Cenke Xu

Note: This webpage is for my friends who might have seen my recent papers on arXiv. Sometimes papers can be too technical and complicated to read, that is why I designed these webpage to explain the basic ideas of my papers in a less technical, and hopefully more understandable way, with more introduction of background that do not fit the standard format of academic papers. Besides, I am never a good writer or speaker, so I hope this webpage can help myself convey the ideas of my work.

1, Quantum Spin Hall, triplet Superconductor, and topological liquid on the honeycomb lattice , Cenke Xu, arXiv:1010.0455

The original motivation of this paper was to look for a fully gapped spin liquid phase on the honeycomb lattice, which is adjacent to a "Neel" like phase after a direct second order transition . A recent famous quantum Monte carlo simulation ( reference ) on the Hubbard model on the honeycomb lattice explicitly demonstrated the existence of such senario.

Why is such senario nontrivial?

First of all, let us think about whether a completely featureless gapped liquid state on the honeycomb lattice is possible at all, for a half-filled spin-1/2 electron system. By completely featureless, I mean it does not break any symmetry, and also has no topological degeneracy. If you try it, you probably will immediately realize that it is not easy at all. However, A completely featureless gapped liquid state on the honeycomb lattice is not against any well-known theorem, because it does have two sites per unit cell (there is a famous theorem proved by Matthew B. Hastings , which states that any spin-1/2 system on two dimensional lattice with one particle per unit cell has to be either gapless or have ground state degeneracy). Then this leads to a broad curiosity that whether such state can in principle exist on the honeycomb lattice.

The essence of such state is "breaking no symmetry", therefore we should first know the symmetry of the Hubbard model. Although the Hubbard model obviously has the SU(2) spin symmetry, the famous work by C. N. Yang and S. C. Zhang showed that the Hubbard model on any bipartite lattice at half-filling Actually has an extra SU(2) charge symmetry, which mixes particle and hole states. So overall speaking, the actual symmetry of the Hubbard model on the honeycomb lattice is SO(4) ~ SU(2)*SU(2) symmetry. For instance, on every site there can be four different electron states: empty, spin-up, spin-down, double-filled. Spin-up and spin-down are a SU(2)_spin doublet, while emplty and double-filled states are a SU(2)_charge doublet.

Now we have identified the symmetry of the Hubbard model on the honeycomb lattice, then the phrase "breaking no symmetry" implies that such liqui state has a full SO(4) symmetry. In fact, if we soften this criterion, and only require this state to have SU(2) spin symmetry, then it is actually not so difficult to show that the featureless state does exist on the honeycomb lattice, or equivalently:

For extended Hubbard model without SO(4) symmetry, there can be a fully gapped featureless state on the honeycomb lattice.

This conclusion comes from a lunch discussion I had with Leon Balents . In short, one can construct a featureless fully gapped ground state if the SO(4) symmetry is absent in the Hamiltonian. (you can try this construction yourself)

We have now established the importance of the SO(4) symmetry in order to correctly understand the fully gapped liquid state obtained by their QMC, then the first task we should do is classify all the order parameters on the honeycomb lattice with the representation of the SO(4) symmetry, and this is the first part of this paper. It turns out that, this classification of order parameters leads to quite interesting results: for instance, the two types of topological order, the quantum spin Hall and triplet superconductor, is unified as one representation of the SO(4) symmetry group, and the SO(4) symmetry will transform one topological order to the other.

How does the classification of order parameter relate to our original goal of searching for fully gapped spin liquid state? It turns out that depending on the microscopic parameters, the SO(4) symmetry group can have two types of ground states with different symmetry breakings. One of these two types of ground states has ground state manifold [S^2*S^2]/Z_2, which means that both spin and charge have manifold S^2. Therefore both spin and charge manifolds can have Skyrmion like topological defects. Because both quantum spin Hall and triplet superconductor are topological, the spin and charge Skyrmions will carry nontrivial quantum numbers: spin Skyrmion carries charge, while charge Skyrmion carries spin i.e. spin and charge view each other as topological defects.

Now we can draw a two dimensional global phase diagram with two tuning parameters : the mass gaps for spin and charge Skyrmions. Now when both Skyrmions condense, the system enters a Z_2*Z_2 liquid phase, with 16 fold degeneracy on the torus. While when charge Skyrmion is condensed while spin Skyrmion is gapped, The system will have both Neel and spin nematic order, and the ground state manifold is SO(3)/Z_2. This Z_2*Z_2 liquid phase and SO(3)/Z_2 ordered phase are presumably the featureless spin liquid phase and Neel ordered phase observed in quantum Monte Carlo simulation. And our formalism proposes that the transition between these two is a 3d O(4) transition.

2, Majorana liquids: the complete fractionalization of the electron , Cenke Xu, Subir Sachdev arXiv:1004.5431

2, Emergent Gravity at a Lifshitz Point from a Bose Liquid on the Lattice , Cenke Xu, Petr Horava arXiv:1003.0009