They study the ground state of Heisenberg model on the isotropic triangular lattice with ring exchange. This model is relevant to the Mott insulating phase of two different organic charge transfer salts which may have a spin liquid ground state.
The authors perform variational Monte Carlo for Gutzwiller-projected BCS states. (These are RVB=resonating valence bond states, proposed by P.W. Anderson in 1987). A mean-field theory on these states gives a BCS state with broken time-reversal symmetry (known as the chiral spin liquid). The form of the fermion pairing function is d+id (d_x^2-y^2 + i d_xy) which belongs to the E representation of the C_6v point group symmetry of the lattice.
A few highlights from the paper:
- For a range of ring exchange strengthes (0.1
- The authors suggest that under pressure (increasing t/U) the Mott insulator will be destroyed leading to a superconducting state with the same d_x^2-y^2 pairing. This is qualitatively different to what one gets with a mean-field RVB theory of the model without the ring exchange (described in this PRL by Ben Powell and myself).
- The theory cannot explain why the observed low temperature specific heat of kappa-ET2-(CN)3 is weakly dependent on magnetic field.
- The "Amperean pairing" theory proposed earlier by Lee, Lee, and Senthil does not have this problem but has difficulty describing the superconducting state which develops under pressure.
Several issues not addressed in the paper are:
1. Exact diagonalisation calculations on small lattices (Ref. 10) give a different spin liquid ground state, one with an energy gap to triplet excitations, and many singlet excitations inside the gap.
2. The NMR relaxation rate 1/T_1 for kappa-ET2-(CN)3 has a power law temperature dependence consistent with gapless excitations.
3. The estimate of the impurity scattering rate (~1.5 K) can be compared to independent estimates see other organic charge transfer salts. (See Table I in this PRB, I think the estimates there suggest the authors proposal that the scattering rate may be even larger than their estimate is unlikely).
4. The real materials may have a small spatial isotropy. The Heisenberg model on the anisotropic triangular lattice can also have a spin liquid ground state, without the need for ring exchange (see for example, this paper).
Hi Ross,
ReplyDeleteFirst I should point out that our variational result is that the projected nodal d-wave state actually has lower energy than the projected d+ id or f-wave states in a range of ratio of four particle ring exchange (J_4) to nearest neighbor exchange (J_2).
Re some of your specific comments.
1. The exact diagonalization leaves room for gapless spin liquid states for sufficiently big values of J_4/J_2.
2. The NMR relaxation in the insulator is not described by the usual exponential relaxation but rather fits to a streched exponential. This cautions against any simple interpretation of the measured temperature dependence of T_1.
3. It is indeed interesting to compare with the estimated impurity scattering rate in other organics - however we should keep in mind that for us the pertinent rate is the spinon scattering rate and this may well be different from the electron scattering rate in a proximate superconductor.
4. There are many different kinds of spin liquid phases and correspondingly different kinds of Hamiltonians that will stabilize them. I am not sure about the extent of the evidence for a spin liquid phase for just the nearest neighbor Heisenberg model on anisotropic triangular lattice that you study. I would guess though that any spin liquid that appears in that model is distinct from the gapless ``projected d-wave" state we describe in our paper.
Best wishes,
Senthil.