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.
Outline:
1. Introduction to quantum magnetism and spin liquid.
2. 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.
Conventional Anti-ferromagnet (AF):
Louis Néel
1970 Nobel Prize
Cliff Shull
1994 Nobel Prize
strongly-correlated electron system
example: Hi Tc cuprate.
One hole per site: should be a metal according to band theory.
Mott insulator.
t
Undoped CuO2 plane:
Mott Insulator due to
e- - e- interaction
Virtual hopping induces
AF exchange J=4t2/U
CuO2 plane with doped holes:
La3+  Sr2+: La2-xSrxCuO4
Competing visions of the antiferromagnet
“….To describe antiferromagnetism, Lev landau and
Cornelis Gorter suggested quantum fluctuations to mix
Neel’s solution with that obtained by reversal of
moments…..Using neutron diffraction, Shull confirmed
(in 1950) Neel’s model.
……Neel’s 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
|  | 
Classical
Quantum
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?
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 S=1/2?
1 dimensional chain:
Energy per bond of singlet trial wavefunction is
-(1/2)S(S+1)J = -(3/8)J vs. -(1/4)J for AF.
Spin liquid is more than the
absence of Neel order.
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 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!
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.
Two routes to spin liquid:
1.Geometrical frustration:
spin ½ Heisenberg model
on Kagome, hyper-kagome.
2. Proximity to Mott
transition.
Introduce fermions which carry spin index
Constraint of single occupation,
no charge fluctuation allowed.
Two ways to proceed:
1. Numerical: Projected trial wavefunction.
Extended Hilbert space: many to one representation.
2. Analytic: gauge theory.
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.
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.
General problem of compact gauge field coupled to fermions.
Mean field (saddle point) solutions:
1. For cij real and constant: fermi sea.
2. For cij complex: flux phases and Dirac sea.
3. 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.
Z2 gauge theory: generally gapped. Several exactly soluble examples.
(Kitaev, Wen)
U(1) gauge theory: gapless Dirac spinons or Fermi sea.
Hermele et al (PRB) showed that deconfinement is possible if number of
Dirac fermion species is large enough. (physical problem is N=4). Sung-sik
Lee showed that fermi surface U(1) state is always deconfined.
Stability of gapless Mean Field State against
non-perturbative effect.
1) Pure compact U(1) gauge theory :
always confined. (Polyakov)
• U(1) instanton
2) Compact U(1) theory +
large N Dirac spinon :
deconfinement phase
F
[Hermele et al., PRB 70, 214437 (04)]
3) Compact U(1) theory +
Fermi surface :
Ф
more low energy fluctuations
deconfined for any N.
(Sung-Sik Lee, PRB 78, 085129(08).)
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.
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).]
Physical meaning of gauge field:
gauge flux is gauge invariant
b=
xa
Fermions hopping around a plaquette
picks up a Berry’s 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)
Three examples:
1. Organic triangular lattice near the Mott transition.
2. Kagome lattice, more frustrated than triangle.
3. Hyper-Kagome, 3D.
We are not talking about spin glass, spin ice etc.
Kagome lattice.
Herbertsmithite : Spin ½ Kagome.
Mineral discovered in
Chile in 1972 and
named after H. Smith.
Spin liquid in Kagome system. (Dan Nocera, Young Lee etc. MIT).
Curie-Weiss T=300, fit to high T expansion gives J=170K
No spin order down to mK (muSR, Keren and co-workers.)
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)
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.
How to understand Huse-White result?
Gapped Z2 spin liquid.
1. 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.
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.
Mendels group PRL 2008
Mendels group, PRL 2012
Large single crystals available (Young Lee’s group at MIT).
Neutron scattering possible. Science 2012.
Projected Dirac S(k). Serbyn and PAL.
Q2D organics k-(ET)2X
ET
dimer model
X
Mott insulator
X = Cu(NCS)2, Cu[N(CN)2]Br,
Cu2(CN)3…..
t t
t’
anisotropic triangular lattice
t’ / t = 0.5 ~ 1.1
Q2D antiferromagnet
k-Cu[N(CN)2]Cl
t’/t=0.75
6
10
5
10
0 kbar
3.5 kbar
Resistance ()
4
10
4 kbar
Q2D spin liquid
k-Cu2(CN)3
3
10
4.5 kbar
2
10
1
10
0
10
8 kbar
-1
10
5.5 kbar
5 kbar
-2
10
-3
10
1
10
T (K)
100
t’/t=1.06
No AF order down to 35mK.
