Phase competition and anomalous thermal evolution in high-temperature superconductors

The interplay of competing orders is relevant to high-temperature superconductivity known to emerge upon suppression of a parent antiferromagnetic order typically via charge doping. How such interplay evolves at low temperature—in particular at what doping level the zero-temperature quantum critical point (QCP) is located—is still elusive because it is masked by the superconducting state. The QCP had long been believed to follow a smooth extrapolation of the characteristic temperature $T^{*}$ for the strange normal state well above the superconducting transition temperature. However, recently the $T^{*}$ within the superconducting dome was reported to unexpectedly exhibit back-bending likely in the cuprate ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. Here we show that the original and revised phase diagrams can be understood in terms of weak and moderate competitions, respectively, between superconductivity and a pseudogap state such as $d$-density or spin-density wave, based on both Ginzburg-Landau theory and the realistic $t\text{−}{t}^{\prime }\text{−}{t}^{\prime \prime }\text{−}J\text{−}V$ model for the cuprates. We further found that the calculated temperature and doping-level dependence of the quasiparticle spectral gap and Raman response qualitatively agrees with the experiments. In particular, the $T^{*}$ back-bending can provide a simple explanation of the observed anomalous two-step thermal evolution dominated by the superconducting gap and the pseudogap, respectively. Our results imply that the revised phase diagram is likely to take place in high-temperature superconductors.

Figure 1

Phase diagrams of the competing SC (yellow), CO (cyan), and coexisting (pink) states in Ginzburg-Landau theory. (a) The original type, where $x_{\text{QCP}}^{}>x_{\text{OP}}$, for $q=1$. (b) The revised type, where $x_{\text{QCP}}^{}<x_{\text{OP}}$, for $q=0.2$. The other parameters are ${\alpha }_s\left(x\right)=10(x-0.3)$, ${\alpha }_d\left(x\right)=27(x-0.22)$, $\beta =2$, and $g=1.2$ used in Ref. [46] for cuprates. Lower panels are our fits to the experimental data (various symbols) on Ba-122 iron-pnictides [43, 44, 45] using (c) Eq. (3) with ${\alpha }_s\left(x\right)=10(x-0.13)$, ${\alpha }_d\left(x\right)=50(x-0.068)$, $q=0.23$, $\beta =2$, $g=1.1$, and (d) Eq. (4) with ${\alpha }_s\left(x\right)=10(x-0.125)$, ${\alpha }_d\left(x\right)=26(x-0.079)$, $q=0.4$, $p=0.3$, $\beta =2$, $g=1.4$.

Figure 2

Phase diagrams of the competing SC (yellow), CO (cyan), and coexisting (pink) states in Ginzburg-Landau theory using Eq. (3) for fixed $q=0.4$. (a) The original type, where $x_{\text{QCP}}^{}>x_{\text{OP}}$, for $g=0.6$. (b) The revised type, where $x_{\text{QCP}}^{}<x_{\text{OP}}$, for $g=1.2$. (c) Complete phase separation for $g>2$. The other parameters are ${\alpha }_s\left(x\right)=10(x-0.13)$, ${\alpha }_d\left(x\right)=65(x-0.069)$, $\beta =2$. (d) Phase diagram in terms of interaction ${\alpha }_d(x,T)/{\alpha }_s(x,T)$ vs $g/\beta$. The light gray, and heavy gray arrows demonstrate phase transition, which occurs in the underdoping (overdoping), and in the intermediate doping region of revised phase diagram, respectively.

Figure 3

Schematic of the DDW order. Solid and hollow circles are for the A and B sublattice, respectively. Signs of the DDW orders are denoted by the arrows as described in the Appendix.

Figure 4

Phase diagram of the extended $t\text{−}J\text{−}V$ model in the color codes of yellow (SC), cyan (DDW), pink (coexisting). (a) Results in the slave-boson approximation for $t^{\prime }=-0.25$, $t^{\prime \prime }=0.1$, $J=0.35$, and $V=0.12$; (b) Results in the renormalized mean-field theory approximation for $t^{\prime }=-0.2$, $t^{\prime \prime }=0.1$, $J=0.3$, and $V=0.095$. Besides the conventional underdoping (UD) and overdoping (OD) region, an intermediate doping (ID) region where the back-bending of $T_d$ occurs is marked out.

Figure 5

Model parameter dependence of the phase diagram in the color codes of yellow (SC), cyan (DDW), pink (coexisting). (Top) The $V$ dependence for $t^{\prime }=-0.25$. (Bottom) The $t^{\prime }$ dependence for $V=0.12$. $t^{\prime \prime }=0.1$ and $J=0.35$ for all the figures.

