Abstract
The collective mode spectrum of a symmetrybreaking state, such as a superconductor, provides crucial insight into the nature of the order parameter. In this work, we study two collective modes which are unique to unconventional superconductors that spontaneously break time reversal symmetry. We show that these modes are coherent and underdamped for a wide variety of timereversal symmetry breaking superconducting states. By further demonstrating that these modes can be detected using a number of existing experimental techniques, we propose that our work can be leveraged as a form of “collective mode spectroscopy” that drastically expands the number of experimental probes capable of detecting timereversal symmetry breaking in unconventional superconductors.
Introduction
There is a rich and constantly expanding taxonomy of unconventional superconducting phases, and increasingly sophisticated probes are needed to distinguish one such phase from another. A defining feature of any superconducting state is its collective mode spectrum, which encodes the dynamics of the order parameter. The collective modes of conventional (phononmediated) superconductors are wellestablished, consisting of two modes which correspond to fluctuations in the amplitude and phase of the order parameter. The first, called the Higgs mode^{1,2}, is massive and resides at the edge of the quasiparticle continuum^{3,4}, while the second, called the Anderson–Bogoliubov–Goldstone (ABG) mode^{5,6}, is massless in a neutral system, in accordance with Goldstone’s theorem, but is lifted to the plasma frequency in the presence of longranged Coulomb interactions^{7}, and is thus indistinguishable from the usual plasmon in real materials.
However, systems with more complex order parameters can exhibit a rich collective mode spectrum featuring other modes, such as additional Higgs modes in anisotropically gapped (e.g., dwave) superconductors^{8}, or Leggett modes in multiband superconductors^{9}. The presence of these additional modes in the spectrum can then be taken as a fingerprint of the underlying order parameter symmetry. That is, the dynamics of the order parameter can be studied to gain insight into its equilibrium structure. Such a scheme has only recently been proposed in the context of anisotropically gapped superconductors, where the spectrum of nonequilibrium Higgs modes can be used to deduce the orbital symmetry of the order parameter^{10,11}.
In this article, we generalize this notion of “collective mode spectroscopy” to a particularly exotic class of unconventional superconductors, namely those which spontaneously break timereversal symmetry in addition to global U(1) symmetry at the superconducting transition. These timereversal symmetry breaking (TRSB) superconducting states are the subject of considerable current interest, and are believed to be realized in a number of bulk materials, including Sr_{2}RuO_{4}^{12,13}, UPt_{3}^{14,15}, URu_{2}Si_{2}^{16}, UTe_{2}^{17,18}, PrOs_{4}Sb_{12}^{19,20}, and Kdoped BaFe_{2}As_{2}^{21}, as well as engineered structures such as Bi/Ni bilayers^{22} and SnTe nanowires^{23}. Moreover, recent theoretical proposals have suggested that such states may also be realized in moiré heterostructures^{24,25}, and could be generically engineered in twisted bilayers of anisotropically gapped superconductors^{26,27}.
In what follows, we identify two collective modes, the “generalized clapping modes,” which are unique to TRSB superconductors and subsequently derive their spectrum from a generally applicable microscopic weakcoupling theory. Our first key finding is that these modes are coherent collective excitations for a wide variety of TRSB states, even those which have point or line nodes in the superconducting gap. Owing to the universality of these modes in TRSB superconductors, they may serve as clear spectroscopic signatures which identify a TRSB superconducting state. To this end, we discuss a variety of existing experimental probes that can couple to these modes, and hence can be used as means to detect TRSB superconductivity in quantum materials. We pay special attention to the superconducting state of Sr_{2}RuO_{4}, where the generalized clapping mode spectrum could distinguish the two current leading candidate order parameters.
Results
Generalized clapping modes
TRSB superconducting states are characterized by a doublydegenerate complex multicomponent order parameter of the form Δ = Δ_{1} ± iΔ_{2} (see supplementary Note 1), and can be divided into two classes: (1) systems where both components Δ_{1} and Δ_{2} belong to the same multidimensional irreducible representation (irrep) of the crystalline point group, in which case ∣Δ_{1}∣ = ∣Δ_{2}∣ is required by symmetry, and (2) “mixed symmetry” systems where the two components belong to different irrep’s, which can arise due to either an accidental degeneracy between two pairing channels or two successive superconducting transitions (see Supplementary Note 1), in which case the ratio ∣Δ_{1}∣/∣Δ_{2}∣ is unconstrained.
Given this internal orbital structure of the order parameter, we would expect that in addition to the usual ABG and Higgs modes, there should be a massive mode corresponding to fluctuations of the relative phase between the two order parameter components around its equilibrium value of ±π/2. Further, we expect that there should be a second amplitude mode which corresponds to fluctuations of the relative amplitude ∣Δ_{1}∣/∣Δ_{2}∣ so there are a total of four real modes.
Historically, similar modes corresponding to “internal vibrations of the structure of the order parameter” were first recognized in ^{3}HeA^{28}, one of which was named the “clapping mode.” This was subsequently extended to the twodimensional chiral pwave (p + ip) superconductor^{29}, where there are two such clapping modes which are degenerate and reside at a frequency \({{\Omega }}=\sqrt{2}{{{\Delta }}}_{0}\), with Δ_{0} the magnitude of the order parameter.
As argued above, we can anticipate analogous modes, which we call the generalized clapping modes, for any multicomponent TRSB state on symmetry grounds alone^{30}. Although it is intuitively obvious that the generalized modes should exist in principle, there is no reason to expect a priori that they are not overdamped by quasiparticle excitations. For a fully gapped superconductor, this requires that the frequencies of both modes lie below the quasiparticle continuum or, in the case of a nodal quasiparticle gap function, requires that the spectral function of each mode retains sharp features despite the presence of nodal quasiparticles. To establish that the generalized clapping modes are coherent, and thus experimentally detectable, for a generic TRSB state requires a derivation of their spectrum starting from a microscopic theory, which we furnish below. By studying the generalized clapping mode spectrum for a wide variety of TRSB order parameters we are able to establish the general features of these modes, which ultimately enable their use in the collective mode spectroscopy of real materials.
