# Question about trivializing an SPT phase via group extension.

+ 5 like - 0 dislike
252 views

Consider a d (spatial) dimensional SPT phase with an on-site symmetry G, classified by some non-trivial cocycle  $\omega^{d+1}(\{g_i\}) \neq \delta \mu^d(\{g_i\})$, $[\omega^{d+1}(\{g_i\})] \in H^{d+1}(G,U(1))$, . In a recent paper, Wang, Wen and Witten construct gapped boundaries via a suitable group extension $1 \longrightarrow H \overset{i}{\longrightarrow} K \overset{r}{\longrightarrow} G \longrightarrow 1$ such that the cocycle fro $H$ defined via pullback is trivial $r^*\omega^{d+1}(\{h_i\}) = \omega^{d+1}(\{r(h_i)\})= \delta \mu^d(\{h_i\})$. The gapped boundary corresponds to an $H$ invariant theory but with $K$ gauged so that the global symmetry is $H/K \cong G$ as required. The following sentence they say however confuses me:

" By definition, two states in two different $G$-SPT phases cannot smoothly deform into each other via deformation paths that preserve the $G$-symmetry. However, two such $G$-SPT states may be able to smoothly deform into each other if we view them as systems with the extended $H$-symmetry and deform them along the paths that preserve the $H$-symmetry. "

To me, it sounds like the two sentences contradict each other.
Q1) If there is an $H$ invariant deformation path to connect the system to a trivial state, then does that not automatically give us a $G$ invariant deformation path?
Q2) Does sentence 2 somehow only apply to the boundary rather than the bulk?

asked Sep 14, 2017

Please give a background reference. What is SPT?

What is SPT?

### SPT phase = symmetry-protected topological phase

I apologize for the delayed response. The response by Xiao-Gang Wen in this post in physics stack exchange gives a nice explanation on what a Symmetry Protected Topological (SPT) phase is and is not. https://physics.stackexchange.com/questions/135398/definition-of-short-range-entanglement

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.