Lorentz symmetric dispersion relation from a random Hamiltonian

We match the density of energy eigenstates of a local field theory with that of a random Hamiltonian order by order in a Taylor expansion. In our previous work we assumed Lorentz symmetry of the field theory, which entered through the dispersion relation. Here we extend that work to consider a generalized dispersion relation and show that the Lorentz symmetric case is preferred, in that the Lorentz symmetric dispersion relation give a better approximation to a random Hamiltonian than the other local dispersion relations we considered.

Figure 1

A plot of the density of energy eigenvalues for a random Hamiltonian using Eq. (1) and a field theory using Eq. (2) matching the zeroth and first order terms in a Taylor expansion around $E_0$ (the vertical line).

Figure 2

The exponents of the $E$ factor (solid) and $m$ factor (dashed) in Eq. (35) evaluated at $d\in \{1,2,3,4\}$ (the height of the curve increases with increasing $d$). Note that the $E$ exponent decreases with increasing $g$ while the $m$ exponent grows.

Figure 3

The entropy $S(E_0,g)$ for $d=3$ for $m/m_c=0.01$, 0.1, 1, 10 and 100 in order of lowest to highest curves. We see that taking $m=m_c$ removes the $g$ dependence of $S$. For $m>m_c$, $S$ increases with $g$, whereas $S$ decreases for $m<m_c$. (Here $E_0={10}^{80}\mathrm{GeV}$ and $k_V={10}^{25}\mathrm{GeV}$.)

Figure 4

The value of the prefactor $P$ [defined in Eq. (38)] as a function of $g$. The three curves show $d=2$, $d=3$ and $d=4$, lowest to highest. The fact that $P(g,d)=O(1)$ for a wide range of values is why we can neglect $P$ when building intuition about the dependence of ${\mathrm{\Delta }}_2$ on $g$.

Figure 5

This figure is similar to Fig. 3 but shows values of $m$ much closer to $m_c$ ($m/m_c=0.9$, 0.95, 1, 1.05 and 1.25). For $m$ so close to $m_c$ the nontrivial shape of the prefactor shown in Fig. 4 becomes visible and can lead to local minimum somewhat away from $g=0$. That shifts the value of $m$ for which $g=0$ is preferred to a value slightly higher than $m_c$.

Figure 6

$N_H^{-1}dN/dE$ for random Hamiltonians (dashed lines) and field theory (solid lines) matching the zeroth and first order terms in an expansion around $E_0$ as a function of $E/E_0$ for $g=0$ (upper panel), $g=1$ (center) and $g=2$ (lower). The Lorentz symmetric ($g=0$) case gives the broadest overlap, illustrating the main conclusion of this paper. The plots correspond to $d=3$, $\beta =2$, $\gamma =1/2$, $k=10-3E_M$, $E_0=8E_M$ and $m=E_M$. The parameters $N_H$ and $E_S$ were solved for as explained in Sec. 2.