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 CuO 2 plane: Mott Insulator due to e - e interaction Virtual hopping induces AF exchange J=4t 2 /U CuO 2 plane with doped holes: La 3+



Sr 2+ : La 2-x Sr x CuO 4

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.

” Lev Landau Jacques Friedel, Obituary of Louis Neel, Physics today, October,1991.

|   |  Quantum Classical

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 c ij becomes a compact gauge field a ij and i l becomes the time component.

on link ij

Compact U(1) gauge field coupled to fermions.

General problem of compact gauge field coupled to fermions.

Mean field (saddle point) solutions: 1. For c ij real and constant: fermi sea.

2. For c ij 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.

• 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.

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 ~ T

2/3 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= x a 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 2012 Mendels group PRL 2008

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)

2

X

ET X dimer model Mott insulator X = Cu(NCS) 2 , Cu[N(CN) 2 ]Br, Cu 2 (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

10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 1 0 kbar 3.5 kbar 4 kbar 4.5 kbar 8 kbar 5.5 kbar 5 kbar 10 T (K) 100 Q2D spin liquid k -Cu 2 (CN) 3 t ’ /t=1.06

No AF order down to 35mK.

J=250K.

From Y. Nakazawa and K. Kanoda, Nature Physics 2008.

Something happens around 6K.

Partial gapping of spinon Fermi surface due to spinon pairing?

g is about 15 mJ/K^2mole 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.

ET 2 Cu(NCS) 2 9K sperconductor ET 2 Cu 2 (CN) 3 Insulator spin liquid

Importance of charge fluctuations

Fermi Liquid Metal Charge fluctuations are important near the Mott transition even in insulating phase Mott transition I n s u l a t o r Heisenberg model 120 ° AF order U/t Numeric.[Imada and co.(2003)] Spin liquid state with ring exchange.

[Motrunich, PRB72,045105(05) ] J ~ t 2 /U + J’ ~ t 4 /U 3 + …

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 c ij becomes a compact gauge field a ij and i l becomes the time component.

on link ij

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: over damped 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 self energy:

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

Maxwell term.

½ filled Landau level with 1/r interaction. Expansion parameter: e= z b -2.

Limit N  infinity, e  0, e N 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=pl 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, 2k f 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?

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, Mishmash…C. Xu, arXiv 2013.

3 2 5 4 8 7 6 inhomogeneous 13 C NMR relaxation rate 1 0 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: 2k F 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 S i .S

j 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 T c =100K, t=.4eV, T c /t=1/40 .

Organic superconductor T c =12K, t=.05eV, T c /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.