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Local quasiparticle excitations and topological quasiparticle excitations To understand and classify anyonic quasiparticles in topologically ordered states, such as FQH states, it is important to understand the notions of local quasiparticle excitations and topological quasiparticle excitations. First let us define the notion of ``particle-like'' ...


26

Ghostly Lie algebra cohomology Let $\mathfrak{g}$ be our Lie algebra and $V_\rho$ a representation space with representation map $\rho : \mathfrak{g} \to \mathrm{End}(V_\rho)$. $V_\rho$ is, by the action through the representation, naturally a $\mathfrak{g}$-module (people missing the ring structure in $\mathfrak{g}$ - just embed it into the universal ...


25

Very loosely speaking the reasoning is this. Imagine a two band system in which the fermi sea has one filled band with Chern number $n$ and another system with $N$ filled bands but also with Chern number $n$. Physically they have the same topological properties (for example the same Hall conductance, edge states and so on), but cannot be deformed ...


21

Sometimes you can The obvious example is a purely dipolar charge which has been displaced from the origin, such as a dipolar gaussian $$ \rho(\mathbf r) =p_z(z-z_0)\frac{e^{-(\mathbf{r}-\mathbf{r}_0)^2/2\sigma^2}}{\sigma^5(2\pi)^{3/2}} . $$ This system is neutral, and it has a nonzero dipole moment $p_z=\int z\rho(\mathbf r)\mathrm d\mathbf r$ along the $...


21

A good analogy for the difference between the two can be given in terms of two other examples of anomalies, that are possibly more familiar. Consider a field theory with a global symmetry, take $U(1)$ for simplicity. At the classical level, the equations of motion lead to the existence of a conserved current (Noether's theorem). At the quantum level, the ...


17

In the cases when the gauge group is disconnected, both choices of defining the physical space as a the quotient of the field space by the whole gauge group $\mathcal{A}_{physical} = \frac{\mathcal{A} }{ \mathcal{G}}$ or by its connected to the identity component $\mathcal{A}_{physical} = \frac{\mathcal{A} }{ \mathcal{G}_0}$ are mathematically sound. In the ...


16

The two kinds of trace anomalies are related but distinct. The first one that you refer to is the anomaly in Weyl transformations that occurs when you put a CFT on a curved background. The CFT is still exactly conformally invariant in flat space, but this symmetry is broken by the background gravitational field. It's useful to think about CFTs in two ...


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-...


15

A conformal field theory is a quantum field theory which is invariant under conformal transformations. Due to this invariance, correlation functions must obey linear equations called conformal Ward identities. Conformal blocks are not just solutions of the conformal Ward identities, but actually elements of a particular basis of solutions. Let us focus on ...


15

Is there a group of (paid) researchers that work on M-Theory 24/7, hoping that someday they'll finally unify physics? Or is it more like a thing that passionate people do in their spare time? Virtually all serious physics research is done by full time professionals on salaries funded by grants or the institutions they are associated with, or both, or by ...


14

I) Here we discuss the problem of defining a connection on a conformal manifold $M$. We start with a conformal class $[g_{\mu\nu}]$ of globally$^{1}$ defined metrics $$\tag{1} g^{\prime}_{\mu\nu}~=~\Omega^2 g_{\mu\nu}$$ given by Weyl transformations/rescalings. Under mild assumption about the manifold $M$ (para-compactness), we may assume that there ...


14

There are different categories of topological superconductors. I’m guessing that you are referring to the time-reversal invariant (class DIII) ones, in 2D or 3D. Yes, it is possible to distinguish the surface/edge states of 3D/2D topological superconductors from the bulk. I'm not talking about designing some intricate experimental technique to separate out ...


14

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 ...


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 ...


12

The trick here in "observing a quantum jump" is that this is not equivalent to doing a strong measurement that "collapses" the wavefunction. The archetypal quantum jump is a two level system with a ground state $\lvert G\rangle$ ("Ground")and an excited state $\lvert D\rangle$ ("Dark") (I'm naming the state kets like ...


11

Part b) is a big mathematical physics topic in its own right. The divergent tail of an asymptotic series is not garbage, rather it contains a lot of information that together with some additional information can be used to compute non-perturbative effects. A general introduction to this topic is given here. There are different approaches possible, some ...


