Breaking electrons apart in condensed
matter physics
T. Senthil
(MIT)
Group at MIT
Other main collaborators
Predrag Nikolic
Dinesh Raut
O. Motrunich (now at KITP)
A. Vishwanath
L. Balents (UCSB)
Matthew P.A Fisher (KITP)
Subir Sachdev (Yale)
D. Ivanov (Zurich)
Conventional condensed matter physics:
Landau’s 2 great ideas
1. Theory of fermi fluids
(electrons in a metal, liquid He-3, nuclear matter, stellar
structure,……..)
2. Notion of ``order parameter’’ to describe phases of
matter
related notion of spontaneously broken symmetry
basis of phase transition theory
Fermi liquid theory
• Electrons in a metal:
quantum fluid of fermions
• Inter-electron spacing ~ 1 A
 Very strong Coulomb
repulsion ~ 1-10 eV.
Fermi surface
ky
But effects dramatically weakened
due to Pauli exclusion.
Important `quasiparticle’ states near
Fermi surface scatter only weakly
off each other.
Describes conventional metals
extremely well.
kz
kx
Filled states unavailable for
scattering
Order parameter
• Example - ferromagnetism
Ferromagnet:
Spins aligned
Paramagnet:
Spins disordered
Increase temperature
• Spontaneous magnetization: `order parameter’.
• Ordered phase spontaneously breaks spin rotation symmetry.
Notion of order parameter and symmetry breaking
Powerful unifying framework for thinking generally about
variety of ordered phases (eg: superfluids,
antiferromagnets, crystals, etc).
Determine many universal properties of phases
- eg: rigidity of crystals, presence of spin waves in
magnets, vortices in superfluids,…….
Phase transitions -Theoretical paradigm
• Critical singularities: long wavelength fluctuations of
order parameter field.
• Landau-Ginzburg-Wilson: Landau ideas +
renormalization group
- sophisticated theoretical framework
Modern quantum many-electron physics
• Many complex materials studied in last two decades
DEFY understanding within Landau thinking
Examples:
1. One dimensional metals (Carbon nanotubes)
2. Quantum Hall effects
3. High temperature superconductors
4. Various magnetic ordering transitions
in rare-earth alloys
Need new ideas, paradigms!!
Well-developed
theory
??!!
High temperature superconductors
Parent insulator
La 2 CuO4
remove electrons
La
O
Cu
Superconductor
at relatively high
temperatures
Complex phase diagram
T
Insulating
antiferro
magnet
Another
strange
metal
Strange non-Fermi
Liquid metal
Fermi liquid
Superconductor
x = number of
doped holes
T = 0 phase transitions in rare earth alloys
• Examples: CePd2Si2, CeCu6-xAux, YbRh2Si2,……
Magnetic metal
Fermi liquid
Pressure/B-field/etc
(Quantum) critical point with striking non-fermi liquid physics
unexpected in Landau paradigm.
In search of new ideas and paradigms
• Most intriguing – electron breaks apart!!
(Somewhat) more precise: Fractional quantum numbers
Excitations of many body ground state have quantum
numbers that are fractions of those of the underlying
electrons.
Fractional quantum numbers
• Relatively new theme in condensed matter physics.
• Solidly established in two cases
d = 1 systems (eg: polyacetylene, nanotubes, …..),
d = 2 fractional quantum Hall effect in strong magnetic
fields
Broken electrons in d = 1
• Remove an electron from a d = 1 antiferromagnet
Removed electron
Broken electrons in d = 1
• Remove an electron from a d = 1 antiferromagnet
Removed charge
Spin domain wall
= removed spin
Broken electrons in d = 1
• Remove an electron from a d = 1 antiferromagnet
Removed charge
Spin domain wall
= removed spin
Broken electrons in d = 1
• Remove an electron from a d = 1 antiferromagnet
Removed charge
Spin domain wall
= removed spin
Broken electrons in d = 1
• The charge and spin of the removed electron move
separately – the electron has broken!
