Spin-Glass Superconductivity

• Spin-glass superconductivity is a state where superconductivity coexists with a frozen, disordered magnetic configuration lacking long-range order.
• This phenomenon is observed in systems such as Eu-based pnictides, Co-/Ni-doped BaFe₂As₂, and nickelates, exhibiting reentrant and cluster spin-glass behaviors.
• Experimental studies reveal non-monotonic transport, frequency-dependent AC susceptibility, and distinct local versus bulk order, informing theoretical models like multiband Eliashberg and RKKY exchange.

Spin-glass superconductivity denotes a regime in which a superconducting condensate coexists with a spin-glass magnetic state or a cluster-spin-glass texture: a frozen, disordered magnetic configuration with no long-range periodic order, slow relaxation, history dependence, and memory effects. In the materials literature this regime appears in several distinct settings, including Eu-based iron pnictides, Co- and Ni-doped BaFe$_2$As$_2$, ruthenocuprate magneto-superconductors, rare-earth borocarbides, and multiple nickelate platforms; in each case the central issue is the coexistence of superconductivity with glassy magnetism rather than with a conventional ferromagnetic or antiferromagnetic phase (Zapf et al., 2013, Dioguardi et al., 2013, Saykin et al., 3 Jun 2025).

1. Definition and conceptual scope

A spin glass is characterized by frozen, randomly oriented spins below a freezing temperature $T_{SG}$, without long-range magnetic order. A standard order parameter is the Edwards–Anderson parameter

$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$

which measures long-time memory of a local spin orientation (Ummarino, 2020). In superconducting materials, spin-glass superconductivity refers to a state in which a superconducting condensate forms while a localized-moment subsystem freezes into such a glassy configuration, or in which finite antiferromagnetic or ferromagnetic domains freeze into a cluster glass inside a superconducting matrix (Dioguardi et al., 2013).

Several subtypes recur in the literature. A reentrant spin glass is a phase in which glassiness develops below an already ordered magnetic phase; EuFe$_2$(As$_{1-x}$P$_x$)$_2$ is identified explicitly in this sense, with canted $A$-type antiferromagnetism at $T_{cAFM}$ followed by spin-glass freezing of transverse Eu components at $_2$0 (Zapf et al., 2013). A cluster spin glass consists of finite-size magnetic regions that are internally ordered or nearly ordered but do not establish coherent long-range order across the sample; this language is central to Co- and Ni-doped BaFe$_2$1As$_2$2, where frozen antiferromagnetic domains or short-range incommensurate antiferromagnetism coexist with superconductivity (Dioguardi et al., 2013, Lu et al., 2014).

The concept also has a broader theoretical extension. Weak quenched disorder can convert a pair-density-wave or spin-density-wave state into a glass phase that retains Edwards–Anderson-type frozen disorder while supporting composite order such as spin nematicity or charge-$_2$3 superconductivity; this is a related but not identical route to a glassy superconducting state (Mross et al., 2015).

2. Experimental phenomenology and diagnostics

The defining phenomenology combines superconducting signatures with slow magnetic dynamics. In dc magnetization, spin-glass freezing usually appears as pronounced ZFC–FC or ZFC–FCC hysteresis below a characteristic temperature, together with strong time dependence. In EuFe$_2$4P$_2$5, for example, the low-$_2$6 anomaly below $_2$7 K is accompanied by ZFC/FCC splitting and a relaxation time $_2$8 s at 25 K, whereas the higher-temperature Eu ordering at $_2$9 K shows minimal thermal hysteresis (Zapf et al., 2013).

AC susceptibility is a primary diagnostic because the freezing temperature shifts with frequency. In canonical interacting spin glasses this shift follows a Vogel–Fulcher law,

$T_{SG}$0

and this behavior is reported explicitly for EuFe$T_{SG}$1P$T_{SG}$2, YRu-1222, and EuRu-1222 (Zapf et al., 2013, Kumar et al., 2012, Kumar et al., 2011). In NMR studies of Ba(Fe$T_{SG}$3Co$T_{SG}$4)$T_{SG}$5As$T_{SG}$6, glassiness instead appears through stretched-exponential recovery,

$T_{SG}$7

with $T_{SG}$8, reflecting a broad distribution of local relaxation rates and hence of local magnetic environments (Dioguardi et al., 2013).

