Questions tagged [topological-order]

Topological order is a new kind of order in quantum matter, which corresponds to pattern of long-range quantum entanglement. See http://en.wikipedia.org/wiki/Topological_order

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Chiral symmetry in Su-Schrieffer-Heeger (SSH) model

We know that the hamiltonian SSH model in the presence of on-site potential(V) can be written on the basis of the Pauli matrix. $$h(k)=V\sigma_0+h_x\sigma_x+h_y\sigma_y,$$ and the term V breaks the ...
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Numerical solution to Harper's equation

I want to learn how to solve Harper's equation and plot the Hofstader butterfly spectrum numerically. Can anyone suggest references/links where this is done?
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Why topologically non-trivial materials are robust againist any external perturbations or defects?

Topologically non-trivial materials are insensitive to perturbations or defects. How can I prove it mathematically? I thought of making the first-order perturbation term zero. $$\left< \psi \right|...
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What is the relation between non-local order parameters and topological phases?

I know of several definitions of phases of matter: The first is the "old" one, Landau theory and symmetry breaking. In this definition we pick a local order parameter $m$ (as far as I can ...
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Do quasicrystals exhibit topological order?

According to the book Quantum Information meets Quantum Matter, ordered phases can either be described by a Landau free energy: symmetry-breaking ordered phases; or there are different ways in which ...
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Staggered Zeeman field in topological magnetic insulators

I was reading the following paper. However, I do not understand a crucial part of their argumentation. They add a parity (P) and time (T) symmetry breaking term to the Hamiltonian in eq (2). Then they ...
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Where do interactions enter the composite fermion theory in the fractional quantum Hall effect?

The question is, in short, where in the composite fermion argument are electron-electron interactions used? I know that interactions, namely Coulomb repulsion, between electrons are crucial in ...
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One question about topological excitation in quantum many body system

I attended a lecture given by Professor Wen Xiaogang. In the lecture, Prof.Wen gave an example of topological excitation: For a state $$(\uparrow\downarrow)(\uparrow\downarrow)(\uparrow\downarrow)(\...
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Anyon statistics in a lattice Moore-Read state

I'm trying to understand this paper1, in particular the remark after Eq. 26. Let me rephrase the problem. According to the paper, one can write the Berry phase as $$ \theta_B=i\oint_\Gamma\frac{1}{C}\...
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Quantum Hall effects with an additional uniform unit flux on a compact manifold

I have two questions: Let us imagine that we have an integer quantum Hall system with electric Hall conductance as $\sigma_\text{H}$ on a two-dimensional (spatial) torus with size $L_1\times L_2$. If ...
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“Axial” gauge in the $Z_2$ lattice gauge theory

I am reading the paper by Fradkin and Susskin on the lattice gauge theory (Order and disorder in gauge systems and magnets). In section III. C, where they were trying to introduce the duality ...
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References on what kind of order does the integer quantum Hall effect exhibit

I'm looking for references to understand how exactly does the integer quantum Hall effect (IQHE) escapes Landau's symmetry breaking description. Any recommendations would be appreciated, thanks!
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Different sectors of the Ising gauge theory

The Hamiltonian of Quantum 2D Ising gauge theory is given by: $$ H=-\sum_p \prod_{i\in \square}\sigma^z_i -g \sum_{i\in \text{links}} \sigma^x_i$$ This $H$ is invariant under the local symmetries: $$ ...
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Aharonov-Bohm effect in topological insulator in a square lattice

Does the presence of the Aharonov-Bohm (AB) effect break Time-reversal symmetry (TRS) for spinless systems in a topological square lattice? As we know that TRS protects the edge states in Topological ...
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Topology in quantum materials

So far I have learned about topological quantum material, my understanding is that topological order in a quantum material is the way the eigenvectors of the Hamiltonian of the system aligned. So if I ...
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Deducing fusion rules of non-abelian fluxons

I have been reading about non-abelian fluxons in John Preskill's lectures notes on topological quantum computing and I do not understand how he deduced the fusion rules for fluxons in the example he ...
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Hopf algebras vs Fusion categories for topological order

Disclaimer: Before I begin with the question I want to warn that some people would argue that it is a math question and not a physics question. However, it finds it origins in the study of topological ...
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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 ...
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How do topological insulators violate Nielsen-Ninomiya Theorem?

I am under the impression that topological insulators have a distinguishing characteristic where they have an odd number of Dirac points that intersect band gaps at the Fermi energy. However, this ...
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What is the difference between order and correlation?

The concepts of correlation and order are ubiquitous in statistical physics and condensed matter but I have yet to find a reference that makes an order in the confusing terminology. As far as I ...
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Bosonic SPT phases with time reversal and a $Z_2$ symmetry

Consider a bosonic system with time reversal symmetry $\mathcal{T}$ and a unitary on-site $\mathbb{Z}_2$ symmetry. Suppose the symmetry is realized in a special way such that $$\mathcal{T}^2= (-1)^B$$ ...
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Why the intrinsic entanglement in a topologically ordered state lowers the entanglement entropy?

The entanglement entropy of the topologically ordered state exhibits the universal additive constant to the area law: $$S \sim \alpha L^{d-1} - \gamma $$ The topologically ordered states are long-...
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What is the “BCS” ground state of a p-wave superconductor?

$\newcommand{\Ket}[1]{\left|#1\right>}$ In BCS theory (I always understood this as referring only to s-wave pairing), the ground state wavefunction takes the form $\Ket{BCS} = \prod_k(u_k + v_k c_{...
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Definition for long range entanglement (LRE) by generalized local unitary (gLU) and generalized stochastic local (gSL) transformations

I am studying this book: Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems (https://arxiv.org/abs/1508.02595). In chapter 7, it introduces ...
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What are gapless superconductors?

