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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Kosterlitz-Thouless transition and renormalisation group theory

I'm trying to understand the Kosterlitz-Thouless transition in 2d systems. There is a section in Altland and Simons' Condensed Matter Field Theory that discusses the phenomenon, but I don't really ...
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Is it possible that the speed of sound in some strange quantum material is faster than the speed of light in vacuum [closed]

If we believe the spacetime structure(including the limit of speed) could emerge from some vacuum structure(such as string-net condensation),then, is it possible that the speed of mode of excitation ...
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Does a topological charge always need to be an integer

Does a topological charge always need to be an integer, I see many papers where people talk about non-integer topological charges due to boundary conditions. According to the formula for the ...
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What is the BKT transition in non-equilibrium systems?

There is a useful page on the BKT (or KT) transition here which describes the role of vortices in 2d equilibrium systems. I am interested in what happens in 2d non-equilibrium driven-dissipative ...
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Examples of Simple Hamiltonians Giving Different Phases?

I am trying to study about simple condensed matter models that I can simulate numerically and use to calculate some topological invariant that defines a (topological) phase. My interest is in ...
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BKT transition: nature of topological transition

BKT-transition is one of the most well-known topological transition in $O(2)$ model.But I misunderstand the physical interpratation of this transition. I started from the low-temperature expansion of ...
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166 views

Basics of topological order and its relation to entanglement

What is a topological order that drives a topological phase transition? How is it different from say magnetic ordering or the superfluid ordering? What is its relation with entanglement? Please ...
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Do Chern Insulators (QAHE) have topological order (long-range quantum entanglement)?

I know IQHE is a example having "invertible" topological order from Professor Wen's definition. And Topological Insulators is SRE because of necessary of underlying symmetry protection. After that, ...
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spinless and time reversal symmetry breaking of p-wave pairing in topological superconductors

In the context of Majorana zero modes, I often hear that the p-wave pairing is effectively 'spinless' and time reversal symmetry broken. I understand that s-wave and p-wave refer to the spin portion ...
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PEPS expression of Toric Code ground state

It is well known that the ground state of 2D Toric Code system can be represented by $D=2$ PEPS (Projected Entangled Pair States, i.e. the tensors in the 2D network are connected by 2-dimensional ...
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89 views

One-dimensional $SU(3)$ Heisenberg Model, the non-linear sigma model, $\theta$-term

Let's consider a one dimensional $SU(N)$ antiferromagnetic Heisenberg Model with an irreducible representation and its conjugate on alternating sites, such that they correspond to a Young tableaux ...
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Can superconductors undergo a BKT transition?

In the article by Kosterlitz and Thouless (1973) they write in the abstract: "This type of phase transition (BKT) cannot occur in a superconductor for reasons that are given". Later in the paper they ...
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A good instruction on Symmetry enriched Topological phases

I am looking for a good introduction to SETs, and topologically ordered phaeses it should be something describing first principles and gives a good explanation on the basics and the logic of this ...
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What is so topological about topological phase transitions?

I am studying the KT-transition, which is called a topological phase transition. The phase transition is driven by vortices in a 2-D superfluid, where it is shown that at a critical temperature $T_c$ ...
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Why is the composite fermion not included in the anyon contents of FQH topological orders?

For example, both the $\nu=1/3$ Laughlin state and the Moore-Read state has a simple interpretation in terms of composite fermions, which are bound states of an electron and two fluxes. Both the ...
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Are fractional quantum hall effect system symetry enriched topological phases?

In the papers I review they first start to talk about topologically ordered phases of matter. Their standard example of it is FQHE. Than they give another set examples which are quantum spin liquids, ...
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Are interacting symmetry protected topological (SPT) phases and symmetry enriched topological (SET) phases must be gapped?

I wonder are interacting SPT and SET phases gaped? Can we have a SET or interacting SPT phase in a semi metal?
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Topological Thouless pumping

I can't understand why the number of electrons pumped per cycle of a quantum pump is protected topologically. As for me this number is an integer in any case, because the number of electrons in ...
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Symmetry Protected Topological (SPT) phases of spin-1 chains

Let's consider this family of 1D spin-1 of hamiltonians: $$\sum_{i}[S^x_{i}S^{x}_{i+1}+S^y_{i}S^{y}_{i+1}+\lambda S^z_{i}S^{z}_{i+1} + D(S^{z}_{i})^2].$$ If I understand it right, these models have: ...
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Is 2D toric code dual to something?

I can understand 2D toric code as a quantum Z2 gauge theory defined on a lattice. Is this model dual to some simpler spin model? A bit of motivation to clarify my intention: I know 3D classical Ising ...
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Experimental confirmation of Majorana modes in Kitaev chain

I'm confused about majorana modes at the edge of Kitaev chain, what do we seek in experiment? When we first define this one we write the creation and annihilation operators as: $$a^{+}=\frac{1}{2}(\...
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Topological materials and fractionalized excitations

I've been told several times that topological materials (such topological insulators) must have "fractionalized" excitations. Equivalently, if a material does not have fractionalized excitations, it ...
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Is it possible to directly derive the $K$ matrix for a topological order described by a gauge-theory Hamiltonian?

To be concrete, Let us consider a $Z_2$ gauge theory in the deconfined phase coupled to matter field, \begin{align} S_{Z_2}=\beta\sum_{\vec r\mu}\phi(\vec r)U_\mu(\vec r)\phi(\vec r+\vec e_{\mu}) + K \...
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“Weak” and “Strong” topological insulators

For translationally invariant systems, we can define some topological invariant based on the translational symmetry, which is referred to "weak" topological invariant. For example, according to Kitaev'...
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Is the string-net model Hermitian?

