Quantum Spin Liquid

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Quantum Spin Liquid Patrick Lee MIT
Collaborators M. Serbyn, A. Potter, T.
Senthil N. Nagaosa X-G Wen Y. Ran Y. Zhou M.
Hermele T. K. Ng T. Grover .
Supported by NSF.
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• Outline
• Introduction to quantum magnetism and spin
  liquid.
• Why is spin liquid interesting?
• Spin liquid is much more than the absence of
  ordering Emergence of new particles and gauge
  fields.
• 3. Spin liquid in organic compounds and kagome
  lattice.
• 4. Low energy theory fermion plus gauge field.
• 5. Proposals for experimental detection of
  emergent particles and gauge fields.

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Conventional Anti-ferromagnet (AF)
Louis Néel
Cliff Shull 1994 Nobel Prize
1970 Nobel Prize
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strongly-correlated electron systemexample Hi
Tc cuprate.
One hole per site should be a metal according to
band theory. Mott insulator.
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Competing visions of the antiferromagnet
.To describe antiferromagnetism, Lev landau and
Cornelis Gorter suggested quantum fluctuations to
mix Neels solution with that obtained by
reversal of moments..Using neutron diffraction,
Shull confirmed (in 1950) Neels model. Neels
difficulties with antiferromagnetism and
inconclusive discussions in the Strasbourg
international meeting of 1939 fostered his
skepticism about the usefulness of quantum
mechanics this was one of the few limitations of
this superior mind. Jacques Friedel, Obituary of
Louis Neel, Physics today, October,1991.
Lev Landau


?
Quantum
Classical
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Mott against Slater debate
Mott One electron per unit cell. Charge gap is
due to correlation. Antiferromagnetism is
secondary. Mott insulator violate band theory.
Slater Anti-ferromagnetic ground state. Unit
cell is doubled. Then we have 2 electrons per
unit cell and the system can be an insulator,
consistent with band theory.
Can there be a Mott insulator which does not have
AF order?
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P. W. Anderson introduced the RVB idea in
1973. Key idea spin singlet can give a better
energy than anti-ferromagnetic order. What is
special about S1/2? 1 dimensional chain Energy
per bond of singlet trial wavefunction is
-(1/2)S(S1)J -(3/8)J vs. -(1/4)J for AF.
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Spin liquid destruction of Neel order due to
quantum fluctuations.
In 1973 Anderson proposed a spin liquid ground
state (RVB) for the triangular lattice Heisenberg
model.. It is a linear superposition of singlet
pairs. (not restricted to nearest neighbor.)
Spin liquid is more than the absence of Neel
order.
New emergent property of spin liquid Excitations
are spin ½ particles (called spinons), as
opposed to spin 1 magnons in AF. These spinons
may even form a Fermi sea. Emergent gauge field.
(U(1), Z2, etc.) Topological order (X. G. Wen)
in case of gapped spin liquid
ground state degeneracy, entanglement entropy.
More than 30 years later, we may finally have
several examples of spin liquid in higher than 1
dimension!
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It will be very useful to have a spin liquid
ground state which we can study.

• Two routes to spin liquid
• Geometrical frustration spin ½ Heisenberg model
  on Kagome, hyper-kagome.
• Proximity to Mott transition.

Requirements insulator, odd number of electron
per unit cell, absence of AF order. Finally there
is now a promising new candidate in the organics
and also in a Kagome compound.
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Introduce fermions which carry spin index
Constraint of single occupation, no charge
fluctuation allowed.
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Why fermions? Can also represent spin by boson,
(Schwinger boson.)
Mean field theory 1. Boson condensed Neel
order. 2. Boson not condensed gapped state.
Generally, boson representation is better for
describing Neel order or gapped spin liquid,
whereas fermionic representation is better for
describing gapless spin liquids. The open
question is which mean field theory is closer to
the truth. We have no systematic way to tell
ahead of time at this stage. Since the observed
spin liquids appear to be gapless, we proceed
with the fermionic representation.
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Enforce constraint with Lagrange multipier l
The phase of cij becomes a compact gauge field
aij on link ij and il becomes the time component.
Compact U(1) gauge field coupled to fermions.
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General problem of compact gauge field coupled to
fermions.

• Mean field (saddle point) solutions
• For cij real and constant fermi sea.
• For cij complex flux phases and Dirac sea.
• Fermion pairing Z2 spin liquid.

