Several superconducting materials have been discovered to manifest evidence of spontaneous time-reversal symmetry breaking (TRSB) that appears only below their critical superconducting transition temperatures T _c [1–4]. Evidence for such TRSB originates mainly from an enhanced muon-spin relaxation (μSR) and/or polar Kerr effect measurements finding a change in the optical polar Kerr angle below T _c. Additional probes able to detect breaking of time-reversal symmetry in the superconducting phase include Josephson interferometry [5], superconducting quantum interference device (SQUID) magnetometry [6], and polarized and small-angle neutron scattering [7]. These experimental methods detect the internal magnetic fields spontaneously generated in the superconducting state. Explanations for the occurrence of such fields from the perspective of clean (homogeneous) physical effects are either in terms of a TRSB non-unitary spin-triplet pairing state or in terms of multi-component superconducting condensates entering a complex TRSB superposition of two superconducting order parameter components. The latter option is particularly natural for superconducting instabilities condensing in two-dimensional (2D) irreducible representations of the associated crystal point group. Complex superpositions of two symmetry-distinct order parameters break TRSB, and generate persistent supercurrent patterns at material edges, dislocations, or around various defect sites, which produce magnetic fields that may not average to zero locally. Such a multi-component scenario for the origin of TRSB below T _c has been extensively discussed for e.g., Sr₂RuO₄ [4, 8–12], UPt₃ [13, 14], URu₂Si₂ [15], UTe₂ [16, 17], SrPtAs [18], Re [19], Zr₃Ir [20], LaNiC₂ [21], LaNiGa₂ [22], PrOs₄Sb₁₂ [23, 24], Ba_1−x K _x Fe₂As₂ [25, 26], FeSe_1−x S _x [27], and more recently also for the kagome superconductors AV₃Sb₅ (A: K, Rb, Cs) [28–30], 4Hb-TaS₂ [31], CaPd₂Ge₂ [32], skutterudites [33, 34] and Na₂Cr₃As₃ [35]. For most of the compounds listed here, the evidence for TRSB originates from μSR experiments. For further materials displaying TRSB in their superconducting phases, and other types of evidence, we refer to Refs. [1–4].

At present, the status of the precise superconducting ground state remains controversial for many of the materials listed above. This is the case, for example, for the two materials Sr₂RuO₄ and UTe₂, where specific heat data features only a single thermodynamic transition that does not split under uniaxial strain, casting considerable doubt on the multi-component nature of their superconducting condensates [36–39]. Therefore, it appears timely and important to pursue other possibilities for the origin of TRSB detected solely inside the superconducting phase. This motivates the question of what mechanisms there are for TRSB of superconducting condensates composed of a single component order parameter? Barring non-unitary TRSB spin-triplet order, the answer to this question naturally leads to a study of different forms of spatial inhomogeneities and their influence on the local properties of superconductivity. For example, it is known that some superconductors generate magnetic moments below T _c due to an interplay between the gap structure and electronic correlations [40–46]. As mentioned above, another possibility for TRSB appearing only below T _c includes the generation of localized orbital loops of supercurrents near impurities. The latter is well-known to arise near nonmagnetic disorder sites in complex TRSB multi-component condensates [47–54], but was recently shown to be also present in the strongly disordered regime of single-component superconductors [55, 56]. Likewise, one might expect dislocations and grain boundaries to similarly operate as seeds of localized supercurrents producing sizable internal magnetic fields. Recently, the latter scenario was studied in Refs. [57, 58] as a possible explanation for reconciling the existence of TRSB and a single specific heat transition in Sr₂RuO₄. Indeed for this material, edge dislocations are known to be prevalent in many samples [59]. Finally, one might also envision that time-reversal symmetry is already broken above T _c through the existence of small magnetic regions that are either too dilute or rapidly fluctuating to be picked up by most probes. Only through a superconductivity-enhanced coupling, or a sufficient slowing-down of such preexisting magnetism below T _c, can it be detected, e.g., by entering the muon relaxation time window.

Here, motivated by the above-mentioned developments we survey a number of different mechanisms for inhomogeneity-induced TRSB in superconductors. Our discussion of disorder-induced TRSB refers both to cases where TRSB exists only locally near inhomogeneity sites and where TRS is broken globally, but manifested via induced currents near disorder sites. We discuss these both in view of recent theoretical and experimental developments of unconventional superconductivity.

