Breaking electrons apart in condensed matter physics

1
Breaking electrons apart in condensed matter
physics

• T. Senthil
• (MIT)

Group at MIT Predrag Nikolic Dinesh Raut O.
Motrunich (now at KITP) A. Vishwanath
Other main collaborators L. Balents
(UCSB) Matthew P.A Fisher (KITP) Subir Sachdev
(Yale) D. Ivanov (Zurich)
2
Conventional condensed matter physics Landaus
2 great ideas

• 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

3
Fermi liquid theory

• Electrons in a metal
• quantum fluid of fermions
• Inter-electron spacing 1 A
• Very strong Coulomb
• repulsion 1-10 eV.
• 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.

Fermi surface
kz
ky
kx
Filled states unavailable for scattering
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Order parameter

• Example - ferromagnetism

Ferromagnet Spins aligned
Paramagnet Spins disordered
Increase temperature

• Spontaneous magnetization order parameter.
• Ordered phase spontaneously breaks spin rotation
  symmetry.

5

• 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,.

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Phase transitions -Theoretical paradigm

• Critical singularities long wavelength
  fluctuations of order parameter field.
• Landau-Ginzburg-Wilson Landau ideas
  renormalization group
• - sophisticated theoretical framework

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Modern quantum many-electron physics

• Many complex materials studied in last two
  decades
• DEFY understanding within Landau thinking
• Examples
• One dimensional metals (Carbon nanotubes)
• Quantum Hall effects
• High temperature superconductors
• Various magnetic ordering transitions
• in rare-earth alloys
• Need new ideas, paradigms!!

Well-developed theory
??!!
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High temperature superconductors
Parent insulator
Superconductor at relatively high temperatures
remove electrons
La
O
Cu
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Complex phase diagram
T
Strange non-Fermi Liquid metal
Insulating antiferro magnet
Fermi liquid
Another strange metal
x number of doped holes
Superconductor
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T 0 phase transitions in rare earth alloys

• Examples CePd2Si2, CeCu6-xAux, YbRh2Si2,

(Quantum) critical point with striking non-fermi
liquid physics unexpected in Landau paradigm.
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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.

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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

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Broken electrons in d 1

• Remove an electron from a d 1 antiferromagnet

Removed electron
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Broken electrons in d 1

• Remove an electron from a d 1 antiferromagnet

Removed charge
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Broken electrons in d 1

• Remove an electron from a d 1 antiferromagnet

Removed charge
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Broken electrons in d 1

• Remove an electron from a d 1 antiferromagnet

Removed charge
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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

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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)

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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.

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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, ..)

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All important question

• Are broken electrons restricted to such exotic
  situations
• (d 1 or d 2 in strong magnetic fields)?
• NO!!!

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Recent theoretical progress

• Electrons can break apart in regular solids with
  strong interactions in 2 or 3 dimensions and in
  zero B-fields
• Novel quantum phases with fractional quantum
  numbers (spin-charge 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!!

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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.

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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 havent
  found out!!

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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.
• .

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Phase diagram in d 2
Fractionalized insulator sandwiched between
superfluid and conventional insulator. Fraction
alized phase excitations with half of Cooper
pair charge.
Josephson
Charging energy
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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)
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Broken symmetry versus fractionalization
Goldstone modes (spin waves, phonons, etc) Gauge excitations
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)
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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.

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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)
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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)

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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)
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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
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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)
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Slave particle mean field theory(Coleman, Read,
Kotliar, Lee,.)

• Write electron operator ca bfa

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.
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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)
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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?)

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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, ??
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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 havent been
  checked for yet.

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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?

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Outlook (contd)

• Extreme pessimist
• Why bother? Might not be seen in any material.
• Extreme optimist Might be happening everywhere
  without us knowing (eg in Nickel Sulfide,..)

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Outlook (contd)

• 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 lt 70 years
• Had to await development of new probes like
  neutron scattering

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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?

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Quantum Hall effect

• Hall conductance

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Fractional charge in FQHE

• More pictures

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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?