Transport often shows reentrance or non-monotonic behavior when superconductivity and glassy magnetism overlap. In EuFe$T_{SG}$9(As$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$0P$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$1)$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$2, $$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$3 drops at $$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$4 K, shows a reentrant upturn near 19 K when Eu moments order, and reaches zero only near $$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$5 K, after the glassy Eu phase has developed (Zapf et al., 2013). A closely related optical and transport reentrance is reported in electron-doped EuFe$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$6As$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$7, where superconductivity appears, is partially disrupted by Eu magnetism, and recovers at lower temperature (Baumgartner et al., 2017).

Thermodynamic and local probes are essential because bulk and local views can differ. In Ba(Fe$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$8Co$$q = \lim_{t \rightarrow +\infty} \langle S_i(t) \cdot S_i(0) \rangle,$$9)$_2$0As$_2$1, bulk probes show long-range antiferromagnetism vanishing near $_2$2, while $_2$3As NMR still detects frozen antiferromagnetic domains and glassy dynamics up to $_2$4 inside the superconducting dome (Dioguardi et al., 2013). This distinction between global order and local freezing recurs across the field.

3. Representative material systems

The experimentally documented realizations span several material families and several distinct microscopic architectures.

System	Characteristic spin-glass/superconducting phenomenology	Source
EuFe$_2$5(As$_2$6P$_2$7)$_2$8	Two Eu transitions, reentrant spin glass, coexistence with bulk superconductivity near $_2$9–0.23	(Zapf et al., 2013)
Eu(Fe,Rh/Ir)$_{1-x}$0As$_{1-x}$1	Eu canted $_{1-x}$2-type AFM, spin glass below $_{1-x}$3, optical and resistive reentrance	(Baumgartner et al., 2017)
Ba(Fe,Co/Ni)$_{1-x}$4As$_{1-x}$5	Frozen AF clusters or short-range incommensurate AF identified as cluster spin glass near optimal superconductivity	(Dioguardi et al., 2013, Lu et al., 2014)
Ru-1222 ruthenocuprates	Spin-glass matrix with ferromagnetic clusters in RuO$_{1-x}$6 layers, superconductivity in CuO$_{1-x}$7 planes below $_{1-x}$8 K	(Kumar et al., 2012, Kumar et al., 2011)
$_{1-x}$9Gd$_x$0Ni$_x$1B$_x$2C	Coexistence region of superconductivity and spin-glass state on the $_x$3–$_x$4 phase diagram	(Bud'ko et al., 2010)
Nickelates	Spin-glass onset above $_x$5, Kerr or magnetoresistance hysteresis, glass-like dynamics, and broad spin-freezing backgrounds	(Saykin et al., 3 Jun 2025, Ji et al., 22 Aug 2025, Lin et al., 2021)

In the Eu-based 122 pnictides, the most detailed case is the phase diagram mapped by Zapf et al. for EuFe$_x$6(As$_x$7P$_x$8)$_x$9. Across $_2$0, two Eu transitions are resolved: $_2$1, a canted $_2$2-type antiferromagnetic transition with a ferromagnetic component along $_2$3, and $_2$4, a lower-temperature glass transition of in-plane Eu moments. Superconductivity occurs in a narrower composition range, and around $_2$5–0.2 the spin-glass phase coexists with superconductivity, giving “reentrant spin glass superconductivity” (Zapf et al., 2013). Electron-doped EuFe$_2$6As$_2$7 shows the same hierarchy, with Eu canted antiferromagnetism, spin-glass freezing roughly $_2$8 K lower, and superconductivity that reenters in resistivity and optics (Baumgartner et al., 2017).

