Duke University, Department of Physics, Durham, North Carolina, 27708, USA.
Phys Rev Lett. 2011 Nov 18;107(21):210501. doi: 10.1103/PhysRevLett.107.210501.
We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit. We prove that any two-dimensional Hamiltonian which is a sum of commuting local terms is not topologically ordered at T > 0. We show that such trivial states cannot be used to store quantum information using certain stringlike operators. This definition is not too restrictive, however, as the four dimensional toric code does have a nontrivial phase at nonzero temperature.
我们提出了一个在非零温度下拓扑序的定义，类比于通常的零温度定义，即如果一个态不能通过局部（或近似局部）量子电路变换为乘积态（或接近乘积态），那么这个态就是拓扑有序的，或者说是“非平凡的”。我们证明了任何二维哈密顿量，如果它是由 commuting 局部项的和组成的，那么在 T > 0 时就不是拓扑有序的。我们表明，这样的平凡态不能使用某些类似字符串的算子来存储量子信息。然而，这个定义并不是太严格，因为四维托里码在非零温度下确实有一个非平凡的相。