In this section we discuss three different mechanisms for disorder-generated TRSB in spin-singlet superconductors. Section 2.1 reviews the generation of magnetic fields from disorder in superconducting phases with TRSB in the bulk or generated near spatial inhomogeneities. Section 2.2 discusses another case where disorder in conjunction with electronic interactions and appropriate superconducting gap structures lead to the generation of static magnetic moments. Due to the required gapping of states for these moments to form, this mechanism is tied to the superconducting phase. Finally, Sec. 2.3 surveys a third case where preexisting magnetic local order (fluctuation moments) in the normal state get enhanced (slowed-down) by superconductivity. Clearly these mechanisms do not exhaust the options for disorder-induced TRSB in superconductors where e.g., cases with substantial spin-orbit coupling may also be of interest [60, 61]. We focus on these three options given recent research along similar lines and for their relevance for some of the superconducting materials mentioned above. For a discussion of the possibility of TRSB in homogeneous superconductors originating from conventional electron-phonon coupling, we refer to the review [3] and the references therein. A theoretical proposal for a mechanism has been proposed in Ref. [62] and recent experimental evidence for TRSB in materials where superconductivity is believed to originate from electron-phonon coupling is reported in Refs. [32–34, 63].

In this section we discuss the main current experimental techniques to directly probe TRSB in superconductors, and mention subtleties in the interpretation of experimental results. Many unconventional superconductors are investigated using the muon spin relaxation (μSR) technique where a beam of spin-polarized muons with given energy is directed to the sample [140]. Muons are stopped in the material, implanted at preferred positions in the crystal structure and can be used to detect the magnetic field locally at their positions. Within the lifetime of the muons, their spins precess in the local field. Once the muon decays into a positron and two neutrinos, the direction of the magnetic moment fixes the probability distribution of emitted positrons. The counting rates of the latter in the forward and backward directions of the (original) polarization of the muon beam yield the asymmetry A(t), which is recorded as a function of time.

The Fourier transform of A(t) for measurements on superconductors in a finite field (in the vortex state) is directly related to the magnetic field distribution of the sample and can be used to deduce the temperature dependence of the penetration depth [140–142]. In a zero field experiment, the information about the field distribution is also contained in the asymmetry function, but typically full oscillations are not observed in superconductors because of the smallness of the internal field. Still, an increased relaxation as a function of temperature is taken as signature of the occurrence of an additional internal field that collectively depolarizes the implanted muons. Average fields that are static on the time scale of the muon decay can then be detected down to the scale of μT [140]. From the shape of the asymmetry A(t) at low temperature (when there is evidence for TRSB), it is possible to estimate the average field by the reduction at largest measuring times. A quick initial reduction and no further deviation as a function of time are indications of few muons getting depolarized in a large field. By contrast, an initial weak difference of the depolarization rates that behaves quadratically in time indicates an almost uniform field distribution. Fitting to such behavior can give bounds on the fraction of the sample exhibiting TRSB. Such bounds may be further benchmarked against scanning SQUID microscopy measurement [6, 143, 144]. Depending on the initial energy of the muons, implantation happens on length scales of 10 nm–100 nm from the surface, such that the technique can be considered as bulk-like. By varying the initial energy of the muons, the stopping profiles allow deductions of the TRSB as function of distance from the surface [139]. The μSR technique has been used to claim TRSB in a number of superconductors; for a list see Ref. [3].

Knowledge from ab initio methods about the preferred position within the elementary cell where muons are implanted [145, 146] can facilitate the interpretation of the results, i.e., help decipher whether the positive charge of the muons might cause a perturbation large enough to self-generate a TRSB signal. In addition, the position of the stopping site and the associated form factor can be important for a quantitative description of the relaxation, e.g., in cases where the muon locations may feature enhanced or reduced field effects from purely geometric reasons. The typical magnitude of the internal field at the muon site for a generic interstitial position in the unit cell may be estimated; in the vicinity of local spin-1/2 droplets (Sec. 2.2) or impurity-induced localized orbital currents are of order 1 G [3, 147], which is of the same order as fields reported in experiments.

The second main probe of TRSB is the polar Kerr effect, where the polarization plane of light incident on a sample is rotated upon reflection. The classic example is reflection from a ferromagnetic surface, but in recent years the technique has been employed to detect TRSB in unconventional superconductors [1, 2, 148–151]. The angle θ _K, which is related to the Hall conductivity, can be expressed as θ K ω = 4 π ω I m σ H ω n n 2 − 1 , where n is the complex, frequency-dependent index of refraction, and σ _H(ω) is the Hall conductivity. For the experimental identification of a polar Kerr effect arising from the superconducting state, it needs to be checked that the rotation onsets at T _c, the sign and magnitude of the signals vary after each cool-down in the absence of an external magnetic field, a small training field saturates the rotation angle (but with sign given by the direction of the field), and the temperature dependence of the Kerr angle follows mean-field-like order parameter behavior [148].

