Fermions and spin liquid

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Fermions and spin liquid Patrick Lee MIT
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Conventional Anti-ferromagnet (AF)
Louis Néel
Cliff Shull 1994 Nobel Prize
1970 Nobel Prize
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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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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.)
New 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.) Topolgical order (X. G. Wen)
With doping, vacancies (called holons) becomes
mobile in the spin liquid background becomes
superconductor.
More than 30 years later, we may finally have two
example of spin liquid in higher than 1 dimension!
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In high Tc , the ground states are all
conventional (confined phases). Physics of spin
liquid show up only at finite temperature.
Difficult to make precise statements and sharp
experimental tests. It will be very useful to
have a spin liquid ground state which we can
study.
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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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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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.

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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• Three examples
• Organic triangular lattice near the Mott
  transition.
• Kagome lattice, more frustrated than triangle.
• Hyper-Kagome, 3D.

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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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Spins on triangular lattice in Mott insulator
k-(ET)2X
X- Ground State U/t t/t
Cu2(CN)3 Mott insulator 8.2 1.06
CuN(CN)2Cl Mott insulator 7.5 0.75
CuN(CN)2Br SC 7.2 0.68
Cu(NCS)2 SC 6.8 0.84
Cu(CN)N(CN)2 SC 6.8 0.68
Ag(CN)2 H2O SC 6.6 0.60
I3 SC 6.5 0.58
Half-filled Hubbard model
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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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Magnetic susceptibility, Knight shift, and 1/T1T
C nuclear A. Kawamoto et al. PRB 70, 060510
(04)
H nuclear Y. Shimizu et al., PRL 91, 107001 (03)

• Finite susceptibility and 1/T1T at T0K
  abundant low energy spin excitation (spinon Fermi
  surface ?)

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From Y. Nakazawa and K. Kanoda, Nature Physics,
to appear.
g is about 15 mJ/K2mole
Something happens around 6K. Partial gapping of
spinon Fermi surface due to spinon pairing?
Wilson ratio is approx. one at T0.
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More examples have recently been reported.
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Alternative explanation? Kawamoto et al proposed
that the electrons are localized. With the
specific heat data, we infer a density of
states. Using 2 dim Mott formula to fit the
resistivity, we extract a localization length of
0.9 lattice spacing. This requires very strong
disorder and is highly implausible, given that
under pressure one obtains a good metal with
RRR200. A metallic like thermal conductivity in
this insulator will definitively rule out
localization.
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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)
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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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In the insulator charge degrees of freedom
described by
?
is gapped.

• Compact U(1) gauge theory coupled with spinon
  Fermi surface

ky
kx
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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. (Called the spin boson metal by Matthew
Fisher.)
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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Physical Consequence

• Specific heat C T2/3

Gauge fluctuations dominate entropy at low
temperatures. (See also Motrunich,2005)
Reizer (89)Nagaosa and Lee (90)
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Justification of RPA by large N recent
development.

• Gauge propagator is correct in large N limit.
• (J. Polchinski, Nucl Phys B, 1984)

2. However, fermion Green function is not
controlled by large N. (Sung-Sik Lee, PRB80,
165102(09) )
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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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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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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Thermal conductivity Using the Boltzmann
equation approach, Nave and PAL (PRB 2007)
predicted that for Fermi sea coupled to gauge
field, k/T goes as T-2/3 and then saturate to a
constant at low T due to impurity scattering.
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What about experiment?
Linear T specific heat, not T2/3. Decrease of
1/T1T below about 1K. (stretched exponent decay
in ET, which usually indicates non-intrinsic
behavior, but recent data on dmit shows a
recovery to exponential decay which may indicate
gap opening.
These problem are solved by spinon pairing. U(1)
breaks down to Z2 spin liquid. The gauge field is
gapped.
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.)
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There is evidence to support d wave pairing based
on projected wavefunction study in the presence
of ring exchange term. (Grover et al PRB 2010).
Consistent with thermal conductivity and its
increase with 2T magnetic field. (Zeeman effect
closes the gap).
Earlier it was thought that the 6K peak in ET is
Tc for pairing. This peak is totally insensitive
to magnetic field, in contrast to what is
expected for d wave pairing. This lead us to
propose an exotic pairing between fermions
travelling in the same direction (Amperean
pairing). However, maybe the 6K transition is
something else and the true spinon pairing
happens at 1K, as indicated by 1/T1. The issue of
pairing is currently not well understood.
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Amperean pairing instability. (Sung-Sik Lee, PAL
and T. Senthil, cond-mat) Ampere (1820)
discovered that two wires carrying current in
the same direction attract.
This suggests pairing of electrons moving in the
same direction.
Pair Qp and Q-p
Pair momentum is 2Q, similar to LOFF.
However, phase space is much more restricted than
BCS. In our case the transverse gauge propagator
is divergent for small q and cannot be screened,
leading to log divergence.
Q
Q
LOFF
BCS
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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 ordering
   temperature. Not seen experimentally so far.
3. In gap optically excitation. (Ng and Lee PRL 08)

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Role of gauge field?
Kezsmarki et al.PRB74, 201101(06)
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) Where 1/tw(4/3)
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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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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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New plots from Lhuillier and Sindzingre.
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Predictions T2 specific heat. Linear T spin
susceptibility 1/T1 goes as Th .
Unfortunately current data seems dominated by a
few per cent of local moments.
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T. Imai et al, cond mat
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Beyond mean-field theory

• Mean-field picture massless Dirac fermions
• Include coupling to U(1) gauge field

Long-range resonating valence-bond
state (similar to d-wave RVB discussed in context
of cuprates) Algebraic spin liquid
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Measurable properties mother of many
competing orders
(Hermele, Senthil, M.P.A. Fisher)

• The competing orders are 15 observables (mass
  terms). They have slowly-decaying power-law
  correlations characterized by the same critical
  exponent.
• For the kagome there are three kinds of these

Magnetic orders Valence-bond solid orders S S order
3 triplets 3 singlets 1 triplet
neutrons, NMR, muSR, ... optical phonon lineshape (T-dependence) polarized neutrons
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Magnetic competing orders
NMR relaxation rate
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Valence-bond solid competing orders
M1
M2
Hastings VBS state
M3
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S S competing order

• Corresponds to DM interaction with

Define orientation on bonds
Y. Ran et al computed some of these correlation
functions using projected wave-functions and they
do NOT have the same power law decay. Either
sample size is too small, or projuected
wavefunction does not capture the physics of the
low energy field theory.
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Caveats 1. Dzyaloshinskii- Moriya term
Estimated to be 5 to 10 of AF exchange.
2. Local moments (6) , perhaps from Zn
occupying Cu sites.
3. Singh and Huse proposed a ground state of
36 site unit cell valence bond solid studied by
Nikolic and Senthil.
Perturb in weak bond and set it 1
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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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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.
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 Dirac
spinon (algebraic spin liquid) More experimental
confirmation needed. New phenomenon of emergent
spinons and gauge field may now be studied. If
the same set of tools (slave boson theory,
projected wavefunctions) are successful in
describing the spin liquids, this should
strengthen the case for a spin liquid description
of the pseudogap and superconducting state in the
cuprates.