Electronic nematic phase transition in the presence of anisotropy

We study the phase diagram of electronic nematic instability in the presence of $xy$ anisotropy. While a second-order transition cannot occur in this case, mean-field theory predicts that a first-order transition occurs near Van Hove filling and its phase boundary forms a wing structure, which we term a Griffiths wing, referring to his original work of ${\mathrm{He}}^3-{\mathrm{He}}^4$ mixtures. When crossing the wing, the anisotropy of the electronic system exhibits a discontinuous change, leading to a metanematic transition, i.e., the analog to a metamagnetic transition in a magnetic system. The upper edge of the wing corresponds to a critical end line. It shows a nonmonotonic temperature dependence as a function of the external anisotropy and vanishes at a quantum critical end point for a strong anisotropy. The mean-field phase diagram is found to be very sensitive to fluctuations of the nematic order parameter, yielding a topologically different phase diagram. The Griffiths wing is broken into two pieces. A tiny wing appears close to zero anisotropy and the other is realized for a strong anisotropy. Consequently three quantum critical end points are realized. We discuss that these results can be related to various materials including a cold atom system.

Figure 1

Mean-field results. (a) Schematic phase diagram in the space spanned by $\mu , {\mu }_d$, and $T$. At ${\mu }_d=0$ the $d\mathrm{PI}$ occurs around Van Hove filling, from which $\mu$ is measured. The transition is of second order at high $T$ (solid line) and of first order at low $T$ (double line). The solid circle denotes the TCP. The BI state is realized in the striped region. The wing (colored in orange) stands almost vertically on the plane of $\mu$ and ${\mu }_d$ close to Van Hove filling and vanishes at the ${\mathrm{QCEP}}_1$; the index 1 implies that the system is almost one dimensional. The upper edge of the wing (solid line) is a CEL. (b) Temperature of the CEL ($T_{\mathrm{CEL}}$) as a function of ${\mu }_d$; The wing is projected on the plane of ${\mu }_d$ and $T$. The electron density at $T_{\mathrm{CEL}}$ is also plotted. (c) Jump of the nematic order parameter across the wing at $T=0.001$.

Figure 2

Results in the presence of weak nematic order-parameter fluctuations. (a) Schematic phase diagram. The $d\mathrm{PI}$ phase is slightly suppressed by fluctuations. The wing obtained in Fig. 1 is broken into two pieces: one tiny wing close to ${\mu }_d=0$ and the other wing for a large ${\mu }_d$. The indices of ${\mathrm{QCEP}}_2$ and ${\mathrm{QCEP}}_{1^{\prime }}$ imply that the system can be regarded to be two- and quasi-one-dimensional, respectively, at the QCEP. The dashed line corresponds to a crossover. The BI phase is assumed to be the same as the mean-field result. (b),(c) The wings are projected on the plane of ${\mu }_d$ and $T$ for several choices of the cutoff ${\Lambda }_0$. (d) The critical anisotropy of the hopping integral to obtain the ${\mathrm{QCEP}}_2$ as a function of the ratio of the tricritical temperature and its mean-field value.