Weakcoupling theory
We begin with a single band of fermions subject to the attractive interaction \({V}_{{{{{{{\rm{kk}}}}}}}^{\prime}}={\sum }_{\ell = 1,2}{g}_{\ell }{\chi }_{{{{{{\rm{k}}}}}}}^{\ell }{\chi }_{{{{{{{\rm{k}}}}}}}^{\prime}}^{\ell }\) where g_{ℓ} > 0 are coupling constants and \({\chi }_{{{{{{\rm{k}}}}}}}^{\ell }\) are form factors which encode the orbital symmetry of the interaction. We take these to be real and normalized according to the inner product \(\int \frac{{{{{{\rm{d}}}}}}{\phi }_{{{{{{\rm{k}}}}}}}}{2\pi }\,{\chi }_{{{{{{\rm{k}}}}}}}^{\ell }{\chi }_{{{{{{\rm{k}}}}}}}^{\ell ^{\prime} }={\delta }_{\ell \ell ^{\prime} }\). We assume pairing in the \({S}_{{{{{{\rm{tot}}}}}}}^{z}=0\) sector, but within this sector our results are applicable to both singlet and m = 0 triplet pairing. We treat this system within the imaginarytime pathintegral formalism by introducing a HubbardStratonovich decoupling field Δ^{ℓ} in each pairing channel and integrating out the fermions. One then arrives at the effective action for the order parameters Δ^{ℓ} of
The inverse fermion propagator is \({{\mathbb{G}}}_{k+q,k}^{1}=(i{\omega }_{n}{\xi }_{{{{{{\bf{k}}}}}}}\,{\tau }_{z}){\delta }_{q,0}+{\sum }_{\ell }{{{\Delta }}}_{q}^{\ell }{\chi }_{{{{{{\bf{k}}}}}}}^{\ell }\,{\tau }^{+}+{\sum }_{\ell }{\bar{{{\Delta }}}}_{q}^{\ell }{\chi }_{{{{{{\bf{k}}}}}}}^{\ell }\,{\tau }^{}\) where τ_{i} are the Pauli matrices in Nambu space, \({\tau }^{\pm }=\frac{1}{2}\left({\tau }_{x}\pm i{\tau }_{y}\right)\), and ξ_{k} = k^{2}/2m − μ is the singleparticle energy measured with respect to the Fermi level. We have also combined fermionic/bosonic Matsubara frequencies and momenta into the fourvectors k = (iω_{n}, k) and q = (iΩ_{m}, q), where q corresponds to the centerofmass momentum of the fermion pair and k corresponds to the relative momentum.
We assume g_{1}, g_{2} are such that we find a saddlepoint configuration in which both Δ^{(1)} and Δ^{(2)} are condensed with a relative phase of π/2, breaking timereversal symmetry as discussed above. It will be convenient to change basis from (Δ^{(1)}, Δ^{(2)}) → (Δ^{+}, Δ^{−}) according to \({{{\Delta }}}_{q}^{(1)}{\chi }_{{{{{{\bf{k}}}}}}}^{(1)}+{{{\Delta }}}_{q}^{(2)}{\chi }_{{{{{{\bf{k}}}}}}}^{(2)}={{{\Delta }}}_{q}^{+}{\chi }_{{{{{{\bf{k}}}}}}}^{+}+{{{\Delta }}}_{q}^{}{\chi }_{{{{{{\bf{k}}}}}}}^{}\) where the ± form factors are defined as
and η_{1,2} quantify the relative magnitude of each order parameter component. We choose to normalize them such that \({\eta }_{1}^{2}+{\eta }_{2}^{2}=1\) and can express both in terms of a “mixing angle” as \({\eta }_{1}=\cos \eta \) and \({\eta }_{2}=\sin \eta \).
Expanding around the saddle point with \({{{\Delta }}}_{q}^{+}={{{\Delta }}}_{0}=\,{{\mbox{const.}}}\,\) and \({{{\Delta }}}_{q}^{}=0\), the meanfield equations are
where \({E}_{{{{{{\bf{k}}}}}}}^{2}={\xi }_{{{{{{\bf{k}}}}}}}^{2}+ {{{\Delta }}}_{{{{{{\bf{k}}}}}}}{ }^{2}\) and the angledependent quasiparticle gap function is \({{{\Delta }}}_{{{{{{\bf{k}}}}}}}={{{\Delta }}}_{0}{\chi }_{{{{{{\bf{k}}}}}}}^{+}\). Given a particular set of pairing symmetries and coupling constants, these equations can be solved to determine the equilibrium values of η and Δ_{0} which characterize the condensate.
Effective action for fluctuations
Now, we move on to consider the fluctuations around this saddle point, which we parameterize as
where θ is the ABG phase mode, h is the Higgs mode, and the a and b modes are fluctuations in the relative amplitude and phase of the two order parameter components, i.e., the generalized clapping modes (see Fig. 1a). This parameterization suggests that we can equivalently think of the generalized clapping modes as being fluctuations in the degenerate timereversed Δ^{−} pairing channel, which are formally similar to BardasisSchreiffer modes^{31,32}, as illustrated in Fig. 1b.
To organize our calculations in a manifestly gaugeinvariant way, we minimally couple the system to an external (classical) gauge field and perform a unitary transformation \({{\mathbb{G}}}^{1}\to U{{\mathbb{G}}}^{1}{U}^{{{\dagger}} }\) with \(U={{{{{{\rm{e}}}}}}}^{i\theta {\tau }_{z}}\) such that the ABG mode and gauge field only appear together as the gaugeinvariant vector field V^{0} = A^{0} + ∂_{τ}θ and V = A − ∂θ (where we have set the electron charge equal to one). To quadratic order in these fields and setting q → 0, the action is
In the above, Π^{00} is the electronic compressibility, \({n}_{s}^{ij}\) is the superfluid density, \({{{{{{\mathcal{D}}}}}}}_{h,a,b}\) are the propagators for the Higgs, relative amplitude, and relative phase modes, and \({\tilde{{{\Pi }}}}^{ha}\) and Π^{0b} are linear couplings between the Higgs/relative amplitude and ABG/relative phase modes which are nonvanishing in the q → 0 limit. This action is derived and the correlation functions which appear in it are evaluated in Supplementary Note 2.
