Doping and temperature evolution of pseudogap and spin-spin correlations in the two-dimensional Hubbard model

Cluster perturbation theory is applied to the two-dimensional Hubbard $t-t^{\prime }-t^{\prime \prime }-U$ model to obtain doping and temperature-dependent electronic spectral function with $4\times 4$ and 12-site clusters. It is shown that evolution of the pseudogap and electronic dispersion with doping and temperature is similar and in both cases it is significantly influenced by spin-spin short-range correlations. When short-range magnetic order is weakened by doping or temperature and Hubbard-I-like electronic dispersion becomes more pronounced, the Fermi arc turns into a large Fermi surface and the pseudogap closes. It is demonstrated how static spin correlations impact the overall dispersion's shape and how accounting for dynamic contributions leads to momentum-dependent spectral weight at the Fermi surface and broadening effects.

Figure 1

The electronic spectral function at half-filling at (a) $\beta =10/t$ and (b) $\beta =3/t$ obtained with a 12-site cluster. The chemical potential is at zero energy here and below.

Figure 2

The electronic spectral function at $T=0$ obtained using a $4\times 4$ cluster for different values of doping: (a) $p=0$, (b) $p=0.0625$, and (c) $p=0.25$.

Figure 3

The electronic spectral function map at the Fermi level at $T=0$ obtained using a $4\times 4$ cluster. The value of doping is (a) $p=0.0625$, (b) $p=0.125$, (c) $p=0.1875$, and (d) $p=0.25$.

Figure 4

Spin-spin correlation function defined in Eq. (2) for different values of doping at $T=0$.

Figure 5

(a) The antinodal/nodal ratio $R$ for the spectral weight at the Fermi level, defined in the text for different doping levels; the two dashed lines are linear fits to data within the SPG doping range and outside; smooth transitions between background colors illustrate different doping regimes discussed in the text, (b) the symmetrized spectral function defined by Eq. (3) for a wave-vector ${\mathbf{k}}_F$ in the antinodal direction for different values of doping at zero temperature [finite temperature $\beta =12/t$ was substituted into the Fermi-Dirac function in Eq. (3) to produce smooth ARPES-like curves].

Figure 6

The same as in Fig. 3 for $t^{\prime \prime }=0$.

Figure 7

The same as in Fig. 4 for $t^{\prime \prime }=0$.

Figure 8

The same as in Fig. 5 for $t^{\prime \prime }=0$.

Figure 9

The electronic spectral function at $p=0.167$ obtained using a 12-site cluster at (a) $\beta =24/t$, (b) $\beta =8/t$, and (c) $\beta =4/t$.

Figure 10

The electronic spectral function map at the Fermi level at $p=0.167$ obtained using a 12-site cluster at (a) $\beta ={10}^4/t$, (b) $\beta =24/t$, (c) $\beta =12/t$, (d) $\beta =8/t$, (e) $\beta =6/t$, and (f) $\beta =4/t$.

Figure 11

Spin-spin correlation function defined in Eq. (2) for different values of temperature at $p=0.167$.

Figure 12

(a) The antinodal/nodal ratio $R$ for the spectral weight at the Fermi level, defined in the text for different temperatures, (b) the symmetrized spectral function defined by Eq. (3) for a wave-vector ${\mathbf{k}}_F$ in the antinodal direction at different values of temperature. Both are shown for $p=0.167$.

Figure 13

(a) The schematic of the Fermi surface within the static approximation for three values of doping divided by two Liphsitz transitions, (b) the CPT result using a $4\times 4$ cluster at $T=0$ for similar doping values, (c) the CPT result using a 12-site cluster at $p=0.167$ for different temperatures, (d) sketch of the phase diagram where arrows denote two directions of evolution of the electronic structure studied in this paper, $T_N$ is the AFM Néel temperature, $T_C$ is the superconducting transition temperature, $T^{*}$ is the pseudogap transition line (WPG to NFL transition), and $T^{*1}$ is the SPG to WPG crossover line.

Figure 14

Cluster coverings used for (a) a $4\times 4$ cluster, (b) and (c) a 12-site cluster. In the case of a 12-site cluster, the hopping matrix was averaged among two coverings (b) and (c) similar to calculations of Ref. [50].

Figure 15

The electronic spectral function obtained at $T=0$ and $p=0.167$ with (a) a $4\times 4$ cluster and (b) a 12-site cluster.

Figure 16

Spin-spin correlation functions [Eq. (2)] obtained with two types of cluster at $T=0$ and $p=0.167$.

Figure 17

Charge correlation functions [Eq. (C1)] obtained within a $4\times 4$ cluster for different doping values as indicated in the inset for $U=8,t^{\prime }=-0.2$, and $t^{\prime \prime }=0$.