Effects of frustration on the nonequilibrium dynamics of photoexcited lattice systems

We theoretically investigate the effects of spin frustration on the nonequilibrium dynamics of photoexcited carriers in a half-filled two-dimensional Hubbard model. Using a nonequilibrium generalization of the dynamical cluster approximation, we compare the relaxation dynamics in lattices which interpolate between the triangular lattice and square lattice configuration. To clarify the influence of the density of states of the different lattices, we also consider the corresponding single-site dynamical mean-field theory results. Our study shows that the cooling effect resulting from the disordering of the spin background is less effective in the triangular case because of the frustration. This manifests itself in a longer relaxation time of the photodoped population, as measured by the time-resolved photoemission signal, and a higher effective temperature of the photodoped carriers in the nonthermal steady state after the intra-Hubbard-band thermalization.

Figure 1

(a) Illustration of the triangular lattice in real space and of the direction-dependent electron hoppings. A real-space cluster with $N_c=2\times 2$ sites is marked by the black line. (b) Corresponding reciprocal space representation with $N_c$ cells. The coarse graining around the momentum points $\mathbf{K}$ in the first Brillouin zone (1. BZ) is indicated by the colored areas.

Figure 2

(a) Double occupancy calculated at $T=0.1$ using DCA for different lattice geometries: the tilted square lattice ($t_h^{\prime }=0$, solid green line with diamonds), the square lattice (dashed green line with squares), and the triangular lattice ($t_h^{\prime }=t_h$, solid red line with triangles). For comparison, we also show results from DMFT calculations at the same temperature for the square (black line with squares) and triangular lattice (black line with triangles). The horizontal axis is normalized by the noninteracting bandwidth $W$. (b) Corresponding nearest neighbor spin-spin correlation function calculated using DCA with the same color coding as in panel (a).

Figure 3

Equilibrium spectral functions for different lattice geometries obtained using (a) the DCA and (b) the DMFT approximation. The interaction strengths $U$ and the hopping parameters $t_h$ have been adjusted to roughly match the gap size and the band width. (a) Spectral function for a tilted square lattice with $t_h^{\prime }=0, U=22$ (solid green line) and a triangular lattice (solid red line) with renormalized parameters ($t_h^{\prime }=t_h\to 0.8t_h^0, U=22.5$). For comparison, we also plot the spectral function for a square lattice with $U=22$ (dashed green line). (b) DMFT spectral function for the square lattice (green solid line) with the renormalized parameters ($t_h\to 0.87t_h^0, U=22.6$) and for the triangular lattice (red solid line) with renormalized parameters ($t_h^{\prime }=t_h\to 0.76t_h^0, U=22.56$). The inset shows the noninteracting DoS for the corresponding square and triangular lattices, respectively.

Figure 4

Pulse-induced change in the double occupation. The parameters of the Mott-Hubbard systems are the same as in the equilibrium calculations (see Fig. 3). (a) The DCA evolution of the double occupancy after a short excitation for the tilted square lattice (green line) and the triangular lattice (red line). The inset shows the Fourier transform of the data directly after pumping (with a smooth background subtracted) to illustrate the oscillations with a frequency $\omega \approx U$. (b) The DMFT evolution for the double occupancy on the square lattice (green line) and on a triangular lattice (red line). The excitation strength is chosen such that the same photodoping is achieved for the longest propagation time, namely $\mathrm{\Delta }{t}_h\approx 0.4$ for the tilted square lattice and $\mathrm{\Delta }{t}_h\approx 0.47$ for the triangular lattice in the DCA case. The excitation strengths in the DMFT case are $\mathrm{\Delta }{t}_h\approx 0.55$ and $\mathrm{\Delta }{t}_h\approx 0.45$ for the square and triangular lattices, respectively. The black line in (b) represents the pulse ($t_h\left(t\right)-t_h^0$) in arbitrary units. Horizontal dotted lines in (a) and (b) serve as a guide to the eye.

Figure 5

Nearest-neighbor spin-spin correlation function (normalized by its equilibrium value) for a tilted square lattice (solid green line) and a triangular lattice (solid red line). The excitation protocols are the same as in Fig. 4.

Figure 6

(a),(b) Occupation in the upper Hubbard band for a tilted square lattice with $t_h^{\prime }=0$ (left) and for a triangular lattice with $t_h^{\prime }=t_h$ (right) after photodoping (colored solid lines) obtained using the DCA approximation. The black dashed line corresponds to the spectral function calculated from the retarded component of the Green's function in equilibrium, whereas the green solid line corresponds to the effective inverse temperature ${\beta }_{\text{eff}}$ calculated at $t_p\approx 11$. (c),(d) Effective distribution function calculated at $t_p\approx 11$. (e),(f) Time-dependent photoemission spectrum integrated in the frequency window $[\mathrm{\Omega }-\mathrm{\Delta }\mathrm{\Omega },\mathrm{\Omega }+\mathrm{\Delta }\mathrm{\Omega }]$ [shown by shaded areas in (a),(b) with $\mathrm{\Delta }\mathrm{\Omega }=0.5]$ and normalized to the value at time $t_p=5$ for a tilted square lattice and a triangular lattice, respectively.

Figure 7

(a) Relaxation time (measured in units of $t_h$) extracted from the PES spectral weight at the upper edge of the UHB as a function of the hopping parameter $t_h^{\prime }$. (b) Effective temperature measured at the lower edge of the UHB at time $t\approx 11$.

Figure 8

(a),(b) Occupation in the upper Hubbard band for a square lattice (left) and for a triangular lattice with $t_h^{\prime }=t_h$ (right) after photodoping (colored solid lines) obtained using the DMFT approximation. The black dashed line corresponds to the spectral function calculated from the retarded component of the Green's function in equilibrium, whereas the green solid line corresponds to the effective inverse temperature ${\beta }_{\text{eff}}$ calculated at $t_p\approx 11$. (c),(d) Effective distribution function calculated at $t_p\approx 11$. (e),(f) Time-dependent photoemission spectrum integrated in the frequency window $[\mathrm{\Omega }-\mathrm{\Delta }\mathrm{\Omega },\mathrm{\Omega }+\mathrm{\Delta }\mathrm{\Omega }]$ [shown by shaded areas in (a),(b) with $\mathrm{\Delta }\mathrm{\Omega }=0.5]$ and normalized relative to the value at $t_p=5$ for a square lattice and a triangular lattice, respectively. The excitation protocols are the same as in Fig. 4.