To gain intuition, we begin by considering the familiar case of p + ip pairing in two dimensions. When the p_{x} and p_{y} components of the order parameter occur with equal amplitudes (η = π/4), as is dictated by symmetry in most cases of physical interest, we find that after analytically continuing to realtime the propagators for both the a and b modes have a pole at \({{\Omega }}=\sqrt{2}{{{\Delta }}}_{0}\), i.e., the two modes are degenerate. So, we see that the usual clapping modes previously studied in p + ip superconductors and ^{3}HeA are indeed a special case of the generalized clapping modes a and b studied in this work. Moreover, at this point the gap is isotropic and the couplings \({\tilde{{{\Pi }}}}^{ha}\) and Π^{0b} vanish so that the (generalized) clapping modes decouple from both quasiparticle excitations and other collective modes and thus are infinitely longlived at zero temperature.
In real materials, anisotropic crystal fields can lead to deviations from the equalamplitude η = π/4 state^{33}, making it interesting to consider the collective mode spectrum of the p + ip state for general mixing angles. We plot the spectral functions of the a and b modes in Fig. 2a, b as a function of the mixing angle η and frequency Ω. We see that the two modes split as the mixing angle deviates from π/4, so that for generic mixing angles there are two generalized clapping modes corresponding to the relative amplitude and phase fluctuations.
Twodimensional systems
We will now investigate the generalized clapping mode spectrum for several even parity twocomponent TRSB order parameters which are potentially relevant to experimental systems: the \({d}_{{x}^{2}{y}^{2}}+i{d}_{xy}\) state, originally studied in the context of the cuprate hightemperature superconductors^{34,35}, and now the subject of renewed interest due to its potential relevance to a number of moiré systems^{24,25,36}, other heterostructures^{37}, and its proposed realization in twisted bilayers of cuprates^{26} and other unconventional superconductors^{27}; the s + id_{xy} state, which has long been of interest in relation to the iron pnictide hightemperature superconductors^{38,39,40}; and the \({d}_{{x}^{2}{y}^{2}}+i{g}_{xy({x}^{2}{y}^{2})}\) state which has recently been proposed as the order parameter of Sr_{2}RuO_{4}^{41,42}. The basic properties of each order parameter studied in this work are listed in Table 1.
Because these are all mixed symmetry states where the mixing angle is unconstrained by point group symmetries, it is important to survey the generalized clapping mode spectra over the full range of η. We first turn our attention to the relative phase mode, the spectra of which we plot for each of the above order parameters in Fig. 3a–f. Crucially, we observe that the relative phase mode is a welldefined, coherent mode for all values of the mixing angle, with a sharp spectral function that can be seen in the line cuts in Fig. 3d–f. This even remains true for nodal order parameters, e.g., the d + ig state.
Next, we turn our attention to the relative amplitude mode. As we show in Supplementary Note 2, this mode is coupled to the Higgs mode even at zero momentum. The relative amplitude and Higgs modes then hybridize to form two orthogonal amplitude modes, which we call A_{+} and A_{−}. We calculate the propagators D_{A±}(Ω) for these modes in Supplementary Note 2, which we use to plot the spectral functions of each mode for various pairing symmetries, as shown in Fig. 4a–f.
For all of the pairing symmetries studied, the A_{−} mode resides at or slightly below the gap edge, much like the conventional Higgs mode in a singlecomponent superconductor. More interesting is the A_{+} mode, which lies well below the quasiparticle continuum for a wide range of mixing angles. This lowfrequency amplitude mode represents a second novel collective excitation characteristic of TRSB superconducting states.
We also note that at η = π/4, the \(d+id^{\prime} \) state is fully gapped and chiral, and both generalized clapping modes are degenerate with a frequency of \({{\Omega }}=\sqrt{2}{{{\Delta }}}_{0}\), much like the chiral pwave state in Fig. 2^{43}. As we will show below, this degeneracy is a general feature of chiral order parameters.
Threedimensional systems
We now consider several TRSB superconducting states in threedimensional systems: the 3d p_{x} + ip_{y} state (i.e., a p_{x} + ip_{y} order parameter defined over a spherical Fermi surface), better known as the Anderson–Brinkman–Morel (ABM) state of ^{3}HeA^{44}, which is also a minimal model for a Weyl superconductor, and thus may bear some qualitative similarities to the superconducting state of UTe_{2}^{18} and other candidate Weyl systems; the threedimensional \({d}_{{x}^{2}{y}^{2}}+i{d}_{xy}\) “double Weyl” superconducting state possibly realized in the “hidden order” phase of URu_{2}Si_{2}^{45,46} or SrPtAs^{47,48}; and finally the d_{xz} + id_{yz} state which is a candidate order parameter for Sr_{2}RuO_{4} as well as the hidden order phase of URu_{2}Si_{2}^{49,50}.
All of these states are most naturally considered as equaladmixtures, i.e., with η = π/4, in which case the generalized clapping modes decouple from the Higgs mode. As seen in Fig. 5a–d, the generalized clapping modes are degenerate with one another in all cases. Unlike the twodimensional case, however, the mode frequency is not \(\sqrt{2}{{{\Delta }}}_{0}\). By inspection of the analytic form of the generalized clapping mode propagators (see Supplementary Note 2), it is evident that the mode frequency is only \(\sqrt{2}{{{\Delta }}}_{0}\) if the order parameter is both chiral and isotropically gapped, whereas the ABM and 3d \(d+id^{\prime} \) (d + id) states exhibit point (line) nodes. We also note that this demonstrates that the generalized clapping modes remain coherent even for a system with line nodes in the superconducting gap, i.e., these modes’ survival seems largely insensitive to the nodal structure of the order parameter. Combined with the prior results for the chiral p + ip and \(d+id^{\prime} \) states in twodimensions, we see that chiral order parameters are generically characterized by degenerate generalized clapping modes. In this way, the generalized clapping mode spectrum can be used to differentiate chiral states from mixed symmetry TRSB states, as further discussed below.
