1. 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

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

3. 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)

4. 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:

5. 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

6. 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

7. 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

8. 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

9. 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)

10. 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!

11. Melting of a two-dimensional solid : dissociation of dislocation pairsB 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 …

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

14. 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

15. 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

16. 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)

17. 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

18. The floating phase

19. 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

20. Biblio
Kosterlitz J M (2016). Kosterlitz–Thouless physics: a review of key issues. Rep. Prog. Phys. 79, Balibar S., Alles H., Parshin Alexander Ya. (2005) The surface of helium crystals. Rev. Mod. Phys. 77, 317 Einstein T. L Equilibrium Shape of Crystals Handbook of Crystal Growth, Fundamentals, edited by T. Nishinaga