In Ba-based iron pnictides, the glassy state appears not in a rare-earth subsystem but in Fe-derived antiferromagnetic textures. NMR in Ba(Fe$_2$9Co$A$0)$A$1As$A$2 detects cluster spin-glass behavior and frozen AF domains for $A$3, precisely where bulk superconductivity is present (Dioguardi et al., 2013). Neutron scattering in BaFe$A$4Ni$A$5As$A$6 shows that the short-range incommensurate AF order near optimal superconductivity is not a nesting-driven spin-density wave but is consistent with a cluster spin glass embedded in a superconducting matrix (Lu et al., 2014).

Ruthenocuprates provide a layered version in which the magnetic and superconducting subsystems are spatially distinct. In YRu-1222 and EuRu-1222, the RuO$A$7 layers host spin-glass behavior with homogeneous or non-homogeneous ferromagnetic clusters, whereas superconductivity resides in the CuO$A$8 planes below roughly $A$9–$T_{cAFM}$0 K (Kumar et al., 2012, Kumar et al., 2011). In the borocarbides $T_{cAFM}$1Gd$T_{cAFM}$2Ni$T_{cAFM}$3B$T_{cAFM}$4C ($T_{cAFM}$5 Lu, Y), Gd moments suppress $T_{cAFM}$6, alter $T_{cAFM}$7, and generate a region of coexistence of superconductivity and spin-glass state on the $T_{cAFM}$8–$T_{cAFM}$9 diagram (Bud'ko et al., 2010).

Nickelates have broadened the subject substantially. Bulk infinite-layer LaNiO$_2$00, PrNiO$_2$01, and NdNiO$_2$02 show universal spin-glass behavior in the parent compounds, with frequency-dependent AC susceptibility peaks and memory effects (Lin et al., 2021). Magneto-optical measurements on superconducting La$_2$03Sr$_2$04NiO$_2$05 and Nd$_2$06Sr$_2$07NiO$_2$08 show spin-glass onset temperatures far above $_2$09, irreversible susceptibility, slow aging, and memory effects (Saykin et al., 3 Jun 2025). In bilayer nickelate films $_2$10Ni$_2$11O$_2$12, the low-temperature superconducting phase shows hysteretic magnetoresistance with coalescing minima at zero field, logarithmically slow resistance relaxation, and glass-like dynamics, which the authors identify as signatures of spin-glass superconductivity (Ji et al., 22 Aug 2025).

A useful limiting case is $_2$13-pyrochlore K$_2$14Os$_2$15O$_2$16. Here electrochemical potassium de-intercalation converts a superconducting ground state into a spin-glass-like state. This is not a coexistence regime in the strict sense, but it demonstrates how disorder and frustration can drive a superconducting material into a glassy magnetic state (Lee et al., 2011).

4. Microscopic mechanisms of coexistence

Several microscopic mechanisms recur. In Eu-based pnictides, the Eu$_2$17 moments interact through RKKY exchange mediated by conduction electrons in the Fe(As/P) layers. Intralayer Eu coupling is ferromagnetic, interlayer coupling is antiferromagnetic, and density-functional results indicate nearly degenerate ferromagnetic and antiferromagnetic states. This competition frustrates transverse components and produces reentrant spin-glass freezing below the canted $_2$18-type antiferromagnetic transition (Zapf et al., 2013). The same conduction bands also host superconductivity, so the Eu subsystem both couples to and perturbs the condensate.

One important proposal is that glassiness can facilitate coexistence rather than simply obstruct it. For EuFe$_2$19(As$_2$20P$_2$21)$_2$22, Zapf et al. argue that the freezing of in-plane Eu components destroys coherence between Eu layers, effectively decoupling them. Combined with a ferromagnetic Eu component along $_2$23 and possible spontaneous vortices perpendicular to the FeAs layers, this decoupling may reduce detrimental pair-breaking and support superconductivity in the FeAs layers (Zapf et al., 2013). This suggests that the glassy state is not merely tolerated but can alter the magnetic geometry in a way favorable to the condensate.

