=1) half-integer versus integer spin saga
s = 1/2 (Bethe ’31 , Lieb, Schultz, Mattis ’61, ... )
I no long-range order, gapless excitations (no gap) I power-law decay of correlations Duncan Haldane
quantum antiferromagnetic spin chains (d=1) half-integer versus integer spin saga
s = 1/2 (Bethe ’31 , Lieb, Schultz, Mattis ’61, ... )
I no long-range order, gapless excitations (no gap) I power-law decay of correlations
s = 1 (Haldane ’83, ’85)
I no long-range order, non-zero gap I exponential decay of correlations