Enemy of spin liquid is confinement(p flux state
and SU(2) gauge field leads to chiral symmetry
breaking, ie AF order)
If we are in the de-confined phase, fermions and
gauge fields emerge as new particles at low
energy. (Fractionalization) The fictitious
particles introduced formally takes on a life of
its own! They are not free but interaction leads
to a new critical state. This is the spin liquid.
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Stability of gapless Mean Field State against
non-perturbative effect.
1) Pure compact U(1) gauge theory always
confined. (Polyakov) 2) Compact U(1) theory
large N Dirac spinon deconfinement phase
Hermele et al., PRB 70, 214437 (04) 3)
Compact U(1) theory Fermi surface
more low energy fluctuations deconfined for
any N. (Sung-Sik Lee, PRB 78, 085129(08).)

• U(1) instanton

F
?
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Non-compact U(1) gauge theory coupled with Fermi
surface.
Integrating out some high energy fermions
generate a Maxwell term with coupling constant e
of order unity. The spinons live in a world
where coupling to E M gauge fields are strong
and speed of light given by J. Longitudinal
gauge fluctuations are screened and gapped. Will
focus on transverse gauge fluctuations which are
not screened.
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Physical Consequence

• Specific heat C T2/3

Gauge fluctuations dominate entropy at low
temperatures. Non-Fermi liquid.
Reizer (89)Nagaosa and Lee (90), Motrunich
(2005).
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Physical meaning of gauge field gauge flux is
gauge invariant b x a
Fermions hopping around a plaquette picks up a
Berrys phase due to the meandering quantization
axes. The is represented by a gauge flux through
the plaquette.
It is related to spin chirality (Wen, Wilczek
and Zee, PRB 1989)
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• Three examples
• Organic triangular lattice near the Mott
  transition.
• Kagome lattice, more frustrated than triangle.
• Hyper-Kagome, 3D.
• We are not talking about spin glass, spin ice
  etc.

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Kagome lattice.
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Mineral discovered in Chile in 1972 and named
after H. Smith.
Herbertsmithite Spin ½ Kagome.
Spin liquid in Kagome system. (Dan Nocera, Young
Lee etc. MIT). Curie-Weiss T300, fit to high T
expansion gives J170K No spin order down to mK
(muSR, Keren and co-workers.)
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Spin ½ Heisenberg on Kagome has long been
suspected to be a spin liquid. (P. W. Leung and
V. Elser, PRB 1993)
Projected wavefunction studies. (Y. Ran, M.
Hermele, PAL,X-G Wen)
Effective theory Dirac spinons with U(1) gauge
fields. (ASL)
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White, Huse and collaborators find a gapped spin
liquid using DMRG. Entnglement entropy
calculations (Hong-Chen Jiang and others) show
that their state is a Z2 spin liquid.
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How to understand Huse-White result?
Gapped Z2 spin liquid.

• Slave boson
• Motrunich 2011 projected slave boson mean
  field.
• Proximity to QCP?

