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. <i;j> P P HXY = −J ~si · ~sj = −J cos θi − θj continuous sym. <i;j> <i;j> 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. <i;j> P P HXY = −J ~si · ~sj = −J cos θi − θj continuous sym. <i;j> <i;j> symmetry Ising model; si = ±1; x y XY model; ~si = (si ; si ) = (cos θi ; sin θi ) P HIsing = −J si sj discrete sym. <i;j> P P HXY = −J ~si · ~sj = −J cos θi − θj continuous sym. <i;j> <i;j> Ising modeld ` = 1 XY modeld ` = 2 discrete symmetry d = 1: m(T ; 0) = 0 (T > 0) 8 < 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) 8 < 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) 8 < 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 > ] 1 8 −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 > ] 1 8 −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 > ] 1 8 −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 > ] 1 8 −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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 (rΦ ) + (Φ ) + eff ` 0 2 ` 2 ` 1 2 ) K 2 aτ + 6 A u 2 + (rΦt ) + (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 6ajτj 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 6ajτj 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) -