Charge-density-wave order with momentum $(2Q,0)$ and $(0,2Q)$ within the spin-fermion model: Continuous and discrete symmetry breaking, preemptive composite order, and relation to pseudogap in hole-doped cuprates

We analyze charge order in hole-doped cuprates within the the spin-fermion model. We show that a magnetically mediated interaction, which is known to give rise to $d$-wave superconductivity and charge order with momentum along zone diagonal, also gives rise to charge order with momenta $Q_x=(2Q,0)$ and $Q_y=(0,2Q)$ consistent with the experiments. We show that an instability towards ${\Delta }_k^Q=\langle c_{\mathbf{k}+\mathbf{Q}}^{†}{c}_{\mathbf{k}-\mathbf{Q}}\rangle$ with $\mathbf{Q}=Q_x$ or $Q_y$ is a threshold phenomenon, but the dimensionless spin-fermion coupling is above the threshold, if the magnetic correlation length $\xi$ exceeds a certain critical value. At a critical $\xi$, the onset temperature for the charge order terminates at a quantum-critical point distant from the magnetic one. We argue that the charge order with $Q_x$ or $Q_y$ changes sign under $\mathbf{k}\to \mathbf{k}+(\pi ,\pi )$, but $|{\Delta }_k^Q|\ne |{\Delta }_{k+(\pi ,\pi )}^Q|$. In real space, such an order has both bond and site components; the bond one is larger. We further argue that ${\Delta }_k^Q$ and ${\Delta }_{-k}^Q$ are not equivalent, and their symmetric and antisymmetric combinations describe, in real space, incommensurate density modulations and incommensurate bond current, respectively. We derive the Ginzburg-Landau functional for four-component $U\left(1\right)$ order parameters ${\Delta }_{\pm k}^Q$ with $\mathbf{Q}=Q_x$ or $Q_y$ and analyze it first in mean-field theory and then beyond mean field. Within mean field we find two types of charge-density-wave (CDW) states, I and II, depending on system parameters. In state I, density and current modulations emerge with the same $\mathbf{Q}=Q_x$ or $Q_y$, breaking $Z_2$ lattice rotational symmetry, and differ in phase by $\pm \pi /2$. The selection of $\pi /2$ or $-\pi /2$ additionally breaks $Z_2$ time-reversal symmetry, such that the total order parameter manifold is $U\left(1\right)\times Z_2\times Z_2$. In state II, density and current modulations emerge with different $\mathbf{Q}$ and the order parameter manifold is $U\left(1\right)\times U\left(1\right)\times Z_2$, where in the two realizations of state II, $Z_2$ corresponds to either lattice rotational or time-reversal symmetry breaking. We extend the analysis beyond mean field and argue that discrete symmetries get broken before long-range charge order sets in. For state I, which, we argue, is related to hole-doped cuprates, we show that, upon lowering the temperature, the system first breaks $Z_2$ lattice rotational symmetry ($C_4\to C_2$) at $T=T_n$ and develops a nematic order, then breaks $Z_2$ time-reversal symmetry at $T_t<T_n$ and locks the relative phase between density and current fluctuations, and finally breaks $U\left(1\right)$ symmetry of a common phase of even and odd components of ${\Delta }_k^Q$ at $T=T_{\mathrm{CDW}}<T_t<T_n$ and develops a true charge order. We argue that at a mean field, $T_{\mathrm{CDW}}$ is smaller than superconducting $T_{\mathrm{SC}}$, but preemptive composite order lifts $T_{\mathrm{CDW}}$ and reduces $T_{\mathrm{SC}}$ such that at large $\xi$ charge order develops prior to superconductivity. We obtain the full phase diagram and present quantitative comparison of our results with ARPES data for hole-doped cuprates.

Figure 1

The Fermi surface, Brillouin zone, and magnetic Brillouin zone (dashed line). Hot spots are defined as intersections of the FS with magnetic Brillouin zone. The hot spot pairs 1-2 and 3-4 denote the CDW pairing we consider. They are coupled through the antiferromagnetic exchange interaction peaked at momentum $(\pi ,\pi )$, as shown by the dashed arrows.

Figure 2

The set of linearized equations for CDW vertices constructed out of fermions near hot spots. The momenta $k_0$ and $k_{\pi }=k_0+(\pi ,\pi )$ are in-between the two neighboring hot spots either along the $x$ or along the $y$ direction, chosen such that $k_0\pm Q$ and $k_{\pi }\pm Q$ are right at the hot spots. The solid lines are full fermionic propagators, the wavy lines represent spin-mediated interaction peaked at $(\pi ,\pi )$.

Figure 3

The integrals $S_1$ and $S_2$ as functions of the angle $\phi$. Both integrals vanish at $\phi =\pi /4$ because at this $\phi$ the Landau damping coefficient diverges.

Figure 4

The one-loop ladder diagram and the two-loop nonladder diagram for the renormalization of the CDW vertex. Both diagrams are logarithmical and, parameter-wise, of the same order, however, the numerical prefactor for the two-loop diagram is much smaller.

Figure 5

The diagrams for the coefficients $A_1$ and $A_2$ in the effective action, Eq. (52).

Figure 6

The scaling function $f(\Lambda /A,C^2/AB,A/B)$ from Eq. (64) is plotted as a function of $\Lambda$ for fixed $A=B=1$, $C=2$. The function has a $\Lambda$-dependent oscillating component, but clearly converges to $f=1$ at large $\Lambda$.

