An Introduction to Quantum Order, String-net Condensation, and Emergence of Light and Fermions∗

The existences of light and fermions are two big mysteries in nature. The mysteries are so deep that the questions like, “What are light and fermions?”, “Where do light and fermions come from?”, “Why do light and fermions exist?”, are regarded by many people as philosophical or even religious questions. To appreciate the physical significance those questions let us ask three simpler questions: “What are phonons?”, “Where do phonons come from?”, “Why do phonons exist?”. We know these three questions to be scientific questions and we know their answers. Phonons are vibrations of a crystal. Phonons come from a spontaneous translation symmetry breaking. Phonon exists because the translation-symmetry-breaking phase actually exists in nature. It is quite interesting to see that our understanding of a gapless excitation – phonon – is rooted in our understanding of the phases of matter as symmetry breaking states [1, 2]. However, our picture for massless photons and nearly massless fermions[70] (such as electrons and quarks) is quite different from our picture of gapless phonons. We regard photons and fermions as elementary particles – the building block of our universe. But why should we regard photons and fermions as elementary particles? Why don’t we regard photons and fermions as emergent quasiparticles like phonons? We can view this question from several different points of view. First point of view: Before late 1970’s, we felt that we understood, at least in principle, all the physics about phases and phase transitions based on Landau’s symmetry breaking theory [3, 4]. In such a theory, if we start with a bosonic model, the only way to get gapless excitations is via spontaneous breaking of a continuous