Re-entrant Superconductivity (RSC)

• RSC is defined by a superconducting phase that emerges, disappears, and reappears as a control parameter (e.g., temperature or magnetic field) is varied.
• Key mechanisms include competition between superconducting condensation energy and magnetic or exchange energies, as well as proximity interference effects in layered nanostructures.
• The phenomenon offers practical insights for tuning superconducting devices and understanding phase transitions in complex systems such as heavy fermions, oxide interfaces, and Josephson junctions.

Searching arXiv for recent and foundational papers on re-entrant superconductivity and related mechanisms. Searching arXiv for reentrant superconductivity. Searching for "reentrant superconductivity". arXiv search query: reentrant superconductivity Re-entrant superconductivity (RSC) denotes a non-monotonic superconducting response in which superconductivity appears, disappears, and then reappears as a control parameter is varied. In the systems surveyed here, the relevant control parameter can be temperature, magnetic field, ferromagnet thickness, magnetic flux, carrier density, radio-frequency power, or the balance between competing superconducting instabilities. The term therefore covers several distinct phenomena: temperature-driven partial loss and recovery of superconductivity in ferromagnet–superconductor nanostructures, field-induced high-field superconducting domes in uranium heavy fermions, thickness-driven extinction and recovery in proximity systems, flux-controlled reappearance of Josephson critical current in Corbino junctions, and geometry- or interaction-driven re-emergence of phase coherence in low-dimensional platforms (Singh et al., 2013, Goryo, 30 Dec 2025, Lesser et al., 20 Jan 2026).

1. Definitions and phenomenological forms

RSC is not a single microscopic mechanism but a family of re-entrant behaviors with a common operational structure: a superconducting phase is separated into two disconnected regions by an intervening non-superconducting or normal-like regime. In the HoNi$_5$–NbN–HoNi$_5$ trilayers, the sequence on cooling is metallic normal state, onset of superconductivity near $6$ K, a normal-like resistive interval just below the onset, and then recovery of zero resistance below about $5$ K; the re-entrant interval is bounded by the minimum and subsequent maximum in $R(T)$ (Singh et al., 2013). In UTe$_2$, the canonical form is field-driven: a low-field superconducting phase is suppressed at intermediate field and a second high-field superconducting phase reappears above $\sim 40$ T for a restricted angular range (Zambra et al., 10 Jun 2025). In Corbino Josephson junctions, RSC refers to the flux-driven reappearance of a dissipationless Josephson current at selected vortex numbers after fluxoid quantization has suppressed it at intermediate values (Lesser et al., 20 Jan 2026).

A compact way to organize the phenomenology is to distinguish the variable that generates the re-entrant sequence.

Control parameter	Representative realization	Re-entrant pattern
Temperature $T$	HoNi$_5$–NbN–HoNi$_5$ trilayers (Singh et al., 2013)	normal $_5$0 SC $_5$1 normal-like $_5$2 SC
Magnetic field $_5$3	UTe$_5$4, spin-triplet GL theories (Zambra et al., 10 Jun 2025)	SC $_5$5 normal $_5$6 SC
Ferromagnet thickness $_5$7	Cu$_5$8Ni$_5$9/Nb/Cu$6$0Ni$6$1 (Zdravkov et al., 2013)	SC $6$2 no SC $6$3 SC
Flux / vortex number $6$4	Corbino Josephson junctions (Lesser et al., 20 Jan 2026)	$6$5 for most $6$6, $6$7 periodically
RF power / temperature	granular aluminum Josephson array (Avraham et al., 2 Sep 2025)	SC $6$8 insulator $6$9 SC
Temperature and altermagnetic exchange	altermagnets with dispersive phonons (Iorsh, 24 Mar 2025)	normal at $5$0, SC at finite $5$1, FFLO at low $5$2

A further distinction concerns whether the high- and low-parameter superconducting phases are the same. In the conical-ferromagnet/superconductor bilayers studied with self-consistent BdG theory, superconductivity exists only in an intermediate temperature interval and vanishes both above the upper transition and below the lower one; the low- and high-temperature phases are the same normal state, and the re-entrance is described as strictly reentrant (Wu et al., 2011). By contrast, in twisted trilayer graphene the high-field phase is proposed to be a distinct finite-momentum superconducting state rather than a simple reappearance of the low-field condensate (Lake et al., 2021).