From a theoretical point of view, a finite Kerr angle is directly related to a nonzero Hall conductivity, which requires time-reversal symmetry breaking and breaking of mirror symmetries about all planes parallel to the incident wave vector [152]. Therefore, the order parameter has to be chiral and not simply break TRSB. For example, a pure s + id superconductor is nonchiral and will not generate an intrinsic Kerr signal. By contrast, a one-band p + ip superconductor will produce a weak intrinsic Kerr effect with θ _K arising only from the collective modes of the order parameter [153]. This effect is many orders of magnitude smaller than typical measured rotation angles θ _K of order tens of nanoradians. The intrinsic effect in a chiral multiband superconductor, arising from contributions to σ _H from interband transitions, is larger at optical frequencies (but still difficult to definitively quantify) [154]. On the other hand, if the state is nonchiral, there will be no intrinsic Kerr signal at all.

It is important to note that chirality can occur for complex linear combinations of 1D irreps of the point group, and not necessarily only those states corresponding to 2D irreps that have been traditionally discussed, like p _x + ip _y in tetragonal and d x y + i d x 2 − y 2 in hexagonal symmetries. For example, in tetragonal symmetry, d x 2 − y 2 and d _xy are distinct 1D irreps, but may occur in an accidental linear combination a ⋅ d x y + i b ⋅ d x 2 − y 2 . Such states should also give rise to an intrinsic polar Kerr effect, but nonchiral TRSB combinations like s + id will not.

One of the problems with the interpretation of Kerr measurements is that the size of an intrinsic signal is very hard to estimate, and the size of an extrinsic one may in principle be in fact much larger, but depend on the type of disorder and the nature of the ground state. For example, the division between chiral and non-chiral states with regard to their polar Kerr signals is muddied somewhat by disorder. In p + ip states, an extrinsic Kerr rotation angle orders of magnitude larger than the intrinsic effect may result from the skew-scattering of chiral Cooper pairs from impurities [155, 156], but for d + id and higher order angular momentum states, the contribution is of higher order in ( n V imp 2 ) where V _imp is the impurity potential of point-like impurities with density n [1, 155]. Estimates in both cases are model dependent and require disorder averaging. Impurities in nonchiral TRSB states will also give rise to small signals from induced currents, but after averaging such signals should be substantially smaller, while for chiral d + id states the signal is small but remains finite upon impurity averaging [157]. A similar contribution should therefore be provided by local spatially fluctuating magnetic moments. Whether or not a Kerr signal is observed, and its size, depends on how much averaging of the fluctuations takes place over the laser spot. In both these cases, some kind of ability to “train” the state with an external magnetic field should be observed [148]. Finally, we mention some recent calculations showing that local currents can be created near dislocations, which induce d + ig or s + id states [57, 58]. The Kerr effect arising from these local states has not been properly explored.

In the above we focused on different theoretical mechanisms where magnetism in various guises appear only below the superconducting critical transition temperature T _c and is generated by disorder or other spatial inhomogeneities. This included discussions of localized defect-induced orbital supercurrents, magnetic moment generation from superconductivity and electronic correlations, and the possibility of enhanced RKKY exchange coupling by superconductivity.

Naturally an important question relates to the relevance of these mechanisms to the observation of TRSB below T _c in currently discussed unconventional superconducting materials. In the above, we provided a detailed discussion of the cuprates and the possibility of superconductivity-enhanced spin-polarization. Two other materials of significant current interest are UTe₂ and Sr₂RuO₄. For these compounds, two-component condensates are widely discussed, but inconsistent with specific heat measurements, also under uniaxial strain featuring no signs of double transitions [36–39]. This points to single-component superconductivity which has received recent support from a number of other probes as well [117, 150, 158, 159], but leaves open the question of what causes a TRSB signal in some measurements. We have surveyed several possible answers to this question from the perspective of induced fields within the superconducting state from spatial inhomogeneity.

Resolving the detailed nature of TRSB in superconductors is important since the resolution will have profound implications for our general understanding of unconventional superconductivity. Therefore, we hope that future collective research efforts will continue to be devoted to this task. This includes both improved experiments and better theoretical descriptions, the latter including material-specific studies of the interplay between unconventional pairing states and realistic models for the relevant sources of spatial inhomogeneity. μSR and Kerr experiments on systematically disordered samples, e.g., with strong chemical impurities or Frenkel defects created by electron irradiation, can help to distinguish the extrinsic TRSB effects described here from intrinsic effects, and from each other.