j. alicea, o. motrunich, t. senthil and mpafcolloq/talk21/presentation21.pdf · 7 band theory • s...
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Quantum Choreography: Exotica inside Crystals
• Electrons inside crystals: Quantum Mechanics at room temperature
• Quantum Theory of Solids: Band Theory Metals/Insulators/Semiconductors …
Electrons do their own thing…
• Need revisiting…
Electrons are cooperative “dancers” -exotic quantum dancers
U. Toronto - Colloquia 3/9/2006
J. Alicea, O. Motrunich, T. Senthil and MPAF
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Electrons in Solids:
Quantum mechanics at room temperature -
a=3 AngstromsIons - nuclei with closed
shell of electrons
outer shell of electrons -often are mobile
Energy Scales: Kinetic energy: Quantum zero point motion
Coulomb Interaction: between electrons and ions
Interest: Quantum Dynamics of 1023 electrons
Quantum Ground State Phases
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Quantum Theory of Solids: Standard Paradigm
Landau Fermi Liquid Theory
Landau Theory of Phase Transitions
Accounts for electronic behavior of simple metals, insulators and semiconductors
Provides a framework to understand broken symmetry phases of metals, including -
• superconductors,• ferromagnets,• antiferromagnets,• charge density waves,• spin density waves,…
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Fermi Liquid Theory py
pxFree Fermions
Filled Fermi seaparticle/hole excitations
Interacting Fermions
Retain a Fermi surface Luttingers Thm: Volume of Fermi sea same as for free fermions
Particle/hole excitations are long lived near FS Vanishing decay rate
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Add periodic potential from ions in crystal
• Plane waves become Bloch states
• Energy Bands and forbidden energies (gaps)
• Band insulators: Filled bands
• Metals: Partially filled highest energy band
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Even number of electrons/cell - (usually) a band insulator
Odd number per cell - always a metal
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Landau Theory of Phase Transitions
Order Parameter: A local observable, non-zero in one phase and zero in all others
Example: Electron Hamiltonian in metal
• Superconductor
• Ferromagnet
Landau-Ginzburg-Wilson “Free energy” functional:
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Band Theory
• s or p shell orbitals : Broad bandsSimple (eg noble) metals: Cu, Ag, Au - 4s1, 5s1, 6s1: 1 electron/unit cell
Semiconductors - Si, Ge - 4sp3, 5sp3: 4 electrons/unit cell
Band Insulators - Diamond: 4 electrons/unit cell
Band Theory Works
• d or f shell electrons: Very narrow “bands”
Transition Metal Oxides (Cuprates, Manganites, Chlorides, Bromides,…): Partially filled 3d and 4d bands
Rare Earth and Heavy Fermion Materials: Partially filled 4f and 5f bands
Electrons can ``self-localize”
Breakdown
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Electron “Fractionalization”
What is it?
Quantum Ground state of strongly correlated electronsystem which supports “particle-like” excitations whichcarry fractional quantum numbers
Not “built” from electrons: Q=e, s=1/2
Where does it occur?
• 1d systems of electrons: quantum wires, Carbon nanotubes
• 2d electrons in very strong magnetic fields: Fractional QHE
• Other 2d or 3d systems in ordinary crystals????
Holy Grail
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Mott Insulators:Insulating materials with an odd number of electrons/unit cell
Correlation effects are critical!
Hubbard model with one electron per site on average:
electron creation/annihilationoperators on sites of lattice
inter-site hopping
on-site repulsion
t
U
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Antiferromagnetic Exchange
Spin Physics
For U>>t expect each electron gets self-localized on a site
(this is a Mott insulator)
Residual spin physics:
s=1/2 operators on each site
Heisenberg Hamiltonian:
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High Temperature Superconductors
La2-xSrxCuO4
2d square lattice: Mott insulator with one electron/Cu atom
Remove electrons: dope in holes
Doped Mott insulator becomes ahigh temperature superconductor
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“males and females”
Rules: Electrons are -• homophobes• basically shy Antiferromagent
(boring)
“Quantum Docey Doe”
“Quantum Choreography”
Electrons like to dance
singlet -valence/chemical bond Valence Bond Crystal
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Electrons like to “swing”
“resonate”
foursome
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Electrons like to “swing”
“resonate”
foursome
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Electrons like to “swing”
“resonate”
foursome
“Quantum Grand Right and Left”
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“Quantum Grand Right and Left”
Resonating Valence Bond state (RVB) PW Anderson
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“Quantum Grand Right and Left”
Resonating Valence Bond state (RVB) PW Anderson
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“Quantum Grand Right and Left”
Resonating Valence Bond state (RVB) PW Anderson
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Breakups - break valence bonds
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Separation
Loners - “Spinons” s=1/2 , Q=0 , fermions spin of electronbut no charge
Electron Fractionalization: Spin-Charge separation
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Dejected and ejected: Dope in holes
“Holons” s=0 , Q=e, bosons electron chargebut no spin
Electron Fractionalization: Spin-Charge separation
Kivelson et. al.
