Robust spin and charge excitations throughout the high-$T_c$ cuprate phase diagram from incipient Mottness

The generic phase diagram of lightly hole doped high-$T_c$ cuprates hosts an antiferromagnetic insulating phase with well-defined spin-wave excitations. Contrary to the weak-coupling prediction, these modes persist up to the overdoped metallic regime as intense and dispersive paramagnons. Here we report on our study of the low-energy magnetic and charge excitations within the extended Hubbard model at strong coupling, using a modified $1/N$ expansion method with a variational state serving as the saddle point solution. Despite clear separation of magnetic and Hubbard-$U$ energy scales, we find that incipient Mottness affects qualitatively dispersions and widths of magnetic modes throughout the entire phase diagram. The obtained magnetic and charge dynamical structure factors agree semiquantitatively with recent resonant x-ray and neutron scattering data for ${\mathrm{La}}_{2-\mathit{x}}{\mathrm{Sr}}_{\mathit{x}}{\mathrm{CuO}}_4$ and ${(\mathrm{Bi},\mathrm{Pb})}_2{(\mathrm{Sr},\mathrm{La})}_2{\mathrm{CuO}}_{6+\delta }$ at all available doping levels. The weak-coupling random-phase approximation fails already for underdoped samples, pointing to the nontrivial intertwining of distinct energy scales in cuprate superconductors. The existence of a discrete charge mode which splits off the electron-hole continuum is also predicted.

Figure 1

Calculated quantum spin-fluctuation spectra for (a) and (b) an undoped AF insulator ($n=1$), (c) and (d) a paramagnetic metal ($n=0.88$), and (e) an antiferromagnetic metal ($n=0.88$). The model parameters are provided in Table 1. Dashed lines in (a)–(e) represent the real parts of the quasiparticle pole ${\omega }_p$ (the so-called propagation frequency), obtained by damped-harmonic-oscillator modeling of the numerical data as detailed in Appendix pp4. Experimental energies for ${\mathrm{La}}_2{\mathrm{CuO}}_4$ [INS; actually shifted by ${\mathbf{q}}_{\mathrm{AF}}=(0.5,0.5)$] and ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$ (RIXS) are taken from [48] and [26], respectively. (f) and (g) A direct comparison of the calculated and experimental dispersions for $n=1.0$ and $n=0.88$, respectively. Angle $\theta$ in (c)–(e) parameterizes the Brillouin-zone arc $\mathbf{k}\left(\theta \right)\equiv 0.37(cos\theta ,sin\theta )$.

Figure 2

Calculated imaginary part of the dynamical charge susceptibility $\left|t\right|{\chi }_C^{″}(\mathbf{k},\omega )$ along selected Brillouin-zone contours for LSCO at $n=0.88$ (see Table 1 for parameters). (a) shows discrete charge-excitation mode splitting of the electron-hole continuum as obtained from the $\mathrm{SGA}+1/N$ approach. (b) Analogous results obtained from RPA are displayed for comparison. In the latter situation the mode appears at the upper threshold of the continuum. Angle $\theta$ parameterizes the arc $\mathbf{k}\left(\theta \right)\equiv 0.37(cos\theta ,sin\theta )$. Diamonds are RIXS data for ${\mathrm{La}}_{2-\mathit{x}}{(\mathrm{Br},\mathrm{Sr})}_{\mathit{x}}{\mathrm{CuO}}_4$ at $n=0.875$ [15].

Figure 3

Spin- and charge-dynamic susceptibilities, ${\chi }_S^{+-}$, ${\chi }_S^{zz}$, and ${\chi }_C$, calculated using the present $\mathrm{SGA}+1/N$ approach. Red circles correspond to RIXS data for ${(\mathrm{Bi},\mathrm{Pb})}_2{(\mathrm{Sr},\mathrm{La})}_2{\mathrm{CuO}}_{6+\delta }$ [33]. Note the different energy scales for the spin and charge modes. For computational details see Appendixes pp1 and pp4. The Brillouin zone contour for (a)–(i) is shown at the top. Dashed lines represent the propagation frequency ${\omega }_p$ (see Appendix pp4).

Figure 4

Inverse of the real part of the static ($\omega =0$) (a)–(h) longitudinal spin and (i)–(p) charge susceptibilities for antiferromagnetic (AF) and paramagnetic (PM) states as a function of wave vector. The model parameters are $t=-0.34$, $t^{\prime }=0.25\left|t\right|$, $U=7\left|t\right|$. Calculations were performed using the present $\mathrm{SGA}+1/N$ method [in (a)–(d) and (i)–(l)] and RPA [in (e)–(h) and (m)–(p)]. Temperature and doping level are displayed on top of each panel.

Figure 5

Inverse of the real part of static ($\omega =0$) longitudinal (a)–(h) spin and (i)–(p) charge susceptibilities for antiferromagnetic (AF) and paramagnetic (PM) states as a function of wave vector, calculated using the present $\mathrm{SGA}+1/N$ method at various doping levels. The model parameters are $t=-0.31\mathrm{eV}$, $t^{\prime }=0.25\left|t\right|$, $U=6\left|t\right|$. Temperature is set to $k_{B}T=0.36\left|t\right|$ for all the dopings specified.

Figure 6

Inverse of the real part of static ($\omega =0$) longitudinal (a) spin and (b) charge susceptibilities for the AF state and $n=0.88$, calculated using the present $\mathrm{SGA}+1/N$ approach. The model parameters are $t=-0.34\mathrm{eV}$, $t^{\prime }=0.25\left|t\right|$, $U=7\left|t\right|$. Temperatures in (a) are given in eV. As temperature is lowered, nearly simultaneous dynamical incommensurate spin and charge instabilities of the AF state are generated.

Figure 7

Exemplary fits of the total dissipative part of the susceptibility ${\chi }_{\mathrm{total}}^{″}\equiv {\chi }_{\mathrm{oscillator}}^{″}+{\chi }_{\mathrm{incoh}}^{″}$ (blue solid curve) to the imaginary part of the dynamical susceptibility $\mathrm{Im}{\chi }^{+-}(\mathbf{k},\omega )$ calculated using (a), (c), and (e) $\mathrm{SGA}+1/N$ and (b), (d), and (f) RPA (red circles) for doping $n=0.88$. The wave vectors $\mathbf{k}$ are given in units of $2\pi /a$, with $a$ being the lattice spacing. The damped-oscillator and background partial contributions are marked in green and orange, respectively.

Figure 8

The same as in Fig. 7, but at half-filling ($n=1$). The sharp peaks represent a true magnon excitation in the stable AF phase. Panels (a)–(f) represent the same quantities as the corresponding ones in Fig. 7.