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Questions tagged [symmetry-protected]

Symmetry Protected refers to the symmetry protection by Symmetry Protected Topological order (SPT order). It is beyond the Landau-Ginzburg theory. See https://en.wikipedia.org/wiki/Symmetry_protected_topological_order

Symmetry Protected refers to the symmetry protection by Symmetry Protected Topological order (SPT order). It is beyond the Landau-Ginzburg theory. See https://en.wikipedia.org/wiki/Symmetry_protected_topological_order.

Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states with a symmetry (by contrast: for long-range entanglement see topological order, which is not directly related to the famous EPR paradox). Since short-range entangled states have only trivial topological orders we may also refer the SPT order as Symmetry Protected "Trivial" order.

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Low energy description of Symmetry Enriched Topological phases

Prelude: low energy description of Symmetry Protected Topological (SPT) phases It is known [1] that the low energy effective description of SPT phases, protected by a group $G$ is an invertible field theory (iQFT). Namely, if $A$ is a…
ɪdɪət strəʊlə
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What is topological about topological (Dirac or Weyl) semimetals?

The following is my rough understanding of topological phases of matter (please let me know if it is incorrect.) Topologically ordered phases of matter are topological in the sense that they are determined by their topological excitations, and…
d_b
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Classification of higher Symmetry Protected Topological (SPT) phases

Suppose that we have a $d$ dimensional bosonic SPT phase, protected by some $p$-form symmetry, $G^{[p]}$. Suppose also that it is classified within group cohomology, so that we don't have to run into cobordisms. It is not clear to me what type of…
ɪdɪət strəʊlə
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Time-reversal (explicitly) broken surface of $(3+1)$-dimensional topological insulator

Let us consider the surface of $(3+1)$-dimensional topological insulator, which is protected by the charge conservation $U(1)_Q$ and a time-reversal symmetry $\mathbb{Z}_2^T$. Such a surface, if not breaking $U(1)_Q\rtimes\mathbb{Z}_2^T$ explicitly,…
Yuan Yao
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Relationship between different $Z_{16}$ classifications

I find that there exist two classifications which have a $Z_{16}$ group structure: The sixteen fold way of classifying Majorana fermions, vortex systems appearing in Kitaev's paper on his honeycomb model: https://arxiv.org/abs/cond-mat/0506438. In…
Abhishodh
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Partition functions of descendent SPTs of the Haldane chain

The Haldane chain can be viewed as a $1+1$ D SPT protected by an $SO(3)$ symmetry. If this SPT is put on a triangulated closed manifold $X$, its partition function can be written as $$ e^{i\pi\int_Xw_2(SO(3))} \tag{1}\label{SO(3)} $$ where…
Mr. Gentleman
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Why don't certain decorated domain wall constructions for SPTs lead to spontaneous symmetry breaking?

There is a construction of symmetry protected topological (SPT) states which roughly goes as follows. We start with a $d$-dimensional system with symmetry $\mathbb{Z}_2 \times G$ in the phase where the $\mathbb{Z}_2$ is spontaneously broken. To the…
pianyon
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What are the differences between Haldane phase, non-interacting topological insulator/superconductor, and SPT order?

Haldane phase, and non-interacting topological insulator/superconductor are often regarded as examples of symmetry protected topological (SPT) orders.
Xiao-Gang Wen
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't Hooft twisted torus construction and its relation to characteristic (e.g. Stiefel-Whitney) class

It is known that the $PSU(2) = SO(3)$ and there is an associated global anomaly labeled by the second Stiefel-Whitney class $w_2$. This second Stiefel-Whitney class $w_2$ can detect the 1+1 dimensional Haldane phase (antiferromagnet spin-1 gapped…
wonderich
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Inverse of Haldane phase?

Based on what I have learned so far, Haldane phases are a nontrivial SPT for 1D spin-1 chains. The trivial phase acts as an "identity" under the group of SPT phases ( with stacking as the group operation ). If my understanding is correct, can anyone…
baba26
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Chern-Simons theory with a discrete gauge symmetry

Let us consider a Chern-Simons theory on a $3$-manifold $M$ (can be a spin manifold with a given spin structure if needed) with a discrete-symmetry gauge field e.g. $\mathbb{Z}_n$ symmetry. It can be embedded in a $U(1)$ gauge group with constraints…
Yuan Yao
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In a class of parametrized symmetric Hamiltonians, should its symmetry operator be parametrized the same way?

I would like to ask the following in the context of symmetry-protected topological phase. Consider a class of Hamiltonians parametrized by $\{a_1,a_2,...\}$ denoted by $H(a_1,a_2,...)$. Suppose there is a path $p$ in the parameter space, let me…
git-able
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The ground state degeneracy of a single spin 1/2 under $SU(2)$ spin-rotation symmetry

Let us consider a single spin 1/2 in (0+1) dimension. It is expected that when we add more interaction onto such single spin, the energy gap cannot be opened as long as the Hamiltonian possesses $SU(2)$ spin-rotation symmetry. My question is how to…
Yuan Yao
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