Interplay of charge, spin, and lattice degrees of freedom in the spectral properties of the one-dimensional Hubbard-Holstein model

We calculate the spectral function of the one-dimensional Hubbard-Holstein model using the time-dependent density matrix renormalization group, focusing on the regime of large local Coulomb repulsion, and away from electronic half-filling. We argue that, from weak to intermediate electron-phonon coupling, phonons interact only with the electronic charge, and not with the spin degrees of freedom. For strong electron-phonon interaction, spinon and holon bands are not discernible anymore and the system is well described by a spinless polaronic liquid. In this regime, we observe multiple peaks in the spectrum with an energy separation corresponding to the energy of the lattice vibrations (i.e., phonons). We support the numerical results by introducing a well controlled analytical approach based on Ogata-Shiba's factorized wave function, showing that the spectrum can be understood as a convolution of three contributions, originating from charge, spin, and lattice sectors. We recognize and interpret these signatures in the spectral properties and discuss the experimental implications.

Figure 1

The parameter $\tilde{f}$ is obtained by minimizing the ground-state energy of Hamiltonian (19) with respect to $f$. Panel (a) shows DMRG calculations for two different values of e-ph coupling $g$ (the potential curve obtained for $g=1.8$ has been shifted in energy by an arbitrary constant, which is irrelevant in our description where only the position of the minimum is important). Panel (b) shows $\tilde{f}$ as a function of $g$. All results are for density $N/L=0.75$.

Figure 2

Photoemission spectrum calculated with the ZFA method in the antiadiabatic regime $\left({\omega }_0=2.0\right)$ for different e-ph couplings $g=0.2,0.6,1.0,1.2,1.5,2.0$. Here $L=32$ sites, $U=20$, and filling $N/L=3/4$.

Figure 3

Three cuts at $k=0,k_F,2k_F$ of photoemission spectrum calculated with the ZFA approach [dashed (red) line] and using the tDMRG [solid (black) line].

Figure 4

Photoemission spectrum of the HH model calculated with tDMRG for the same frequency and e-ph coupling values considered in Fig. 2. Here $L=32$ sites, $U=20$, and filling $N/L=3/4$.

Figure 5

Momentum distribution function for the same parameter values as in Fig. 4. Solid (black), dashed (red), dotted (green), dashed-dotted (blue), dashed-dotted-dashed (cyan), short-dashed (magenta), represent respectively $g=0.2,0.6,1.0,1.2,1.5,2.0$. Inset: Luttinger parameter $K$ as a function of $g$, extracted from the electronic density-density correlation function.

Figure 6

Panels (a) and (b) show the spin-spin correlation function from the center of the chain for different $g$'s in logarithmic (absolute value) and linear scale, respectively. Inset of panel (a): slope of a linear fit for the spin-spin correlation in logarithmic scale.

Figure 7

Panel (a) shows the photoemission spectrum for $g=1.0$, and different phonon frequencies ${\omega }_0=0.5,1.0,2.0$. Panel (b) shows three cuts at $k=0,k=k_F$, and $k=2k_F$ of the photoemission spectrum reported in panel (a).