Dung-Hai Lee U.C. Berkeley

Quantum state that never condenses

Condense = develop some kind of order

As a solid develops order, some symmetry is broken.

Spin rotational symmetry is broken !

Ice crystal Superfluid

Neutron star Expanding universe

Examples of order

Metals are characterized by the Fermi surface

Metals do break any symmetry, but they are not stable at zero temperature. Metals always turn into some ordered states with symmetry breaking as T 0.

Different types of Fermi surface instability lead to different order.

Cooper instability superconductivity

Fermi surface nesting instability spin density wave, or charge density wave

Metal

Superconductivity

Charge density wave

Spin density wave

Landau’s paradigm

• Ordered state is characterized by the symmetry that is broken.• All ordered states originate from the metallic state due to Fermi surface instability.

Is it possible for a solid not to develop any order at zero

temperature ?

Insulators with integer filling factor are good candidates

Fermion band insulator Boson Mott insulator

Fermion Mott insulator

Mott insulator

Boson Mott insulator

Insulating due to repulsion between particles.

Examples of electron band insulator

C, Si, Ge, GaAs, …

YBa2Cu3O6 – the parent compound of high temperature superconductor

CuO2 sheet

An example of electron Mott insulator

An example of boson Mott insulator: optical lattice of neutral atoms

Greiner et al, Nature 02

Why are we interested in insulators ?Doping make them very useful !

Most of the time, doping make the particle mobile, hence can conduct.

Doped band insulator

A Silicon chip

Doping Mott insulators has produced many materials with interesting properties.

High Tc superconductors Colossal magneto-resistive materials

Doped YBa2Cu3O6 Doped LaMnO3

Doped Mott insulators

Is it possible that a solid remains insulating after

doping ?

Yes

An interesting fact: all insulators with fractional filling factor break some kind of symmetry hence exhibit some kind of order.

Antiferromagnet Dimmerization

fermion boson

Oshikawa’s theorem

If the system is insulating, and if the filling factor = p/q, the ground state is q-fold

degenerate.

Usually the required degeneracy is achieved by long range order.

Why is uncondensed insulator so rare at fractional filling ?

Can a fractional filled insulator exist without symmetry breaking ?

Oshikawa PRL 2000

It is generally believed that featureless insulators will have very unusual

properties.

Such as fractional-charge excitations …

Anderson’s spin liquid idea

Spin liquid is a featureless insulator (at half filling) with no long range order ! It has S=1/2 excitations (spinons).

+ + . . .

It exists in the parent state of high-temperature superconductors.

Resonating singlet patterns

Anderson, Science 1987

Condensed matter physicists have searched for such insulators for 20

years.

The usual search guide line is “frustration”.

?

Melts crystal order but never changes the C-M position preserve 3-fold degeneracy.

A new idea: symmetry protected uncondensed quantum state

Filling factor =1/3

The Quantum Hall effect

Rxx = VL /I; Rxy = VH /I

The fractional quantum Hall effect

Lee & Leinaas, PRL 2004

One example of this type of state is the fractional quantum Hall liquid

Another example is the quantum dimer liquid

Moessner & Sondhi, PRL, 2001

All existing models in the literature that exhibit uncondensed quantum state conserve the center-of-mass position and momentum.