Tuesday, August 7th Part II
Yong Baek Kim (U. Toronto)
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.
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
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.
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.
S = a * perimeter - Log(2)
This second term has recently been observed in DMRG computations on the Kagome Lattce
If you added explicit translational symmetry breaking as an additional variational term in your variational wavefunction (ie allowing magnetic order) and you found it lowered the energy would this disprove the spin liquid groundstate anstaz?
ReplyDeleteComment: In 1D s=1 spin systems, the AKLT VBS also does not break translational symmetry (therefore it's not necessarily true that a VBS breaks lattice symmetries, at least in d=1).
ReplyDeleteQuestion: Like the gapless spinon spectra measured in the s=1/2 chain, have there been similar neutron scattering experiments to see the gapped excitations in s=1 chains?
Vipin