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This is actually a very tricky question, mathematically. Physicists may think this question to be trivial. But it takes me one hour in a math summer school to explain the notion of gapped Hamiltonian. To see why it is tricky, let us consider the following statements. Any physical system have a finite number of degrees of freedom (assuming the universe is ...


35

1) Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc? Gauge symmetry is a local symmetry in CLASSICAL field theory. This may be why people call gauge symmetry a local symmetry. But we know that our world is quantum. In quantum systems, gauge symmetry is not a symmetry, in the sense ...


24

In the presence of massless chiral fermions, a $\theta$ term in can be rotated away by an appropriate chiral transformation of the fermion fields, because due to the chiral anomaly, this transformation induces a contribution to the fermion path integral measure proportional to the $\theta$ term Lagrangian. $$\psi_L \rightarrow e^{i\alpha }\psi_L$$ $${\...


24

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'' ...


21

I just discovered this very interesting website through Prof Wen's homepage. Thanks Prof Wen for the very interesting question. Here is my tentative "answer": The spontaneous symmetry breaking in the ground state of a quantum system can be defined as the long range entanglement between any two far-separated points in this system, in any ground state that ...


20

The realization of non-Abelian statistics in condensed matter systems was first proposed in the following two papers. G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991) X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991) Zhenghan Wang and I wrote a review article to explain FQH state (include non-Abelian FQH state) to mathematicians, which include the explanations ...


20

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 ...


20

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 ...


19

The distinction between "ordinary" and topological charges comes from the fact that the conservation of the ordinary charges is a consequence of the Noether's theorem, i.e., when the system under consideration possesses a symmetry, then according to the Noether's theorem, the corresponding charge is conserved. Topological charges, on the other hand, do ...


17

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 $...


17

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 ...


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

Free nonabelian gauge theory is the limit of zero coupling. So it is true, in a sense, that free $SU(N_c)$ gauge theory is a theory of $N_c^2 -1$ free "photons." There's a crucial subtlety, though: it's a gauge theory, so we only consider gauge-invariant states to be physical. Thus, already in the free theory, there is a "Gauss law" constraint that renders ...


15

This question posted by Prof. Wen is so profound that I had hasitated to response. However motivated by Jimmy's insightful answer, I eventually decided to join the discussion, and share my immature ideas. 1) Quantum SSB is a non-linear quantum dynamics beyond the description of Schordinger's equation. Regarding the transverse field Ising model mentioned in ...


15

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 ...


14

For researchers who study condensed matter physics (i.e. low-energy physics), it might be helpful to read following books and articles. H. Haug and A. P. Jauho: Quantum Kinetics in Transport and Optics of Semiconductors (Springer, New York, 2007). We can learn the (minimal) essence of Keldysh formalism by reading pp. 35-69 (sections 3 and 4). This article ...


14

I recommend you Chapter 5 (page 150+) of the AdS Bible, http://arxiv.org/abs/hep-th/9905111 Concerning your individual questions, which are mostly answered at the beginning of that Chapter, the additional Virasoro generators correspond to bulk coordinate reparametrizations that preserve the metric at infinity, but they do map the ground state to excited ...


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 ...


13

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 ...


13

Goals one wants to achieve with those two theories are similar. We know that superstring theory is a potential theory of everything. One may want to ask what is the difference between the string-net-liquid approach and the superstring approach? Our understanding of the superstring theory has been evolving. According to an early understanding of the ...


13

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 ...


13

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 ...


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

Elementary particles, like photons and electrons, are not more elementary in the sense that there are underlying theories, such as quantum spin model on lattice, from which they can be derived as an effective approximation (see for example arXiv:hep-th/0302201). In particular, the string-net condensation provides a unified origin for gauge interactions and ...


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