Quantum ether: photons and electrons from a rotor model

TL;DR

This work demonstrates that a purely bosonic lattice can host emergent gauge bosons and fermions in (3+1) dimensions via string-net condensation, effectively realizing a quantum ether. By constructing a 3D rotor model and mapping its low-energy sector to a lattice gauge theory, photons emerge as string-net fluctuations while charges appear as string endpoints; a simple twist converts bosonic charges into fermions. A more controlled regime with a two-sublattice structure and pi-flux yields massless Dirac fermions, giving a gauge theory with multiple Dirac species coupled to a dynamical U(1) field. Collectively, the results illuminate a unified microscopic origin for gauge bosons and fermions and point toward string-net realizations of richer theories, while also highlighting obstacles like chirality in lattice constructions.

Abstract

We give an example of a purely bosonic model -- a rotor model on the 3D cubic lattice -- whose low energy excitations behave like massless U(1) gauge bosons and massless Dirac fermions. This model can be viewed as a ``quantum ether'': a medium that gives rise to both photons and electrons. It illustrates a general mechanism for the emergence of gauge bosons and fermions known as ``string-net condensation.'' Other, more complex, string-net condensed models can have excitations that behave like gluons, quarks and other particles in the standard model. This suggests that photons, electrons and other elementary particles may have a unified origin: string-net condensation in our vacuum.

Figures (6)

• Figure 1: A picture of quantum ether: the fluctuations of the string-nets give rise to gauge bosons (such as photons). The ends of the strings give rise to fermions (such as electrons).
• Figure 2: A picture of the rotor model (\ref{['strnetH']}). The term $Q_{\v I} = (-1)^{\v I}\sum_{\text{legs of }\v I} L_{\v i}^z$ acts on the six "legs" of $\v I$ - that is, the six rotors adjacent to $\v I$, drawn above as filled dots. The term $B_{\v p}= L^+_{1} L^-_{2} L^+_{3} L^-_{4}$ acts on the four rotors, labeled by $1,2,3,4$, along the boundary of the plaquette $\v p$.
• Figure 3: A picture of a closed string state. The rotors along the thick closed curve have angular momentum $L^z = \pm 1$, while all the other rotors (depicted as empty dots) have $L^z = 0$.
• Figure 4: (a) A projection of the $3D$ cubic lattice onto the $2D$ plane. (b) An example of a curve $C$ with a framing $C'$. The links along the string are labeled $1,2,3$. The links which cross $C'$ are the filled dots labeled by $3,4,5,6,7$. The corresponding twisted string operator $W^{\text{tw}}(C)$ is given by $L_1^+ L_2^- L_3^+ (-1)^{L^z_3+L^z_4+L^z_5 +L^z_6+L^z_7}$
• Figure 5: A picture of the $\tilde{B}_{\v p}$ term in the twisted rotor model (\ref{['twistedH']}). Just like any other twisted string operator, the term $\tilde{B}_{\v p} = L_1^+ L_2^- L_3^+ L_4^- (-1)^{L_1^z+L_4^z+L_5^z+L_6^z+L_7^z+L_8^z+L_9^z+L_{10}^z}$ acts on the four rotors, labeled by $1,2,3,4$, along the boundary of the plaquette $\v p$, and the $8$ rotors, labeled by $1,4,5,6,7,8,9,10$, that cross the framing curve.