Fermions and spin liquidPatrick Lee

MIT

Conventional Anti-ferromagnet (AF):

1970 Nobel PrizeLouis Néel Cliff Shull

1994 Nobel Prize

Competing visions of the antiferromagnet

Lev Landau

Quantum

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“….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.

Classical

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.

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.

Spin liquid: destruction of Neel order due to quantum fluctuations.

More than 30 years later, we may finally have two example of spin liquid in higher than 1 dimension!

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.

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.

Introduce fermions which carry spin index

Constraint of single occupation, no charge fluctuation allowed.

Two ways to proceed:

1. Numerical: Projected trial wavefunction.

2. Analytic: gauge theory.

Extended Hilbert space: many to one representation.

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

The phase of ij becomes a compact gauge field aij on link ij and ibecomes the time component.

Compact U(1) gauge field coupled to fermions.

Physical meaning of gauge field:

gauge flux is gauge invariant

b= x a

It is related to spin chirality (Wen, Wilczek and Zee, PRB 1989)

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.

General problem of compact gauge field coupled to fermions.

Mean field (saddle point) solutions:

1. For ij real and constant: fermi sea.

2. For ij complex: flux phases and Dirac sea.

Enemy of spin liquid is confinement:

(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.

Three examples:

1. Organic triangular lattice near the Mott transition.

2. Kagome lattice, more frustrated than triangle.

3. Hyper-Kagome, 3D.

X = Cu(NCS)2, Cu[N(CN)2]Br, Cu2(CN)3…..

Q2D organics -(ET)2X

anisotropic triangular lattice

dimer model

ET

X

t’ / t = 0.5 ~ 1.1

t’

t t

Mott insulator

Spins on triangular lattice in Mott insulator

X- Ground State U/t t’/t

Cu2(CN)3Mott insulator 8.2 1.06

Cu[N(CN)2]Cl Mott insulator 7.5 0.75

Cu[N(CN)2]Br SC 7.2 0.68

Cu(NCS)2SC 6.8 0.84

Cu(CN)[N(CN)2] SC 6.8 0.68

Ag(CN)2 H2O SC 6.6 0.60

I3SC 6.5 0.58

-(ET)2X

t’t t

Half-filled Hubbard model

Q2D spin liquid -Cu2(CN)3

Q2D antiferromagnet -Cu[N(CN)2]Cl

t’/t=1.06

No AF order down to 35mK.

J=250K.

t’/t=0.75

1 10 10010-3

10-2

10-1

100

101

102

103

104

105

106

4.5 kbar

5 kbar

5.5 kbar

8 kbar

4 kbar

3.5 kbar0 kbar

Resis

tance ()

T (K)

Magnetic susceptibility, Knight shift, and 1/T1T

• Finite susceptibility and 1/T1T at T~0K : abundant low energy spin excitation (spinon Fermi surface ?)

H nuclear

[Y. Shimizu et al., PRL 91, 107001 (03)]

C nuclear

[A. Kawamoto et al. PRB 70, 060510 (04)]

Wilson ratio is approx. one at T=0.

is about 15 mJ/K^2moleSomething happens around 6K.

Partial gapping of spinon Fermi surface due to spinon pairing?

From Y. Nakazawa and K. Kanoda, Nature Physics, to appear.

More examples have recently been reported.

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 RRR~200.

A metallic like thermal conductivity in this insulator will definitively rule out localization.

M. Yamashita et al, Science 328, 1246 (2010)

Thermal conductivity of dmit salts.

mean free path reaches 500 inter-spin spacing.

ET2Cu(NCS)2 9K sperconductor ET2Cu2(CN)3 Insulator spin liquid

Importance of charge fluctuations

Heisenberg model120° AF order

Charge fluctuations are important near the Mott transition even in insulating phase

U/t

Fermi LiquidMott

transition

Metal I n s u l a t o r

J ~ t2/U

Numeric.[Imada and co.(2003)]

Spin liquid state

with ring exchange.

