A continuous family of fully-frustrated Heisenberg model on the Kagome lattice
We find that the antiferromagnetic Heisenberg model on the Kagome lattice with nearest neighboring exchange coupling(NN-KAFH) belongs to a continuous family of fully-frustrated Heisenberg model on the Kagome lattice, which has no preferred classical ordering pattern. The model consists of the first, second and the third neighboring exchange coupling . We find that when , the lowest band of , namely, the Fourier transform of the exchange coupling, is totally non-dispersive. Exact diagonalization calculation indicates that the ground state of the spin- NN-KAFH is locally stable under the perturbation of and , provided that . Interestingly, we find that the same flat band physics is also playing an important role in the RVB description of the spin liquid state on the Kagome lattice. In particular, we show that the extensively studied U(1) Dirac spin liquid state on the Kagome lattice can be generated from a continuous family of gauge inequivalent RVB parameters and that the spinon spectrum is not uniquely determined by the structure of this RVB state.
The study of the spin- Kagome antiferromagnetic Heisenberg model(KAFH) has attracted a lot of attention both theoretically and experimentally. It is generally believed that the spin- KAFH with nearest-neighboring exchange(NN-KAFH) is a promising candidate to realize the quantum spin liquid state. Early studies on the NN-KAFH have accumulated a lot of evidences for a quantum spin liquid ground state in this modelED1 (); ED2 (); ED3 (); HTSE1 (); ED4 (); ED5 (). However, the exact nature of the possible spin liquid ground state of the NN-KAFH is still elusiveHTSE2 (); TRG1 (). While DMRG studies tend to imply a fully gapped spin liquid ground stateDMRG1 (); DMRG2 (); DMRG3 (); DMRG4 (), variational studiesVMC1 (); VMC2 (); VMC3 (); VMC4 (); VMC5 (); VMC6 (); VMC7 () and tensor network simulationsTRG2 () seem to prefer a gapless spin liquid ground state. At the same time, the origin of the massive spin singlet excitation in the low energy spectrum of NN-KAFH is still poorly understoodSinglet1 (); Singlet2 (); Singlet3 (); Singlet4 (). These problems have motivated several theoretical suggestions that the spin- NN-KAFH may sit at or be very close to a quantum critical pointQCP1 (); QCP2 (); QCP3 (), where two or even a massive number of phases meet. Perturbation away from NN-KAFH may thus be illuminating for the understanding of the physics of the KAFH in an unified fashion. For example, DMRG studies have found evidence for a chiral spin liquid state when there is more extended exchange couplingsDMRG5 (); DMRG6 (); DMRG7 (); DMRG8 ().
At the semiclassical level, the NN-KAFH is special in that it is fully frustrated in the sense that it has no preferred classical ordering pattern. For a general Heisenberg model of the form , the semiclassical ordering pattern of the system is determined by the Fourier transform of the exchange couplings, , which is in general a matrix for system with a complex lattice. More specifically, the wave vector of the semiclassical ordering is given by the momentum at which the lowest band of reaches its minimum. A model is fully frustrated when the lowest band of is totally non-dispersive. This is impossible on a simple lattice, when is simply a number, since the exact flatness of can only be achieved when for all . However, for system with a complex lattice, one or more bands of can be totally non-dispersive even if is nontrivial. For example, the lowest band of for the NN-KAFH is exactly flat.
In general, such fully frustrated model is realized only as isolated point in the space of Hamiltonian parameters. Perturbation away from such special point will immediately lift the semiclassical degeneracy and choose for the system a particular classical ordering pattern. This is of course not what we prefer if our purpose is to discover more exotic quantum phases. Here we show that the NN-KAFH is not such an isolated point in the space of Hamiltonian parameters, but is within a continuous family of fully frustrated models. More specifically, we show that the flat band in the spectrum of is robust against the introduction of the second and the third neighbor exchange coupling, namely, and as illustrated in Figure 1, provided that . We find that for , the flat band of is always the lowest band and the model is thus always fully frustrated. The study of such a continuous family of fully frustrated model may shed important light on the physics of the NN-KAFH.
We first look into the origin of the flat band in the case of the NN-KAFH. is now a matrix and has the same form as the Hamiltonian matrix of a free electron with hoping integral between nearest neighboring sites. It is given by
in which and are the three vectors connecting nearest neighboring sites of the Kagome lattice(see Figure 1 for an illustration). Using the identity , one can easily prove the existence of a flat band in the spectrum of with eigenvalue . The origin of this flat band can be understood more intuitively in real space. More specifically, it can be attributed to the destructive interference between the hopping amplitudes out of a localized Wannier orbital defined on an elementary hexagon of the Kagome latticeFlat (), as is illustrated in Figure 2. Interestingly, one find that the same destructive interference is effective even if we introduce the second and the third neighbor exchange coupling, provided that . We thus expect the flat band to be robust against the introduction of and , provided that . Indeed, one find that always has as one of its three eigenvalues when .