In electron-doped EuFe$_2$24As$_2$25, the proposed interplay is phrased in scattering terms. The materials are multiband $_2$26 superconductors; Eu magnetism introduces magnetic scattering, and the evolution from canted antiferromagnetism to spin glass modifies the balance between more pair-breaking and less pair-breaking channels. This is used to interpret why superconductivity appears, reenters, and only reaches zero resistance at lower temperature (Baumgartner et al., 2017).

In ruthenocuprates, the coexistence is helped by spatial separation: RuO$_2$27 layers host weak ferromagnetism, spin-glass order, ferromagnetic clustering, and superparamagnetic behavior, while superconductivity resides in CuO$_2$28 planes (Kumar et al., 2012). The authors of the YRu-1222 and EuRu-1222 studies emphasize that superconductivity and magnetism are “seemingly decoupled” or only weakly coupled in a gross sense, even though internal Ru fields likely influence vortices and low-field magnetization (Kumar et al., 2012, Kumar et al., 2011).

In nickelates, the mechanisms are more closely tied to disorder, orbital selectivity, and multi-orbital exchange frustration. In infinite-layer nickelates, the spin-glass state is linked to frozen, time-reversal-breaking magnetic textures of Ni and possibly rare-earth moments, with disorder from Sr substitution, oxygen content, or structural defects acting as a natural source of frustrated couplings (Saykin et al., 3 Jun 2025). In bilayer nickelates, successive oxygen reduction simultaneously suppresses both superconductivity and hysteretic signatures of the glassy state, which the authors connect to selective Ni orbitals near the Fermi level (Ji et al., 22 Aug 2025). In Nd$_2$29Pr$_2$30Ni$_2$31O$_2$32, increasing Nd content shrinks the spin-freezing region and enhances low-temperature spin fluctuations, suggesting a competition between static freezing and dynamic fluctuation regimes that may matter for possible superconductivity (Huangfu et al., 2023).

A general lesson emerges from these examples: strong static uniform magnetism tends to be strongly pair-breaking, whereas spatial separation, short-range clustering, or glassy randomization can weaken coherent internal fields and alter the competition. This is explicit in Eu-based pnictides and implicit in several other material classes.

5. Theoretical descriptions

The most direct theoretical framework in the supplied literature is a multiband Eliashberg treatment of a non-phononic $_2$33 spin-glass superconductor (Ummarino, 2020). In that model, superconductivity is mediated by dynamical antiferromagnetic spin fluctuations, while the spin-glass subsystem enters as a static, temperature-dependent magnetic scattering rate,

$_2$34

This term vanishes at $_2$35 and is maximal at $_2$36, so pair breaking is strongest at low temperature and weakens on warming (Ummarino, 2020).

Within this framework, the competition between pairing and glassy pair breaking produces several nonstandard outcomes. The superconducting gaps can be non-monotonic in temperature; the superfluid density can be suppressed at very low $_2$37 and increase on warming; and sufficiently strong spin-glass pair breaking can generate reentrant superconductivity, including cases with two separate superconducting temperature windows (Ummarino, 2020). These results formalize phenomena that are only qualitatively visible in the Eu-based experimental systems.

A complementary language comes from magnetic pair-breaking theory. For conventional superconductors, Abrikosov–Gor’kov theory gives

$_2$38

where $_2$39 is the pair-breaking parameter (Zapf et al., 2013). In the systems discussed here, the magnetic subsystem is not a dilute set of independent impurities but a dense or clustered network. Still, the same logic is invoked repeatedly: glassy or random internal fields may suppress $_2$40 less efficiently than coherent ferromagnetic order, especially when superconductivity resides in a different subsystem or layer (Zapf et al., 2013, Bud'ko et al., 2010).

