Phases with slowly decaying correlations Jacques Villain 1. The two-dimensional easy plane magnet 2. Superfluid helium films 3. Vortices and phase transition 4. Kosterlitz-Thouless theory and renormalisation group 5. The two-dimensional solid. 6. Thermal roughness of a crystal surface and roughening transition. 7. The two-dimensional isotropic magnet 8. The Ising model with competing interactions

The two-dimensional easy plane magnet Six= s cos fi Siy= s sin fi Nearest neighbour interaction Siz=0 XY model At low temperature: <| F(k)|2>=kBT/(Js2k2a2)

The two-dimensional XY model <cos (fi–fj)>= <exp[i(fi–fj)]>=exp[–<(fi–fj)2>] < Si. Sj>/s2=<cos (fi–fj)> (a/rij)T/Q No conventional long range order at non-vanishing T But « algebraic » correlations Infinite susceptibility at low temperature Therefore there is a phase transition (Wegner 1967)

Superfluid helium films Accessible to experimental investigation (Reppy) A macroscopic number of atoms is in the same state (Bose condensation). They have the same wave function Y(r)= |Y(r)| exp[-if(r)]. Mean current density A superfluid particle at r has velocity and kinetic energy : Energy of the superfluid:

Vortices and phase transition XY model Velocity field in a superfluid Energy density is proportional to 1/r 2 Energy of an isolated vortex: infinite

Vortex pairs Energy of a pair of radius R: B Energy of a pair of radius R: R Boltzmann factor: exp[–W(R)/kBT]=(R/a)–T0/T Diverges for T>T0/4. Transition by mixture of the pairs slightly below T0/4

Kosterlitz-Thouless theory and renormalisation group Principle: elimination degrees of freedom (pairs) of small radius - - + + Analogous to: i) Elimination of electrons in the Born-Openheimer approximation ii) Screening in the three-dimensional Coulomb gas But here one eliminates first pairs of radius<ae, then<ae2, etc. The minimum radius ael goes to infinity. Infinite (semi-)group

After l eliminations, 2 parametres: 1) The force 2pK(l)/r between 2 vortices à distance rel 2) The vortex « fugacity » y(l)>0 (If y=0, there is no vortex pair). In the low temperature phase: y(∞)=0 This implies that K be large enough. K>2/p dy/dl=(2–pK)y dK–1/ dl =4p3y2

The transition is continuous…. K y 0,02 0,04 0,06 0,08 0,6 0,7 0,4 0,8 1 The transition is continuous…. …. But there is a discontinuity of K(∞) In the superfluid, the superfluid density rs(T) jumps from a non-vanishing value to 0. No observable specific heat anomaly (essential singularity)

The two-dimensional solid. Atomic displacements u(r), Fourier transform uk Two-dimensional isotropic medium or triangular lattice: <u-ka u-kg> ∝ 1/k2 when k goes to 0. Bragg lines intensity: ∝ Debye-Waller factor exp(–k2<ui2>/3) =0 in two dimensions Bragg peaks are not delta functions!

Melting of a two-dimensional solid : dissociation of dislocation pairs B A b The shear modulus has a discontinuity! b a The Young modulus has a universal jump! The transition is continuous! (except for possible complications) But this transition does not produce a liquid phase …

To reach the liquid phase another (continuous) transition is necessary An unpairing of disclinations (Halperin, Nelson 1978)

Thermal roughness of a crystal surface and roughening transition. <(zi–zj)2> T ln(rij/a) Valid for a liquid surface (neglecting gravity) or for a crystal surface at a not too low temperature x y z crystal surface at low temperature : smooth! Roughening transition at TR TR depends on the surface orientation Critical behaviour analogous to K-Th. g(TR)a2/TR =pkB/2 Below TR the crystal has facets

JJ Metois and JC Heyraud T > 100 °C indium 20 < T < 40 °C à basse température: de plus en plus de facettes photographies JJ Metois and JC Heyraud CRMC2 Marseille 40 < T < 100 °C T < 10 °C 10 < T < 20 °C Melting temperature 156°C

The two-dimensional isotropic magnet Classical spins with 3 components Jij=J >0 if i and j are neighbours, Otherwise Jij =0. One can try a decoupling: Kosterlitz et Thouless (1973): we would conjecture that the xy model has a phase transition but the Heisenberg model does not”. a) If there are free vortices, correlations decay exponentially b) If the vortex energy is finite, there are free vortices at T m 0 c) Vortex structures with finite energy exist, e.g. Sz(r)=s (R2–r2)1/2/R, Sx(r)=(x/R)s, Sy(r)=(y/R)s, with r=(x,y) and r<R. Confirmed by renormalisation group calculations (Polyakov)

The Ising model with competing interactions J1>0 J0>0 J2<0 Ground state for J1<–2J2 For J1= –2J2 the ground state is degenerate and made of stripes of arbitrary width >2

The floating phase

Conclusion Many two-dimensional systems Magnets with easy magnetisation plane Helium films Two-dimensional solids Crystal surfaces Ising model with competing interactions have extremely strange properties: Power law decay of correlation functions Continuous transition with discontinuities Melting through two continuous transitions These properties are related to topological phenomena Their experimental observation is possible, not always easy

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