Emergence of helicity +/- 2 modes (gravitons) from qubit models

Abstract

The general equivalence principle and the associated diffeomorphism gauge symmetry are regarded as a founding principles of nature. But, one may wonder, can diffeomorphism gauge symmetry emerge as a low energy property of certain topological/quantum order in a qbit model that has no diffeomorphism gauge symmetry? In this paper, we showed that, at least, the linearized diffeomorphism gauge symmetry h_{μν}\to h_{μν} +\prt_μf_ν+\prt_νf_μcan indeed emerge from some qbit models (or quantum spin models). Physically, the emergence of the (linearized) diffeomorphism gauge symmetry implies the emergence of gapless helicity +/- 2 excitations (ie the gravitons). In the first qbit model (called the L-type model), we show that helicity +/- 2 gapless excitations appear as the only type of low energy excitations using a reliable semiclassical approach. The dispersion of the gapless helicity +/- 2 is found to be \eps_k \propto k^3. The appearance of the gapless helicity +/- 2 modes suggests that the ground state of the qbit model is a new state of matter. In the second model (called the N-type model) the collective modes are strongly interacting and there is no reliable approach to understand its low energy dynamics. Using a spin-wave/quantum-freeze approach (which is shown to reproduce the correct emergent U(1) gauge theory in a quantum rotor model), we argue that the second model may contain helicity +/- 2 gapless excitations as the only type of low energy excitations with a linear dispersion \om \propto k.

Figures (7)

• Figure 1: (Color online) The integer coefficients $c^{ab}(\boldsymbol{i},\boldsymbol{r})$ in (\ref{['Scabr']}). A short solid line in the $a$ direction marks a non-zero $c^{aa}(\boldsymbol{i},\boldsymbol{j})$. The thicker lines correspond to $c^{aa}(\boldsymbol{i},\boldsymbol{j})=4$ and thinner lines $c^{aa}(\boldsymbol{i},\boldsymbol{j})=1$. The filled circles mark the non-zero $c^{xy}(\boldsymbol{i},\boldsymbol{i}+\frac{\boldsymbol{x}}{2}+\frac{y}{2})$etc whose values are equal to $-1$. Red represents positive integers and blue negative integers. The action of the scaler constraint operator $S(\boldsymbol{i})$ in eqn. (\ref{['constr1']}) changes $L^{ab}(\boldsymbol{r})$ mod $n_G$. The figure also shows those changes. The short solid lines mark the non-zero changes of $L^{aa}(\boldsymbol{i})$ where thicker lines correspond to a change of $4$ and thinner lines a change of 1. The filled circles mark the non-zero changes of $L^{xy}(\boldsymbol{i}+\frac{\boldsymbol{x}}{2}+\frac{y}{2})$etc . The changes are equal to $-1$.
• Figure 2: (Color online) The integer coefficients $d^{ab}(\boldsymbol{i},\boldsymbol{i}+\boldsymbol{x},\boldsymbol{r})$ in eqn. (\ref{['Vdabr']}). The short solid lines mark the non-zero $d^{aa}(\boldsymbol{i},\boldsymbol{i}+\boldsymbol{x},\boldsymbol{j})$ which are equal to $1$. The filled circles mark the non-zero $d^{xy}(\boldsymbol{i},\boldsymbol{i}+\boldsymbol{x},\boldsymbol{i}+\frac{\boldsymbol{x}}{2}+\frac{y}{2})$etc , which are also equal to $1$. The action of the vector constraint operator $V(\boldsymbol{i},\boldsymbol{i}+\boldsymbol{x})$ in (\ref{['constr2']}) changes $n_G\theta^{ab}(\boldsymbol{r})/2\pi$ mod $n_G$. The figure shows those changes. The short solid lines mark the non-zero changes of $n_G\theta^{aa}(\boldsymbol{i})/2\pi$. The filled circles mark the non-zero changes of $n_G\theta^{xy}(\boldsymbol{i}+\frac{\boldsymbol{x}}{2}+\frac{y}{2})/2\pi$etc . All the changes are equal to 1.
• Figure 3: (Color online) The action of the operator ${\cal R} _{zz}(\boldsymbol{i})$ in eqn. (\ref{['cRzz']}) changes $L^{ab}(\boldsymbol{r})$ mod $n_G$. The short solid lines mark the non-zero changes of $L^{aa}(\boldsymbol{i})$. The thicker lines correspond to an change of 2 and thinner lines a change of 1. The filled circles mark the non-zero changes of $L^{xy}(\boldsymbol{i}+\frac{\boldsymbol{x}}{2}+\frac{y}{2})$etc . The changes are equal to $-1$.
• Figure 4: (Color online) The action of the operator ${\cal R} _{yz}(\boldsymbol{i}+\frac{\boldsymbol{y}}{2}+\frac{\boldsymbol{z}}{2})$ in eqn. (\ref{['cRyz']}) changes $L^{ab}(\boldsymbol{r})$ mod $n_G$. The short solid lines mark the non-zero changes of $L^{aa}(\boldsymbol{i})$ where the changes are equal to $2$. The filled circles mark the non-zero changes of $L^{xy}(\boldsymbol{i}+\frac{\boldsymbol{x}}{2}+\frac{y}{2})$etc , where the bigger circle corresponds to a change of $2$ and smaller circles a change of $\pm 1$. Red represents positive integers and blue negative integers.
• Figure 5: (Color online) The integer coefficients $f^x_{x,cd,\boldsymbol{r}}$ in eqn. (\ref{['sif']}) (see eqn. (\ref{['eqsixx']})). The filled circles mark the non-zero $f^x_{x,cd,\boldsymbol{r}}$etc , which have values $\pm 1$. Red represents positive integers and blue negative integers.