2. Fermion pairing Lu, Ran and Lee classified
projected fermionic pairing state. However,
recent QMC calculation by Iqbal, Becca and
Poilblanc did not find energy gain by pairing.
They found that the Dirac SL is remarkably stable
and has energy comparable to DMRG after two
Lanchoz steps.
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Theoretically, the best estimate (Huse and White)
is that there is a triplet gap of order
0.14J. Experimentally, the gap is much smaller.
Specific heat, NMR (Mendels group PRL2008, 2011,
T. Imai et al 2011). See also recent neutron
scattering. (Y. Lee group, Nature 2012.)
Caveats Heisenberg model not sufficient. 1.
Dzyaloshinskii- Moriya term
Estimated to be 5 to 10 of AF exchange.
QCP between Z2 spin liquid and AF order. (Huh,
Fritz and Sachdev, PRB 2010)
2. Local moments, current understanding is that
15 of the Zn sites are occupied by copper.
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Mendels group PRL 2008
Mendels group, PRL 2012
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Large single crystals available (Young Lees
group at MIT). Neutron scattering possible.
Science 2012.
Projected Dirac S(k). Serbyn and PAL.
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Q2D organics k-(ET)2X
ET
dimer model
X
X Cu(NCS)2, CuN(CN)2Br, Cu2(CN)3..
anisotropic triangular lattice
t / t 0.5 1.1
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Q2D antiferromagnet k-CuN(CN)2Cl
t/t0.75
Q2D spin liquid k-Cu2(CN)3
t/t1.06 No AF order down to
35mK. J250K.
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From Y. Nakazawa and K. Kanoda, Nature Physics
2008.
g is about 15 mJ/K2mole
Wilson ratio is approx. one at T0.
Something happens around 6K. Partial gapping of
spinon Fermi surface due to spinon pairing?
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More examples have recently been reported.
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Thermal conductivity of dmit salts.
mean free path reaches 500 inter-spin
spacing.
M. Yamashita et al, Science 328, 1246 (2010)
However, ET salt seems to develop a small gap
below 0.2 K.
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ET2Cu(NCS)2 9K sperconductor
ET2Cu2(CN)3 Insulator spin liquid
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Importance of charge fluctuations
Mott transition
Fermi Liquid
I n s u l a t o r
Metal
U/t
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Slave-rotor representation of the Hubbard Model
S. Florens and A. Georges, PRB 70, 035114
(04), Sung-Sik Lee and PAL PRL 95,036403 (05)
Q. What is the low energy effective theory for
mean-field state ?
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Effective Theory fermions and rotor coupled
tocompact U(1) gauge field.Sung-sik Lee and P.
A. Lee, PRL 95, 036403 (05)
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3 dim example? Hyper-Kagome.
Okamoto ..Takagi PRL 07
Near Mott transition becomes metallic under
pressure.
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Strong spin orbit coupling. Spin not a good
quantum number but J1/2.
Approximate Heisenberg model with J if direct
exchange between Ir dominates. (Chen and Balents,
PRB 09, see also Micklitz and Norman PRB 2010 )
Slave fermion mean field , Zhou et al (PRL
08) Mean field and projected wavefunction.
Lawler et al. (PRL 08)
Conclusion zero flux state is stable spinon
fermi surface. Low temperature pairing can give
line nodes and explain T2 specific heat.
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Enforce constraint with Lagrange multipier l
The phase of cij becomes a compact gauge field
aij on link ij and il becomes the time component.
Compact U(1) gauge field coupled to fermions.
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Non-compact U(1) gauge theory coupled with Fermi
surface.
Integrating out some high energy fermions
generate a Maxwell term with coupling constant e
of order unity. The spinons live in a world
where coupling to E M gauge fields are strong
and speed of light given by J. Longitudinal
gauge fluctuations are screened and gapped. Will
focus on transverse gauge fluctuations which are
not screened.
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RPA results
1. Gauge field dynamics over-damped gauge
fluctuations, very soft!
2. Fermion self energy is singular.
No quasi-particle pole, or z ? 0.
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Only bosons with q tangent to a given patch
couple. Two patch theory. This is special to 2D.
In 3D bands of tangential points are coupled.
Then all points are coupled.
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Large N Polchinski (94), Altshuler, Ioffe and
Millis (94). N fermions coupled to gauge field.