Altogether, the results we have presented so far provide strong evidence that the generalized clapping modes are always welldefined subgap excitations in TRSB superconducting states. A rigorous proof of this conjecture would be an interesting direction for future research, but is beyond the scope of the present work. At a practical level, this universality is essential to these modes’ application in collective mode spectroscopy.
Having established that the relative phase and amplitude modes are typically welldefined collective excitations in TRSB superconductors, we now propose several means for their experimental detection.
Detection of relative phase modes
From the action (6), we see that the relative mode contributes to the charge density as ρ ~ i∂_{t}b due to the nonvanishing linear coupling Π^{0b} between b and the scalar potential. Since the relative phase mode resides at finite frequency, it is not affected by Coulomb interactions in the usual way and is not lifted up to the plasma frequency (similar to ref. ^{51}).
One may then integrate b out of this action, renormalizing the electronic compressibility Π^{00}(Ω) → Π^{00}(Ω) + δΠ^{00}(Ω) where
is the relative phase mode’s contribution to Π^{00}(Ω). We plot this function for several pairing symmetries in Fig. 6a–d, where we see clear features at the relative phase mode frequency.
Taking into account Coulomb interactions and integrating out the ABG phase mode, the fully renormalized longitudinal dielectric function is
where \({\tilde{{{\Pi }}}}^{00}({{\Omega }})={{{\Pi }}}^{00}({{\Omega }})+\delta {{{\Pi }}}^{00}({{\Omega }})\) is the renormalized compressibility and n_{s} is the superfluid density. For frequencies much smaller than the plasma frequency, the dielectric function is \({\epsilon }_{L}({{\Omega }},{{{{{\bf{q}}}}}})\approx 1+{\tilde{{{\Pi }}}}^{00}({{\Omega }})/{{{{{{\bf{q}}}}}}}^{2}\), which includes a feature at the relative phase mode frequency that can be detected by probes sensitive to the density–density response function, as discussed below.
For most candidate TRSB superconductors, the superconducting gap, and hence the relative phase mode frequency, is in the terahertz regime. The ac electronic compressibility can be directly measured at THz frequencies using existing experimental techniques such as momentumresolved electron energy loss spectroscopy (MEELS)^{52,53,54,55}, which enables direct experimental detection of the relative phase mode.
In such an experiment, the relative phase mode can be distinguished from trivial nonelectronic modes by its dispersion and the fact that the peak corresponding to this mode should vanish above T_{c}. It can be shown that the relative phase mode disperses as \({{{\Omega }}}^{2}={{{\Omega }}}_{\,{{\mbox{b}}}\,}^{2}+\alpha {v}_{F}^{2}{q}^{2}\), where Ω_{b} is the frequency of the mode and α is a constant dependent on the mixing angle and orbital symmetries of the order parameter. In light of these considerations, the observation of a subgap peak in the charge response, measured via MEELS, which disperses with an electronicscale wavelength would constitute smokinggun evidence for the relative phase mode in a TRSB superconductor.
Alternatively, these modes may also be detected by subgap peaks in the microwave power absorption, as shown in refs. ^{56,57}, and via magnetooptical measurements^{58}. In fact, a collective mode of unknown origin has been observed in microwave power absorption measurements of the heavyfermion superconductor UBe_{13}^{59}. To date, there is no evidence that the superconducting state of this system is TRSB, but a split transition has been reported in specific heat measurements of its Thdoped relative U_{1−x}Th_{x}Be_{13}^{60,61}, which is suggestive of a multicomponent order. In light of this, it could be interesting to revisit the order parameter symmetry of these compounds and whether timereversal symmetry is broken. If so, ref. ^{59} might represent the first measurement of a generalized clapping mode.
In addition to the equilibrium probes discussed above, the relative phase mode should also be detectable out of equilibrium using timeresolved THz spectroscopy^{62}. In fact, a recent experiment has claimed to observe the Leggett phase mode in the multiband superconductor MgB_{2} using this technique^{63}. However, another THz study of MgB_{2} attributed the same experimental signatures to other modes, complicating the identification of the Leggett mode^{64}. In any case, further work is necessary to establish the precise experimental signatures of the relative phase mode in such an experiment.
If one were to observe a peak near the relative phase mode frequency in any of the experiments listed above, its origin can be further distinguished with the application of an external magnetic field. For most tetragonal twocomponent superconductors, there is a symmetryallowed linear coupling to an external magnetic field \(\delta f=ig{H}_{z}({\bar{{{\Delta }}}}_{1}{{{\Delta }}}_{2}{{{\Delta }}}_{1}{\bar{{{\Delta }}}}_{2})\)^{65}. In terms of the relative phase φ between Δ_{1} and Δ_{2}, this can be written as \(\delta f=gH\sin \varphi \approx gH\delta {\varphi }^{2}\) where we have assumed that the fluctuations δφ are small around the equilibrium value of φ_{0} = π/2. Then, on very general grounds^{30}, this coupling shifts the relative phase mode frequency from Ω_{b,0} at zero field to \({{{\Omega }}}_{b}^{2} \sim {{{\Omega }}}_{b,0}^{2}+gH\). That is, the frequency of an experimentally observed peak should vary linearly with small applied magnetic fields if it truly corresponds to the relative phase mode, and in this way, one may unambiguously assign an observed peak to the relative phase mode.
Detection of amplitude modes
The A_{±} modes couple to electromagnetic fields in the same manner as the Higgs mode in a conventional singlecomponent superconductor, through the nonlinear coupling δS ~ A^{2}A_{±}. One method to detect amplitude modes using this coupling is via the observation of resonantly enhanced thirdharmonic generation in a nonlinear THz spectroscopy experiment, which has been successfully performed for both conventional^{66} and hightemperature superconductors^{67}. Amplitude modes have also been detected in THz pump, optical probe experiments for conventional superconductors^{68} and cuprates^{69}. However, despite the rapid development of THz spectroscopy of a probe of Higgs modes in superconductors, the impact of charge density fluctuations^{70}, disorder^{71,72,73}, and other modes^{74} on the THz response continue to be actively discussed, and the identification of the Higgs mode in unconventional superconductors remains controversial.
