Symmetry-mixed bound-state order

Near the accidental degeneracy of two symmetry-distinct orders, i.e., where both transition temperatures almost coincide, phase interactions significantly reshape the ordering behavior. Strong attraction between the parent phases are known to induce a joint first-order transition. For weak interactions, I report a second-order transition of both primary orders, preceded by a fluctuation-driven state of matter. Due to symmetry mixing, this antecedent bound-state order exhibits unique signatures, incompatible with either parent phase. Within a field-theoretical formalism, I derive the generic phase diagram for a system with bound-state order, study its response to strain, and evaluate analytic expressions for a specific model. As the realization of a perfect degeneracy between symmetry-distinct orders is accidental by nature, interpretations of that kind—such as $(d+ig)$-superconducting state as a candidate for ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$—have inarguably a weak point. The results presented here raise the plausibility for realizing such a situation by widening the degeneracy point to an extended line. Finally, I discuss how a trainable Kerr effect under strain provides strong evidence for a bound-state phase of symmetry-distinct orders.

Figure 1

Phase diagram of two phases $\langle {\eta }_1\rangle$ (blue) and $\langle {\eta }_2\rangle$ (red) near their degeneracy point $x=0$ for repulsive (a) or attractive (b) phase interaction. Dotted lines indicate the phase boundary of the subordinate order in the absence of interaction. Including fluctuations (c) the symmetry-mixed bound-state order $\mu =\langle -i({\eta }_1^{*}{\eta }_2-{\eta }_1{\eta }_2^{*})/2\rangle$ emerges in the vicinity of the phase intersection (yellow). This stand-alone phase acquires a finite expectation without the appearance of the phases $\langle {\eta }_j\rangle$. For sufficiently strong phase attraction the bound-state order gives way for a first-order transition into a fully ordered state (inset).

Figure 2

Susceptibilities ${\chi }_1\left(T\right)$ (blue), ${\chi }_2\left(T\right)$ (red), and ${\chi }_{\mu }\left(T\right)$ (yellow) upon approaching the ordered phase. Far from the phase degeneracy (a), the order $\langle {\eta }_1\rangle$ sets in first, followed by the instability in the two remaining sectors. Near the degeneracy (b), the symmetry-mixed bound state $\mu$ appears (yellow) while the parent orders remain zero. A renormalization of the susceptibilities induces a simultaneous divergence of ${\chi }_1$ and ${\chi }_2$.

Figure 3

Left: In the phase space of two symmetry-distinct order parameters ${\eta }_j$ in the vicinity of the degeneracy point $x=0$ and with an attractive interaction $g$, this color map indicates which order appears first upon cooling: Far from the degeneracy $\left|x\right|>x^{*}\left(g\right)$, a primary order (either red or blue) appears through a second-order transition. For large $g/u$ the system undergoes a first-order transition (magenta) to a fully ordered state, where all orders become finite at once. For attractive interactions $g<u/2$ the symmetry-mixed bound-state $\mu =\langle -i({\eta }_1^{*}{\eta }_2-{\eta }_1{\eta }_2^{*})/2\rangle$ acquires a finite expectation value, without the appearance of the primary phases (yellow). The right panels show the $Tx$-phase diagram for two specific interaction strengths $g/u=0.4$ and 0.1, respectively.

Figure 4

Qualitative dependence of the transition temperatures for a $({\eta }_1+i{\eta }_2)$ superconductor to an external $B_{1g}$ strain with (left) and without (right) magnetic field $H$. In the first case, the bound-state order $\mu$ is induced by $h_{\mu }=\epsilon H$ with $\epsilon$ the (symmetry-)appropriate strain. This coupling imposes a joint ordering of the components ${\eta }_j$. Without $H$, the system first develops a bound state preserving the $U\left(1\right)$ gauge symmetry, followed by a full $({\eta }_1+i{\eta }_2)$-superconducting state. In this sequence the strain dependence is indirect, through $r_1\left(\epsilon \right)$ in Eq. (8). Above critical strain the superconducting (blue) and the time-reversal-symmetry breaking (red) transitions split. The superconducting transition temperature now directly depends on $r_1\left(\epsilon \right)$ via Eq. (6).