Boulder School Lecture 1 and 2

8/11/2019 Boulder School Lecture 1 and 2

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Fermions and spin liquid

Patrick Lee

MIT


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Conventional Anti-ferromagnet (AF):

1970 Nobel PrizeLouis Nel Cliff Shull

1994 Nobel Prize


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Competing visions of the antiferromagnet

Lev Landau

Quantum

| |

.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 internationalmeeting 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


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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 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.


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In 1973 Anderson proposed a spin liquid

ground state (RVB) for the triangular lattice

Heisenberg model.. It is a linearsuperposition of singlet pairs. (not

restricted to nearest neighbor.)

New property of spin liquid:

Excitations are spin 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 exampleof spin liquid in higher than 1 dimension!


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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 statementsand sharp experimental tests. It will be very useful to

have a spin liquid ground state which we can study.


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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.


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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. Wehave 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

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

Fermions hopping around a plaquette

picks up a Berrys phase due to the

meandering quantization axes. The isrepresented by a gauge flux through

the plaquette.


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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.

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


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Three examples:

1. Organic triangular lattice near the Mott transition.

2. Kagome lattice, more frustrated than triangle.

3. Hyper-Kagome, 3D.


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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


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Spins on triangular lattice in Mott insulator

X- Ground State U/t t/t

Cu2(CN)3 Mott 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)2 SC 6.8 0.84

Cu(CN)[N(CN)2] SC 6.8 0.68Ag(CN)2H2O SC 6.6 0.60

I3 SC 6.5 0.58

k-(ET)2X

t

t t

Half-filled

Hubbard model


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Q2D spin liquid

k-Cu2(CN)3

Q2D antiferromagnet

k-Cu[N(CN)2]Cl

t/t=1.06No AF order down to 35mK.

J=250K.

t/t=0.75

1 10 100

10-3

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

Resistance

()

T (K)


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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. PRB70, 060510 (04)]


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Wilson ratio is approx.

one at T=0.

g 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.


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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

RRR~200.A metallic like thermal conductivity in this insulator will

definitively rule out localization.


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

Thermal conductivity of dmit salts.

mean free path reaches

500 inter-spin spacing.


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


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Importance of charge fluctuations

Heisenberg model

120AF order

Charge fluctuations

are important near

the Mott transition

even in insulating

phase

U/t

Fermi Liquid

Mott

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

+ +


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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)]

L = -1 0 1Constraint :

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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Compact U(1) gauge theory coupled with spinon Fermi

surface

kx

ky

is gapped.

In the insulator charge degrees of freedom described by


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Stability of gapless Mean Field State against

non-perturbative effect.

U(1) instanton

F

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).)


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Non-compactU(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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1. Gauge field dynamics: over-

damped gauge fluctuations, verysoft!

2.Fermionself energy is singular.

RPA results:

No quasi-particle pole, or z 0.


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Physical Consequence

Specific heat : C ~ T2/3

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

Gauge fluctuations dominate entropy

at low temperatures. (See also Motrunich,2005)


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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.


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

Maxwell term.

filled Landau level with 1/r interaction.

Expansion parameter: e=zb-2.

Limit Ninfinity, e0, eN finite gives a controlled expansion.

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

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/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 stillderive 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 atlow T due to impurity scattering.


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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. 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.

A i i i t bilit (S Sik L PAL d T S thil d t)


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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 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 gaugepropagator is divergent for small

q and cannot be screened,

leading to log divergence.

LOFFBCS

Q


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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. Notseen experimentally so far.

3 In gap optically excitation. (Ng and Lee PRL 08)


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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)

Where 1/t=w^(4/3)

Kezsmarki et al.PRB74, 201101(06)

Role of gauge field?


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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 Kagome.

Mineral discovered in

Chile in 1972 and

named after H. Smith.


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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.

Predictions:


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Predictions:

T^2 specific heat.

Linear T spin susceptibility

1/T1 goes as T^h .

Unfortunately current data seems dominated by a

few per cent of local moments.

T. Imai et al, cond mat


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T. Imai et al, cond mat


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Mean-field picturemassless Dirac fermions:

Include coupling to U(1) gauge field

Beyond mean-field theory

Index =1,,4 labels 2 nodes and spin

=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


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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 samecritical exponent.

For the kagome there are three kinds of these:(Rantner & Wen)

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

Magnetic ordersValence-bond

solid ordersS 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

M1

+ +

--

M2

+

+-

-

M3

+

+

-

-

M1

M2

M3

Detect by scaling (neutron scattering, NMR):

NMR relaxation rate:


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Valence-bond solid competing orders

M1

M2

M3

M1 M2 M3

Hastings VBS state


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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.

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.


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Okamoto ..Takagi

PRL 07

3 dim example?

Hyper-Kagome.

Near Mott transition: becomes metallic under pressure.


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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.


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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.

Doping of an organic Mott insulator.


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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 maynow 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.

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