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\section*{symmetry protected trivial order}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{symmetryprotected_trivial_order}{}\section*{{Symmetry-protected trivial order}}\label{symmetryprotected_trivial_order}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples_of_spt_order}{Examples of SPT order}\dotfill \pageref*{examples_of_spt_order} \linebreak
\noindent\hyperlink{group_cohomology_theory_for_spt_phases}{Group cohomology theory for SPT phases}\dotfill \pageref*{group_cohomology_theory_for_spt_phases} \linebreak
\noindent\hyperlink{spt_phases_in_free_fermion_systems}{SPT phases in free fermion systems}\dotfill \pageref*{spt_phases_in_free_fermion_systems} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{reviews}{Reviews}\dotfill \pageref*{reviews} \linebreak
\noindent\hyperlink{classification_for_bosonic_spt_phases}{Classification for bosonic SPT phases}\dotfill \pageref*{classification_for_bosonic_spt_phases} \linebreak
\noindent\hyperlink{classification_for_free_fermion_spt_phases}{Classification for free fermion SPT phases}\dotfill \pageref*{classification_for_free_fermion_spt_phases} \linebreak
\noindent\hyperlink{early_discovery_articles}{Early discovery articles}\dotfill \pageref*{early_discovery_articles} \linebreak
\noindent\hyperlink{other_articles}{Other articles}\dotfill \pageref*{other_articles} \linebreak
\noindent\hyperlink{conference_and_seminar_cycles}{Conference and seminar cycles}\dotfill \pageref*{conference_and_seminar_cycles} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Symmetry Protected Trivial order (SPT order)} (also known as \textbf{Symmetry Protected Topological order}) is a new kind of order in zero-temperature [[solid state physics|states of matter]] that have a [[symmetry]] and a finite energy gap. The SPT order has the following defining properties:

\begin{enumerate}%
\item distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a [[phase transition]], if the deformation preserves the symmetry.
\item however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation.

\end{enumerate}
Using the notion of quantum entanglement, we can say that SPT states are short-range [[entanglement|entangled states]] with a symmetry.\newline Using the notion of [[topological order]], we can say that SPT states are symmetric states with trivial [[topological order]].

\hypertarget{examples_of_spt_order}{}\subsection*{{Examples of SPT order}}\label{examples_of_spt_order}

The first example of SPT order is the Haldane phase of spin-1 chain. It is a SPT phase protected by the $SO(3)$ spin [[rotation group]] symmetry. Another example of SPT order is the [[topological insulator]] of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry.

\hypertarget{group_cohomology_theory_for_spt_phases}{}\subsection*{{Group cohomology theory for SPT phases}}\label{group_cohomology_theory_for_spt_phases}

Recently, it was shown that the bosonic SPT orders are described by [[group cohomology]] theory: \textbf{d+1D SPT states with on-site symmetry G are labeled by the elements in group cohomology class $H^{d+1} [G, U(1)]$.} It was also shown that the fermionic SPT orders are described by group super-cohomology theory.

So the group (super-)cohomology theory may allow us to classify all SPT orders even for interacting systems, which include interacting topological insulator/superconductor.

\hypertarget{spt_phases_in_free_fermion_systems}{}\subsection*{{SPT phases in free fermion systems}}\label{spt_phases_in_free_fermion_systems}

Free fermion system can also have non-trivial SPT phases, such as [[topological insulator]]s and topological superconductors. Those free fermion SPT phases are classified by K-theory.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[protection from quantum corrections]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Related entries: [[TQFT]], [[topological order]], [[group cohomology]], [[entanglement]]

\hypertarget{reviews}{}\paragraph*{{Reviews}}\label{reviews}

\begin{itemize}%
\item \href{http://en.wikipedia.org/wiki/Symmetry_protected_topological_order}{SPT order}

\end{itemize}
\hypertarget{classification_for_bosonic_spt_phases}{}\paragraph*{{Classification for bosonic SPT phases}}\label{classification_for_bosonic_spt_phases}