The KAFH is fully frustrated when the above flat band becomes the lowest band of . In Figure 3, we plot the band structure of for several values of . From the plot we see that the flat band is the lowest band of when . The NN-KAFH is thus within a continuous family of fully frustrated models, rather than an isolated point in the space of Hamiltonian parameters. The existence of such a continuous family of fully frustrated Heisenberg model on the Kaogme lattice offers a much broader playground for the search of quantum spin liquid state on the Kagome lattice. In particular, it is interesting to know to what extend the singular physics of the NN-KAFH will be modified by perturbation inside the space of this continuous family of fully frustrated models.
For this reason, we have performed exact diagonalization calculation for the model on a finite cluster of 36 sites. We have only considered the identity representation. To diagnose the stability of the ground state of the NN-KAFH under the perturbation of and , we have calculated the second derivative of the ground state energy with respect to and . We find that the ground state energy of the system behaves rather smoothly when . This is in stark contrast with the situation when , for which a sharp peak in is observed at the NN-KAFH point. We note that previous study find that the second derivative of the ground state energy with respect to the six-site ring exchange coupling also exhibits a sharp peak at the NN-KAFH pointQCP3 (). These results indicate that the perturbation corresponding to , and are all strongly relevant at the NN-KAFH point, but the joint perturbation of and is irrelevant when . This give us the hope that the spin liquid ground state of the NN-KAFH may form a phase of finite range of stability in the space of Hamiltonian parameters, rather than being an isolated critical point.
Interestingly, we find that the flat band physics discussed above is important not only for the determination of the classical ordering pattern in the KAFH, it also plays an important role in the RVB description of the possible spin liquid state on the Kagome lattice. Here we will concentrate on the extensively studied U(1) Dirac spin liquid on the Kagome latticeVMC1 (); VMC2 (); VMC3 (); VMC4 (). This state is constructed from Gutzwiller projection of the ground state of the following mean field Hamiltonian
Here are phase factors introduced to ensure that a -flux is enclosed in each unit cell. More specifically, they are translational invariant in the direction, but will change sign when translated by one unit in the direction, if the unit cell indices of site and in the direction differ by an odd number. Here and are the two basis vectors of the Kagome lattice, as is illustrated in Figure 5. on the 18 independent bonds of the first unit cell, as is shown in Figure 5, all equal to 1.
Just as in the case of , a non-dispersive band at emerges in the spectrum of when . The origin of such a flat band can also be attributed to the destructive interference between the hopping amplitudes out of a local orbital defined on an elementary hexagon of the Kagome lattice. More interestingly, one find that the ground state of at half-filling is independent of the value of when and . This striking result can be understood by rewriting as , and noting the fact that when . Here denotes the part of the Hamiltonian in that is proportional to . The inclusion of with will thus only modify the eigenvalues of , but not change its eigenvectors. In Figure 6, we plot the spectrum of for different values of . One find that for , the Fermi level always crosses the Dirac point and the occupied states is independent of the value of . For outside this range, the Fermi level will move into the spectral continuum and the Dirac spin liquid state will be destroyed.
We note that while the ground state of is independent of when , the excitation spectrum is strongly -dependent. Thus the same U(1) Dirac spin liquid state is compatible with very different spinon spectrum. The lack of connection between the ground state property and the low energy spectrum is very unusual for system with a local Hamiltonian. It is interesting to see if this is a genuine property of the KAFH, or just an artifact of the RVB description. At the same time, the nearly redundant nature of the RVB parameters around the U(1) Dirac spin liquid state will greatly complicate their optimization procedure. More specifically, the variational energy as a function of the RVB parameters will in general exhibit a long, narrow and curved valley with a very flat bottom in the space of variational parameters. In a forthcoming paper, we will show that notwithstanding such a numerical challenge, we succeed to find a gapped spin liquid state as the most favorable variational ground state of the spin- NN-KAFH in the thermodynamical limit. Different from previous variational studies on the NN-KAFHVMC1 (); VMC2 (); VMC3 (); VMC4 (), the state we find has very large RVB parameters on both the second and the third neighboring bonds.
In summary, we find that there is a continuous family of fully frustrated Heisenberg models on the Kagome lattice, within which the extensively studied NN-KAFH is but an ordinary point. This discovery greatly enlarge the playground for the search of exotic physics in the KAFH. We find that the flat band physics that is responsible for the fully frustrated nature of the spin model is also playing an important role in the RVB description of the spin liquid state on the Kagome lattice. In particular, we find that extensively studied U(1) Dirac spin liquid can be generated from a continuous family of gauge inequivalent RVB parameters, which indicates that the spinon spectrum is not uniquely determined by the structure of this RVB state.
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