Brad Marston: "Do gapless spin liquids exist in 2D ?"

Brad Marston started by saying that the topic of his talk is a controversial one. He said he will present some works done in collaboration with Ookie Ma and Arun Paramekanti.

The first thing he mentioned was the Matt Hasting's theorem for two dimensional spin systems, namely a generalization of the Lieb-Shulz-Matthis theorem in 1D. The theorem applies to the systems with half-odd-integer spin per unit cell and it says the ground state of the system can only be one of two types.

1) When there is an excitation gap about the ground state, the ground state has to be degenerate. The examples are dimerized states and Z2 spin liquids.

2) If there is no gap, then the ground state is a U(1) spin liquid.

Thus the non-generate ground state with an excitation gap is not possible.

Brad mentioned there are some interesting materials with half-odd-integer spins per unit cell, where possible signatures of spin liquid phases may have been discovered.

There examples are as follows.

1) Cs2 Cu Cl4 (Radu Coldea's neutron scattering experiments)

Cu spin-1/2 moments reside on an anisotropic triangular lattice. The lattice can be regarded as a square lattice with an additional strong diagonal bond only in one diagonal direction or a coupled spin-chain system with the chains along the diagonal direction. The exchange coupling J along the diagonal or chains is three times stronger than the inter-chain J'. DM-interaction also exists and it is about J/10; this leads to highly anisotropic response to external magnetic field.

At this point, Andrey Chubukov asked how people know about the magnitudes of J and J'. Brad said that one can go to the ferromagnetic state by applying external magnetic field, look at the
magnon spectra, and get the exchange couplings. John Mydosh asked whether this problem is related to the BEC of spin excitations. Bard replied that it may be so in the high field limit.

Brad then mentioned that the ground state below T < 1K has a magnetic long range order. At intermediate temperatures, neutron scattering sees continuum of excitations that could be due to deconfined spinons of a spin liquid phase (other interpretation may also be possible).

2) \kappa-(ET)2 Cu2 (CN)3 (Kanoda's experiment)

Effective spin-1/2 moments reside on almost isotropic triangular lattice. Spin-liquid-like behavior is seen in the insulating phase near the Metal-Insulator transition. Charge fluctuation may be important and lead to substantial ring-exchange type contributions. No long range order is seen down to 20mK (NMR). There are anomalies in thermodynamic quantities around 5K.

3) Zn Cu3 (OH)6 Cl2 (Herbertsmithties, Young Lee's experiments and many others)

Cu spin-1/2 moments reside on the isotropic Kagome lattice. No order is seen down to 50mK (NMR, \muSR). Power law in T is seen in the specific heat.

Then Bard switched the gear and started the discussion on possible instabilities of gapless spin liquid phases. This is mainly because there have been some proposals advocating that Herbertsmithites may be an example of 2D gapless spin liquid phases. He wants to see whether this really works and there is any alternative ground state or not.

He listed the following instabilities or other possible ground states.

1) Long range spin order
2) Various forms of dimerization
3) More exotic candidates; spin-nematic, time-reversal-symmetry breaking, charge-conjugate symmetry in the case of the SU(N) magnets.
4) Spin liquid with an extended Fermi surface.
There may be spin-triplet (V. Galitski + Y. B. Kim) or Amperian pairing (S. lee+P. A. Lee).

Brad talked about an interesting theoretical paper by Y. Ran, M. Hermele, X. G. Wen, and P. A. Lee. This is a variational calculation of various candidate ground states. The paper claimed that the ground state seems to be a gapless spin liquid where spinons have a Dirac spectrum. It can be obtained by putting the (fictitious) \pi-flux in each hexagon, to be seen by the spinons. Then they performed the projection (remove double-occupancy) on the mean-field wavefunction.

\Psi_{variational} = P_G \Psi_{MF}

Their variational energy is remarkably good in the sense that it is quite close to the result of
the exact diagonalization, namely

E_{\pi flux state} = - 0.42866(2) J

E_{ED} = - 0.43 J

But Brad believes that the true ground may not be a spin liquid and Y. Ran et al's calculation may have some problems. First he mentioned his old work with C. Zeng (1991), where they realized that the ground state of the Heisenberg model in the 1/N expansion is some kind of dimerized states. Basically the system wants to maximize the number of hexagons with (non-touching) three dimers. One can show that at least 18 site unit cell is needed to achieve this. Thus one ends up with the dimerization patterns with the unit cell of some multiples of 18 sites.

Then he mentioned a recent work by Singh and Huse, where dimer expansion is used to obtain the dimerized state with the unit cell of 36 sites. The energy of this state is

E_{dimer} = - 0.432 J

In order to see how good Y. Ran et al's \pi-flux state is, Brad examined the effect of small number of dimerized bonds on the \pi-flux state. With 5% dimerization (calculation done on the 12 X 32 X 3 = 432 sites), he gets

E_{5% dimer} = - 0. 42860(3) J

which is not terribly different from Y. Ran et al's result. This implies that energy land scape is quite flat ! Brad also mentioned a problem with Y. Ran et al's calculation, namely they imposed the anti-periodic boundary condition (by inserting a fictitious flux) in one of two directions. This breaks the rotational symmetry. Apparently this has a big effect; in fact this is the way that the degeneracy was broken. Brad said there exists 10% dimerization in their numerical calculation because of this choice of boundary condition.

Brad stressed that according to Singh this is the first time example where dimer expansion
converges so nicely; this may be a strong evidence that the ground state is indeed a dimerized state with a huge unit cell.

Brad then mentioned a work by White and Singh (Physical Review Letters 2000). They looked at a "kagome-strip" or coupled chains where there are "crossed" inter-chain couplings. The ground state turns out to be dimerized and the gap is very small (gap = 0.01 J). Brad looked at this problem and found the detailed ordering pattern of the dimerization (which turns out to be rather complicated).

I asked whether these results are in contradiction with Y. Ran et al's argument based on the PSG analysis, where they claimed all physically relevant perturbations about the \pi-flux phase seem to be irrelevant and the spin liquid phase is stable. Brad replied that perhaps the gauge field fluctuation is so strong that the PSG of the mean field states are not that useful.