Figure 1

The zeroth moment of the self-energy versus the density normalized to the exact value $U^2\frac{n(2-n)}{4}$. These data are in two dimensions with $U=10,\hspace{0.5em}W=2$. The inset shows the $k$ independence of the sum rule along the (11) direction for the case $n=0.04$, with variations in the sixth significant figure.Reuse & Permissions

Figure 2

High-frequency (UHB) density of states in two dimensions, $n=1/20$. As the hopping is decreased from $t=\frac{1}{2},\frac{1}{8},\frac{1}{32}$ at a fixed $U=10$, the UHB consists of a sharp peak and a broad $k$-independent background that maintains a width of $O(U)$. The sharp $k$-dependent peak does narrow as the hopping decreases. In Fig. 3, the broad UHB of the full band can be seen with the $G_1$ from Eq. (3) superimposed. Here, the UHB feature is broadened by $\eta$ and the LHB is suppressed for clarity.Reuse & Permissions

Figure 3

Two dimensions, $U=10$, $W=2.5$, and $n=1/20$. The spectral function at three values of the wave vector $(0,0),(\frac{\pi }{2},\frac{\pi }{2}),(\pi ,\pi )$ in blue (square), red (circle) and gold (triangle) colors. Aside from the quasiparticles, we observe three features emerging in each spectral function. Most obvious is the UHB feature that lies at a $\omega \approx O(U)$ and integrates to a weight of $n/2+O(n^2)$. This feature is dramatically broadened in the self-consistent $G$ also becoming less $k$ dependent. On each edge of the quasiparticle band, we observe small dispersing features. Reference15 has previously identified the negative frequency feature as a two-hole antibound state, while Ref. 24 has discussed a particle-hole antibound state just above the quasiparticle band. These features are essentially unchanged in going from $G_1$ from Eq. (3) to the exact $G$.Reuse & Permissions

Figure 4

Two dimensions, $U=0.25$, $U=10$, $W=2$, and $n=0.049$. The local ${\rho }_{\Sigma }(\omega )$ divided by $U$ versus $\omega$. The two chosen values of $U$ are in the weak-coupling (blue (circle) $U=0.25$) and strong-coupling (red (square) $U=10$) ranges, respectively. We see at the lowest temperatures that the the self-energy curves overlap when scaled by $U$ displaying a characteristic quadratic dip at the chemical potential.Reuse & Permissions

Figure 5

The local self-energy spectrum in two dimensions, $U=10$, $W=2$, and $n=0.05$. The log-scale plot shows the full scale of the UHB. The inset highlights the quadratic minimum at low energies. The quadratic minimum drops below the scale of $\eta$, so it can be said to represent an infinite lifetime.Reuse & Permissions

Figure 6

The two-dimensional DOS, i.e., the momentum-averaged spectral function ${\rho }_G(\mathrm{k⃗},\nu )$ and the momentum-averaged ${\rho }_{\Sigma }(\mathrm{k⃗},\nu )$. The LHB feature is the sharp peak near $\omega ~0$. The UHB feature in the DOS is nearly invisible here but lies just below the feature in ${\rho }_{\Sigma }$ scaled down by a factor of $\omega {}^2$. The real part of the self-energy for $\omega ⩾0$ initially drops linearly with frequency over a range $W\ll \omega ~\frac{U}{2}$, as required in the limit of extreme correlations (Refs. 7 and25). It then flips at the threshold of the UHB, rising across the range of the UHB until, at the highest energy, it begins to decay down toward the Hartree term at infinite energy.Reuse & Permissions

Figure 7

The integrated spectral weight over the UHB is called $m_3(k)$. It is plotted here for $\frac{W}{U}=$ 1.6, 0.56, 0.196, 0.0686, 0.024, and 0.0085. In this case, $n=0.15$. We observe that the weight of the UHB exceeds $n/2$ by $O(n^2)$ and becomes flat as $U/W$ tends to infinity.Reuse & Permissions

Figure 8

From the convolution structure of ${\rho }_{\Sigma }(k,\omega )$, we see that the local objects of $\Sigma$ and $\Gamma$ are related by the ratio $n/2$ when $\omega >W$ for all values of $W/U$. In the strong-coupling limit, where the upper band is essentially independent of $k$, this relationship will be approximately true for each wave vector. On the negative frequency side, the thermal function acts differently such that the ratio for $\omega <-W$ is approximately $(1+\frac{n}{2})$.Reuse & Permissions

Figure 9

One-dimensional $U=10$, $W=0.56$ (dashed line), $W=0.196$ (solid line), $n=0.15$, and T $=$ 0.005. Here, $m_1$ is essentially the zero-temperature quasiparticle occupation, while $m_2$ accounts for the LHB particle addition spectrum. The sharp step in occupation occurs precisely at the Luttinger Fermi surface, which satisfies the Luttinger-Ward sum rule. The sum of $m_1$ and $m_2$ is less than one due to the weight transferred to the upper band. The total lower band weight approaches $1-n/2$ as $U/W$ goes to infinity.Reuse & Permissions

Figure 10

The doublon dynamics breaks into two regimes: a sharp decay at early times followed by a long exponential tail. The magnitude of the initial decay depends strongly on the density. In the limit $n\to 0$, the initial decay disappears, indicating that the UHB is comprised of sharp features only in the limit of vanishing density. In the right panel, the $U$ dependence of the long-time decay is shown, it slows down, and is finally limited by the level broadening $\eta$ assumed in our numerics. Here, $\gamma$ consists of a UHB piece and a LHB piece that interfere, causing oscillations. For large $U/W$, the LHB piece becomes small and, hence, there is no interference.Reuse & Permissions

Figure 11

The inset shows that $\gamma (1,t_r)$ goes to zero at early times since there is no mechanism to hop at small times. On a longer time scale, we see the development of an exponential decay. The small magnitude of the correlation is due to fact that the UHB is largely $k$ independent.Reuse & Permissions

Figure 12

Doublon decay on a cubic lattice with $U=15$ and $W=12$. The shape of $|{\gamma }_U({\mathit{Ut}}_r)|{}^2$ (red dashed curve) initially deviates slightly from the exact numerical result (blue curve) due to the neglect of the ${\gamma }_L$ term, which decays much more quickly than the UHB contribution.Reuse & Permissions

Figure 13

Doublon decay on a cubic lattice with $U=5$ and $W=12$. In the case $U<W$, it is much more difficult to find an exact analytical form, so only the numerical result is displayed.Reuse & Permissions

Figure 14

Two theoretical calculations, from ladder diagrams of Eq. (19) in two dimensions (Solid) and the exact two-particle solution from Eq. (21) and Eq. (A10) in two and three dimensions (long dash and short dash). These are compared to the experiment Eq. (22) in three dimensions (dot dash), scaled to coincide at weak coupling by a factor $26.4$. The theory and experiment are in very different limits of physical parameters, but have a similar shape, except at large $U/t$.Reuse & Permissions