In Kitaev and Kong's paper, they define the Hermitian inner product on morphism spaces in Eq. (11). My question is that: Given that F symbols satisfy the pentagon identity, does that the string-net ...
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Linking phase of flux lines and excitation energy of monopole

I am reading this paper and on the left-hand side of pp.10 it states the following relation between linking phase and excitation energy of monopole: Now the $\theta = \pi$ term in the bulk implies ...
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What does the Chern number physically represent?

In 2D the Chern number can be written as $$C_m=\frac 1{2\pi}\int_{BZ}\Omega_m(\mathbf k)\cdot d^2 \mathbf k$$ where we are integrating over the Brillouin zone. In 2D this is equivalent to finding ...
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Relation of SPT phases with different boundary conditions

Using the definition that two SPT phases are distinct if they can't be connected by a symmetric finite depth local unitary, how does one relate systems with different boundary conditions? For example,...
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54 views

How do I understand different realizations of symmetry in the absence of fractionalization?

To use a simple example to ask my question, consider the two dimensional toric code with a $Z_2$ global symmetry acting in two ways: The most boring trivial way possible. By permuting the charge and ...
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107 views

Models for non-universal topological quantum computation

Anyon models do not lead in general to universal topological quantum computation (= existence of a universal set of quantum gates) when only the braiding is used for implementing gates. The Fibonacci ...
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Significance of topology in topologically ordered systems

The topology on which a lattice is placed plays an important role in topologically ordered systems, for example in toric code the degeneracy in the ground states is given by $4^{g}$ where $g$ is the ...
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360 views

Is gauge structure necessary for topological order?

All of the famous theories with topological order have some kind of gauge structure, for example the $Z_2$ lattice gauge theory or in QHE. So my question is that whether the existence of a kind of ...
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If we cool a topologically ordered system down to zero temperature, will it end up in a pure or mixed ground state?

If a quantum system with degenerate ground states is fully ergodic at zero temperature, then it is maximally mixed over the ground-state (GS) manifold; i.e. its density matrix $\rho$ is the projection ...
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Relation between cobordism and unitary fusion category classification of TQFT/ SPT phase

In the introduction part of Gaiotto's paper (https://arxiv.org/pdf/1712.07950.pdf), he says "in the context of topological field theory, homotopy-theoretic ideas also lead to the classification of ...
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What determines if a quasiparticle is a spinon or a vison?

Across topological literature, I've seen references to spinons and visons. In Kitaev's famous "Anyons in an exactly solved model" paper, he mentions that visons are "spinless bosons" whereas spinons ...
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125 views

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 ...
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In what sense are quasiholes and quasiparticles “excitations” in Fractional Quantum Hall (FQH) systems?

In the theory of Fractional Quantum Hall states, one often sees quasi-holes and quasi-electrons (or quasi-particles) being called "excitations" from the ground state initially given by Laughlin (...
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Toric Code model with an extra projector? (Levin-Gu)

In the seminal work by Levin and Gu in 2012 ( Braiding statistics approach to symmetry-protected topological phases ) they give a concrete prescription for how to gauge a global symmetry to a local ...
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Why, the Valence Bundle is non-trivial and the complete Bloch Bundle isn't?

I'm thinking about the bundles defined in Topological Insulators, and I took that sentence, present in many sources, as true: The topology of the Valence Bundle is non-trivial, i.e., the Valence ...
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Qualitative understanding of Hamiltonian Terms for Quantum Phases

I have been reading up on topological order and quantum phases which are continually being discovered in condensed matter systems. (Here's a great article...https://www.quantamagazine.org/physicists-...
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What's the tensor network representation for local ground state?

It's well known that many topological phases can be represented using matrix product state and PEPS. Example, toric code, $H=-\sum_{v}A_v-\sum_p B_p $. My question is: What's the tensor network ...
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How do spectrum gaps relate to topological protected states?

In particular, I want to understand what fundamental (mathematical) structure gives rise to topological mechanical metamaterials, or topological protected states in general. According to a recent PNAS ...
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How to define the 'gapped ' and 'gapless' states?

In the former Phys.SE post Can gapped state and gapless state be adiabatically connected to each other?, I saw different answers from Norbert Schuch and Xiaogang Wen. I am confused by the question: ...
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How to calculate Edge states of Topological insulators

Topological insulators are novel state of matter in which bulk is insulator and edges are gapless. How do we calculate these gapless states? I am reading the following PRL Feng Liu and Katsunori ...
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Different definitions of topological phases

When doing classification of topological phases, one need to formalize the problems mathematically. But, it seems that there are two not obviously equivalent ways to describe topological phases. In ...
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Machine Learning toric code ground states and phase transition under perturbation

I was wondering if the following is a viable method using machine learning and neural networks to get to the ground states of the toric code and also understand the phase transition in the presence of ...
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Can gapped state and gapless state be adiabatically connected to each other?

Is it possible that we can construct a gapped state and a gapless state which are adiabatically connected? Here adiabatically connected I mean: there exists a class of Hamiltonians $H(g)$ with ...
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266 views

Effective Hamiltonian Method used in Kitaev Chain

Kitaev's paper Unpaired Majorana Fermions in 1D Quantum Wires (https://arxiv.org/abs/cond-mat/0010440) is famous as a promising experimental proposal for realizing topologically-robust zero-energy ...
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Ergodicity in the Quantum Dimer Model

Background: The Quantum Dimer Model is a lattice model, where each configuration is a covering of the lattice with nearest-neighbour bonds, like in the figure on the left: Each of the configurations ...
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Derivation related to gauge theory of quantum spin antiferromagnets

I was reading Chapter 5 of the book by N. Nagaosa on Quantum Field Theory in Strongly Correlated Electronic Systems. In the first section, he introduced the complex-field ($z_{\alpha}$) representation ...