Pseudogap, charge order, and pairing density wave at the hot spots in cuprate superconductors

We address the timely issue of the presence of charge ordering at the hot spots in the pseudogap phase of cuprate superconductors in the context of an emergent SU(2) symmetry which relates the charge and pairing sectors. Performing the Hubbard-Stratonovich decoupling such that the free energy stays always real and physically meaningful, we exhibit three solutions of the spin-fermion model at hot spots. A careful examination of their stability and free energy shows that, at low temperatures, the system tends towards the coexistence of a charge density wave (CDW) and the composite order parameter made up of the diagonal quadrupolar density wave and pairing fluctuations of K. B. Efetov et al. [Nat. Phys. 9, 442 (2013)]. The CDW is sensitive to the shape of the Fermi surface, in contrast to the diagonal quadrupolar order, which is immune to it. SU(2) symmetry within the pseudogap phase also applies to the CDW state, which therefore admits a pairing density $p$-wave counterpart breaking time-reversal symmetry.

Figure 1

Schematic Fermi surface of cuprate superconductors in the first Brillouin zone of a square lattice. The eight hot spots are depicted by filled (red) circles labeled with the $(\Sigma ,\Lambda ,L)$ space. The angle $\theta$ is defined between the velocity vector at the [111] hot spot and the $x$ axis. The experimentally observed CDW vector ${\mathbf{Q}}_x$ connects the $L_2$ sector, whereas the diagonal QDW/SC vector $Q_1$ does not.

Figure 2

Typical form of the order parameters in the CE phase as a function of the frequency $\omega$ and temperature $T$: $B_{1x/1y}$ (a, b), representing QDW/SC; and $B_{2x/2y}$ (c, d), representing CDW/PDW order. Note that the CDW/PDW component is one order of magnitude smaller than the QDW/SC solution. We take $g_1$ slightly bigger than $g$ in order to stabilize the CDW sector. The actual figure corresponds to $g_1=20$, $g_2=30$, $v=6$, $m_a=0.1$, $\gamma =3$, and $W=2\pi$, where $W$ is the bandwidth of integration in k space and $\theta =0.1$.

Figure 3

Typical form of the order parameters in the CE phase as a function of the frequency $\omega$ and velocity angle $\theta$: $B_{1x/1y}$ (a, b), representing QDW/SCl and $B_{2x/2y}$ (c, d), representing CDW/PDW order. Note that the CDW/PDW component is one order of magnitude smaller than the QDW/SC solution. Parameters are the same as in Fig. 2.

Figure 4

Stability conditions [left-hand side of Eq. (30)] as a function of the frequency $\omega$ and velocity angle $\theta$ for the three possible solutions in the $B_1$ and $B_2$ directions: pure QDW/SC solution [${\mathrm{a}}_1$ (dir. $B_1$) and ${\mathrm{a}}_2$ (dir. $B_2$)]; pure CDW/PDW solution [${\mathrm{b}}_1$ (dir. $B_1$) and ${\mathrm{b}}_2$ (dir. $B_2$)]; and CE solution (${\mathrm{c}}_1$ and ${\mathrm{c}}_2$). Note that the pure CDW/PDW solution is always unstable, while the CE solution is stable. Parameters are the same as in Fig. 3.

Figure 5

Free energy $F$ of the three MF solutions as a function of $\theta$ and $T$: pure CDW/PDW (blue area), pure QDW/SC (green area), and CE (magenta area). Note that the free energy for the CE solution is slightly lower than that for both the QDW/SC and the CDW/PDW solutions. Parameters are the same as in Fig. 2.

Figure 6

Schematic phase diagram for high-temperature cuprate superconductors as a function of the temper-ature $T$ and hole doping $x$. The different types of order are antiferromagnetic (AFM), quadrupole density wave (QDW), pairing density wave (PDW), charge density wave (CDW), and superconductivity (SC).

Figure 7

Typical solution for the pure QDW/SC solution, as a function of $\left({\epsilon }_n,T\right)$ in (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$) and as a function of $\left({\epsilon }_n,\theta \right)$ in (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Values of the parameters are $g_1=20$, $g_2=30$, $v=6$, $m_a=0.1$, $\gamma =3$, and $W=2\pi$. The velocity angle is $\theta =0.1$ for (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$), whereas the temperature is $T=1$ for (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Note the very feeble dependence on the Fermi angle for this solution.