The A_{+} mode in TRSB superconductors is attractive on account of its low frequency, which should make it less heavily damped than conventional Higgs modes and easier to disentangle from the charge density fluctuations which onset at the gap edge^{75}. Nonetheless, in light of the aforementioned controversy surrounding the THz detection of conventional Higgs modes, further work is necessary to establish the detailed THz response of the amplitude modes discussed in our work to facilitate comparison with potential future experiments.
Alternatively, the amplitude modes can be detected using microwave spectroscopy, or within linear response as a resonance in the optical conductivity in the presence of a background dc supercurrent^{76}, as has been demonstrated in NbN^{77}.
Like the relative phase mode discussed above, the observation of the lowfrequency A_{+} amplitude mode is a direct signature of the TRSB superconducting state. Thus, the detection of this mode represents a different avenue in the emerging field of “Higgs spectroscopy”^{10,11,67} where nonequilibrium amplitude oscillations are used to gain insight into the symmetry of the condensate. However, in contrast to these pioneering works, generalized clapping modes can be detected in equilibrium and using a wide variety of experimental probes, as discussed above.
Discussion
To summarize, we have shown that fluctuations in the relative amplitude and phase of twocomponent TRSB order parameters are welldefined collective modes for a wide class of model order parameters with both spinsinglet and spintriplet pairing (and possibly mixedparity systems as well). Moreover, even for nodal gap functions, we have found that these modes are not overdamped by lowenergy quasiparticles in the T → 0 limit. The frequency of each mode depends strongly on both the orbital symmetries of the two order parameter components and the relative amplitude of the two components in equilibrium.
Further, we have proposed a number of means to experimentally detect generalized clapping modes: the relative phase mode can be directly detected via measurement of the ac electronic compressibility (using e.g., MEELS), the amplitude modes can be detected using ultrafast and nonlinear THz spectroscopy as well as optical conductivity measurements (in the presence of a dc supercurrent), and both modes can be detected in microwave power absorption measurements. The observation of these modes in a given material would constitute robust evidence of a twocomponent TRSB order parameter.
Our work enables a variety of existing experimental techniques including ultrafast and nonlinear optics, electron scattering, and microwave spectroscopy to be used as direct probes of TRSB in unconventional superconductors. These measurements can be thought of as a form of “collective mode spectroscopy,” where one obtains information about the structure of the order parameter from its collective mode spectrum. Given the relative scarcity of probes directly sensitive to TRSB in superconductors (previously limited to only Kerr rotation and muon spin relaxation), this represents a substantial expansion of the experimental tools available to characterize these exotic TRSB superconducting states.
Detection of the generalized clapping modes also offers the unique ability to estimate the relative magnitudes of each order parameter component through the frequency at which the mode resides (as seen in all of our results, the phase mode frequency is maximum for an equal admixture of order parameter components, and decreases as one component or the other becomes dominant). Moreover, observing these modes using ultrafast THz techniques would allow one to assess TRSB superconductivity in driven nonequilibrium superconductors. The exploration and manipulation of generalized clapping modes also represent a distinct direction in the wider, rapidly developing field of THz control of unconventional superconductors and their collective excitations^{74,78,79,80,81,82,83}.
As mentioned above, the generalized clapping mode spectrum can be used to unambiguously distinguish chiral TRSB states from nonchiral states, as a generic TRSB superconductor has two nondegenerate generalized clapping modes, whereas it is only in the special case of a chiral state that the two modes are degenerate.
This capability is particularly wellsuited to clarify the structure of the order parameter in Sr_{2}RuO_{4}—a problem of tremendous current interest. At the time of writing, the two leading candidate order parameters for this system are the mixed symmetry d + ig state (see Figs. 3c, f and 4e, f) and the chiral d + id state (see Fig. 5d), where the generalized clapping mode spectrum is plotted for the cylindrical Fermi surface relevant to Sr_{2}RuO_{4}). Quasiparticle interference^{84} and thermal transport^{85} measurements have demonstrated that the order parameter exhibits vertical line nodes along the zone diagonals (i.e., along the [110] direction), consistent with the d + ig scenario, and counter to the horizontal line nodes expected for a d + id state. In contrast, recent muon spin relaxation measurements failed to detect a splitting of the critical temperature under hydrostatic pressure^{86}, which is only consistent with the chiral d + id order. Given these seemingly conflicting results, a measurement of the generalized clapping mode spectra, via e.g., microwave absorption measurements, could provide crucial insight into the chirality, or lack thereof, of the order parameter, and help the community converge on one candidate order parameter over the other.
Beyond its utility as a form of spectroscopy, the detection of generalized clapping modes is also interesting from a fundamental physics perspective, as the analog of the clapping mode in ^{3}HeA has yet to be realized in any electronic system. Previously, the existence and detection of clapping modes were considered primarily in p + ip superconductors^{29,87,88}, which have proven elusive to realize experimentally^{89}. This work opens up a number of other materials platforms^{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27} as candidate systems to finally realize these exotic collective modes in a solidstate system.
Finally, we speculate that our work may also be extended to the enigmatic pseudogap phase of the cuprate hightemperature superconductors where a nonzero Kerr rotation has been reported^{90}, suggesting the existence of a TRSB phase above the superconducting transition. It has recently been suggested^{36,91,92,93,94} that the relative phase between two order parameter components can acquire a phase stiffness before either order parameter becomes phase coherent and condenses. In this scenario, the relative phase mode studied in this work could persist even above the superconducting transition, representing a distinctive collective excitation in a TRSB metallic phase.
Code availability
All relevant code used in this study is available from the corresponding authors upon reasonable request.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
References
Anderson, P. W. Randomphase approximation in the theory of superconductivity. Phys. Rev. 112, 1900 (1958).
Schmid, A. The approach to equilibrium in a pure superconductor the relaxation of the Cooper pair density. Phys. der kondensierten Mater. 8, 129 (1968).
Volkov, A. F. & Kogan, S. M. Collisionless relaxation of the energy gap in superconductors. Zh. Eksp. Teor. Fiz. 65, 2038–2046 https://www.osti.gov/biblio/4356386 (1973).