Minimal 2 patch model. Sung-Sik Lee, (PRB80
165102 (09)
Plus opposite patch with e -gt -e Note curvature
of patch is kept.
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It was believed that 1/N expansion is systematic,
and D has no further singular correction, but
Fermion G might. Sung-Sik Lee showed that 1/N
expansion breaks down.
This term is dangerous if it serves as a cut-off
in a diagram. He concludes that an infinite set
of diagrams contribute to a given order of 1/N.
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Recent progress Metlitski and Sachdev PRB82,
075127 (10) They did loop expansion anyway and
found no log correction to boson up to 3 loops,
but for fermion self-energy
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Solution double expansion. (Mross, McGreevy,Liu
and Senthil).
Maxwell term.
½ filled Landau level with 1/r interaction.
Expansion parameter ezb-2. Limit N ? infinity,
e? 0, eN finite gives a controlled expansion.
Results are similar to RPA and consistent with
earlier e expansion at N2. The double expansion
is technically easer to go to higher order.
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Conclusion No correction to boson z3/2.
For the gauge field problem, h is positive and
sub-leading. RPA is recovered to 3 loop.
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Sung-Sik Lee, arXiv 2013, co-dimension expansion.
2 patch theory fails for d gt 2. Therefore cannot
do conventional epsilon expansion. Instead, keep
FS to be a line and extend the dimension
perpendicular to it to d-1.
He finds an expansion about d2.5.
Results are consistent with Mross et al No
correction to boson D to 3 loops. Correction to
fermions for the nematic problem
For the gauge field problem, h is positive and
sub-leading. RPA is recovered to 3 loop.
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How non-Fermi liquid is it?
Physical response functions for small q are Fermi
liquid like, and can be described by a quantum
Boltzmann equation. Y.B. Kim, P.A. Lee and X.G.
Wen, PRB50, 17917 (1994)
Take a hint from electron-phonon problem.
1/tplT, but transport is Fermi liquid.
If self energy is k independent, Im G is sharply
peaked in k space (MDC) while broad in frequency
space (EDC). Can still derive Boltzmann equation
even though Landau criterion is
violated.(Kadanoff and Prange). In the case of
gauge field, singular mass correction is
cancelled by singular landau parameters to give
non-singular response functions. For example,
uniform spin susceptibility is constant while
specific heat gamma coefficent (mass) diverges.
On the other hand, 2kf response is enhanced.
(Altshuler, Ioffe and Millis, PRB 1994). May be
observable as Kohn anomaly and Friedel
oscilations. (Mross and Senthil)
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What about experiments?
Linear T specific heat, not T2/3.
Thermal conductivity Nave and Lee, PRB 2007.
If second term due to impurity dominates, we have
k/T goes to constant, in agreement with expt.
Numerically the first term due to gauge field
scattering is very close to expt at 0.2 K. Then
we may expect small upturn and small deviation
from linearity.
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NMR on dmit. Stretched exponential decay at low
T. Is there a nodal gap?
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Evidence for phase transition at 6K in ET.
Spinon pairing? U(1) breaks down to Z2 spin
liquid. The gauge field is gapped.
Thermal expansion coefficient Manna et al.,
PRL 104 (2010) 016403
What kind of pairing? One candidate is d wave
pairing. With disorder the node is smeared and
gives finite density of states. k/T is universal
constant (independent on impurity conc.) However,
singlet pairing seems ruled out by smooth
behavior of spin susceptibility up to 30T. More
exotic pairing? Amperean pairing, SS Lee,PL,
Senthil. (PRL). Other suggestions time reversal
breaking, Barkeshli, Yao and Kivelson, arXiv
2012, quadratic band touching, MishmashC. Xu,
arXiv 2013.
inhomogeneous
NMR Relaxation rate Shimizu et al., PRB 70
(2006) 060510
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Open issues on organic spin liquids Nature of
the small gap in ET vs no gap in
dmit. Explanation of the low temperature NMR,
field induced broadening of nmr and MuSR
line. Is it U(1) or Z2? If U(1) ,where is the
evidence for gauge fluctuations? What is the
nature of the phase transition at 6K in ET and
possibly 4K in dmit? Quantum critical point
between spin liquid state with spinon Fermi
surface and metal. Non-Fermi liquid metal?
Effective field theory charge carried by xy
bosonic model (21 dim) and spinons coupled to
gauge field. (S-S. Lee and PAL, PRL 2005).
Critical theory described by T. Senthil (PRB
2008).
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Other experiments? How to see spinon Fermi
surface?
Angle resolved photo-emission (ARPES) (Evelyn
Tang, PL and Matthew Fisher, also Pujari and
Lawler, arXiv 2012)
Electron spectrum convolution of fermion with
boson with gap D.
Location of the lowest threshold traces out the
spinon Fermi surface.
Another idea 2kF Friedel oscillations may be
observable by STM. Mross and Senthil, PRB 2010.
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How to see gauge field?
Coupling between external orbital magnetic field
and spin chirality. Motrunich, see also Sen and
Chitra PRB,1995.

1. Quantum oscillations? Motrunich says no. System
   breaks up into Condon domains because gauge field
   is too soft.
2. Thermal Hall effect (Katsura, Nagaosa and Lee,
   PRL 09). Expected only above spinon pairing
   temperature. Not seen experimentally so far.
   (perhaps due to Meissner effect of spinon
   pairing)
3. In gap optically excitation. Electric field
   generates gauge electric field. (Ng and Lee PRL
   08)
4. Ultra-sound attenuation, (Yi Zhou and P. Lee,
   PRL 2011)
5. Direct coupling to neutron using DM term in
   Herbertsmithite. (Lee and Nagaosa, PR 2013)

