Outline• Concept of geom etric frustration and exam ples.
• G eneral therm odynam ic properties of frustrated m agnets.
• G round state degeneracy.
• E nergy barriers and order by disorder.
• Frustration in quantum m agnets.
• QQuantum
uantum states: spin liquids and valence bond order.
• D im ensional reduction and fractional excitations.
• K lein m odels, checkerboard lattice and quantum spinspin - ice.
• E lectrodynam ics of spinons.
• Perturbations aw ay from the optim ally frustrated point.
• Sum m ary

G eom etric Frustration
Geometric frustration arises when the spins cannot satisfy all the interactions
simultaneously:

FMAF
AFAFAFAF

AF

AF

Non-frustrated
case

Geometricalfrustration

?

AF?
AFAF
Frustration induced
by disorderE xam ples of frustrated lattices
1D2D

NaCu2O2

Azurite: Cu3(CO3)2(OH)2

CuFeO2

SrCu2(BO3)2
SrCr9Ga3O19

3D

Dy2Ti2O7

SrL2O4

G eneral Susceptibility

G round State D egeneracy
Classical n.n. AF Heisenberg Hamiltonian:For q mutually interacting spins:

where.

The minimum energy condition L=0imposes three constraints Lx= Ly = Lz for
spins with n=3 components. The numberof constrains is K=n. The number ofdegrees of freedom excluding the global
rotations is F=q (n-1)-n. So, up to global
rotations, the dimension of the groundstate manifold is D = F-K=q(n-1)-2n.
Note that for n=3, we get D=0 for thetriangle (q=3) and D=2 for the tetrahedron(q=4).
φθ

q=2

q=3

q=4-R. Moessner, Can. J. Phys. 79, 1283 (2001).

G round State D egeneracy
For a lattice built up of frustrated units theHamiltonian can be written as:

where α runs over all N units. If the latticeconsists of corner-sharing units of q spinswith n components we have:

Here we have assumed that the constraints
are independent. If n=3, the dimension of
the ground state manifold becomes
extensive at q=4, i.e., the pyrochlore
antiferromagnet has an extensive groundstate dimension. In general, D becomes
extensive for n>3 q>4.

It turns out that the Kagome lattice also has
an extensive D because of dependent
constraints.

E nergy barriers and order by disorder
The topology of the ground state is another important property. Is it possible to connect
two ground states continuously or there are energy barriers separating them. When thetemperature becomes finite, the state of equilibrium is the one that minimizes the freeenergy. Whereas the internal energy of every ground state is the same, the number ofaccessible low energy states (entropy) may be much higher for one of them. In thissituation the system effectively spends all the time around this particular ground state.
In general, this is an ordered state and its selection is known as order by disorderbecause it is stabilized by thermal fluctuations.
Phase diagram for n component spins
arranged in corner-sharing units of q.
Ordered states, provided they exist,
are selected for q≤4 and n≤3 with the
exception of q=4 and n=3 which ismarginal.

R. Moessner et. al., Phys. Rev. Lett. 80, 2929
(1998).

Quantum Frustration
Quantum fluctuations can also stabilize an ordered state out of the mechanism oforder by disorder. In some cases, this ordered state has no classical analog. Let us
consider the following simple example of a spin 1/2 Heisenberg model on a chainwith nearest and next-nearest neighbor interactions.

J2J2J22J2J2J1J1αJ1J11J23J2J2For J1=2J2, this Hamiltonian can be expressed as:

projects the spins ofwhere α denotes a triangular unit. The operator
the triangular plaquette α onto thesubspace. This model is known as
the Majumdar-Ghosh Hamiltonian.
-C. K. Majumdar and D. K. Ghosh, J. Math. Phys.10, 1399 (1969).

Quantum Frustration
Therefore a state containing one single per triangular plaquette is a ground state.
There are only two possible ground states:

This ordered quantum state breaks spontaneously the Z2 symmetry of HMG and it isknow as valence bond solid or crystal. The elementary low energy excitations aredeconfined spinons (S=1/2 excitations) that carry a topological defect called soliton:
SolitonAnti-Soliton

Confinem ent of fractional excitations in higher D
Valence bond crystal ground state of a 2D Hamiltonian (broken Z4 symmetry):Confinem ent in 2D

Quantum Frustration in 2D
Klein generalized this idea to other lattices (D. J. Klein, J. Phys. A: Math. Gen. 15, 661(1982). Let’s consider the following case for thesquare lattice:

J1J2αFor J2=J1/2 and K= J1/8 we obtain:
C. D. Batista and S. A. Trugman, Phys. Rev. Lett. 93, 217202 (2004).

M assive ground state degeneracy

Where L is a linear
dimension of thesystem.

