To clarify the origin of the pseudogap and strange metal states as well as their mutual relationship in cuprate superconductors, a comprehensive study on the spectral function, Fermi surface, resistivity, and dynamical spin susceptivity of the Hubbard model on the square lattice has been conducted by means of the ladder dual-fermion approximation with an electron self-energy correction similar in spirit to Moriyaesque $\lambda$ correction. It is found that the appearance of these two states requires that the characteristic hole concentration below which the Mott-Heisenberg and Slater mechanisms of electron localization occurs $p_{\mathrm{MS}}$ nearly coincides with the hole concentration where the Van Hove singularity (VHS) point, i.e., the renormalized quasiparticle energy ${\tilde{\varepsilon }}_{\mathit{X}}^{*}$ at the X point $\mathit{k}=(\pi ,0)$, is in the vicinity of the Fermi level. When this condition is met, ${\tilde{\varepsilon }}_{\mathit{X}}^{*}$ is pinned at which the nesting condition of the antiferromagnetic (AFM) fluctuations is fulfilled almost everywhere on the Fermi surface in a wide range of the hole concentration in a metallic state, i.e., the strange metal state. The spin fluctuations of the strange metal state are nearly quantum critical and the dynamical spin susceptivity is well described by overdamped spin wave having the $\omega /T$ scaling with the relaxation rate at the Planckian limit, $\hslash \mathrm{\Gamma }\approx 2k_{\mathrm{B}}T$. Because of these distinctive features of the strange metal state, the $\mathit{k}$ dependence of scattering rate of electrons is small and electrons behave as the marginal Fermi liquid, where the imaginary part of their self-energy has Im ${\mathrm{\Sigma }}_{\mathit{k}}\left(\omega \right)=Tf(\omega /T)$ scaling, resulting in $T$-linear resistivity. In contrast, the pseudogap state is magnetically in the renormalized classical regime and the pseudogap is formed near the X point where the nesting condition of the short-range AFM order is fulfilled. Although the main origin of these two states is the AFM fluctuation, the end point of the pseudogap phase $p^{*}$ is placed near $p_{\mathrm{MS}}$ and the strange metal state is present not just around the quantum critical point of the AFM fluctuation $p_{\mathrm{QCP}}$ at low temperatures as expected in the Hertz-Millis-Moriya theory but extends toward $p^{*}$ due to the pinning of the VHS point. These results are consistent with the resistivity in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO), where the linear coefficient is maximum at $p^{*}$, the first-order-like sudden drop in the intensity of the angle resolved photoemission spectra at the Brillouin zone boundary at $p^{*}$ found in ${(\mathrm{Bi},\mathrm{Pb})}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+d}$ and the extended quantum critical behavior of spin fluctuations at low temperatures in the inelastic neutron scattering experiments of LSCO in the strange metal state.

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• Antiferromagnetism
• Electrical conductivity
• Flat bands
• Nesting
• Quantum criticality
• Van Hove singularity
• Dynamical mean field theory
• Hubbard model