56

Symmetry of the superconducting gap First of all, a bit of theory. Superconductivity appears due to the Cooper paring of two electrons, making non-trivial correlations between them in space. The correlation is widely known as the gap parameter $\Delta_{\alpha\beta}\left(\mathbf{k}\right)\propto\left\langle c_{\alpha}\left(\mathbf{k}\right)c_{\beta}\left(-\...


32

They're variants, different kinds of quantum field theory, but they're not mutually exclusive. The different adjectives you mention separate quantum field theory to "pieces" in different ways. The different sorts of variants you mention are being used and studied by different people, the classification has different purposes, the degree of usefulness and ...


24

The absence of physical excitations in 3 dimensions has a simple reason: the Riemann tensor may be fully expressed via the Ricci tensor. Because the Ricci tensor vanishes in the vacuum due to Einstein's equations, the Riemann tensor vanishes (whenever the equations of motion are imposed), too: the vacuum has to be flat (no nontrivial Schwarzschild-like ...


17

The three classes of QFTs you are referring to are distinguished by different symmetry assumptions (Poincare invariance, conformal invariance, and volume-preserving diffeomorphism invariance) and different background spacetimes (Minkowski, Riemann curve (or families of them), and arbitrary manifolds). Moreover, Wightman axioms only characterize the vacuum ...


16

I would like to particularly address this nice question relating the Hamiltonian formulation of this superconducting state (via Bogoliubov-de Gennes (BdG) equation) to the low energy quantum field theory, especially the Topological Quantum Field Theory (TQFT). What is a $p_x+i p_y$ superconductor: It is a chiral $p$-wave superconductor. It is an ...


16

Let me first answer your question "is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?" (1) Topological superconductors, by definition, are free fermion states that have time-reversal symmetry but no U(1) symmetry (just like topological insulator always have time-...


16

As you say yourself, indeed every connection on a bundle is locally given by a Lie algebra valued 1-form and in general only locally. Let's say this more in detail: for $X$ any manifold, a $G$-principal connection on it is (in "Cech data"): a choice of good open cover $\{U_i \to X\}$; on each patch a 1-form $A_i \in \Omega^1(U_i)\otimes \mathfrak{g}$; on ...


13

The Atiyah-Segal axioms and generally the axioms of FQFT formalize the Schrödinger picture of quantum physics: to a codimension-1 slice $M_{d-1}$ of space one assigns a vector space $Z(M_{d-1})$ -- the (Hilbert) space of quantum states over $M_{d-1}$; to a spacetime manifold $M$ with boundaries $\partial M$ one assigns the quantum propagator which is the ...


13

A quote from http://en.wikipedia.org/wiki/Symmetry_protected_topological_order : The SPT order (for both frermionic and bosonic systems) has the following defining properties: Distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry. However, they all can be ...


12

The "topological" in "topological order" and the "topological" in "topological insulator" have different meanings. The 'topological' in topological order means 'robust against ANY local perturbations'. The "topological" in "topological insulator" means 'robust against some local perturbations that respect certain symmetry'. In fact the properties of ...


12

Any gapped field theory flows in the infrared to a TQFT which describes the set of operators which are neither screened nor confined. See this paper for a very clear point of view on this: https://arxiv.org/abs/1307.4793 . In gapped gauge theories like QCD, this is equivalent to specifying the dyon content of the theory ( https://arxiv.org/abs/1305.0318 ). ...


10

The relation is very deep and has a rich mathematical structure, so (unfortunately) most stuff will be written in a more formal, mathematical way. I can't say anything about Donaldson theory or Floer homology, but I'll mention some resources for Chern-Simons theory and its relation to the Jones Polynomial. There is first of all the original article by ...


10

After stating the solution, I'll try to give some physical insights to the best of my knowledge and some more references. The dimension of the required state space is given by the Verlinde formula, having the following form for a general compact semisimple Lie group $G$ on a Riemann surface with genus $g$ corresponding to the level $k$: $$ \mathrm{dim} V_{...


10

Let's start with a simpler example: $1+1$ dimensional gravity. This is actually a pretty important example to understand because all of string theory takes place in this framework!Within $1+1$ dimensional gravity, just as in $2+1$ dimensions, there are no local interactions. Thus, it seems like a pretty good starting place to talk about this kind of stuff. ...


