Real-Space Visualization of Remnant Mott Gap and Magnon Excitations

We demonstrate the ability to visualize real-space dynamics of charge gap and magnon excitations in the Mott phase of the single-band Hubbard model and the remnants of these excitations with hole or electron doping. At short times, the character of magnetic and charge excitations is maintained even for large doping away from the Mott and antiferromagnetic phases. Doping influences both the real-space patterns and long timescales of these excitations with a clear carrier asymmetry attributable to particle-hole symmetry breaking in the underlying model. Further, a rapidly oscillating charge-density-wave-like pattern weakens, but persists as a visible demonstration of a subleading instability at half-filling which remains upon doping. The results offer an approach to analyzing the behavior of systems where momentum space is either inaccessible or poorly defined.

Figure 1

Charge (top) and spin (bottom) density response as a function of frequency at $(\pi ,\pi )$ for half-filling. The blue (red) lines represent the real (imaginary) parts of ${\chi }^{(c/s)}(\pi ,\pi )$ and the dashed lines denote the baseline (or zero). The inset shows the 18A real-space Betts cluster employed in this study [26].

Figure 2

Evolution of (a) charge and (b) spin susceptibility ${\chi }^{(c/s)}(\mathbf{r},t)$ at half filling in the real-space cell. Color represents the intensities on a logarithmic scale. Characteristic time scales for charge and spin response $2\pi /U$ and $2\pi /J$ are marked in the panels, respectively. (c,d) Featured frames of ${\chi }^{(c,s)}(\mathbf{r},t)$ at representative times, respectively. The real-space cluster has been overlaid on the first frames to indicate the position of each site for clarity.

Figure 3

(a) Comparison of ${\chi }^{(c)}(\mathbf{r},t)$ for 11.1% hole- (left) and electron-doping (right). (b) Comparison of ${\chi }^{(s)}(\mathbf{r},t)$ for 11.1% hole- (left) and electron-doping (right).

Figure 4

(a) Charge and (b) spin density response functions ${\chi }^{(c/s)}(\mathbf{r},t)$ at different real-space coordinates and doping levels. The various blue (red) shaded lines represent progressively longer range response functions on the same (opposite) AFM sublattice. The grey dashed line marks the characteristic time scale $2\pi /U$ ($2\pi /J$) for charge (spin). The green arrows highlight the corresponding time for each frame in Fig. 3.