Tuesday, August 7th Part II

     Yong Baek Kim (U. Toronto)

      A Lightning review of quantum spin liquids

 

      Blogged by Andrey Chubukov

Hello, Yong Baek is ready to go. 

This will be a theorists (romantic) viewpoint on quantum spin liquids. The first issue is: what is a quantum spin liquid? Point -- it is different from a cooperative paramagnet.

In a classical cooperative paramagnet, there is a large number of equally unhappy classical ground states.

Classical spin liquid in a frustrated system -- paramagnetic state in between mean-field and the actual Kurie temperatures (the actual one is reduced due to frustration). Still, excitations are coherent (gapped spin waved) at intermediate T. 

Quantum spin liquid -- no coherent excitations, elementary excitations are S=1/2 spinons. Example -1D S=1/2 antiferromaget. Neutron scattering shows spinon continuum with a well-defined boundary. 

Higher dimensions.  Consider possible states with <S> =0. One possibility is valence bond solid state. This one breaks translational symmetry.  But one can make  a superposition    (resonating valence bonds, RVB).  As long as elementary excitations are spinons, the state is a quantum spin liquid. It can be in the form of short-range RVB, with exponential correlations, or in the form of long-range RVB, with power-law correlations.  At this stage, the talk is interrupted by many questions about the nature of RVB state, on the interplay between spin-spin correlations and correlations of spin pairs. 

Yong Baek next discussed examples -- systems which show insulating behavior in resistivity, yet the susceptibility is finite, and specific heat shows power-law behavior.  The materials are the same ones which  Hide Takagi discussed in his talk.    

Some of materials, mostly the ones on Kagome lattices,  are strong Mott insulators (Mot gap is >1eV). Others, like systems on triangular lattices, are close to metal-insulator transition. In terms of spin model., smaller charge gap implies that the spin interaction is more complex than the Heisenberg one (more long ranged).  The reasoning then is that the system looses long-range magnetic order before it becomes a metal. 

Many questions at this point.

One question is what constitutes the proof that a spin liquid excists in D >1.  Yong-Baek's point is that there is strong experimental evidence, but so far no "smoking gun" experiments. Theoretical evidence is also strong, both numerical and analytical  (blogger agrees that numerical evidence is strong, less certain about analytical evidence).  

Again, many questions, from all corners. 

Next item -- a "micky mouse" version of the theory (Yong Baek's terminology).   He began with variational wave function in slave-particle approach. Example: write spins in terms of fermions and replace Heisenberg interaction by 4-fermion interaction. Then use mean-field.   One can classify all possible mean-field states and characterize them using the notations of "topological" and "quantum" order.  One can further introduce a projected wave functionto get rid of doubly occupied states. This procedure partly (but only partly) takes care of the local constraint.

Next item: gauge theory. Yong-Baek discussed U(1) spin liquid (no anomalous <ff> term). Then there is U(1) gauge invariance. For Z_2 spin liquid (<ff> \neq 0), the only transformation allowed is Z_2 (f_{i,\alpha} ->  +- f_{i, \alpha}). For U(1) spin liquids, projected wave function is Gutzviller projector applied to free fermion wave function, for Z_2 spin liquid  the projection acts on BCS wave function.

SECOND TALK

Yong Baek started  by considering excitations in Z_2 spin liquid. His point is that one has to take a wave function of a BCS superconductor, formally set U to be infinite and use Gutzviller projection. This way, one obtains excitations with exactly zero charge and S=1/2. .   He went on to discuss that there are two  different states, one can be viewed as a projected state with no flux, another as a projected state with a flux. 

Predictions:  

1. U(1) spin liquid. At the mean-field level, this is like free fermions with a particular band structure.

Beyond mean-field, the powers of specific heat change. For example, in 2D case interaction with a gauge field changes C(T) from O(T) to T^{2/3}.  The discussion about the accuracy of T^{2/3} analysis emerged. 

2. Z_2 spin liquid.  Because of Z_2, there are visons -- topological defects,  with transform locally from one topological sector to the other.   

