Archive for 2010年11月

Kane-Mele-Hubbard Model

2010年11月29日 留下评论

Recently two preprints posted on arXiv studied Kane-Mele-Hubbard model(Kane-Mele model+Hubbard type on-site repulsion) at half-filling using QMC. There is no sign problem due to particle-hole symmetry. Basically, at finite U topological insulating phase undergoes a phase transition to antiferromagnetically ordered phase.

Particle-hole symmetry and interaction effects in the Kane-Mele-Hubbard model
Dong Zheng, Congjun Wu, Guang-Ming Zhang

Correlation Effects in Quantum Spin Hall Insulators: a Quantum Monte Carlo Study
M. Hohenadler, T. C. Lang, F. F. Assaad


Sign problems in Fermionic QMC

2010年11月23日 留下评论

Well-known cases free of sign problem:

  • Negative-U Hubbard model.  Determinantal QMC has no sign problem if one decouples the on-site interaction in the right way(charge channel).
  • Hubbard model(positive U) at half-filling. Particle-hole symmetry prevents negative probability from happening. Another way to see this is that it can be mapped onto negative-U Hubbard model.  Not restricted to square lattice, any biparticle lattice is OK. For example honeycomb lattice.

Congjun Wu proves a general theorem which gives a sufficient condition under which QMC has no sign problem. Basically, there must be some anti-unitary symmetry(like time-reversal symmetry for spin 1/2). The proof was particularly for determinantal QMC. It requires that for arbitrary Hubbard-Stratonovich field configuration, the resulting determinant  must have certain anti-unitary symmetry, which means that eigenvalues come in complex conjugate pairs.  So the determinant is always positive. Notice that the determinant is generally a time-ordered product, it’s not necessarily Hermitian.


Research note 11/16/10

2010年11月17日 留下评论
  1. Surface of 3D topological insulator+s-wave superconductor.  Time-reversal invariant. In the original proposal by Fu and Kane, the induced pairing is assumed to have only s-wave component. In fact, there is also p-wave component. Later it has been shown that even in the presence of p-wave pairing, non-Abelian topological order still persists. See Phys. Rev. B 82, 144505 (2010)
  2. 2D Spin-orbit coupled semiconductor+Zeeman field in z direction+s-wave SC(proposed by Sau et. al.). The same question can be asked that what p-wave pairing does to the non-Abelian topological order. The answer is still that it doesn’t do anything. In a nice paper Phys. Rev. B 82, 184525(2010) by Ghosh et. al., it is shown that the Z_2 invariant that characterize the non-Abelian topological order(essentially the parity of Chern number) only depends on Hamiltonian at k=0 where triplet pairing vanishes.

Zero modes, Chern number and all that

2010年11月17日 留下评论

1. 在量子Hall态中Chern数等于边缘激发态的支数。这点可以从Laughlin的flux insertion argument得到。这个关系对一般的时间反演破缺的绝缘体(包括超导体)也成立。

2. 超导体中vortex的零模个数。因为电荷共轭(或者叫粒子-空穴)对称性的关系,vortex的零模个数由Chern数模2给出。类似的是一维超导体两端的零模。可以用dimensional reduction从2维的Z分类得到一维的Z_2分类。见Qi, Hughes, Zhang的文章。


Counting Majorana zero modes in superconductors
Luiz Santos, Yusuke Nishida, Claudio Chamon, Christopher Mudry

Topological Majorana and Dirac Zero Modes in Superconducting Vortex Cores
Rahul Roy

Z_2 index theorem for Majorana zero modes in a class D topological superconductor
T. Fukui, T. Fujiwara


arXiv Digest 11/15/10

2010年11月16日 留下评论

2010年11月5日 留下评论

Observation of topological order in a superconducting doped topological insulator

这是最新一期Nature Physics上的文章,标题十分ambitious,宣称在拓扑绝缘体里看到了topological order。究其内容,其实是发现铜掺杂的硒化铋在一定条件下会进入超导态。至于具体是什么样的超导态似乎没有任何观测数据。此外还宣称在里面找到了涡旋结构,因此很有可能存在Majorana费米子。我觉得在做这些claim之前,最好把超导的配对对称性等问题都弄清楚。

Update:Liang Fu已经对这个体系做了一个odd parity topological superconductor的理论。见Phys. Rev. Lett. 105, 097001 (2010)