Gauge dynamics of kagome antiferromagnets - phys.virginia.edu · Martinez et. al. 1994 Broholm et....
Transcript of Gauge dynamics of kagome antiferromagnets - phys.virginia.edu · Martinez et. al. 1994 Broholm et....
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Gauge dynamics of kagome antiferromagnets
Michael J. Lawler
(Binghamton University,
Cornell University)
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Outline
• Introduction to highly frustrated magnets
• Constrained spin models
– Dirac’s generalized Hamiltonian mechanics
– Degrees of freedom counting
– Edge states?
• Simulations of spin waves in kagome AFM
• Conclusions
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The problem of highly frustrated magnetism
• Nearest neighbor model just selects a low energy subspace!
Consider Na4Ir3O8
3D kagome
lattice
J~300 S=1/2
Okamoto et. al. PRL 2007
defines the “cooperative paramagnet” (Villain, 1979)
Na4Ir3O8, Okamoto et. al. 2007
Nu
mb
er
of
stat
es
Lowering energy
S=1/2
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Cooperative paramagnets
Consider Na4Ir3O8
3D kagome
lattice
J~300 S=1/2
Okamoto et. al. PRL 2007
ΘCW = -650 K
Na4Ir3O8, Okamoto et. al. 2007
temperature range of the cooperative paramagnetic
Frustration parameter:
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Unusual glassy dynamics? SrCa8Ga4O19, S=3/2 kagome AFM
Broholm et. al. 1990 Martinez et. al. 1994
Does not obey hyper-scaling relations
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Herbertsmithite: A quantum spin liquid?
CLASSES OF NEW MAGNETIC SYSTEMS
Highly frustrated “Ring exchange”?
Herbertsmithite
Volborthite
Na4Ir3O8
Organic triangular lattice
Some materials influenced
by frustration some
by “ring exchange”
Han et. al., 2012
Neutron scattering at 0.75 meV
• No magnetic order down to 50 mK
• Continuum of spin excitations at low energies
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DMRG Ground State Yan et. al., Science, 2011 Depenbrock et. al., 2012 Jiang et. al., 2012
Strong numerical evidence that the spin ½ ground state
is a “Z2 spin liquid”
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Definition of quantum spin liquid
• Experimental definition:
– No sign of magnetic ordering
– No sign of “freezing” or glassy behavior
– Odd number of half-odd-integer spins in unit cell
• Theoretical definition:
– A state with long range entanglement between the spins
Why does frustration produce a quantum spin liquid phase?
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Constrained spin models
So the low energy subspace of states obeys
Lets then focus on the simpler model
On the kagome lattice, we can write
Lagrange multipliers
Lawler, 2013
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Spin origami Shender et. al, 1993
Constrained spin model is that of a fluctuating membrane!
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Constrained Hamiltonian Mechanics
Follow Dirac, and fix the Lagrange multipliers hn by
This is a linear algebra problem!
We can invert and solve for hn.
Otherwise, some combinations of hn remain arbitrary!
If
Dirac, 1950,1958
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Gauge dynamics
• A “gauge theory” in mechanics is one with multiple solutions to its equations of motion.
• Example: Maxwell electrodynamics
– There are many solutions to the scalar and vector potential
– The electric and magnetic fields evolve the same way for each solution
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The single triangle model
The constraints obey
This model has constraints
and Hamiltonian
hx, hy and hz are arbitrary!
So
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Map all solutions
Spin origami construction
Physical observables evolve the same way independent of the choice of the arbitrary functions
i
1
2
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Degrees of freedom counting
• How many physical observables are there?
– Dirac discovered where
• D: the number of unconstrained coordinates
• M: the number of constraint functions φm
• NL: the number of arbitrary Lagrange multipliers
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Two polarizations of light
• Consider electricity and magnetism
– D = 8
– M = 2
– NL = 2 (the above two constraints commute)
So Ncanonical = 8 – 2 – 2 = 4 two polarizations of light!
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Back to kagome: pbc’s
Ncanonical = 0!
For every spin configuration that satisfies the constraints:
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Edge states?
Open boundary conditions
Ncanonical = number of dangling triangles
But a local mechanical object requires a position and a momentum coordinate!
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Chern-Simon’s electrodynamics
• Similar to “doubled” Chern-Simon’s electrodynamics in 2 spatial dimensions
• Changes only the statistics of particles
• Quantum model has long range entanglement
• Proposed to govern Z2 spin liquids (Xu and Sachdev, 2009)
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Ordinary Kagome antiferromagnets
Now consider an ordinary kagome antiferromagnet with Hamiltonian
How is the discovered gauge dynamics important here?
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Spin waves
100
200
300
400
0
W
Wave vector Γ Κ Μ Γ
Ener
gy
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What do the eigenmodes corresponding to gauge modes look
like?
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Motion of spins along the side edges
Blue Dots: Spin A
Green Dots: Spin B
Orange Dots : Spin C
Size of dots is proportional to motion of the spins
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What do the eigenmodes corresponding to canonical modes
look like?
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Motion along side edges – Proposed canonical edge states
Blue Dots: Spin A
Green Dots: Spin B
Orange Dots : Spin C
Size of dots is proportional to motion of the spins
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Motion of bulk - Folding
Blue Dots: Spin A
Green Dots: Spin B
Orange Dots : Spin C
Size of dots is proportional to motion of the spins
Just a global spin rotation mode!
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Simulation of edge excitations
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Energy in the “gauge” modes
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Energy in the “canonical” modes
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Conclusions
• Spins constrained to classical ground states of HFMs obeys a kind of electrodynamics.
• Conjecture: frustration is important for the formation of a quantums spin liquid phase.
Nu
mb
er
of
stat
es
Lowering energy
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Strongly correlated metals
• Some strongly correlated metals are also gauge theories.
• Examples:
– Double occupancy constraint implies
– No nearest neighbor constraint of spinless fermions