Figure 6

The effects of $t^{\prime }$ on the pre-back-bending of $T_d$ in the extended $t\text{−}J\text{−}V$ model. The DDW and SC orders are decoupled for (a) $t^{\prime }=0$ and (b) $t^{\prime }=-0.35$. They are coupled for (c) $t^{\prime }=0$ and (d) $t^{\prime }=-0.35$. ${\mathrm{T}}_d^{\prime }$ is the characteristic temperature for the incommensurate DDW as discussed in main text. $t^{\prime \prime }=0$, $J=0.3$, and $V=0.15$ for all.

Figure 7

The effects of $t^{\prime }$ on the pre-back-bending of $T_d$ in the extended $t\text{−}J\text{−}V$ model. The DDW and SC orders are decoupled for (a) $t^{\prime }=0$ and (b) $t^{\prime }=-0.25$. They are coupled for (c) $t^{\prime }=0$ and (d) $t^{\prime }=-0.25$. ${\mathrm{T}}_d^{\prime }$ is the characteristic temperature for the incommensurate DDW as discussed in main text. $t^{\prime \prime }=0.1$. $J=0.35$, and $V=0.12$ for all.

Figure 8

Thermal evolution of the SC and DDW order parameters and measured antinodal gap for three distinct doping levels: (a) underdoping $x=0.11$, (b) overdoping $x=0.18$, and (c) intermediate doping $x=0.135$. The legends of SC and DDW indicate the SC- and DDW-dominated regions, respectively. Solid lines are for the SC (black) and DDW (green) order parameters at the Fermi surface along the antinodal line. The open circles are for the peak energies extracted from the spectral functions; HEP and LEP stand for the high- and low-energy peaks, respectively. The size of the circles scales with the peak intensity. (d) Temperature evolution of the spectral functions at the intermediate doping $x=0.135$. The peak positions are marked by triangles with red for HEP and blue for LEP. The model parameters are $t^{\prime }=-0.25$, $t^{\prime \prime }=0.1$, $J=0.35$, and $V=0.12$. The Fermi energy is fixed at 0.

Figure 9

Thermal evolution of the Raman response in the ${\mathrm{B}}_{1g}$ channel at intermediate doping $x=0.135$. (a) The results for the broadened resolution of $\eta =0.04$. The peak positions are indicated by the triangles. (b)–(d) is Raman response at three typical temperature with high resolution $\eta =0.003$. $T=0.3T_c$ for (b), $T=0.7T_c$ for (c), and $T=0.9T_c$ for (d). The intraband and interband components $R^{\mathrm{tra}}$ and $R^{\mathrm{ter}}$ is denoted by blue, and red dashed lines, respectively. $R$ is the sum of $R^{\mathrm{tra}}$ and $R^{\mathrm{ter}}$ denoted by solid black line. (e) Temperature dependence of the energy of ${\mathrm{B}}_{1g}$ Raman response peak. The circles track the peak energy shown (b)–(d) with high resolution, the intensity is marked by size. The solid line is the peak energy with broadened resolution extracted from (a). SC and DDW denote the SC- and DDW-dominated regions, respectively, as described in text. (f) Theoretical results under broadened resolution at different doping, together with the experimental Raman data in cuprate Hg-1201 [53, 54] and near optimally doped iron-pnictide Ba-122 [59]. $t^{\prime }=-0.25$, $t^{\prime \prime }=0.1$, $J=0.35$, and $V=0.12$.

Figure 10

The $V$ dependence of the phase diagram in the color codes of yellow (SC), cyan (SDW), pink (coexisting) for $t^{\prime }=-0.25$, $t^{\prime \prime }=0.1$, and $J=0.35$.

Figure 11

(a) Phase diagram and (b) Raman response in the extended $t\text{−}J\text{−}V$ model with an $s$-wave-like pseudogap SDW instead of the $d$-wave-like DDW. Symbols in (b) are experimental data extracted from cuprates Hg1201 [53, 54]. $t^{\prime }=-0.25$, $t^{\prime \prime }=0.1$, $J=0.25$, and $V=0$.

Figure 12

(a) The denominator of RPA DDW charge susceptibility $\mathcal{D}\left(q\right)$ with different parameters. (a) $t^{\prime }=0$, $t^{\prime \prime }=0$, $J=0.3$, and $V=0.15$; (b) $t^{\prime }=-0.35$, $t^{\prime \prime }=0$, $J=0.3$, and $V=0.15$; and (c) $t^{\prime }=-0.25$, $t^{\prime \prime }=0.1$, $J=0.3$, and $V=0.12$. The temperature is fixed at $1\times {10}^{-4}$.