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Condensation of holons - Superconductivity
Doping an RVB “spin liquid” - route to high Tc superconductivity?
Liberating the electron of its spin/statistics, allows the bosonic charge e to condense
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Doped high Tc Superconductor
• undoped Mott state is a boring AFM, NOT an RVB spin liquid
• AFM destroyed with low doping -“Pseudogap phase”
• Pseudogap an RVB spin liquid?
• How can one tell?
Topological Order
Simplest RVB spin liquid has “hidden” Z2 symmetry,
In addition to the spinons and holons which carrya Z2 “electric charge”, there are particles carrying Z2“magnetic charge” - “visons”
vison
spinon or holonphase change upon encircling vison
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Topological Order:
Four fold degeneracy ona torus - vison/no visonthreading holes of torus
phase change for holon upon encircling hole
vison
Essence of:
• Superconductor - Expulsion of Magnetic field
• Z2 RVB Spin liquid - Expulsion of visons (“Z2 magnetic field”)
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Vison detection:
RVB Liquid
holons
spinons
visons
superconductor
condense
BCS quasiparticle
hc/2e vortex
Tune from pseudogap into superconductor
(Senthil, MPAF)
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cool sc in non-zero B-field, trapping hc/2e flux
Set-up: Superconductor with Hole
prepare vison detect vison
heat into pseudogapand turn off B-field
cool back into sc -nucleate hc/2e vortex if vison is trapped
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Vison detection experiment K. Moler et. al. Nature 414, 887-889 (2001)
Single crystal of very underdopedYBa2Cu3O6.35 with Tc= 10 K
Drill 10 micron diameterhole in 40 x40 (micron)2
sample
Measure trapped flux with scanning SQUID magnetometer
Cycle T into/out of sc
No trapped flux - ie. vison escaped from hole
Conclusion: Pseudogap apparently not Z2 spin liquid
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Avoid symmetry breaking without hole doping?
Mott insulator - symmetry breaking instability - unit cell doubling
eg. AFM order in undoped cuprates
Geometrical Frustration
J J
J
Triangular plaquette of antiferromagnetically coupledspins cannot all be “satisfied”
Oftentimes the system can still find a way to order, but not always. Example: Coplaner3-sublattice arrangement ontriangular lattice -
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• Gap to all excitations in the bulk
• Ground state degeneracy on a torus
• “Fractionalization” of Quantum numbers
• Decoherence free Quantum computing
RVB StateFree “spinon”, with s=1/2
• Gapless Excitations
• “Critical” Power Laws
• No free particle description
• “Topological Order”
• “Algebraic or critical Spin Liquid”
Spin liquids come in two varieties:
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30T0.62KLRO “Spin liquid”0Phase Diagram:
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• Magnetic Bragg peaks at Q
• Continuum Inelastic Scattering near Q
Power law scaling: Suggestive of “critical” spin liquid ??
R. Coldea et al. PRL 86, 1335 (2001)Y. Shimizu et al. PRL 91, 107001 (2003)
Neutron Scattering from Cs2CuCl4
Spiral Order at low T
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• “Algebraic vortex liquid” (AVL)– “Critical Phase” with no free particle description– No broken symmetries - rather an emergent global SU(4) symmetry– Power-law correlations– Stable gapless spin-liquid (no fine tuning)
Theoretical approach to s=1/2 Triangular AFM
• Duality from spins to “vortices”• Fermionize vortices via Chern-Simons flux attachment• Flux smeared mean-field treatment• Dirac equation with U(1) gauge field and SU(4) flavor symmetry (QED3)
TTKT
Spiral AVL
0
Weak instability to spiral coplanerdriven by spin-orbit interactions
(Alicea, Motrunich, MPAF)
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Comparison with Cs2CuCl4Key prediction:
Same for all
ky
+Q-Q
“J”
“G”
Inelastic Neutron Scattering
1st B-zone of triangular lattice
R. Coldea, D. A. Tennant, and Z. TylczynskiPhys. Rev. B 68, 134424 (2003)
Experiments find same scaling exponent “near” G, J– but energy scans are NOT at fixed k– closer look between J and G would be good
kx
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Summary
• Materials with one d or f shell electron per atom are often insulating -Mott insulators - in contrast to band theory predictions.
• If not symmetry broken, the ground state of such a Mott insulator is guaranteed to be an exotic “spin liquid”
• Spin liquids come in two varieties - “topological” and “critical”
• Cuprate pseudogap is not the (simplest) Z2 topologically orderedspin liquid. So what is it??
• Frustrated AFM (eg triangular) show promise for observing “critical”spin liquids
• “Exotic” quantum world awaits discovery and exploration!
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