Figure 7

The diagrammatic representation of the prefactors $I_1$ and $I_2$ in Eq. (65).

Figure 8

The two order parameters responsible for stripe or checkerboard order.

Figure 9

The diagrammatic representation of the quartic terms in the effective action.

Figure 10

States I and II in the parameter space of ${\tilde{\beta }}_m$ and ${\overline{\beta }}_m$.

Figure 11

(a) The structure of density and current modulations in state I. The regions of higher and lower fermionic density are shown by darker and lighter color, respectively. The direction of the current is shown by arrows. The current vanishes when the density fluctuation has either the highest or the lowest value. An oscillating current gives rise to an oscillating magnetic field, whose values are shown by dots and crosses. (b) Current loops, formed by connecting oscillating currents in the bulk via the regions of higher local charge density on the surface.

Figure 12

Two possible real-space structures of charge order in state II. (a) A checkerboard charge-density order together with oscillating currents along both $x$ and $y$ directions. (b) A stripe charge-density order together with an oscillating current along orthogonal direction.

Figure 13

The plot of $F\left({\upsilon }^{*}\right)$ from Eq. (120) as a function of ${\upsilon }^{*}$ with ${\beta }^{*}=0.3$.

Figure 14

The plot of $F\left({\upsilon }^{*}\right)$ from Eq. (120) as a function of ${\upsilon }^{*}$ with ${\beta }^{*}=0.55$.

Figure 15

Ladder equation for $\overline{{\rm Y}}={\Delta }_1^Q{\left({\Delta }_2^Q\right)}^{*}$. Of the two terms in the right-hand side, one contains $\overline{{\rm Y}}$, another ${\overline{{\rm Y}}}^{*}$. For imaginary $\overline{{\rm Y}}$, there is a sign change between these two terms.

Figure 16

The behavior of $r_1$ and $r_0$ as a function of ${\alpha }_{+}$.

Figure 17

Phase diagram for state I. (a), (b) The behavior of superconducting $T_{\mathrm{SC}}$ (a) and the onset temperatures for charge order $T_n$, $T_t$, and $T_{\mathrm{CDW}}$ (b), when superconductivity and charge order are treated independent of each other. $T_n$ is the preemptive nematic transition temperature, and $T_t$ is the temperature below which a $q=0$ order emerges, breaking time-reversal symmetry. (c) The full phase diagram, which includes the competition between superconductivity and charge order. ${\mathrm{QCP}}_1$ and ${\mathrm{QCP}}_2$ are quantum-critical points towards SDW and CDW order, respectively.

Figure 18

The theoretical spectral function at $\omega =0$ for a state with strong CDW fluctuations, which we model by introducing CDW orders ${\Delta }_x$ and ${\Delta }_y$, but keeping a finite lifetime of fermions on the FS. The Fermi arcs, terminating at hot spots, are clearly visible.

Figure 19

The position of the peak in spectral function at $\omega =0$ around the hot spot 1 (red line). The saturation of color indicates the spectral weight of the peak. The spectrum to the right of the hot spot 1 is pushed out of BZ boundary by large enough CDW order parameter $\Delta$ used in the plot.

Figure 20

Interpretation of the ARPES data around antinodal areas. (a) Contribution from the domain I with $Q=Q_y$. (b) Contribution from the domain II with $Q=Q_x$. (c) The combined spectral peaks from both domains seen by ARPES.

Figure 21

The ratio of the eigenvalues ${\lambda }_1$ and ${\lambda }_2$ for even and odd in $k$ CDW order parameters ${\Delta }_1^Q$ and ${\Delta }_2^Q$, respectively, as a function as the momentum integration range $\delta$ of integration around the hot spots. Note the change of the behavior at $\delta =2Q$. We set $T=1\mathrm{meV}$.

Figure 22

Fermions which contribute to diagonal bond order and $d$-wave superconducting order. Filled and empty circles denote particle and hole states, respectively.

Figure 23

The diagrammatic expressions for the fully renormalized vertices in superconducting and bond order channels.

Figure 24

The onset temperatures for SC and BO $T_{\mathrm{SC}}$ and $T_{\mathrm{BO}}$, respectively, as functions of dimensionless FS curvature $\tilde{\kappa }$ at the onset of SDW order (magnetic ${\xi }^{-1}=0$). We set $\tilde{g}=0.1$. Superconducting $T_{\mathrm{SC}}$ is not affected by the curvature, while $T_{\mathrm{BO}}$ decreases but remains finite. In analytical consideration, $T_{\mathrm{BO}}/T_{\mathrm{SC}}$ was found to scale as ${\tilde{\kappa }}^{-2}$ at large enough curvature. We show this functional behavior by a dashed line.

Figure 25

$T_{\mathrm{SC}}$ and the onset temperature $T_{\mathrm{BO}}$ for BO with diagonal $\mathbf{Q}$ as functions of the magnetic correlation length. We set $\tilde{\kappa }=0.14$ and $\tilde{g}=0.1$. The red dashed line is ${\tilde{\xi }}_{\mathrm{cr}}$, given by Eq. (F20).

Figure 26

A schematic phase diagram for the (artificial) case when the only two competing states are SC and BO with a diagonal $\mathbf{Q}$. The light blue region is the pseudogap phase, which combines SC fluctuations at large $x$ (smaller $\xi$) and fluctuations between SC and BO at small $x$ (large $\xi$).