Figure 8

Typical solution for the pure CDW solution, as a function of $\left({\epsilon }_n,T\right)$ for (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$) and as a function of $\left({\epsilon }_n,\theta \right)$ for (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Values of the parameters are $g_1=20$, $g_2=30$, $v=6$, $m_a=0.1$, $\gamma =3$, and $W=2\pi$. The velocity angle is $\theta =0.1$ in (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$), whereas the temperature is $T=1$ in (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Note that although the $T$ dependence of the solution is different from that in Fig. 7, especially where the magnitude of the solution is concerned, its angular dependence is very similar to that depicted in Fig. 7.

Figure 9

Typical solution for the pure CDW/PDW solution, as a function of $\left({\epsilon }_n,T\right)$ in (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$) and as a function of $\left({\epsilon }_n,\theta \right)$ in (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Values of the parameters are $g_1=20$, $g_2=200$, $v=6$, $m_a=0.1$, $\gamma =3$, and $W=2\pi$. The velocity angle is $\theta =0.1$ in (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$), whereas the temperature is $T=1$ in (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Note the strong dependence on the Fermi angle for this solution.

Figure 10

Typical solution for the pure QDW/SC solution, as a function of $\left({\epsilon }_n,T\right)$ in (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$) and as a function of $\left({\epsilon }_n,\theta \right)$ in (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Values of the parameters are $g_1=20$, $g_2=200$, $v=6$, $m_a=0.1$, $\gamma =3$, and $W=2\pi$. The velocity angle is $\theta =0.1$ in (${\mathrm{a}}_1$) and (${\mathrm{a}}_2$), whereas the temperature is $T=1$ in (${\mathrm{b}}_1$) and (${\mathrm{b}}_2$). Note the striking similarities between this solution and the one in Fig. 7.

Figure 11

Typical solutions for the QDW/SC ($B_{1x}$ and $B_{1y}$) and CDW ($B_{2x}$ and $B_{2y}$) in the CE phase. Values of the parameters are $g_1=20$, $g_2=200$, $v=6$, $m_a=0.1$, $\gamma =3$, and $W=2\pi$. The velocity angle is $\theta =0.1$ in (${\mathrm{a}}_1$), (${\mathrm{a}}_2$), (${\mathrm{b}}_1$), and (${\mathrm{b}}_2$), whereas the temperature is $T=1$ in (${\mathrm{c}}_1$), (${\mathrm{c}}_2$), (${\mathrm{d}}_1$), and (${\mathrm{d}}_2$). Note the contrast with the CE solution in Fig. 2. Here the CDW/PDW component is one order of magnitude larger than the QDW/SC component and the $\theta$ dependence of the CDW/PDW component [(${\mathrm{d}}_1$) and (${\mathrm{d}}_2$)] is minimal compared to that of the QDW/SC component [(${\mathrm{c}}_1)$ and (${\mathrm{c}}_2$)].

Figure 12

Stability conditions, (30), for the pure QDW/SC solution as a function of $\left({\epsilon }_n,\theta \right)$ at $T=1$ K [(${\mathrm{a}}_1$) dir. $B_1$ and (${\mathrm{a}}_2$) dir. $B_2$]. Note that although the limit $J_2\gg J_1$ is very unfavorable to this solution, it is still stable.

Figure 13

Stability conditions, (30), for the pure CDW/PDW solution as a function of $\left({\epsilon }_n,\theta \right)$ at $T=1$ K [(${\mathrm{a}}_1$) dir $B_1$ and (${\mathrm{a}}_2$) dir. $B_2$]. Note that although the limit $J_2\gg J_1$ is energetically very favorable to this solution, there is a direction of instability (dir. $B_1$) at low temperatures, indicating instability towards the CE solution at lower temperatures.

Figure 14

Stability conditions, (30), for the pure CE solution as a function of $\left({\epsilon }_n,\theta \right)$ for $T=1$ K. (${\mathrm{a}}_1$) Direction $B_1$ and (${\mathrm{a}}_2$) direction $B_2$. Note that the CE solution is stable in both directions at low temperatures.

Figure 15

Free energy of the three mean-field solutions. Pure CDW/PDW (brown area), pure QDW/SC (dark-blue area), and CE (neon-colored areas). Values of the parameters are $g_1=20$, $g_2=200$, $v=6$, $m_a=0.1$, $\gamma =3$, $W=2\pi$, and $\theta =0.1$.