2. Principal microscopic mechanisms

One broad class of RSC arises from competition between superconducting condensation energy and magnetic or exchange energy. In HoNi$5$3–NbN–HoNi$5$4, the re-entrant interval appears only when the trilayer ordering scales satisfy $5$5, and its origin is attributed to a delicate balance between the magnetic exchange energy and the condensation energy in the interfacial regions of the trilayer (Singh et al., 2013). A related temperature-driven competition appears in YBa$5$6Cu$5$7O$5$8 microstructured particles, where bulk superconductivity coexists with surface-dominated ferromagnetism amplified by particle size reduction and grain boundaries; the 385 nm particles show $5$9 K and $R(T)$0 K, with superconductivity suppressed again at low temperature (López-Romero et al., 2020). In dirty BCS superconductors with magnetic impurities, mean-field equations for the gap $R(T)$1 and impurity parameter $R(T)$2 yield normal, unperturbed superconducting, perturbed superconducting, perturbed gapless superconducting, and impurity phases; within that structure, both re-entrant superconductivity and Jaccarino–Peter compensation appear (Borycki et al., 2010).

A second class is proximity-effect interference. In F/S/F trilayers of Cu$R(T)$3Ni$R(T)$4/Nb/Cu$R(T)$5Ni$R(T)$6, the FFLO-like state in the ferromagnet causes the singlet pair amplitude to oscillate and decay away from the S/F interface, so interference between the two S/F interfaces produces oscillatory $R(T)$7 and, in favorable cases, extinction and recovery of superconductivity as the ferromagnet thickness increases (Zdravkov et al., 2013). The same general theme appears in Ho-type conical-ferromagnet/superconductor bilayers, but there the interference is temperature dependent: self-consistent BdG calculations show that for a range of $R(T)$8 the pair amplitude and condensation free energy vanish both above the upper transition and below the lower one, while the entropy reveals a temperature window where the superconducting state is less ordered than the normal state (Wu et al., 2011).

A third class is field-driven reorganization of competing superconducting instabilities. A minimal Ginzburg–Landau theory with two coupled spin-triplet order parameters $R(T)$9 introduces an internal Josephson coupling $_2$0 that favors a spinful instability with $_2$1, and a linear field coupling $_2$2 that favors a spin-polarized instability with $_2$3. The lower instability eigenvalue

$_2$4

can therefore vanish at low field, re-enter at high field, and generate a non-monotonic $_2$5 without invoking Jaccarino–Peter compensation, FFLO states, or special band structures (Goryo, 30 Dec 2025). A closely related microscopic BdG analysis shows that for a spin-triplet superconductor with $_2$6-vector, spin–orbit vector, and Zeeman field mutually perpendicular, spin–orbit interaction suppresses superconductivity at weak Zeeman fields and enhances it at strong Zeeman fields; the superconducting instability is correlated with even-frequency components, while instability is correlated with odd-frequency Cooper pairs (Sano et al., 16 Feb 2026).

A fourth class is driven by fluctuation physics in strongly correlated magnets. In uranium heavy fermions, the field-induced superconducting dome is associated with magnetic fluctuations near first-order magnetic transitions. The phenomenological theory for URhGe, UCoGe, and UTe$_2$7 identifies a central asymmetry: magnetizations along the easy and hard axes have opposite effects on superconductivity, and the high-field RSC is induced by the fluctuation parallel to the applied magnetic field (Feng et al., 2020). In UTe$_2$8, magnetotropic measurements at 4 K reveal a very large decrease in magnetotropic susceptibility, implying a giant increase in transverse susceptibility; the three superconducting phases, including the high-field re-entrant phase, surround the region of enhanced transverse susceptibility in the field-angle phase diagram, and the strongest transverse susceptibility lies near the critical end point of the high-field metamagnetic transition (Zambra et al., 10 Jun 2025).