J=250K.
From Y. Nakazawa and K. Kanoda, Nature Physics 2008.
g is about 15 mJ/K^2mole
Something happens around 6K.
Partial gapping of spinon Fermi
surface due to spinon pairing?
Wilson ratio is approx.
one at T=0.
More examples have recently been reported.
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.
ET2Cu(NCS)2 9K sperconductor
ET2Cu2(CN)3
Insulator spin liquid
Importance of charge fluctuations
Charge fluctuations
are important near
the Mott transition
even in insulating
phase
Fermi Liquid
Mott
transition
Metal
Numeric.[Imada and co.(2003)]
Spin liquid state
with ring exchange.
[Motrunich, PRB72,045105(05)]
Heisenberg model
120° AF order
Insulator
U/t
+
J ~ t2/U
+…
J’ ~ t4/U3
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)]
Constraint :
L=
-1
0
1
Q. What is the low energy effective theory for mean-field state ?
Effective Theory :
fermions and rotor coupled to
compact U(1) gauge field.
Sung-sik Lee and P. A. Lee, PRL 95, 036403 (05)
3 dim example?
Hyper-Kagome.
Okamoto ..Takagi
PRL 07
Near Mott transition: becomes metallic under pressure.
Strong spin orbit coupling.
Spin not a good quantum number but J=1/2.
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
T^2 specific heat.
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.
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.
RPA results:
1. Gauge field dynamics: overdamped gauge fluctuations, very
soft!
2. Fermion self energy is singular.
No quasi-particle pole, or z  0.
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.
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 -> e
Note curvature
of patch is kept.
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.
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 selfenergy:
Solution: double expansion. (Mross, McGreevy,Liu and Senthil).
Maxwell term.
½ filled Landau level with 1/r interaction.
Expansion parameter: e=zb-2.
Limit N  infinity, e 0, eN finite gives a controlled expansion.
Results are similar to RPA and consistent with earlier e expansion at N=2.
The double expansion is technically easer to go to higher order.
Conclusion:
No correction to boson: z=3/2.
For the gauge field problem, h
is positive and sub-leading.
RPA is recovered to 3 loop.
Sung-Sik Lee, arXiv 2013, co-dimension expansion.
2 patch theory fails for d > 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 d=2.5.
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.
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/t=plT, 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)
What about experiments?
Linear T specific heat, not T^2/3.
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.
NMR on dmit.
Stretched exponential decay at low T. Is there a nodal gap?
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?
8
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.
6
More exotic pairing? Amperean pairing, SS Lee,PL, Senthil.
(PRL). Other suggestions: time reversal breaking,
Barkeshli, Yao and Kivelson, arXiv 2012, quadratic band
touching, Mishmash…C. Xu, arXiv 2013.
0
inhomogeneous
C NMR
relaxation rate
13
7
5
4
3
2
1
inhomogeneous
0 1 2 3 4 5 6 7 8 9 10
Temperature (K)
(a)
NMR Relaxation rate
Shimizu et al., PRB 70 (2006) 060510
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 (2+1
dim) and spinons coupled to gauge field. (S-S. Lee and PAL, PRL
2005). Critical theory described by T. Senthil (PRB 2008).
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.
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)
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.
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)=w^2*(1/t)
Power law is found by Elsasser…Dressel, 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.
Recent terahertz data by Nuh Gedik group at MIT, Pilon et al. arXiv 1301. on
Herbertsmithite.
Potter, Senthil and Lee, recently identified several mechanisms for in gap
absorption in Herbertsmithite.
All proportional to w^2 with varying coefficients for the U(1) spin liquid.
1. Electric field couple to gauge electric field. (Ioffe-Larkin)
Physical meaning of gauge electric field is the gradient of singlet bond.
a. 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.
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.
It turns out that
Is proportional to the gauge electric field.
This is a more physical way to understand the coupling via the gauge field.
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
For the special case of Dirac spinons:
Order of magnitude is in agreement with Gedik’s experiment.
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.
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.
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?
We expect that fluctuations of the z component of S1 contains
information of the fluctuation of the scalar chirality.
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.
Metal- insulator transition by tuning U/t.
U/t
AF Mott insulator
Cuprate superconductor
Tc=100K, t=.4eV, Tc/t=1/40.
Organic superconductor
Tc=12K, t=.05eV, Tc/t=1/40.
metal
x
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).
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.