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Yi Zhou and P. Lee, PRL 2011
1. Spinon coupling to phonon is the same as
electron-phonon coupling in the long wave length
limit.
2. For transverse ultra-sound, the rapid fall
phenomenon, well known for SC, can be a signature
of fermion pairing and the existence of gauge
field.
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Role of gauge field?
With T. K. Ng (PRL 08) Gapped boson is
polarizible. AC electromagnetic A field induces
gauge field a which couples to gapless fermions.
Predict s(w)w2(1/t)
Power law is found by ElsasserDressel, Schlueter
in ET (PRB 2012) but for w larger than J. Need
low frequency data.
Recent terahertz data by Nuh Gedik group at MIT,
Pilon et al. on Herbertsmithite.
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Recent terahertz data by Nuh Gedik group at MIT,
Pilon et al. arXiv 1301. on Herbertsmithite.
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Potter, Senthil and Lee, recently identified
several mechanisms for in gap absorption in
Herbertsmithite. All proportional to w2 with
varying coefficients for the U(1) spin liquid.

• Electric field couple to gauge electric field.
  (Ioffe-Larkin)
• Physical meaning of gauge electric field is
  the gradient of singlet bond.

• Purely electronic. (Ng-Lee, PRL 2007)
• Bulaevskii et al PRB 2008.

b. Magneto-elastic coupling.
2. Modulation of the DM term. Couple to the spin
current in the x direction. Expect smaller
magnitude.
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Bulaevskii, Batista, Mostovoy and Khomskii. PRB
78, 024404 (2008).
Perturbation in t/U of the Hubbard model and
project to the spin sector.
E.P provide the coupling of light to the spin
degree of freedom.
Is proportional to the gauge electric field.
It turns out that
This is a more physical way to understand the
coupling via the gauge field.
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Recall that gauge magnetic field is the spin
chirality.
What is the physical meaning of the gauge
electric field? (Potter et al, appendix)
For a triangle this reduces to
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For the special case of Dirac spinons
Order of magnitude is in agreement with Gediks
experiment.
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Magneto-elastic coupling.
Displacement of the Cu ions within the unit cell
modulates the exchange J.
The symmetry of the modulation of Si.Sj is the
same as the purely electronic mechanism for the
Kagome lattice.
Numerically this gives the same order of
magnitude as the purely electronic mechanism.
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Modulation of the DM term due to motion of the
oxygen ions in the unit cell.
It is interesting and it couples to the spin
conductivity. However, this is estimated to be
smaller in magnitude.
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Using neutron scattering to measure spin
chirality in Kagome lattices. P. A. Lee and N.
Nagaosa, arXiv.
Gauge flux is proportional to scalar spin
chirality. How to measure its fluctuation
spectrum? Maleev, 1995 neutron measurement of
vector chirality. Shastry-Shraiman, 1990 Raman
scattering. Limited to small q. Wingho Ko and
PAL,2011, RIXS, limited energy resolution.
Savary and Balents PRL 2012, (also O. Benton, O.
Sikora and N. Shannon, PRB 2012) showed that
neutron scattering couples to gauge fluctuations
in the spin ice problem, where spin-orbit
coupling is dominant. Can something similar work
for the weak spin-orbit case?
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We expect that fluctuations of the z component of
S1 contains information of the fluctuation of
the scalar chirality.
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A more formal argument
Let be a state which carries chirality
and has no matrix element to couple to neutron
scattering. To first order in DM, it becomes
Intermediate state is triplet. We assume triplet
gap larger than singlet.
We predict that neutron scattering contains a
piece which contains information on the scalar
chirality fluctuations.
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Metal- insulator transition by tuning U/t.
U/t
AF Mott insulator
Cuprate superconductor
Tc100K, t.4eV, Tc/t1/40.
Tc12K, t.05eV, Tc/t1/40.
metal
x
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Doping of an organic Mott insulator. Also
talk by Yamamoto yesterday.
Superconductivity in doped ET, (ET)4Hg2.89Br8,
was first discovered Lyubovskaya et al in 1987.
Pressure data form Taniguchi et al, J. Phys soc
Japan, 76, 113709 (2007).
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Conclusion There is an excellent chance that the
long sought after spin liquid state in 2
dimension has been discovered experimentally.
organic spinon Fermi surface Kagome and
Hyper-Kagome. More experimental confirmation
needed. New phenomenon of emergent spinons and
gauge field may now be studied.
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