3. Interface-driven and proximity-driven nanostructures

The HoNi$_2$9–NbN–HoNi$\sim 40$0 trilayer is a prototypical temperature-driven interface RSC system. The effect occurs only when the NbN layer is thin ($\sim 40$1 nm) and each HoNi$\sim 40$2 layer lies in the intermediate range $\sim 40$3; for $\sim 40$4 nm the re-entrant minimum–peak structure in $\sim 40$5 is clear, while for $\sim 40$6 nm and $\sim 40$7 nm it disappears (Singh et al., 2013). The field dependence is sharply anisotropic: in-plane fields $\sim 40$8–1500 Oe scarcely affect the re-entrant structure, whereas out-of-plane fields broaden the transition, shift the minimum and peak to lower temperature, and monotonically suppress $\sim 40$9 until re-entrance is quenched. Transport current has a similarly non-monotonic effect: the re-entrant feature strengthens as $T$0 rises to $T$1 A/cm$T$2, then weakens and broadens at higher $T$3, with a sharp drop in effective activation energy above $T$4 A/cm$T$5. Thermally activated flux flow analysis gives $T$6 with $T$7 for a bare 10 nm NbN film, $T$8 for a HoNi$T$9/NbN bilayer, and $_5$0 for the symmetric trilayer, indicating a progression from plastic flux creep toward single-vortex pinning as ferromagnetic layers are added on both sides (Singh et al., 2013).

The Cu$_5$1Ni$_5$2/Nb/Cu$_5$3Ni$_5$4 trilayers grown on CoO$_5$5 illustrate thickness-driven RSC in superconducting spin-valve architectures. The key series AF-FSF5a, with $_5$6 nm, shows clear re-entrant superconductivity: just after fabrication, superconductivity disappears for $_5$7 nm and remains absent down to 380 mK over a wide thickness interval, but at larger ferromagnet thicknesses superconductivity reappears (Zdravkov et al., 2013). The effect is interpreted in the standard proximity language of oscillatory singlet pair amplitudes in the weak ferromagnet CuNi. The authors emphasize that large $_5$8 oscillations or RSC are favorable because they indicate strong interference of the pair amplitude, the regime in which a superconducting spin valve would be most sensitive to the relative orientation of the ferromagnetic layers. CoO$_5$9 is introduced to exchange-bias one CuNi layer, and the central result is that the FFLO-like proximity state remains undisturbed by the antiferromagnetic oxide layer, even though the realized exchange bias is too weak for a robust antiparallel configuration (Zdravkov et al., 2013).

Self-consistent BdG calculations for conical-ferromagnet/superconductor bilayers extend the interface picture into a genuinely thermodynamic temperature-reentrant regime. For a fixed superconducting thickness $_5$0 and Ho-like conical magnet with $_5$1, $_5$2, and $_5$3, the computed $_5$4 oscillates with a period set by the spiral period, and around $_5$5 a lower critical line $_5$6 appears beneath the upper critical line $_5$7 (Wu et al., 2011). For $_5$8, the pair amplitude one coherence length into S vanishes at both $_5$9 and $_5$00, while the condensation free energy is negative only for intermediate temperatures. The entropy difference changes sign inside the superconducting interval, so that the superconducting state is less ordered than the normal state over part of that interval, a feature that sharply distinguishes this strict re-entrance from the usual BCS ordering pattern (Wu et al., 2011).

4. Uranium heavy fermions and spin-triplet field-induced phases

U-based heavy-fermion superconductors display the most prominent field-induced RSC. In UTe$_5$01, superconductivity at low field is suppressed around $_5$02–15 T depending on orientation, and a high-field superconducting phase reappears above $_5$03 T for a restricted band of angles around the crystallographic $_5$04 axis; the high-field dome is often called SC3, with an intermediate SC2 phase in some angle ranges (Zambra et al., 10 Jun 2025). The normal-state magnetotropic susceptibility at 4 K provides a direct magnetic precursor of that dome: for $_5$05, the inferred transverse susceptibility $_5$06 becomes more than 30 times larger than the longitudinal $_5$07 by 40 T, and the strongest transverse susceptibility is found near the critical end point of the high-field metamagnetic transition. This supports the view that critical transverse fluctuations of a field-induced magnetic order parameter may provide the pairing mechanism for the high-field phase, in analogy with URhGe and UCoGe but without pre-existing zero-field ferromagnetism (Zambra et al., 10 Jun 2025).