How are superconducting materials classified as gapped or gapless, also is this same as saying that a superconductor is conventional or unconventional? Could you explain how this is linked to topology ...
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Valence Bond Solid order paramter

I'm confused about the valence bond solid (VBS) in condensed matter literature. The idea is a lattice is covered by spin singlets and thus spin rotational invariant. It seems that it's commonly ...
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Must the energy of a topological corner state in a 2D material vanish?

It seems that all the literature (just to name a few: Phys. Rev. Lett. 124, 166804, Phys. Rev. Research 2, 013330, Phys. Rev. Lett. 123, 073601, and Phys. Rev. Lett. 123, 256402) says yes to the ...
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Numerical Berry curvature for bosons

I am trying to numerically compute the Berry Curvature for a generic quadratic Bosonic Hamiltonian of the form $$H = \sum_{ij} A_{ij} b_{i}^\dagger b_j + \frac{1}{2} \sum_{ij}\left( B_{ij} b_i b_j + \...
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Why are energy levels half filled in SSH model?

The SSH model describes states of electrons in a polyacetylene chain, which is modeled as a lattice with two orbitals per site. Now, in many articles it is claimed that in the ground state, half of ...
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Edge-mode operator for semi-infinite SSH chain

I am asked to show that the SSH-Hamiltonian for a semi-infinite chain with intralattice e interlattice hopping $u$ and $t$, respectively, given by $\hat{\mathcal{H}}=\sum_{n=1}^{\infty}u\hat{c}_{n,A}^...
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Charge conjugation symmetry operation on single-particle Hamiltonian

How can I show that given the second-quantized Hamiltonian of a system of non interacting fermions $\hat{\mathcal{H}}=\sum_{\alpha, \beta}\hat{\Psi}_{\alpha}^{\dagger}H_{\alpha\beta}\hat{\Psi}_{\...
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Energy cost for pair of vortices

How should I derive the expression for the energy associated with the formation of a pair of vortices located at $\vec{r}_{1}$ and $\vec{r}_{2}$ in the classical two-dimensional $XY$-model, given by ...
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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 ...
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Connection between Wilson loops and fusion rules in $Z_2$ topological order

I'm looking for references (reviews, original articles, lecture notes, etc.) that discuss the connection between the expectation value of Wilson loops (the "disorder parameter" of the system) and ...
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Non-local order parameter for Kosterlitz–Thouless transition

It is known, that there's no local order parameter in Kosterlitz-Thouless transition. Is the order parameter in Kosterlitz-Thouless transition non-local?
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How to get algebraic PSG solutions once we got the constraints?

The question is more technical than conceptual. I've been trying to understand the classification of spin liquids as done by Prof.Wen. I have got the constraints on IGG(Invariant gauge group) elements ...
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Chern-Simons term in Coulomb or radiation gauge

In some of the literature (for example, below Eq. (A3) of this paper), the following is claimed to be the Chern-Simons term in the Coulomb gauge: \begin{equation} 2a_0(\partial_1a_2-\partial_2a_1) \...
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Experimental progress: Topological phases of matter

It's been: 43 years since Leinaas & Myrheim's seminal paper 38 years since Wilczek coined the term anyon 29 years since Moore & Read's paper on non-Abelions in the Fractional Quantum Hall ...
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One dimensional phase transitions

Due to R. Peierls argument there is not phase transitions is one dimensional lattice systems. Argument in $d=1$ goes like that: flipping of one spin in system of N spins will lead to change of free ...
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Alternatives for calculating topological invariants in topological materials

My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, ...
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How many Anyons can be allowed in a state

For fermions, a state allows only one fermion to exist . For bosons, there can be infinite number of bosons in one state . But for anyons, how many can a state allow?How do we come to this conclusion?
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Gauging a symmetry-protected topological (SPT) phase

In this answer, it is said that gauging the symmetry which protects a symmetry-protected topological (trivial) phase gives something "morally very similar" to a phase with a topological order. What ...
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Book recommendation for statistical physics

Before I start a PhD in Quantum Information I would like to study a bit of statistical physics. In particular I am interested in superfluids, critical phenomena, topological phases of matter and all ...
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Non-physicality of 'strings' in Kitaev's anyon model

I was reading Kitaev's paper on arXiv (arXiv:quant-ph/9707021, 'Fault-tolerant quantum computation by anyons') and was wondering if someone could clear something up for me about the non-physicality of ...
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Is there SSB in IQHE?

The Wikipedia page on FQHE mentions that the discovery of FQH states is significant partly because it shows the limits of Landau's symmetry breaking theory, since different FQH states all have the ...
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Gauged DIII superconductor and Z_2 topological order

I keep hearing that a gauged 2D topological superconductor with preserved time-reversal symmetry that belongs to the DIII class is equivalent to the Z_2 gauge theory (Z_2 topological order with ...
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Research: Mott insulator and topological order

I'm an experimentalist who is mainly focusing on strongly correlated electron systems (SCES), in particular Metal-insulator (Mott) transitions in the classical example $V_2 O_3$. Recently I decided to ...
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Is Chern number still well defined with band touching?

Consider a 2 band system in 2d with band crossing on a ring. The coupling opens a gap. If the coupling is zero at some points of the ring, the band is still touching at these points. The berry ...
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What are $U(n)$ or $\mathbb{Z}_2$ quantum spin liquids?

Quantum spin liquid is a state of matter in which spins are correlated and fluctuate even at zero temperature. My question is about these terms in general. When we say that a state or a quasi-...
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What are emergent gauge fields in condensed matter physics?

My background: I have a very little knowledge about topological insulators. Medium level knowledge of Quantum mechanics and linear algebra. Almost no knowledge about Field Theories. I have studied ...

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