Next point -- how theory analysis is related to experiments on organic materials.  Yong Baek made an argument (originally due to Sethil) that near metal-insulator transition specific heat scales as T^{2/3} only at the lowest T. At higher T, it behaves as T log T.   There is another crossover line near the transition, at higher T. This last one is for transport.   Yong-Baek listed several specific predictors for transport which seem to be in quite reasonable agreement with the data. 

With respect to Kagome Herbersmithite material -- most recent numerical calculations for S=1/2 nearest-neighbor Heisenberg model are consistent with Z_2 spin liquid. Experiments show different behavior, what probably indicates that the underlying spin model is more complex than nearest-neighbor Heisenberg.

Final item on the agenda -- spin ice.

Overall impression of the blogger -- this is VERY rapidly developing field, with impressive achievements over the last few years.  Great talk(s). 

Post talk.  During the discussion with students, Yong Baek discussed the importance of entanglement entropy in characterizing spin liquids. He pointed out that the entanglement entropy of a Z2 spin liquid takes the form

S = a * perimeter - Log(2)

This second term has recently been observed in DMRG computations  on the Kagome Lattce

     

Tuesday, August 7th

Hide Takagi, ISSP Tokyo and Riken

An overview of quantum spin liquid: from toy of theorists to reality.

Blogging by Andy Schofield: Good morning blog watchers! We are good to go...

This talk will be about how experimentalists think about spin liquid states - with the theoretical detail following from Yong Baek Kim. Leon Balents' Nature paper (2010) provides a nice introduction. A decade ago, this topic was a toy for theorists but now we have more than one compound which seem to realized this: (2D triangular, Kagome and 3D hyperkagome).


The phases of correlated matter are many and various (solid, liquid, gas, superfluid). The electron in a material has multiple degrees of freedom (charge, spin, orbital) which behave almost independently so one quantum number could show one phase while the other degree of freedom is in another. We will be dealing with a charge solid, spin liquid phase.

We begin with the square lattice antiferromagnet with long range magnetic order. In 1972, Phil Anderson suggested that in a frustrated triangular system where geometry means the is no obvious way of minimizing the energy of every spin independently, then adding strong quantum effects (of spin 1/2) then this may lead to a spin-liquid state. There are many types of quantum spin liquid, but in essence they can be viewed as a superposition of spin-singlet pairs: RVB - resonating valence bonds). These can be short ranged singlets (where there could be a gap to making the first excitation) or longer range which could make the spin liquid gapless, with a spin-Fermi surface and even a transition into chiral ordering.

Question: what is a spin Fermi surface? Answer: excitations reminiscent of a fermi surface in momentum space
Question: How do you know long range RVB is gapless? Answer: experimentalist view point is of short range correlations so the decay is exponential, by contrast with long range single pairing: I expect we will hear more from Y.B. Kim later I guess.

Geometrical frustration: 2D triangular lattice or its 3D version - the pyrochlore lattice of corner sharing tetrahedra - are the exemplars. To think abut creating this: Start with a cubic lattice looked along the [111] direction then this has a triangular arrangement of atoms. NiO is an example (NaCl structure). The (111) plane is a triangular lattice of same atoms - the next layer is also triangular. This forms a stacked layer of triangles of the same atom: Ni then O in 2D planes. In NiO we have 8 equivalent directions [111], [1,-1,1] etc so the layers are intermixed. However we can space the layers out with cations: (Li_0.5 Ni_0.5) O. This then separates the Ni and O layers with a layer of Li to isolate.
Question (by AJS): How do you know the Li layer will order like this? Answer the more different the ionic radius the more likely they are to order rather than form a mixture.
Question: How do the Li atoms know which of the 8 planes to choose? Answer - long range ionic forces tend to drive this ordering.
Question: What is known about the orbital structure in the plane? Do electrons only hop locally? Answer: minimally you need to do p-p, d-p and d-d neighbour hopping.


Having made triangles we will now see that the pyrochlore is also related to a NaCl structure. Take the Na atoms in NaCl in one unit cell then it forms a tetrahedron. You then can construct a pyrochlore by connecting those tetrahedra (will not include all the Na atoms though). You can then make a Kagome by using the planes perpendicular to [111] in a pyrochlore. It alternates Kagome and triangular layers. So you can then use cation ordering to separate and isolate these layers.