Discussion via [[higher dimensional WZW models]] is in

\begin{itemize}%
\item Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, [[Xiao-Gang Wen]], \emph{Symmetry protected topological orders and the group cohomology of their symmetry group}, Phys. Rev. B 87, 155114 (2013) \href{http://arxiv.org/abs/1106.4772}{arXiv:1106.4772}; A short version in Science \textbf{338}, 1604-1606 (2012) \href{http://dao.mit.edu/~wen/pub/dDSPTsht.pdf}{pdf}

\end{itemize}
\hypertarget{classification_for_free_fermion_spt_phases}{}\paragraph*{{Classification for free fermion SPT phases}}\label{classification_for_free_fermion_spt_phases}

\begin{itemize}%
\item [[Daniel S. Freed]], [[Gregory Moore|Gregory W. Moore]], \emph{Twisted equivariant matter}, Annales Henri Poincar\'e{} December 2013, Volume 14, Issue 8, pp 1927--2023 \href{http://arxiv.org/abs/1208.5055}{arxiv/1208.5055} (uses [[equivariant K-theory]])


\item Alexei Kitaev, \emph{Periodic table for topological insulators and superconductors}, Proc. L.D.Landau Memorial Conf. ``Advances in Theor. Physics'', June 22-26, 2008, Chernogolovka, Russia, \href{http://arxiv.org/abs/0901.2686}{arxiv/0901.2686} (uses [[K-homology]], [[Bott periodicity]] etc.)



\end{itemize}
\hypertarget{early_discovery_articles}{}\paragraph*{{Early discovery articles}}\label{early_discovery_articles}

\begin{itemize}%
\item Zheng-Cheng Gu, [[Xiao-Gang Wen]], Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order , Phys. Rev. B80, 155131 (2009);
\item Frank Pollmann, Erez Berg, Ari M. Turner, Masaki Oshikawa, Symmetry protection of topological order in one-dimensional quantum spin systems , Phys. Rev. B85, 075125 (2012).
\item Xie Chen, Zheng-Xin Liu, [[Xiao-Gang Wen]], 2D symmetry protected topological orders and their protected gapless edge excitations Phys. Rev. B 84, 235141 (2011);
\item Zheng-Cheng Gu, [[Xiao-Gang Wen]], \href{http://arxiv.org/abs/1201.2648}{Symmetry-protected topological orders for interacting fermions -- fermionic topological non-linear sigma-models and a group super-cohomology theory}

\end{itemize}
\hypertarget{other_articles}{}\paragraph*{{Other articles}}\label{other_articles}

\begin{itemize}%
\item Michael Levin, Zheng-Cheng Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012), arXiv:1202.3120.
\item Yuan-Ming Lu, Ashvin Vishwanath, Theory and classification of interacting `integer' topological phases in two dimensions: A Chern-Simons approach, Phys. Rev. B 86, 125119 (2012), arXiv:1205.3156.
\item Davide Gaiotto, Theo Johnson-Freyd, \emph{Symmetry protected topological phases and generalized cohomology}, \href{https://arxiv.org/abs/1712.07950}{arxiv/1712.07950}

\end{itemize}
\hypertarget{conference_and_seminar_cycles}{}\paragraph*{{Conference and seminar cycles}}\label{conference_and_seminar_cycles}

\begin{itemize}%
\item seminar in Koeln \href{http://www.thp.uni-koeln.de/trebst/Lectures/2012-TopoSeminar.html}{Topological states of matter}
\item Topological Phases of Matter: Simons Center, June 10-14, 2013, videos \href{http://scgp.stonybrook.edu/archives/3464}{available}
\item A. Kitaev, \emph{On the classification of short-range entangled states}, \href{http://scgp.stonybrook.edu/archives/7874}{video}

\end{itemize}
[[!redirects symmetry protected trivial order]] [[!redirects symmetry protected trivial orders]] [[!redirects symmetry-protected trivial order]] [[!redirects symmetry-protected trivial orders]] [[!redirects symmetry protected topological order]] [[!redirects symmetry protected topological orders]] [[!redirects symmetry-protected topological order]] [[!redirects symmetry-protected topological orders]] [[!redirects SPT order]] [[!redirects SPT orders]]

[[!redirects SPT state]] [[!redirects SPT states]] [[!redirects SPT phase]] [[!redirects SPT phases]]



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