Critical phenomena at the complex tensor ordering phase transition

Igor Boettcher and Igor F. Herbut

Phys. Rev. B 97, 064504 – Published 6 February 2018

Abstract

We investigate the critical properties of the phase transition towards complex tensor order that has been proposed to occur in spin-orbit-coupled superconductors. For this purpose, we formulate the bosonic field theory for fluctuations of the complex irreducible second-rank tensor order parameter close to the transition. We then determine the scale dependence of the couplings of the theory by means of the perturbative renormalization group (RG). For the isotropic system, we generically detect a fluctuation-induced first-order phase transition. The initial values for the running couplings are determined by the underlying microscopic model for the tensorial order. As an example, we study three-dimensional Luttinger semimetals with electrons at a quadratic band-touching point. Whereas the strong-coupling transition of the model receives substantial fluctuation corrections, the weak-coupling transition at low temperatures is rendered only weakly first order due to the presence of a fixed point in the vicinity of the RG trajectory. If the number of fluctuating complex components of the order parameter is reduced by cubic anisotropy, the theory maps onto the field theory for frustrated magnetism.

Received 15 December 2017

DOI:https://doi.org/10.1103/PhysRevB.97.064504

Physics Subject Headings (PhySH)

Critical phenomena Superconducting fluctuations Superconducting phase transition

Unconventional superconductors

Renormalization group

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsParticles & Fields

Authors & Affiliations

Igor Boettcher and Igor F. Herbut

Department of Physics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada

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Vol. 97, Iss. 6 — 1 February 2018

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Figure 1
Flow of the quartic couplings according to Eqs. (32, 33, 34) in the plane spanned by $λ_{3} = 0$ . The flow equations in the plane $λ_{3} = 0$ correspond to those of the $N$ -component model discussed in Sec. 3c for $N = 5$ . We parametrize $λ_{1, 2}$ in terms of $u = 6 (λ_{1} + λ_{2})$ and $v = 12 λ_{2}$ . This parametrization is useful because (i) the coupling $v$ cannot change sign and (ii) the quartic potential is stable if and only if $u > 0$ . The arrows of stream lines point towards the infrared, so that $b$ increases along the flow. We observe a runaway flow towards $u < 0$ for all generic initial values of $λ_{1, 2}$ . This signals that fluctuations induce a first-order phase transition. Trajectories in the vicinity of the interacting fixed point $(u_{★}, v_{★}) = (\frac{ɛ}{3}, 0)$ are shown in red.
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Figure 2
Fixed-point structure of the three-dimensional theory with action $S$ as a function of $K$ . The plot shows the couplings $λ_{1 ★} / ɛ$ (blue, continuous line), $100 λ_{2 ★} / ɛ$ (red, short-dashed line), and $100 λ_{3 ★} / ɛ$ (orange, long-dashed line). For better visibility, we magnify $λ_{2 ★}$ and $λ_{3 ★}$ by a factor of 100. To determine the couplings, we use the full $K$ -dependent expressions (D29, D30, D31), which yield Eqs. (38)–(40) for small $K$ . The eigenvalues of the stability matrix at the fixed point are $θ_{1} = - ɛ$ and positive $θ_{2, 3} = O (ɛ)$ for all values of $K$ .
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Figure 3
Strong-coupling transition in isotropic Luttinger semimetals. Upper panel: the solid lines, from top to bottom, show the scale evolution of the running couplings $λ_{1} (b)$ (blue line), $λ_{3} (b)$ (magnified by a factor of 10, orange line), $λ_{2} (b)$ (red line). They rapidly enter the region of $(-, -, +)$ in Table 1 and the stability criterion is violated because $λ_{1} + λ_{2} + \frac{4}{3} λ_{3} < 0$ for $b > b_{0} \sim 1$ , indicating a fluctuation-induced first-order transition. The dashed lines show the result of integrating the flow for $K = 0$ (same color scheme), which constitutes a reasonable approximation. Lower panel: the solid line represents the evolution of the couplings $u = 6 (λ_{1} + λ_{2})$ and $v = 12 λ_{2}$ projected onto the plane with $λ_{3} = 0$ (see Fig. 1 for comparison). Arrows point towards the infrared. The dashed line again constitutes the flow when setting $K = 0$ . In the latter case, the instability is indicated by $u < 0$ . In the plots we chose $T / Λ^{2} = 0.3$ as the sole input parameter and set $ɛ = 1$ .
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Figure 4
Weak-coupling transition in isotropic Luttinger semimetals. The color and line scheme is chosen as in Fig. 3 for the strong-coupling transition. Therefore, we focus on pointing out the differences here. Upper panel: the flow enters the regime of $(-, +, +)$ in Table 1 after a relatively long RG time $b$ . An instability is signaled by $λ_{1} + \frac{4}{3} λ_{3} < 0$ for $b > b_{0} \sim 10$ . Due to the large value of $b_{0}$ , the fluctuation-induced first-order transition is only weak. Lower panel: the slowing down of the flow before entering the region of instability can be explained by the flow trajectory being close to the interacting fixed point (red mark). This proximity, in contrast to the flow in Fig. 3, is induced by the small initial couplings. The dashed line representing the flow for $K = 0$ deviates more significantly since the interacting fixed point for $K = 0$ is further to the right in the plot (see Fig. 2). In the plots we choose $μ / Λ^{2} = 0.4$ and $T / μ = 0.005$ , which are values motivated from half-Heusler superconductors [18], and set $ɛ = 1$ .
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Figure 5
Loop integrations that yield the mean-field expressions for the coefficients of the free energy in Eqs. (E1, E2, E3, E4, E5, E6, E7, E8). Here, a straight line represents a fermion propagator and a wiggly line denotes an insertion of $Δ_{a}$ or $Δ_{a}^{*}$ with vertices $γ_{45} γ_{a}$ [18]. The contribution depicted in (a) results in the terms $r, a_{1, 2}$ and $X, Y, Z$ that are quadratic in the boson field, whereas diagram (b) generates the quartic contributions $q_{1, 2}$ .
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