Cascade of vestigial orders in two-component superconductors: Nematic, ferromagnetic, $s$-wave charge-$4e$, and $d$-wave charge-$4e$ states

Cascade of vestigial orders in two-component superconductors: Nematic, ferromagnetic, $s$ -wave charge- $4 e$ , and $d$ -wave charge- $4 e$ states

Matthias Hecker, Roland Willa, Jörg Schmalian, and Rafael M. Fernandes

Phys. Rev. B 107, 224503 – Published 5 June 2023

Abstract

Electronically ordered states that break multiple symmetries can melt in multiple stages, similarly to liquid crystals. In a partially melted phase, known as vestigial phase, a bilinear made out of combinations of the multiple components of the primary order parameter condenses. Multicomponent superconductors are thus natural candidates for vestigial order since they break both the $U (1)$ -gauge and also time-reversal or lattice symmetries. Here, we use group theory to classify all possible real-valued and complex-valued bilinears of a generic two-component superconductor on a tetragonal or hexagonal lattice. While the more widely investigated real-valued bilinears correspond to vestigial nematic or ferromagnetic order, the little explored complex-valued bilinears correspond to a vestigial charge- $4 e$ condensate, which itself can have an underlying $s$ -wave, $d_{x^{2} - y^{2}}$ -wave, or $d_{x y}$ -wave symmetry. To properly describe the fluctuating regime of the superconducting Ginzburg-Landau action and thus access these competing vestigial phases, we employ both a large- $N$ and a variational method. We show that while vestigial order can be understood as a weak-coupling effect in the large- $N$ approach, it is akin to a moderate-coupling effect in the variational method. Despite these distinctions, both methods yield similar results in wide regions of the parameter space spanned by the quartic Landau coefficients. Specifically, we find that the nematic and ferromagnetic phases are the leading vestigial instabilities, whereas the various types of charge- $4 e$ order are attractive albeit subleading vestigial channels. The only exception is for the hexagonal case, in which the nematic and $s$ -wave charge- $4 e$ vestigial states are degenerate. We discuss the limitations of our approach, as well as the implications of our results for the realization of exotic charge- $4 e$ states in material candidates.

Received 2 March 2023
Revised 2 May 2023
Accepted 3 May 2023

DOI:https://doi.org/10.1103/PhysRevB.107.224503

Physics Subject Headings (PhySH)

Multiband superconductivity Nematic order Order parameters Phase transitions Superconducting fluctuations Superconducting order parameter Superconducting phase transition

Unconventional superconductors

Landau-Ginzburg theory Large-N expansion in field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Matthias Hecker ¹, Roland Willa ^2,3, Jörg Schmalian^2,4, and Rafael M. Fernandes¹

¹School of Physics and Astronomy, University of Minnesota, Minneapolis 55455 Minnesota, USA
²Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany
³Institute of Systems Engineering, School of Engineering, HES-SO Valais-Wallis, Sion, Switzerland
⁴Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany

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Vol. 107, Iss. 22 — 1 June 2023

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Images

Figure 1
Mean-field phase diagram, in the parameter space spanned by the quartic Landau coefficients of Eq. (12), of the two-component superconducting pairing states of the (a) tetragonal case $D_{4 h}$ (1) and (b) hexagonal case $D_{6 h}$ (2). In each phase, we show the relationship between the two components of the superconducting order parameter, $Δ = (Δ_{1}, Δ_{2})$ , as well as all nonzero bilinears defined in Eqs. (7) and (8). Each phase is labeled by the additional symmetry that they break aside from the $U (1)$ gauge symmetry, which could be time-reversal symmetry (TRS) or a point-group symmetry. In the regions labeled “unstable” (dark gray), the action is unbounded.
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Figure 2
Large- $N$ phase diagrams for the leading and subleading vestigial-order instabilities associated with a primary two-component superconducting phase in a system with tetragonal $D_{4 h}$ symmetry (a), (b) and hexagonal $D_{6 h}$ symmetry (c), (d). As in the mean-field phase diagram of Fig. 1, the parameter space is that spanned by the quartic Landau coefficients of Eq. (12). The vestigial charge- $4 e$ instabilities are always subleading with respect to either the vestigial nematic or the vestigial ferromagnetic instability, except in the region $v > 0$ of the hexagonal phase diagram, where the vestigial $s$ -wave charge- $4 e$ phase and the vestigial nematic phase are degenerate.
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Figure 3
Variational phase diagrams for the leading and subleading vestigial-order instabilities associated with a primary two-component superconducting phase in a system with tetragonal $D_{4 h}$ symmetry (a), (b) and hexagonal $D_{6 h}$ symmetry (c), (d). The parameter space is that spanned by the quartic Landau coefficients of Eq. (12). The only difference with respect to the large- $N$ phase diagrams of Fig. 2 is the existence of regions where there is no vestigial instability (white region) or no subleading vestigial instability (light gray region). Outside of these regions, the variational and large- $N$ results agree with each other.
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Figure 4
Variational phase diagrams of Fig. 3 for the tetragonal $D_{4 h}$ case with the phase boundaries $| U_{n} | / U_{A_{1 g}} = 3 - d$ (dotted line) and $| U_{n} | / U_{A_{1 g}} = (6 - 2 d) / (6 - d)$ (dashed line) that determine the three different regimes for the coupled vestigial and superconducing transitions (see main text): two split second-order transitions (labeled type-I split); two split transitions with one of them first order (labeled type-II split); one simultaneous first-order transition. For concreteness, here we set $d = 2.4$ .
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