Kulik, I. O., EntinWohlman, O. & Orbach, R. Pair susceptibility and mode propagation in superconductors: a microscopic approach. J. Low. Temp. Phys. 43, 591 (1981).
Bogoliubov, N. N., Tolmachev, V. V. & Shirkov, D. V. A new method in the theory of superconductivity. Fortsch. Phys. 6, 605 (1958).
Anderson, P. W. Coherent excited states in the theory of superconductivity: gauge invariance and the Meissner effect. Phys. Rev. 110, 827 (1958b).
Anderson, P. W. Plasmons, gauge invariance, and mass. Phys. Rev. 130, 439 (1963).
Barlas, Y. & Varma, C. M. Amplitude or Higgs modes in dwave superconductors. Phys. Rev. B 87, 054503 (2013).
Leggett, A. J. Numberphase fluctuations in twoband superconductors. Prog. Theor. Phys. 36, 901 (1966).
Schwarz, L. et al. Classification and characterization of nonequilibrium Higgs modes in unconventional superconductors. Nat. Commun. 11, 287 (2020).
Schwarz, L. & Manske, D. Theory of driven Higgs oscillations and thirdharmonic generation in unconventional superconductors. Phys. Rev. B 101, 184519 (2020).
Xia, J., Maeno, Y., Beyersdorf, P. T., Fejer, M. M. & Kapitulnik, A. High resolution polar Kerr effect measurements of Sr_{2}RuO_{4}: evidence for broken timereversal symmetry in the superconducting state. Phys. Rev. Lett. 97, 167002 (2006).
Luke, G. M. et al. Timereversal symmetrybreaking superconductivity in Sr_{2}RuO_{4}. Nature 394, 558 (1998).
Schemm, E. R., Gannon, W. J., Wishne, C. M., Halperin, W. P. & Kapitulnik, A. Observation of broken timereversal symmetry in the heavyfermion superconductor UPt_{3}. Science 345, 190 (2014).
Luke, G. M. et al. Muon spin relaxation in UPt_{3}. Phys. Rev. Lett. 71, 1466 (1993).
Schemm, E. R. et al. Evidence for broken timereversal symmetry in the superconducting phase of URu_{2}Si_{2}. Phys. Rev. B 91, 140506 (2015).
Ran, S. et al. Nearly ferromagnetic spintriplet superconductivity. Science 365, 684 (2019).
Hayes, I. M. et al. Weyl superconductivity in UTe_{2}. https://www.science.org/doi/10.1126/science.abb0272 (2020).
LevensonFalk, E. M., Schemm, E. R., Aoki, Y., Maple, M. B. & Kapitulnik, A. Polar Kerr effect from timereversal symmetry breaking in the heavyfermion superconductor PrOs_{4}Sb_{12}. Phys. Rev. Lett. 120, 187004 (2018).
Aoki, Y. et al. Timereversal symmetrybreaking superconductivity in heavyfermion PrOs_{4}Sb_{12} detected by muonspin relaxation. Phys. Rev. Lett. 91, 067003 (2003).
Grinenko, V. et al. Superconductivity with broken timereversal symmetry inside a superconducting swave state. Nat. Phys. 16, 789 (2020).
Gong, X. et al. Timereversal symmetrybreaking superconductivity in epitaxial bismuth/nickel bilayers. Sci. Adv. 3, https://doi.org/10.1126/sciadv.1602579 (2017).
Trimbleet, C. J. et al. Josephson detection of time reversal symmetry broken superconductivity in SnTe nanowires. https://doi.org/10.1038/s4153502100359w (2020).
Xu, C. & Balents, L. Topological superconductivity in twisted multilayer graphene. Phys. Rev. Lett. 121, 087001 (2018).
Liu, C.C., Zhang, L.D., Chen, W.Q. & Yang, F. Chiral spin density wave and d + id superconductivity in the magicangletwisted bilayer graphene. Phys. Rev. Lett. 121, 217001 (2018).
Can, O. et al. Hightemperature topological superconductivity in twisted doublelayer copper oxides. Nat. Phys. 17, 519 (2021).
Volkov, P. A., Wilson, J. H. & Pixley, J. H. Magic angles and currentinduced topology in twisted nodal superconductors. Preprint at https://arxiv.org/abs/2012.07860 (2020).
Wölfle, P. Orderparameter collective modes in ^{3}He−A. Phys. Rev. Lett. 37, 1279 (1976).
Tewordt, L. Collective order parameter modes and spin fluctuations for spintriplet superconducting state in Sr_{2}RuO_{4}. Phys. Rev. Lett. 83, 1007 (1999).
Balatsky, A. V., Kumar, P. & Schrieffer, J. R. Collective mode in a superconductor with mixedsymmetry order parameter components. Phys. Rev. Lett. 84, 4445 (2000).
Bardasis, A. & Schrieffer, J. R. Excitons and plasmons in superconductors. Phys. Rev. 121, 1050 (1961).
Allocca, A. A., Raines, Z. M., Curtis, J. B. & Galitski, V. M. Cavity superconductorpolaritons. Phys. Rev. B 99, 020504 (2019).
Sauls, J. A., Wu, H. & Chung, S. B. Anisotropy and strongcoupling effects on the collective mode spectrum of chiral superconductors: application to Sr_{2}RuO_{4}. Front. Phys. 3, 36 (2015).
Laughlin, R. B. Magnetic induction of \({{d}}_{{{x}}^{2}{{y}}^{2}}+{{{{{{\rm{id}}}}}}}_{{{{{{\rm{xy}}}}}}}\) order in highT_{c} superconductors. Phys. Rev. Lett. 80, 5188 (1998).
Tewari, S., Zhang, C., Yakovenko, V. M. & Das Sarma, S. Timereversal symmetry breaking by a (d + id) densitywave state in underdoped cuprate superconductors. Phys. Rev. Lett. 100, 217004 (2008).
Brydon, P. M. R., Abergel, D. S. L., Agterberg, D. F. & Yakovenko, V. M. Loop currents and anomalous hall effect from timereversal symmetrybreaking superconductivity on the honeycomb lattice. Phys. Rev. X 9, 031025 (2019).