Question: you have made the lattice, but how do you make the AFM coupling? Answer: the oxide helps but also the detail of the transition metal ion.
Question: Isn't the ground state of a triangular lattice the 120 degree phase? Answer: higher order interactions can destabilize this to a spin liquid state.
Question: It still looks 3D. Yes - you need the cation ordering (a combination of magnetic and non-magnetic layer).
These are the principles we can use to make lots of interesting frustrated lattices. Pyrochlores form in the spinel structure (AB_2O_4) and (A_2B_2O_7) where A and/or B can be magnetic. We also want to make these highly quantum so ideally spin 1/2.
Question: what about honeycomb lattice? This is not frustrated but bipartite for nearest neighbour - but can be frustrated with large next neighbour interaction - see later.

How do we capture the signature of a quantum spin liquid?  Look at the susceptiblity. At high temperature it is Curie-Weiss like whose intercept (of 1/chi) on the negative T=0 (-T_cw) axis gives the bare scale of the interaction. In 3D ordered systems you usually see a kink roughly at +T_cw which is the Neel ordering temperature T_N. The ratio f = T_N/T_cw indicates the degree of frustration. A gapless spin-liquid would have no kink but one still needs to look for the absence of order with neutrons, NMR, muSR, ...In the gapless case 1/chi as T->0 would be constant (like a metal) but in a gapped spin-liquid then 1/chi would diverge exponentially. Specific heat and thermal conductivity are other important probes.

Question: you cannot get to T=0 so how do you know if it is ordered/gapped etc?  
Answer: compare to the scale of J (which is usually 100-1000K) and we can get to experimental temperatures of 100mK so much less than J.

Question: what about electrical transport properties?  
Answer: today we will be confined to insulators - though many of these systems are actually quite weak insulators.

Question: clarifying the distinction between long range order and long range RVB.
Answered above I hope.

Question: If you have a valence bond solid how do you distinguish that from the gapped spin liquid?  Answer: the crystal structure will show the distortion of the dimers.

Candidates for some quantum spin liquids: 2D triangular - the best candidate is the organic materials of Kanoda's group like kappa-(BEDT-TTF)_2-Cu2(CN)_3. The two parts of this planar molecule forms dimers with one electron per dimer with pairs which form a 2D layers. They are called charge transfer salts. This forms a S=1/2 Mott insulator which looks triangular although the couplings are not identical so it is a t, t' model.




See susceptibility (Y. Shimizu et al, PRL, 91, 5th Sept 2003) - no anomaly for AFM ordering. NMR showed no internal magnetic fields down to 32mK with interaction strength of 100K - so system is paramagnetic. A reference compound shows a Neel temperature of 30K and you see it in NMR.

Question: does the dip in the susceptibility seen in this material indicate short range RVB? Could do.
Specific heat C/T v T^2 seems to show an intercept (ie like an itinerant metal - except it is an insulator) with a value of 13mJ/mol K^2 and a Wilson ratio of 1. [letter to Nature, Satoshi Yamashita et al]. So this could be evidence of a spinon Fermi surface.

Question (by AJS): do the spinons form a Fermi liquid - with a T^2 scattering rate perhaps visible in thermal transport. Answer: issue is complicated and not resolved (will discuss this afternoon).
Question (by AC): ratio of t and t'? Answer: about 1.4
This represents 25% of my talk but time is up, This afternoon we will be able to discuss issues and better compounds.

OK - here we go. The afternoon session begins...