[Motrunich, PRB72,045105(05)] 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)]

L = -1 0 1Constraint :

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)

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

θ

kx

ky

is gapped.

In the insulator charge degrees of freedom described by

Stability of gapless Mean Field State against non-perturbative effect.

• U(1) instanton

F

Ф

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).)

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.

1. Gauge field dynamics: over-damped gauge fluctuations, very soft!

2. Fermion self energy is singular.

RPA results:

No quasi-particle pole, or z 0.

Physical Consequence

Specific heat : C ~ T2/3

[Reizer (89);Nagaosa and Lee (90)]

Gauge fluctuations dominate entropy

at low temperatures. (See also Motrunich,2005)

Justification of RPA by large N: recent development.

1. 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.

Solution: double expansion. (Mross, McGreevy,Liu and Senthil).

Maxwell term.

½ filled Landau level with 1/r interaction.

Expansion parameter: zb-2.

Limit N infinity, N finite gives a controlled expansion.

Results are similar to RPA and consistent with earlier expansion at N=2.

The double expansion is technically easer to go to higher order.

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, 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)

Thermal conductivity:

Using the Boltzmann equation approach, Nave and PAL (PRB 2007) predicted that for Fermi sea coupled to gauge field, /T goes as T^-2/3 and then saturate to a constant at low T due to impurity scattering.

What about experiment?

Linear T specific heat, not T^2/3.

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. /T is universal constant (independent on impurity conc.)

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.

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 Q+p and Q-p

Q

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.

LOFFBCS

Q

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)

With T. K. Ng (PRL 08)

Gapped boson is polarizible. AC electromagnetic A field induces gauge field a which couples to gapless fermions.

Predict ^2*(1/

Where 1/^(4/3)

Kezsmarki et al.PRB74, 201101(06)

Role of gauge field?

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.)

Herbertsmithite : Spin ½ Kagome.

Mineral discovered in Chile in 1972 and named after H. Smith.

Projected wavefunction studies. (Y. Ran, M. Hermele, PAL,X-G Wen)

Effective theory: Dirac spinons with U(1) gauge fields. (ASL)

New plots from Lhuillier and Sindzingre.

Predictions:

T^2 specific heat.

Linear T spin susceptibility

1/T1 goes as T^

Unfortunately current data seems dominated by a few per cent of local moments.

T. Imai et al, cond mat

• Mean-field picture – massless Dirac fermions:

• Include coupling to U(1) gauge field

Beyond mean-field theory

Index α=1,…,4 labels 2 nodes and spinµ=0,1,2 labels space-time direction

“Long-range” resonating valence-bond state(similar to d-wave RVB discussed in context of cuprates)

“Algebraic spin liquid”

Measurable properties: “mother of many competing orders”

•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:

(Rantner & Wen)

(Hermele, Senthil, M.P.A. Fisher)

Magnetic ordersValence-bondsolid orders

S × S order

3 triplets 3 singlets 1 triplet

neutrons, NMR, muSR, ...

optical phonon lineshape (T-dependence)

polarized neutrons

Magnetic “competing orders”

M1

+ +

--

M2

+

+-

-

M3

+

+

-

-

M1

M2

M3

Detect by scaling (neutron scattering, NMR):

NMR relaxation rate:

Valence-bond solid “competing orders”

M1

M2

M3

M1 M2 M3

Hastings’ VBS state

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.

Caveats:

1. Dzyaloshinskii- Moriya term:

Estimated to be 5 to 10% of AF exchange.

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

2. Local moments (6%) , perhaps from Zn occupying Cu sites.

Okamoto ..Takagi

PRL 07

3 dim example?

Hyper-Kagome.

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.

Metal- insulator transition by tuning U/t.

U/t

x

AF Mott insulator

metal

Cuprate superconductor

Organic superconductor

Tc=100K, t=.4eV, Tc/t=1/40.

Tc=12K, t=.05eV, Tc/t=1/40.

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).

Doping of an organic Mott insulator.

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.