9

Band inversion is a necessary but not sufficient condition for topological insulators (TIs). For band TIs you need to evaluate the topological (or $\mathbb{Z}_{2}$) invariant defined by Fu, Kane and Mele in Eq. (2) of: Liang Fu, Charles L. Kane, and Eugene J. Mele. “Topological insulators in three dimensions.” Physical Review Letters 98, no. 10 (2007): ...


9

How to obtain this braiding matrix from Non-Abelian Chern-Simon theory? To obtain braiding matrix $U^{ab}$ for particle $a$ and $b$, we first need to know the dimension of the matrix. However, the dimension of the matrix for Non-Abelian Chern-Simon theory is NOT determined by $a$ and $b$ alone. Say if we put four particles $a,b,c,d$ on a sphere, the ...


9

Consider the finite dimensional unitary representations $\alpha,\beta,\gamma$ of the given compact group $G$ on corresponding vector spaces $V_1,V_2,V_3$. Let $|i\rangle_j,i=1,\dots,n_j$ be an orthnormal basis of $V_j$ where $dim V_j=n_j$. Then $\{|i\rangle_1\otimes|j\rangle_2\otimes|k\rangle_3\}$ forms an orthonormal basis of $V=V_1\otimes V_2 \otimes V_3$....


9

A conformal transformation is one which alters the metric up to a factor, i.e. $$g_{\mu\nu}(x)\to\Omega^2(x)g_{\mu\nu}(x)$$ A field theory described by a Lagrangian invariant up to a total derivative under a conformal transformation is said to be a conformal field theory. These transformations include Scaling or dilations $x^\mu \to \lambda x^\mu$ ...


9

The sphaleron is kind of the opposite of the instanton, and kind of the same. Let's make that statement precise: An instanton is a local minimum of the action that mediates vacuum tunneling (link to an answer of mine how and why instantons do that). The sphaleron sits in-between the vacua, in a certain sense, it is the instanton "in the middle of tunneling":...


8

(sorry I don't have enough reputation to make a comment): This question is very broad/vague, as indeed algebraic/differential topology (symplectic geometry of course) is completely used in theoretical physics, in particular for Topological QFTs. From a physicist's perspective, start with Nakahara's Geometry, Topology, and Physics. Surgery, cobordism, and ...


8

Irving Segal, the obnoxious and disliked genius mathematician of MIT, once went around asking physicists « ¿what is a quantum field? » As he tells the story, only Enrico Fermi gave him an answer (after pausing for a little thought). « The occupation number formalism.» What does this mean. What Fermi meant is that for a given particle, say an electron, ...


8

Another interesting application is that Chern-Simons Theory in 3d is equivalent to General Relativity in 3 space-time dimensions. GR in 3 dimensions is quantisable and following a nice playground for quantum gravity. http://ncatlab.org/nlab/show/Chern-Simons+gravity has a nice reading list about that topic at the References. Maybe a good start is "Edward ...


8

It's not the making as opposed to verifying of topological superconductors that is difficult experimentally. One of the most useful techniques in identifying topological properties of a material is Angle-Resolved Photoemission Spectroscopy (ARPES). ARPES can independently image the bulk and surface modes of a 3-D solid with very good energy and momentum ...


8

This is explained in Section 3 of Witten's "Quantum Field Theory and the Jones Polynomial." The idea is to locally parametrize a three-manifold by $M\times \mathbf{R}$, where $M$ is some two-dimensional manifold and $\mathbf{R}$ is the time direction that we are quantizing along. Once we do this, we can fix temporal gauge, where the time component $A_0$ of ...


8

There are maybe three different stages to be distinguished and to be understood here: first: maybe part of the question is why an $n$-dimensional QFT should assign numbers to closed $n$-dimensional manifolds, and vector spaces to closed $(n-1)$-dimensional manifolds. That is what I had replied to in that other discussion linked to above: the vector spaces ...


7

Topological order is a new kind of order in zero-temperature phase of quantum spins, bonsons, and/or electrons. The new order corresponds to pattern of long-range quantum entanglement. Topological order is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations. However, topological orders ...


7

Olaf has already given most of the references I would recommend. But in the case of Chern-Simons theories and knot theory, there are two (plus one) other very nice references. These are all written by physicist to physicists, so no modular functors, Cobordisms and so on. 1) Marcos Marino - Chern-Simons Theory and Topological Strings (arXiv:hep-th/0406005v4) ...


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