Yang, Z., Qin, S., Zhang, Q., Fang, C. & Hu, J. π/2Josephson junction as a topological superconductor. Phys. Rev. B 98, 104515 (2018).
Platt, C., Thomale, R., Honerkamp, C., Zhang, S.C. & Hanke, W. Mechanism for a pairing state with timereversal symmetry breaking in ironbased superconductors. Phys. Rev. B 85, 180502 (2012).
Lee, W.C., Zhang, S.C. & Wu, C. Pairing state with a timereversal symmetry breaking in FeAsbased superconductors. Phys. Rev. Lett. 102, 217002 (2009).
Maiti, S. & Hirschfeld, P. J. Collective modes in superconductors with competing s and dwave interactions. Phys. Rev. B 92, 094506 (2015).
Kivelson, S. A., Yuan, A. C., Ramshaw, B. & Thomale, R. A proposal for reconciling diverse experiments on the superconducting state in Sr_{2}RuO_{4}. npj Quantum Mater. 5, 43 (2020).
Ghosh, S. et al. Thermodynamic evidence for a twocomponent superconducting order parameter in Sr_{2}RuO_{4}. Nat. Phys. https://doi.org/10.1038/s4156702010324 (2020).
Hsiao, W.H. Universal collective modes in twodimensional chiral superfluids. Phys. Rev. B 100, 094510 (2019).
Leggett, A. J. A theoretical description of the new phases of liquid ^{3}He. Rev. Mod. Phys. 47, 331 (1975).
Li, G. et al. Bulk evidence for a timereversal symmetry broken superconducting state in URu_{2}Si_{2}. Phys. Rev. B 88, 134517 (2013).
Yano, K. et al. Fieldangledependent specific heat measurements and gap determination of a heavy fermion superconductor URu_{2}Si_{2}. Phys. Rev. Lett. 100, 017004 (2008).
Fischer, M. H. et al. Chiral dwave superconductivity in SrPtAs. Phys. Rev. B 89, 020509 (2014).
Biswas, P. K. et al. Evidence for superconductivity with broken timereversal symmetry in locally noncentrosymmetric SrPtAs. Phys. Rev. B 87, 180503 (2013).
Thalmeier, P. & Takimoto, T. Signatures of hiddenorder symmetry in torque oscillations, elastic constant anomalies, and fieldinduced moments in URu_{2}Si_{2}. Phys. Rev. B 83, 165110 (2011).
Rau, J. G. & Kee, H.Y. Hidden and antiferromagnetic order as a rank5 superspin in URu_{2}Si_{2}. Phys. Rev. B 85, 245112 (2012).
Bittner, N., Einzel, D., Klam, L. & Manske, D. Leggett modes and the AndersonHiggs mechanism in superconductors without inversion symmetry. Phys. Rev. Lett. 115, 227002 (2015).
Vig, S. et al. Measurement of the dynamic charge response of materials using lowenergy, momentumresolved electron energyloss spectroscopy (MEELS). SciPost Phys. 3, 026 (2017).
Mitrano, M. et al. Anomalous density fluctuations in a strange metal. Proc. Natl Acad. Sci. USA 115, 5392 (2018).
Husain, A. A. et al. Crossover of charge fluctuations across the strange metal phase diagram. Phys. Rev. X 9, 041062 (2019).
Husain, A. A. et al. Coexisting Fermi liquid and strange metal phenomena in Sr_{2}RuO_{4}. Preprint at https://arxiv.org/abs/2007.06670 (2020).
Hirschfeld, P. J., Wölfle, P., Sauls, J. A., Einzel, D. & Putikka, W. O. Electromagnetic absorption in anisotropic superconductors. Phys. Rev. B 40, 6695 (1989).
Hirschfeld, P. J., Putikka, W. O. & Wölfle, P. Electromagnetic power absorption by collective modes in unconventional superconductors. Phys. Rev. Lett. 69, 1447 (1992).
Yip, S. K. & Sauls, J. A. Circular dichroism and birefringence in unconventional superconductors. J. Low. Temp. Phys. 86, 257 (1992).
Feller, J. R., Tsai, C.C., Ketterson, J. B., Smith, J. L. & Sarma, B. K. Evidence of electromagnetic absorption by collective modes in the heavy fermion superconductor UBe_{13}. Phys. Rev. Lett. 88, 247005 (2002).
Ott, H. R., Rudigier, H., Fisk, Z. & Smith, J. L. Phase transition in the superconducting state of U_{1x}Th_{x}Be_{13} (x=0–0.06). Phys. Rev. B 31, 1651 (1985).
Stewart, G. R. UBe_{13} and U_{1−x}Th_{x}Be_{13}: unconventional superconductors. J. Low. Temp. Phys. 195, 1 (2019).
Krull, H., Bittner, N., Uhrig, G. S., Manske, D. & Schnyder, A. P. Coupling of Higgs and Leggett modes in nonequilibrium superconductors. Nat. Commun. 7, 11921 (2016).
Giorgianni, F. et al. Leggett mode controlled by light pulses. Nat. Phys. 15, 341 (2019).
Kovalev, S. et al. Bandselective thirdharmonic generation in superconducting MgB_{2}: evidence for Higgs amplitude mode in the dirty limit. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.L140505 (2020).
Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239 (1991).
Matsunaga, R. et al. Lightinduced collective pseudospin precession resonating with Higgs mode in a superconductor. Science 345, 1145 (2014).
Chu, H. et al. Phaseresolved Higgs response in superconducting cuprates. Nat. Commun. 11, 1793 (2020).
Matsunaga, R. et al. Higgs amplitude mode in the BCS superconductors Nb_{1−x}Ti_{x}N induced by terahertz pulse excitation. Phys. Rev. Lett. 111, 057002 (2013).
Katsumi, K. et al. Higgs mode in the dwave superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+x} driven by an intense terahertz pulse. Phys. Rev. Lett. 120, 117001 (2018).
Cea, T., Castellani, C. & Benfatto, L. Nonlinear optical effects and thirdharmonic generation in superconductors: Cooper pairs versus Higgs mode contribution. Phys. Rev. B 93, 180507 (2016).