• ``Spin-charge separation’’ the rule in d = 1 metals
Quantum Hall effect
• Confine electrons to two
dimensions
• Turn on very strong magnetic
fields
• Make the sample very clean
• Go to low temperature
 Extremely rich and weird
phenomena
(eg: quantization of Hall
conductance)
Fractional charge
• If flux density (in units of flux quantum) is commensurate
with electron density, get novel incompressible electron
fluid.
• Excitations with fractional charge (and statistics) appear!
(Experiment: Klitzing, Tsui, Stormer, Gossard,……
Theory: Laughlin, Halperin, …………)
Physics Nobel: 1985, 1998.
All important question
Are broken electrons restricted to such exotic situations
(d = 1 or d = 2 in strong magnetic fields)?
Inspiration: Very appealing ideas on cuprate
superconductors based on 2d avatars of spin-charge
separation (Anderson, Kivelson et al, P.A Lee et al, …..)
All important question
Are broken electrons restricted to such exotic situations
(d = 1 or d = 2 in strong magnetic fields)?
NO!!!
Recent theoretical progress
Electrons can break apart in regular solids with strong interactions in 2
or 3 dimensions and in zero B-fields
1. Novel quantum phases with fractional quantum numbers (spincharge separation)
(Many people: Anderson, Read, Sachdev, Wen, TS, Fisher, Moessner, Sondhi,
Balents, Girvin, Misguich, Motrunich, Nayak, Freedman, Schtengel,……..)
2. Novel phase transitions described by fractionalized excitations
separating two conventional phases.
(TS, Vishwanath, Balents, Sachdev, Fisher ,Science March 04)
Complete demonstrable breakdown of Landau paradigms!!
Some highlights
• Theoretical description of fractionalized phases
(eg: nature of excitation spectrum)
• Concrete (and simple) microscopic models showing
fractionalization
• Prototype wavefunctions for fractionalized ground states
• Precise characterization of nature of ordering in the
ground state: replace notion of broken symmetry.
Where might it occur?
Always a hard question: hints from theory

Frustrated quantum magnets with paramagnetic
ground states

``Intermediate’’ correlation regime – neither potential
nor kinetic energy overwhelmingly dominates the other.
(i) Quantum solids near the melting transition
(ii) Mott insulators that are not too deeply into the insulating regime

Possibly in various 3d transition metal oxides

Perhaps even very common but we just haven’t found out!!
One specific simple model – small superconducting islands
on a regular lattice (quantum Josephson junction array)
Motrunich, T.S,
Phys Rev Lett 2002
• Competition between Josephson coupling and charging energy:
H = HJ + Hch
• Josephson : Cooper pairs hop between islands to delocalize
• Charging energy: prefer local charge neutrality, i.e localized Cooper
pairs.
• Superconductivity if Josephson wins, insulator otherwise.
Phase diagram in d = 2
Fractionalized phase:
excitations with half of Cooper
pair charge.
Josephson
Fractionalized insulator
sandwiched between superfluid
and conventional insulator.
Charging energy
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Stiffness (crystal rigidity,
persistent superflow,…)
Topological defects (vortices,
dislocations, etc)
Hartree-Fock mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Tools to detect (Bragg scattering,
Josephson, etc)
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Stiffness (crystal rigidity,
persistent superflow,…)
Topological defects (vortices,
dislocations, etc)
Hartree-Fock mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Tools to detect (Bragg scattering,
Josephson, etc)
Gauge excitations
Why gauge?
• Relic of glue that confines broken pieces together in
conventional phases.
- Analogous to quark confinement.
Conventional phases: Broken pieces (like quarks) are
bound together by a confining gauge field.
Fractionalized phases: Gauge field is deconfined; liberates
the fractional particles.
