I am now learning the topological classification and have some problems on the paper (https://arxiv.org/abs/1303.1843) I am reading now.

In this paper, there is a statement in the second part ( II. REFLECTION SYMMETRY IN BAND INSULATORS ) :

We focus on electronic insulators, i.e., systems with a conserved U(1) charge, but a similar tight-binding formalism can be developed for BdG Hamiltonians of topological superconductors, since both kinds of Hamiltonians can be treated as noninteracting systems.

The part I don't understand is the relation between electronic insulators and conserved U(1) charge

To the best of my knowledge, the U(1) charge conservation means that if we make the following transformation :

$\psi^{\dagger} \left( {\bf r} \right) \to e^{i\theta}\psi^{\dagger} \left( {\bf r} \right)$

$\psi \left( {\bf r} \right) \to e^{-i\theta}\psi \left( {\bf r} \right)$

then the Hamiltonian $H$ won't change. This means that the particle number of the system will conserve since it can be shown that $\left[ {H,N} \right] = 0$ where $N$ is the number operator defined by $N = \int {d{\bf{r}}{\psi ^\dagger }\left( {\bf{r}} \right){\psi ^{}}\left( {\bf{r}} \right)} $

However, I don't know why this property has some relation with the electronic insulator

Is there any suggested reference for me to explore this topic more? I would be extremely grateful for any suggestion!!


  • $\begingroup$ Your formulas refer to the U(1) symmetry, not charge. I haven't read the paper, but I suspect that the authors might mean the conditions where charges don't cross the boundaries of an object thus making this object non-conductive. $\endgroup$ – safesphere Jan 8 '18 at 0:15
  • $\begingroup$ @safesphere Thanks for your reply. Can you explain more about the difference between U(1) symmetry and U(1) charge? I tried to find some relevant lecture notes but failed. Can you suggest some reference that I can read more about U(1) ? Thanks! $\endgroup$ – ocf001497 Jan 8 '18 at 4:36
  • $\begingroup$ I'll leave the details to the experts, but essentially U(1) means that "circles are round" and when you put it on top of the Lorentz invariance of Special Relativity ("local gauge"), you get electromagnetism both classical and quantum. So the U(1) symmetry is never broken, as it is the foundation of this universe. As stated in comments below, it may be convenient for calculations to consider it broken instead of accounting for complex boundary conditions. I already have guessed what the authors may have meant by the"U(1) charge", but I'll leave the full answer to the experts. $\endgroup$ – safesphere Jan 8 '18 at 5:01

The authors want to distinguish between types of gapped systems. An ordinary insulator with an ordinary gap, and a superconductor which has a gap but with broken U(1) symmetry. In fact, metals also respect U(1) symmetry, though this may or may not hold in the strongly interacting case.

  • $\begingroup$ The U(1) symmetry is never broken. The question is about the U(1) charge, not symmetry. $\endgroup$ – safesphere Jan 8 '18 at 0:09
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    $\begingroup$ For the true microscopic Hamiltonian, yes U(1) is never broken. However, the effective BCS Hamiltonian for superconductors does break this symmetry. This gives rise to the Higgs mode as pointed out by Phil Anderson many decades ago. So my answer still stands. $\endgroup$ – KF Gauss Jan 8 '18 at 4:05
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    $\begingroup$ I would like to emphasize that most, if not all, studies of topology in solid state systems these days are done on effective Hamiltonians (tight binding etc.) So breaking of U(1) symmetry is a very useful way to understand things, even though it not true strictly speaking. $\endgroup$ – KF Gauss Jan 8 '18 at 4:08
  • $\begingroup$ @user157879 Thanks for the reply. Can you illustrate more about the relation between gap system and the U(1) symmetry/charge ? Since I am novice for this research field, I would like to explore more about U(1) charge/symmetry, can you further suggest some reference ? Thanks! $\endgroup$ – ocf001497 Jan 8 '18 at 4:38
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    $\begingroup$ U(1) symmetry basically means charge is conserved (number of electrons does not change). This is the general interpretation you would get from quantum field theory. For models of ordinary insulators or simple metals, charge is still conserved. This has nothing to do with gapped or not gapped. For the BCS model of superconductors, charge is no longer conserved, so U(1) symmetry is broken. Microscopically, U(1) symmetry is always conserved, but the BCS model and tight binding models are not microscopically correct, so U(1) can be broken. $\endgroup$ – KF Gauss Jan 8 '18 at 7:37

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