Theory of the nodal nematic quantum phase transition in superconductors

We study the character of an Ising nematic quantum phase transition deep inside a $d$-wave superconducting state with nodal quasiparticles in a two-dimensional tetragonal crystal. We find that, within a $1∕N$ expansion, the transition is continuous. To leading order in $1∕N$, quantum fluctuations enhance the dispersion anisotropy of the nodal excitations and cause strong scattering, which critically broadens the quasiparticle (qp) peaks in the spectral function, except in a narrow wedge in momentum space near the Fermi surface where the qps remain sharp. We also consider the possible existence of a nematic glass phase in the presence of weak disorder. Some possible implications for cuprate physics are also discussed.

Figure 1

(Color online) Quantum critical point at $\lambda ={\lambda }_c$, separating the nodal nematic phase for ${\lambda }^{-1}<{\lambda }_c^{-1}$ from the symmetric phase, and its quantum critical fan. $T_c$ and $T_n$ are the superconducting and nematic critical temperatures and the (purple) wedge corresponds to the thermal critical regime.

Figure 2

(Color online) The nematic mode spectral function $\tilde{B}(\omega ,\vec{p})$ plotted as a function of energy $\omega ∕\Lambda$ at momentum $\vec{p}=(0.05\Lambda ∕v_F,0.15\Lambda ∕v_F)$, where $\Lambda$ is a UV cutoff and $v_F$ is the Fermi velocity. (a) Isotropic case $v_F=v_{\Delta }=v$. (b) Anisotropic case $v_F∕v_{\Delta }=19.5$. Notice how the singular feature at $\omega =v\sqrt{(p_x^2+p_y^2)}=0.16\Lambda$ for an isotropic dispersion splits into two features at $\omega =E_1\left(\vec{p}\right)\sim 0.05$ and $\omega =E_2\left(\vec{p}\right)\sim 0.15$ for an anisotropic dispersion. See Eq. (3.9).

Figure 3

(Color online) Momentum distribution of the nodal nematic QCP spectral function at an energy $\omega =-9\mathrm{meV}$ for $v_F=0.508\mathrm{eV}{(\pi ∕a)}^{-1}$, $v_{\Delta }=0.026\mathrm{eV}{(\pi ∕a)}^{-1}$; $v_F∕v_{\Delta }=19.5$. Here, the momentum $\vec{p}$ is measured with respect to nodal point ${\vec{K}}_1=(K,K)$ in a (rotated) local coordinate system, so that $p_x$ and $p_y$ lie, respectively, along the nodal and tangential directions. (a) Contour plot. ${\theta }_c$ is the critical angle at which a well defined qp peak departs from the incoherent continuum. For $\theta <{\theta }_c$, the qp is well defined. (b) and (c) are line cuts along the nodal direction and tangential direction, respectively.

Figure 4

(Color online) EDCs taken at two points in momentum space. (a) At $\vec{p}=(-9\mathrm{meV}∕v_F,0)$. (b) At $\vec{p}=(0,9\mathrm{meV}∕v_{\Delta })$. These are the momenta corresponding to the position of the peak in Fig. 3b and to the threshold in Fig. 3c, respectively.