Takagi begins by asking why no 120 degree structure forming on the triangle? The answer appears to be the proximity to the Mott transition with higher order exchange processes leading to longer range interactions. Optical conductivity reveals a charge gap of order 0.1eV so this is indeed low.. We next turn to the thermal conductivity which seems to vanish quadratically (maybe cubically) in T (in contrast with kappa going like 1/3 C_v v l which would suggest linearly vanishing in T). Also mean free path l = 3a. So really hard to explain what is going on. The other puzzle is from NMR which shows an anomaly at about 6K - also seen in thermal expansion. Applying pressure you can close the gap and see superconductivity emerge in the metallic phase - with a Tc of 6K. So is the 6K anomaly the same/or connected to the superconductivity in the metallic phase.
We now have a new generation of spin liquid candidates - which will also challenge this bloggers ability to type: EtMe3Sb[Pd(dmit_2]_2 (see T. Itou, Nature Physics (2010)). Very similar physics with a fermi liquid type properties of chi and specific heat with a Wilson ratio of order 1. Thermal conductivity now looks much more like one we could understand. kappa/T looks like specific heat with mean free paths of order microns (like a metal!!). This is data from Matsuda's group in Kyoto - and is truly a thing of beauty.
However at high temperatures the data looks strange with a 1K anomaly  in T_1^-1. It is again not clear what is happening. A phase transition of a spinon fermi surface.
Question: Is there something happening above 10K - any opinion on that? Perhaps a precursor to the 1K anomaly.
Question: What about optical conductivity - is there a Drude peak? Resistivity is diverging as T->0 so there are no charged carriers - so the thermal conductivity is begin carried by non-charged particles.
A further candidate is NiGa2S4: an S=1 triangular antiferromagnet (Satoru Nakatsuji). However there is spin-freezing in NMR at 8.5K so not a spin liquid but there may be some interesting dynamics (T_CW ~ 100K).
We now move to the S=1/2 Kagome system: Hybertsmithite ZnCu_3(OH)_6Cl_2 [M.P.Shores, J. A. Chem. Soc. 127, 13462 (2005).] claimed to be a perfect system for this. J=180K but with no signature of ordering in neutron scattering nor in muSR. There are signatures of defects: Zn and Cu may intermix which can mess the system up so disorder is a matter of debate. Is there a spin-gap? Looking at the susceptibility (intrinsic using NMR to avoid the impurity contribution). The data is ambiguous at this stage since you can fit to gapped or ungapped right now so not yet resolved.
A new Kagome system: Volborthite Cu3V2O7(OH)2.2H2O but it has 2 Cu sites in the Kagome plane so it is not quite an ideal system. However Cu and V are very different so it makes a clean system (almost no hint of impurity contribution to the susceptibility). There is some anomalous magnetisation with a tiny transition at 1K. This structure actually looks more like a 1D chain with a bridge because of a distortion which means not all coppers are equivalent.
A candidate S=1/2 Kagome: Vesignieite BaCu3V2O8...lost it.
So we have lots of interesting candidates for Kagome but it is complicated by disorder and distortion when compared to the organic systems above.
Question (by AC): are there examples of Kagome systems with S>1/2? Yes - many and they tend to show classical order.
Third example: hyper-kagome lattice: Na4Ir3O8. Ir^4+ oxide. It is a B-cation ordered spinel with non-magnetic sodium occupying part of the pyrochlore lattice. So it forms a network of corner shared triangles rather than tetrahedra in a sort of twisted version of a Kagome. The ion has 5d5 so expect S=/12 but actually spin-orbit is strong so there may be that j is a better quantum number. The lattice itself has chirality too based on the orientation of these connected triangles.
Question (by AJS): is this spin isotropic? Good question - and answer is that we cannot exactly tell yet but there are examples of irridates which are very isotropic.
This is a Mott insulator with J=650K, a clear S=1/2 moment and no ordering in chi down to 1.8K and down to 4K in neutrons. The Knight shift shows chi being constant at T=0 (different from bulk susceptibility at low T but that just indicates the presence of impurities). 1/T1 and 1/T1T show power-law decay with 1/T1T constant below 10K which implies a gapless excitation spectrum. At low T there is no evidence for ordering in specific heat but does suggest a small gamma of 1 mJ/mol Ir K^2 but here the Wilson ratio is of order 40. However we are quite close to magnetism in this system and cannot rule out some indication of this.
Question (by AC): but I missed this question - sorry. Andrey perhaps you can add it at some point...? Answered by Yong Baek - spin orbit affects the specific heat.
Finally an alternative routes to spin-liquids but flashed too fast for the blogger to do anything...a part from a reference to spin-ice, charge analogues of spin-liquids. Nature always tries to lift degeneracy. In summary: clean systems close to a metal-insulator transition seems to be the key to spin-liquid formation and stabilisation.
Question: about the 6K anomaly above. The challenge is to measure stuff under pressure so it is very tough but it looks smoothly connected to 6K transition in the superconductor.
Question (by AJS): is there hope for an orbital liquid? Very interesting - may NaLi could be but there is nothing firmly established.