# Tagged Questions

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### Homotopy Theory for Topological Insulators

I'm trying to understand topological insulators in terms of homotopy invariants. I understand that in 2 spatial dimensions, we have Chern insulators since $$\pi_2(S^2) = \mathbb{Z}$$ One subtlety that ...
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### Quantum ising/heisenberg model, states representation

I am working with a hamiltonian which looks like this (Heisenberg model) I have made a program which computes this hamiltonian using Pauli matrices (spin 1/2). My working space is then the tensor ...
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### How to calculate the string order parameter (for Haldane phase) in density matrix renormalization group?

The ground state of the spin-1 chain is the Haldane phase, which is known to be a symmetry protected topological phase and cannot be detected by conventional order parameter (beyond the Landau-...
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### What is the difference between Bosonic and Fermionic symmetry protected topological phases (SPT)

I am reading the paper Braiding statistics approach to Symmetry Protected Topological Phases'' by Levin and Gu. In this paper two spin models considered describe spin-1/2 particles in (1+2) ...
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### Paradox in topological phase of SSH model

Consider the SSH model, i.e. the dimerized tight-binding model with Hamiltonian $$H = \sum_i (t+\delta t) c^\dagger_{Ai} c_{Bi} + (t-\delta t) c_{A(i+1)}^\dagger c_{Bi} + \text{h.c.}.$$ This describes ...
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### What does it mean for a topological phase to be “symmetry protected”?

I have seen some very nice and enlightening awnsers to questions related to topological order and insulators, such as here, or here. However, I'm still puzzled by the concept of "symmetry protection" ...
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### effective field theory of the projective semion model

The "projective semion" model was considered in http://arxiv.org/abs/1403.6491 (page 2). It is a symmetry enriched topological (SET) phase. There is one non-trivial anyon, a semion $s$ which induces a ...
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### what is the difference between the trivial SPT phase and nontrivial SPT phase?

I read some paper and found two terms "trivial SPT" and "nontrivial SPT". I am wondering what is the difference between trivial SPT and nontrivial SPT. Thank you! SPT stands for symmetry-protected ...
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### A naive question about topologically ordered wavefunction?

Topological entanglement entropy (TEE, proposed by Levin, Wen, Kitaev, and Preskill) is a direct characterization of the topological order encoded in a wavefunction. Here I have some confusions, and ...
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### Compute $Z_2$ Invariant of 2D Topological Insulators without Computing the Eigenstates

For 2D Time-Reversal Invariant systems ($T H(\vec{k}) T^{-1} = H(-\vec{k})$), there is a formula by Fu-Kane-Mele in order to determine whether the system belongs to either one of distinct topological ...
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### What is the physical mechanism of the topological phase transition driven by temperature?

The topological property of topological insulators (TIs) is characterized by the non-trivial topological invariants of their band structures, such as $Z_{2}$ topological invariants. While it's clearly ...
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### What is the different between topological order and Landau's order in a system

I have thought about topological order for a long time, but I am still confused it. Roughly speaking in my understanding, the topological state is the eigen-state of some special symmetry such time ...
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### Must a symmetric phase and a symmetry-breaking phase be different?

When I was reading this paper, it urged me to ask whether a symmetric phase and a symmetry breaking phase must be different? As I am considering quantum phase transitions, let $H(g)$ be a general ...
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### Interacting Chiral topological invariants using Green function

We can calculate the topological invariants for 1D interacting topological insulators as $n=\frac{\text{Tr}}{2\pi i}\oint_cG\partial_kG^{-1}$ where as for interacting chiral topological ...
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### Weaker Formulations of Bulk-boundary Correspondence for Interacting Systems

From this post, it seems that bulk-boundary correspondence does not hold in general for interacting systems. What is meant by bulk-boundary correspondence there appears to be the existence of robust (...
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### What can we learn from a band structure diagram?

Other than the band gap and its magnitude, what are the things that we can immediately learn about the properties of the material just by glancing at its band structure? Can we say something about ...
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### How to distinguish between a topological state from from a non-topological one?

How to distinguish between a topological state from from a non-topological one? Is there any standard procedure for identifying the topological features of a given hamiltonian? In general what are the ...