D im ensional Reduction
Local Z2 order parameter on each diagonal zig-zag chain:

There is no effective
coupling between orderparameters on different
chains!

Where L is a linear
dimension of thesystem.

The dimensionality is spontaneously reduced in the low energy sector of HK !
-C. D. Batista and Z. Nussinov, Phys. Rev. B 72, 045137 (2004).

D econfined fractional excitations in 2D

D econfined fractional excitations in 2D

Perturbations

T
One dimensional behavior andspinon excitations

QPTg

Checkerboard Latticewhere. We can rewrite H as:αFor K=Kc= 4J/5 , H becomes a Klein Hamiltonian:Z. Nussinov, C. D. Batista, B. Normand and S. A. Trugman, cond-mat/0602528.

Six V ertex Representation
VertexRepresentation

Ice rule leads tosix zero-divergence
configurations

LineRepresentation
The six vertex model is exactly solvable!
[R. J. Baxter, Exactly Solved Models
in Statistical Mechanics, (AcademicPress, London, 1982).]
We will assume that the dimer
coverings, although non-orthogonal,
they are linearly independent.

D econfined Spinon E xcitations

E ntropy D riven Critical Correlations
VertexRepresentation
We denote the sites of the dual
lattice L D by r and the link
between the sites r and r+eµ by
the pair r,µ where eµ is aprimitive vector of L D . We
introduce the quantity
onthe links of the dual lattice (sitesof the original lattice L). The
charge is +1/2 if the arrow flows
from A to B and -1/2 if the arrowflows from B to A..

A sublattice of dual L
B sublattice of dual L

The local zero
divergence condition
can now be written as:e1Then, the operators:

r e2
1/2

-1/2

1/2

-1/2

are the local U(1)symmetrytransformations of thetheory underconsideration.

L. B. Ioffe and A. Larkin, Phys. Rev. B 40, 6941 (1989); D. A. Huse et al, Phys. Rev. Lett. 91, 167004 (2003).

E ntropy D riven Critical Correlations
We introduce now an
oriented angular momentum
variable on the links of L D :

e1
r e2

where σr=1(-1) if r ∈ A (B ).
By introducing the lattice
gradient:

We can rewrite the ice-rules as an explicit
zero divergence condition on the vector
field:The last equation is theGauss law ofelectrodynamics in absenceof external charges:

Then, we can identify thevariable with an electricfield Eµ(r).
Vertex configurations that violate the icerule should then carry a charge:

E ntropy D riven Critical Correlations
We now conjecture that thelong range fluctuations of thecoarse grained field E isgoverned by the probability
distribution:

e1
r e2.
Configurations that locallyminimize the (coarsegrained) field strength
maximize the number of flippableplaquettes. The last equation implies thatthe gauge field is in a Coulomb phase. It is
now straightforward to deduce the long
distance correlator:

The introduction of spinons(monomers) and defects
violates the local zero
divergence constraint. The
corresponding charge is
given by the divergence ofE:E ntropic Coulom b Interaction Betw een Spinons
The spinon-spinon interaction
is determined by the change inthe total entropy as a functionof the distance between two
monomers inserted in thelattice. The critical dipolarfluctuations of the underlying
dimer field produce aneffective Coulomb interaction
which arises from the local
conservation law of zero
divergence:Plasm a of Spinons

Q= 1

Q=-2



Q= 1

Perturbations and D im er M odels

This Rokhsar-Kivelson (RK) term is the minimal process that connects twodifferent dimer coverings. The low energy theory in the presence of aperturbation that moves the system away from the Klein point is a dimer model
on a non-orthogonal dimer basis.

Perturbations and D im er M odels
The RK process cancels out exactly for the most natural perturbation: K≠Kc=
4J/5. In this case, the next minimal processes that contributes with a non-zero
matrix element are:

Therm ally D riven D econfinem entTDilute plasmaof spinonsVBC I

VBC IIKc

Classical T=0 Critical PointKSum m ary-Strongly frustrated systems have a very high low temperature entropy. There is an optimallyfrustrated case for which this entropy is accumulated at T=0, i.e., the ground state becomes massively
degenerated.
- In this situations, thermal and quantum fluctuations play a fundamental role in determining the low
energy physics of the system under consideration.
- Thermal fluctuations can select an ordered state out of the mechanism of “order by disorder”.
- Quantum fluctuations can even select a “ quantum” ordered state with no classical analog like thevalence bond crystal.
- The massive ground state degeneracy can also appear in quantum systems even when the quantum
fluctuations are fully incorporated. These situations can lead to extremely interesting low energy
phenomena like: dimensional reduction, fractional excitations in higher dimensions, an emergent
electrodynamics for spinons and defects.
- Perturbations away from the optimally frustrated point are usually relevant for stabilizing someparticular ordering at T=0.