Time 
Speaker 
Title 
Chair: YingJer Kao 
9:3011:00 
XiaoGang Wen 
Symmetry protected topological order and group cohomology theory I
Symmetry protected topological (SPT) states are a new kind of gapped
states that do not break any symmetry and have no topological order.
Despite having no symmetry breaking order and having no topological
order (ie no fractionalized topological excitations and no robust gapless boundary excitations),
SPT states can still be nontrivial new states of matter. I will present
(1) a systematic theory of SPT order based on nonlinear sigma model
and its topological terms.
(2) ways to probe/measure SPT orders, and the connection to anomalies
(3) mechanism to generate SPT orders
Slide, Video I, II, III, IV 
11:0011:30 
 Break  
11:3012:15 
TzuChieh Wei 
Topological entanglement and minimally entangled states via the geometric measure of entanglement II
I will introduce a geometric measure of entanglement (GE) and use it to quantify entanglement in toric code ground states in certain basis. It turns out that the geometric entanglement exhibits a bulk law and an additional constant, the latter being the identical to the topological entanglement entropy. The entanglement itself can actually be used to obtain the four minimally entangled states (MES), which form a special basis in the ground space and can be used to obtain the modular matrices. Next, I will examine how the topological entanglement via GE can be used to detect the transition from the Z_2 topological phase to a trivial phase by applying a string tension to the toric code. Such a string tension frustrates large loop configurations in the toric code ground state, and when it is sufficiently large, the wavefunction becomes a trivial product state. Somewhere between the exact toric code ground state and the trivial product state, there must exist a phase transition. I will present numerical results to extract the "topological" part of the geometric entanglement as the string tension varies. In the large system limit, it is indeed constant inside the topological phase and sharply drops to zero in the trivial phase, and it correctly identifies the transition point. It turns out that the transition point can be obtained analytically by mapping to the classical 2D Ising model and agrees with the numerics.
Slide, Video I, II 
12:1513:30 
 Lunch (not catered) 
Chair: Mikio Nakahara 

TzuChieh Wei 
Topological entanglement and minimally entangled states via the geometric measure of entanglement III
I will introduce a geometric measure of entanglement (GE) and use it to quantify entanglement in toric code ground states in certain basis. It turns out that the geometric entanglement exhibits a bulk law and an additional constant, the latter being the identical to the topological entanglement entropy. The entanglement itself can actually be used to obtain the four minimally entangled states (MES), which form a special basis in the ground space and can be used to obtain the modular matrices. Next, I will examine how the topological entanglement via GE can be used to detect the transition from the Z_2 topological phase to a trivial phase by applying a string tension to the toric code. Such a string tension frustrates large loop configurations in the toric code ground state, and when it is sufficiently large, the wavefunction becomes a trivial product state. Somewhere between the exact toric code ground state and the trivial product state, there must exist a phase transition. I will present numerical results to extract the "topological" part of the geometric entanglement as the string tension varies. In the large system limit, it is indeed constant inside the topological phase and sharply drops to zero in the trivial phase, and it correctly identifies the transition point. It turns out that the transition point can be obtained analytically by mapping to the classical 2D Ising model and agrees with the numerics.
Slide, Video I, II 

 Break  

Zhengcheng Gu 
Vortexline condensation in three dimensions: A physical mechanism for bosonic topological insulators
Bosonic topological insulators (BTI) in three spatial dimensions are symmetry protected topological (SPT) phases with U(1)⋊ZT2 symmetry, where U(1) is boson particle number conservation, and ZT2 is timereversal symmetry with T2=1. BTI were first proposed based on the group cohomology theory which suggests two distinct root states, each carrying a Z2 index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, leading to a new BTI root state beyond the group cohomology classification. Nevertheless, it is still unclear what is the universal physical mechanism for BTI phases and what kinds of microscopic Hamiltonians can realize them. In this paper, we answer the first question by proposing a universal physical mechanism via vortexline condensation in a superfluid, which can potentially be realized in realistic systems, e.g., helium4 or cold atoms in optical lattices. Using such a simple physical picture, we find three root phases, of which two of them are classified by group cohomology theory while the other is beyond group cohomology classification. The physical picture also leads to a "natural" bulk dynamic topological quantum field theory (TQFT) description for BTI phases and gives rise to a possible physical pathway towards experimental realizations. Finally, we generalize the vortexline condensation picture to other symmetries and find that in three dimensions, even for a unitary Z2 symmetry, there could be a nontrivial Z2 SPT phase beyond the group cohomology classification.
Slide, Video I, II, III 