Silaev, M. Nonlinear electromagnetic response and Higgsmode excitation in BCS superconductors with impurities. Phys. Rev. B 99, 224511 (2019).
Tsuji, N. & Nomura, Y. Higgsmode resonance in third harmonic generation in NBN superconductors: multiband electronphonon coupling, impurity scattering, and polarizationangle dependence. Phys. Rev. Res. 2, 043029 (2020).
Seibold, G., Udina, M., Castellani, C. & Benfatto, L. Third harmonic generation from collective modes in disordered superconductors. Phys. Rev. B 103, 014512 (2021).
Gabriele, F., Udina, M. & Benfatto, L. Nonlinear terahertz driving of plasma waves in layered cuprates. Nat. Commun. 12, 752 (2021).
Cea, T., Castellani, C. & Benfatto, L. Nonlinear optical effects and thirdharmonic generation in superconductors: Cooper pairs versus Higgs mode contribution. Phys. Rev. B 93, 180507 (2016).
Moor, A., Volkov, A. F. & Efetov, K. B. Amplitude Higgs mode and admittance in superconductors with a moving condensate. Phys. Rev. Lett. 118, 047001 (2017).
Nakamura, S. et al. Infrared activation of the Higgs mode by supercurrent injection in superconducting NbN. Phys. Rev. Lett. 122, 257001 (2019).
Vaswani, C. et al. Light quantum control of persisting Higgs modes in ironbased superconductors. Nat. Commun. 12, 258 (2021).
Yang, X. et al. Lightwavedriven gapless superconductivity and forbidden quantum beats by terahertz symmetry breaking. Nat. Photonics 13, 707 (2019).
Mootz, M., Wang, J. & Perakis, I. E. Lightwave terahertz quantum manipulation of nonequilibrium superconductor phases and their collective modes. Phys. Rev. B 102, 054517 (2020).
Müller, M. A., Volkov, P. A., Paul, I. & Eremin, I. M. Collective modes in pumped unconventional superconductors with competing ground states. Phys. Rev. B 100, 140501 (2019).
Müller, M. A., Volkov, P. A., Paul, I. & Eremin, I. M. Interplay between nematicity and BardasisSchrieffer modes in the shorttime dynamics of unconventional superconductors. Phys. Rev. B 103, 024519 (2021).
Müller, M. A. & Eremin, I. M. Signatures of BardasisSchrieffer mode excitation in thirdharmonic generated currents. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.144508 (2021).
Sharma, R. et al. Momentumresolved superconducting energy gaps of Sr_{2}RuO_{4} from quasiparticle interference imaging. Proc. Natl Acad. Sci. USA 117, 5222 (2020).
Hassinger, E. et al. Vertical line nodes in the superconducting gap structure of Sr_{2}RuO_{4}. Phys. Rev. X 7, 011032 (2017).
Grinenko, V. et al. Unsplit superconducting and time reversal symmetry breaking transitions in Sr_{2}RuO_{4} under hydrostatic pressure and disorder. https://doi.org/10.1038/s41467021241768 (2021).
Higashitani, S. & Nagai, K. Electromagnetic response of a k_{x} ± ik_{y} superconductor: effect of orderparameter collective modes. Phys. Rev. B 62, 3042 (2000).
Chung, S. B., Raghu, S., Kapitulnik, A. & Kivelson, S. A. Charge and spin collective modes in a quasionedimensional model of Sr_{2}RuO_{4}. Phys. Rev. B 86, 064525 (2012).
Pustogow, A. et al. Constraints on the superconducting order parameter in Sr_{2}RuO_{4} from oxygen17 nuclear magnetic resonance. Nature 574, 72 (2019).
Xia, J. et al. Polar Kerreffect measurements of the hightemperature YBa_{2}Cu_{3}O_{6+x} superconductor: evidence for broken symmetry near the pseudogap temperature. Phys. Rev. Lett. 100, 127002 (2008).
Bojesen, T. A., Babaev, E. & Sudbø, A. Time reversal symmetry breakdown in normal and superconducting states in frustrated threeband systems. Phys. Rev. B 88, 220511 (2013).
Bojesen, T. A., Babaev, E. & Sudbø, A. Phase transitions and anomalous normal state in superconductors with broken timereversal symmetry. Phys. Rev. B 89, 104509 (2014).
Zeng, M., Hu, L.H., Hu, H.Y., You, Y.Z. & Wu, C. Phasefluctuation induced timereversal symmetry breaking normal state. Preprint at https://arxiv.org/abs/2102.06158 (2021).
Grinenko, V. et al. Bosonic Z_{2} metal: spontaneous breaking of timereversal symmetry due to Cooper pairing in the resistive state of Ba_{1−x}K_{x}Fe_{2}As_{2}. https://doi.org/10.1038/s41567021013509 (2021).
Acknowledgements
The authors thank Matteo Mitrano and Eugene Demler (Harvard), Dmitri Basov (Columbia), Manfred Sigrist (ETH Zürich), and Roman Lutchyn (Station Q) for insightful discussions about this work. We also thank Charlotte Bøttcher, Marie Wesson, Uri Vool, Yuval Ronen, Zachary Raines, Andrew Allocca, and Zhiyuan Sun for fruitful discussions through various iterations of this study, as well as Rafael Haenel for valuable comments on the manuscript. This work is primarily supported by the Quantum Science Center (QSC), a National Quantum Information Science Research Center of the U.S. Department of Energy (DOE). N.R.P. is supported by the Army Research Office through an NDSEG fellowship. J.C. is an HQI Prize Postdoctoral Fellow and gratefully acknowledges support from the Harvard Quantum Initiative. A.Y. is partly supported by the Gordon and Betty Moore Foundation through Grant GBMF 9468 and by the National Science Foundation under Grant No. DMR1708688. P.N. is a Moore Inventor Fellow and gratefully acknowledges support through Grant GBMF8048 from the Gordon and Betty Moore Foundation.
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Poniatowski, N.R., Curtis, J.B., Yacoby, A. et al. Spectroscopic signatures of timereversal symmetry breaking superconductivity. Commun Phys 5, 44 (2022). https://doi.org/10.1038/s42005022008190
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DOI: https://doi.org/10.1038/s42005022008190
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