10

There are answers in the note by Polchinski linked by Matt, and an article by Shankar in Review of Modern Physics: Renormalization-group approach to interacting fermions. Just to flesh out was it meant by "stability" and "Fermi surface". The Fermi-liquid can be thought of as a phase characterized by several properties: arbitrarily long-lived, gapless ...


10

As you note the construction of asymptotic states breaks down in a CFT since there is no mass gap. It is therefore necessary to introduce an IR regulator by using, eg, dimensional regularisation. The full scattering amplitude will then depend on this regulator. However, it is possible to construct physical observables that do not depend on the regulator. ...


9

I feel that I finally understand the physical meaning of composite (ie non-simple) objects like $\phi\oplus\phi$. It is explained in the section II of my paper with Tian Lan arxiv.org/abs/1311.1784 . We know that putting a few anyons (ie the objects in tensor category) on a Riemann surface may generate degenerate states (ie the fusion space of the objects ...


9

Bei Zeng and I wrote a paper http://arxiv.org/abs/1406.5090 , which addresses this question: A symmetry breaking phase for finite group G is a gLU equivalent class formed by symmetric many-body states that have GHZ entanglement. In other words, a symmetry breaking phase is a set of symmetric states $U_g \Psi = \Psi$ up to a phase, $g \in G$, and those ...


9

First impressions based on a quick read of the preprint: I'm out of my depth on this! I couldn't tell you if their derivation is correct, but assuming that it is: They don't treat real QCD. They study SU(2) YM without quarks. The authors claim they can do real QCD and get the same result, but this is not demonstrated in the paper (they defer this to a later ...


9

The tachyon mode in the open string spectrum is an indication that as a perturbation theory it describes the perturbation about an unstable vacuum. In 1999 Ashoke Sen realized that -- since the open string propagates with its endpoints on the space-filling D25-brane -- that instability must be the instability of the D25, which wants to decay to a "true ...


8

I'll answer the relation between string theory and $E(8)$ -- a common appearance of $E(8)$ in string theory is in the gauge group of Type HE string theory $E(8)\times E(8)$ (see here for an explanation why). But it's interesting physically because it embeds the standard model subgroup. $$SU(3)\times SU(2)\times U(1)\subset SU(5)\subset SO(10)\subset E(6)\...


8

Your question is very interesting. I would like to mention something along the line of your question, but perhaps from another viewpoint. Recently there are some better understanding along the thinking between (1)"whether a theory is free from anomaly (the anomaly matching condition satisfied)," (2)"whether the symmetry of a theory is on-site symmetry," ...


8

In this context, a "current" is an object obeying an affine Lie algebra, also called current algebra and a special case of a Kac-Moody algebra. It is an algebra formed by unit weight operators: take for example a current $J^a(z)$, where $a$ is a label and $z$ is a complex coordinate. The algebra is given by $$[J^a_n,J^b_m]=i{f^{ab}}_cJ^c_{n+m}+mkd^{ab}\...


8

Topological degeneracy is only defined in the thermodynamic limit on a closed manifold. The ground state degeneracy of a finite-sized system or on an open manifold is not "topological", and can not be called topological degeneracy. Considering your examples. (1) The ground state degeneracy is ill-defined with open boundary condition. Because there might be ...


8

The definition you are using in your question is the one that everybody who does rigorous perturbative renormalization uses. The particular choice of method BPHZ vs. Epstein-Glaser, etc. doesn't matter. They both give you the renormalized $n$-point correlation functions as formal power series in either $\hbar$ (a bit more canonical) or the renormalized ...


7

Larry Harson is right; you should read Beneke's Physics Report. But I think I can make your reading easier by clearing up a misconception: The name 'renormalon' is a bit misleading. Renormalons (like instantons) aren't real physical things. They don't appear in the Lagrangian, and they don't correspond to any physical state. They are not auxiliary ...


7

The adiabatic theorem is required to derive the Berry phase equation in quantum mechanics. Therefore the adiabatic theorem and the Berry phase must be compatible with one another. (Though geometric derivations are possible, they usually don't employ quantum mechanics. And while illuminating what is going on mathematically, they obscure what is going on ...


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