Cascade of phase transitions in the vicinity of a quantum critical point

We study the timely issue of charge order checkerboard patterns observed in a variety of cuprate superconductors. We suggest a minimal model in which strong quantum fluctuations in the vicinity of a single antiferromagnetic quantum critical point generate the complexity seen in the phase diagram of cuprates superconductors and, in particular, the evidenced charge order. The Fermi surface is found to fractionalize into hot spots and antinodal regions, where physically different gaps are formed. In the phase diagram, this is reflected by three transition temperatures for the formation of pseudogap, charge density wave, and superconductivity (or quadrupole density wave if a sufficiently strong magnetic field is applied). The charge density wave is characterized by modulations along the bonds of the CuO lattice with wave vectors connecting points of the Fermi surface in the antinodal regions. These features, previously observed experimentally, are so far unique to the quantum critical point in two spatial dimensions and shed a different light on the interplay between strongly fluctuating critical modes and conduction electrons in high-temperature superconductors.

Figure 1

(a) Brillouin zone and Fermi surface. Quantum critical paramagnons single out eight hot spots that we organize in two quartets ($L=1$ and $L=2$). (b) Extended model of hot spot (red) and antinodal states (blue). Nonsingular paramagnons with wave vectors ${\mathbf{K}}_1,...,{\mathbf{K}}_4$ mediate the interaction between hot spot and antinodal states. (c) Cooper pair generation at antinodes $\mathbf{A}$ and $\mathbf{B}$.

Figure 2

Checkerboard charge order for the pseudogap suborder of (a) QDW and (b) antinodal CDW. Modulation vectors ${\mathbf{Q}}_i$ giving the periods ${\mathbf{R}}_i=2\pi {\mathbf{Q}}_i/{\left|{\mathbf{Q}}_i\right|}^2$ are shown in the inset. (c) Qualitative dependence of the superconducting and charge order gaps on the position on the Fermi surface (HS = hot spots, AN = antinodes).

Figure 3

Dimensionless charge density gap ${\overline{b}}_{\mathrm{CDW}}=b_{\mathrm{CDW}}/{\Gamma }_{\mathrm{CDW}}$ as a function of dimensionless temperature $\overline{T}$ interpolated to the frequency $\varepsilon =0$. A CDW order appears below the temperature $T_{\mathrm{CDW}}\approx 0.09{\Gamma }_{\mathrm{CDW}}$.

Figure 4

Qualitative phase diagram summarizing the results of Ref. [27] and the present work for zero magnetic field. Close to the antiferromagnetic (AF) QCP, ${\xi }_{\mathrm{AF}}^{-2}=0$, and upon lowering the temperature, the systems develops first at $T^{*}$ the instability toward the fluctuating pseudogap state (PG) characterized by the order parameter of Eq. (2). At lower temperatures $T<T_{\mathrm{CDW}}<T^{*}$, strong superconducting fluctuations induce a transition toward charge density wave (CDW) formed at the antinodes. Finally, below $T_c$, the particle-particle suborder of the pseudogap prevails due to curvature effects and establishes $d$-wave superconductivity.