Domain structures in quantum graphity

Quantum graphity offers the intriguing notion that space emerges in the low-energy states of the spatial degrees of freedom of a dynamical lattice. Here we investigate metastable domain structures which are likely to exist in the low-energy phase of lattice evolution. Through an annealing process we explore the formation of metastable defects at domain boundaries and the effects of domain structures on the propagation of bosons. We show that these structures should have observable background-independent consequences including scattering, double imaging, and gravitational lensing-like effects.

Figure 1

A graph representation of the state vector $|l_{12}{l}_{13}{l}_{14}{l}_{23}{l}_{24}{l}_{34}⟩=|101101⟩$ which has for 4 vertices and edges.

Figure 2

Two isomorphically equivalent representations of a local minimum graph state for $v_0=3$, $R\ge 7.1$: (a) Honeycomb representation [$a=(0,0)$, $b=(-1,0)$, $c=(0.5,0.87)$, $d=(2,0)$, $e=(0.5,–0.87)$] and (b) brick representation [$a=(0,0)$, $b=(-1,0)$, $c=(0,1)$, $d=(1,0)$, $e=(0,-1)$]. $a$, $b$, $c$, $d$, $e$ are examples of the respective graph’s vertex labeling schemes.

Figure 3

A graphical representation of edge hopping term $H_X=g_X\sum _{qrs}{m}_{rs}{b}_{rq}^{†}{b}_{qs}$. In this example edge $|l_{rq}⟩$ is destroyed and edge $|l_{qs}⟩$ created. This local interaction is only allowed because there is a edge between vertex $r$ and $s$ i.e. $m_{rs}|l_{rs}⟩=1$.

Figure 4

(a) Two domains form independently. Where they meet is an unstable antiphase boundary defect. (b) After the quenching process the boundary defect is frozen into a metastable amorphous state.

Figure 5

Results of Metropolis algorithm applied to a toroid lattice (major diameter: 100 vertices, minor diameter: 9 vertices) with an unstable antiphase domain boundary [Fig. 4a] at quenching temperature $T=0$. Other parameter values: $v_0=3$, $r=7.2$, $p=1$, $g_{\mathrm{V}}/g_{\mathrm{P}}={10}^5$, $L_{\max }=10$, $\mathrm{\text{iterations}}=300$. (a) Plot of the lattice energy averaged over 300 samples at each $t$ step. The average energy converges to $9.1\times {10}^{7}E/g_{\mathrm{P}}$. This is higher than the ground state energy $8.8\times {10}^{7}E/g_{\mathrm{P}}$ (dotted line), meaning the lattice is more likely to settle into a local minimum with defects rather than the ground state. (b) The energy probability distribution of the lattice at $t={10}^5$, when most of the samples have reached a metastable local minimum. As the lattice has been quenched, the distribution does not follow the Boltzmann distribution for finite $T$, but instead resembles the typical energy distribution of quenched metal alloys.

Figure 6

(a) An extract of a metastable lattice at the domain boundary after quenching. (b) Energetically nearby states accessible from configuration (a), by nearest neighbor edge hopping. The dotted lines represent hopping edges. Other neighborhood states (not shown) are of much higher energies as more than one vertex has $v\ne v_0$. Calculation of $H_{\mathrm{loop}}$ shows that the states in (b) are higher energy states then configuration (a). Therefore as configuration (a) is energetically lower than its set of neighboring states, configuration a) forms a metastable state.

Figure 7

Simulation of a localized boson as it scatters at a metastable amorphous domain boundary defect. (a) The boson is initialized at ${\mathbf{x}}_0=(100,80)$, with $\mathbf{\sigma }=(10,10)$ and $\mathbf{k}=(0,-2)$ so that it has ${\mathbf{v}}_g^{\pm }/\kappa =\pm (0,1)$. Most of the boson wave packet propagates as ${\mathbf{v}}_g^{+}$. Inset: Zoomed depiction of the amorphous defect line which runs from vertex (0,100) to (100,100). (b) Snapshot of the system at $t=100/\kappa$ showing that the amorphous defect has caused the boson to scatter; there is some probability that the boson is also localized on the domain boundary for a finite time.

Figure 8

Two domains formed independently with different orientation, labelled as domain-$\pi /2$ and domain-0. At their intersection is a domain boundary defect.

Figure 9

Isoenergy contours of the dispersion properties of domain-0 (left panel) and domain-$\pi /2$ (right panel). A mode with $E/\kappa =2$ (bolded contour lines) and ${\mathbf{k}}_0=(-1.6,-0.5)$ in domain-0 has ${\mathbf{v}}_g^{+}/\kappa =(0.88,0.83)$ as indicated by the arrow. Neglecting edge effects, phase matching will mean that this mode will couple to propagating mode ${\mathbf{k}}_{\pi /2}=(-1.0,-0.5)$ in domain-$\pi /2$, which will have ${\mathbf{v}}_g^{+}/\kappa =(1.6,0.3)$ (indicated by the arrow).

Figure 10

Simulation of the propagation of a boson wave packet over time. Different time instances, $t$, are superimposed; each instance is labeled with $(\kappa t,M)$. For clearer presentation, populations are multiplied by factor M so that the peak value at each time instant is approximately the same. (a) The boson is initialized in domain-0 with ${\mathbf{k}}_0=(-1.6,-0.5)$. As the boson propagates through the domain boundary it undergoes refraction with angle of refraction $|{\theta }_R|\approx {8.8}^o$. (b) The boson is initialized in domain-$\pi /2$ with $k_{\pi /2}=(0.84,-1.1)$. As this mode can not couple to resonant modes in domain-0 the boson will be completely reflected. Inset: Zoomed depictions of the domain boundary.

Figure 11

(a) The intersection of four domains (bottom, left, top, right) with orientation $\alpha =0$, $\pi /2$. (b) Isoenergy contours at $E/\kappa =2$. The blue contour corresponds to domain-$0$ (bottom and top domains) and the green contour corresponds to domain-$\pi /2$ (left and right domains). The arrows represent group velocities. A mode originating in the bottom domain with $\mathbf{k}=(\pi /6,-\pi /2)$ propagates with $v_g^{+}/\kappa =(\sqrt{-3}/2,\sqrt{3}/2)$ (arrow 1). It couples to the propagating mode $\mathbf{k}=(0,-\pi /3)$ in the left domain (arrow 2). At the second domain boundary it will couple to the propagating mode $\mathbf{k}=(-\pi /2,-\pi /6)$ in the top domain (arrow 3). This trajectory will converge with its symmetric (about the $y$-axis) mode producing a lensing-like effect.

Figure 12

Simulation of boson Gaussian wave packets with a) $\mathbf{k}=(\pm \pi /8,-\pi /8)$, b) $\mathbf{k}=(\pm \pi /6,-\pi /6)$, c) $\mathbf{k}=(\pm \pi /5,-\pi /5)$ at the four domain structure of Fig. 11. The arrows indicate the group velocity ($v_g^{+}$) in each domain. As the boson crosses the domain boundaries it undergoes refraction to converge on the far side. The foci points for the different modes of the boson wave packet form a focal line. Note that there are also modes (the faint light blue pulses) associated with $v_g^{-}$ which propagate in the opposite direction (i.e. downward) and are reflected off the hard wall boundary. The notation $(\kappa t,M)$ follows Fig. 10.