A further extension is the theory of SDW and PDW glasses. Weak disorder destroys translational long-range order in incommensurate spin-density-wave or pair-density-wave states, but can leave a glass phase with Edwards–Anderson order together with conventional symmetry breaking. In the PDW case this can produce a glass retaining uniform charge-$_2$41 superconductivity, while in the SDW case it leaves spin-nematic order and glassy frozen disorder (Mross et al., 2015). This does not describe local-moment spin-glass superconductivity directly, but it broadens the theoretical meaning of a superconducting glass state with spin-glass characteristics.

6. Distinctions, controversies, and open questions

One recurring question is whether glassiness is detrimental, neutral, or even favorable to superconductivity. In EuFe$_2$42(As$_2$43P$_2$44)$_2$45, the authors conclude that “the development of superconductivity is supported by the decoupling of magnetic Eu layers in the glass phase” (Zapf et al., 2013). In contrast, the magneto-optical nickelate work concludes that there is “no simple, direct connection between SG order and the mechanism of superconductivity” because large changes in Kerr onset and magnitude can occur with only modest changes in $_2$46 (Saykin et al., 3 Jun 2025). These are not mutually exclusive statements, but they underscore that the functional role of glassiness is material-specific.

A second distinction concerns local versus bulk order. In Ba(Fe$_2$47Co$_2$48)$_2$49As$_2$50, bulk probes indicate that long-range AF order ends near $_2$51, whereas NMR still sees frozen AF clusters and glassy dynamics up to $_2$52 inside the superconducting state (Dioguardi et al., 2013). In BaFe$_2$53Ni$_2$54As$_2$55, neutron scattering shows that the incommensurate AF phase near optimal superconductivity is short-range and glassy rather than a homogeneous spin-density wave (Lu et al., 2014). These results caution against equating the disappearance of bulk long-range order with the disappearance of slow magnetism.

Nickelates add a further controversy concerning disorder and strain. In Nd$_2$56Sr$_2$57NiO$_2$58, films on SrTiO$_2$59 and LSAT exhibit very different Kerr onset temperatures and magnitudes while $_2$60 remains comparatively similar, suggesting that static spin-glass order is highly sensitive to crystallinity and substrate-induced disorder (Saykin et al., 3 Jun 2025). In bilayer nickelates, oxygen reduction suppresses both superconductivity and hysteretic glass-like transport, suggesting a more positive correlation between the two (Ji et al., 22 Aug 2025). A plausible implication is that “spin-glass superconductivity” in nickelates may cover more than one regime: one in which glassiness is a proximate background state, and another in which it is more directly entwined with the zero-resistance phase.

The open experimental agenda is clear in several of the source papers. For Eu-based pnictides, detailed neutron scattering, $_2$61SR, and NMR are needed to image the Eu spin texture and the superconducting state below $_2$62 (Zapf et al., 2013). For the Co- and Ni-doped BaFe$_2$63As$_2$64 systems, local probes remain essential for separating homogeneous coexistence from mesoscopic phase separation (Dioguardi et al., 2013, Lu et al., 2014). For nickelates, combined Kerr, $_2$65SR, RIXS, NMR, STM, and strain-controlled studies are needed to resolve how static spin freezing, dynamic spin fluctuations, and superconductivity are connected (Saykin et al., 3 Jun 2025, Huangfu et al., 2023). The broader theoretical problem is equally unsettled: quantitative models are still needed for RKKY-mediated glassiness in a superconducting environment, for multiband pair breaking by glassy dense-moment subsystems, and for the orbital-selective character of nickelate spin-glass superconductivity (Ummarino, 2020, Ji et al., 22 Aug 2025).

Spin-glass superconductivity therefore names not a single mechanism but a class of intertwined states in which superconductivity survives in the presence of frozen, slowly relaxing, and often spatially inhomogeneous magnetism. The most mature examples show that glassiness can emerge either from frustrated rare-earth moments, from clustered Fe antiferromagnetism, from Ru-layer magnetic textures, or from disorder-frustration backgrounds in nickelates. Across these cases, the central scientific question is how a condensate remains coherent when the magnetic sector does not order conventionally, but instead freezes into a glass.