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Gauge excitations
Stiffness (crystal rigidity,
persistent superflow,…)
Robustness to all perturbations
Topological defects (vortices,
dislocations, etc)
Hartree-Fock mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Tools to detect (Bragg scattering,
Josephson, etc)
Robustness to all perturbations
(gauge rigidity)
• Gauge excitations preserved for arbitrary local
perturbations to the Hamiltonian (including ones that
break symmetries)
• Stable to dirt, random noise, coupling to lattice
vibrations, etc. (``Topological/quantum order’’ – Wen)
• Protected against decoherence by environment
(Potential application to quantum computing – Kitaev)
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Gauge excitations
Stiffness (crystal rigidity,
persistent superflow,…)
Robustness to all local
perturbations
Topological defects (vortices,
dislocations, etc)
Fractional charge
Hartree-Fock mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Tools to detect (Bragg scattering,
Josephson, etc)
Fractional charge: defects in gauge field
configuration
• Fractional charges carry the gauge charge that couples
to the gauge field - hence defects in the gauge field
(as in ordinary electromagnetism)
Electric charge
Electric field lines
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Gauge excitations
Stiffness (crystal rigidity,
persistent superflow,…)
Robustness to all local
perturbations
Topological defects (vortices,
dislocations, etc)
Fractional charge
Hartree-Fock mean field theory
Slave particle mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Tools to detect (Bragg scattering,
Josephson, etc)
Slave particle mean field theory
(Coleman, Read, Kotliar, Lee,….)
• Write electron operator cα = b†fα
Charged spinless boson
(``holon’’)
Neutral spinful fermion
(``spinon’’)
Replace microscopic Hamiltonian with equivalent non-interacting
Hamiltonian for holons and spinons with self-consistently
determined parameters.
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Gauge excitations
Stiffness (crystal rigidity,
persistent superflow,…)
Robustness to all local
perturbations
Topological defects (vortices,
dislocations, etc)
Fractional charge
Hartree-Fock mean field theory
Slave particle mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Coexistence with conventional
broken symmetry
Tools to detect (Bragg scattering,
Josephson, etc)
Coexistence
(Balents, Fisher, Nayak, TS)
• Fractionalization may coexist with conventional broken
symmmetry
(eg: fractionalized magnet, fractionalized superfluid,…)
Important implication: Presence of conventional order may
hide more subtle fractionalization physics.
(Is Nickel Sulfide fractionalized?)
Broken symmetry versus fractionalization
Goldstone modes (spin waves,
phonons, etc)
Gauge excitations
Stiffness (crystal rigidity,
persistent superflow,…)
Robustness to all local
perturbations
Topological defects (vortices,
dislocations, etc)
Fractional charge
Hartree-Fock mean field theory
Slave particle mean field theory
Coexistence of different broken
symmetries (magnetic
superconductors, supersolids,etc)
Coexistence with conventional
broken symmetry
Tools to detect (Bragg scattering,
Josephson, etc)
Flux memory, noise, ??
Detecting the gauge field
• Largely an open problem in general !!
• In some cases can use proximate superconducting
states to create and then detect the gauge flux
(TS, Fisher PRL 2001; TS, Lee forthcoming)
Cuprate experiments (Bonn, Moler) find no evidence for Z2 gauge flux
expected for one possible phase with spin-charge separation.
Other possibilities exist and haven’t been checked for yet.
Outlook
• Theoretical progress dramatic (rapid important developments
every year)
• But no unambiguous experimental identification yet (though many
promising candidates exist)
• Theoretically important answer to 0th order question posed by
experiments:
Can Landau paradigm be violated at phases and phase transitions of
strongly interacting electrons?
Outlook (cont’d)
• Extreme pessimist:
Why bother? Might not be seen in any material.
Extreme optimist: Might be happening everywhere without
us knowing (eg: in Nickel Sulfide,…..)
Outlook (cont’d)
• Strong need for probes to tell if fractionalized (completely
new experimental toolbox).
Ferromagnetism (relatively rare)– known for centuries
Antiferromagnetism (much more common) – known only
for < 70 years
Had to await development of new probes like neutron
scattering
Questions for the future
Will these ideas ``solve’’ existing mysteries like the
cuprates?
Will they have deep implications for other branches of
physics (much like ideas of broken symmetry did)?
See X.-G. Wen, Origin of Light for some suggestions.
Will they form the basis of quantum computing technology?
Quantum Hall effect
• Hall conductance
Fractional charge in FQHE
• More pictures
Outline
•
Some basic ideas in condensed matter physics
•
Complex new materials – crisis in quantum many body physics! New ideas
needed!
•
Why break the electron?
•
What does it mean?
How can you tell? Why should anyone care?