The phenomenological theory for URhGe, UCoGe, and UTe$_5$08 makes the field-direction dependence explicit. In the low-field ferromagnetic superconductors URhGe and UCoGe, the easy-axis magnetization $_5$09 and the field-direction magnetization $_5$10 enter the renormalized superconducting coefficients with opposite effects; transverse field $_5$11 initially suppresses low-field SC, but near the first-order magnetic transition the non-unitary couplings $_5$12 and $_5$13 favor a high-field triplet state with Cooper-pair spin polarization $_5$14, thereby generating the re-entrant dome (Feng et al., 2020). In UTe$_5$15, the same formalism yields a field-induced non-unitary high-field state near the metamagnetic jump in $_5$16, and the abrupt truncation of RSC at the first-order jump follows because the static magnetization terms overwhelm the fluctuation-enhanced pairing terms once $_5$17 becomes too large (Feng et al., 2020).

A distinct, more kinematic explanation was proposed for UTe$_5$18 on the basis of its quasi-two-dimensional Fermi surface. Using corrugated cylindrical sheets with small interlayer hopping $_5$19, Mineev argued that for $_5$20 the cyclotron scale $_5$21 can exceed $_5$22, effectively confining electrons to quasi-two-dimensional motion and suppressing orbital pair-breaking at high field; because equal-spin triplet pairing is assumed, Pauli limiting is absent, so superconductivity can re-emerge and $_5$23 tends back toward its zero-field value at large $_5$24 (Mineev, 2020). This suggests that not all uranium-based RSC must be fluctuation-driven; a plausible implication is that magnetic fluctuations and quasi-two-dimensional orbital physics are complementary rather than mutually exclusive ingredients in UTe$_5$25.

5. Low-dimensional, geometric, and topological realizations

In the (110)-oriented LaTiO$_5$26–KTaO$_5$27 interface, RSC appears in a two-dimensional oxide superconductor under an in-plane magnetic field. The characteristic feature is a resistance peak at $_5$28 T that is nearly independent of gate voltage and temperature within the superconducting regime: at fixed intermediate temperature the sample is superconducting at low $_5$29, becomes resistive near $_5$30 T, and then recovers superconductivity at larger $_5$31 (Maryenko et al., 2 Oct 2025). The proposed mechanism combines a van Hove-like extended saddle point, strong Dresselhaus-like spin–orbit coupling, and a field-driven reshaping of the Fermi surface near the Brillouin-zone boundary. The resulting competition is between an enhanced density of states, which tends to increase $_5$32, and reduced pairing efficiency caused by Fermi-surface asymmetry and finite-$_5$33 pairing, which lowers $_5$34; numerically this yields a non-monotonic $_5$35 with a minimum at $_5$36 T (Maryenko et al., 2 Oct 2025). The paper explicitly argues against simple Pauli limiting and against an in-plane vortex interpretation, because the peak field is gate independent and the thickness inferred from a vortex-onset estimate would be unrealistically large for an oxide interface.

Corbino Josephson junctions realize a flux-driven and geometry-controlled form of RSC. In a circular Corbino junction, fluxoid quantization enforces integer winding $_5$37, and for any $_5$38 the critical current vanishes; there is no re-entrance (Lesser et al., 20 Jan 2026). In non-circular junctions with $_5$39 corners, however, the phase profile acquires a corner harmonic, and analytical expansion of

$_5$40

shows that in a trivial junction the critical current is nonzero only when $_5$41. On a 3D topological-insulator surface with chiral Majorana modes, the effective coupling $_5$42 and vortex-dependent boundary conditions halve that period for even $_5$43, giving re-entrance already at $_5$44 (Lesser et al., 20 Jan 2026). The resulting period halving is proposed as a marker of topological superconductivity in non-circular Corbino devices.

Granular aluminum provides a many-body realization in a naturally occurring Josephson-junction array tuned by RF power. At low temperatures the grAl nanobridge exhibits giant Shapiro steps characteristic of a single effective Josephson junction, indicating coherent phase locking across many junctions. Increasing RF power reduces the effective Josephson coupling $_5$45, and the system can be driven from a superconducting stiff-phase regime into a strongly insulating phase; remarkably, at elevated temperature improved screening suppresses the charging energy so that superconductivity reappears, producing an RF-tuned SC $_5$46 insulator $_5$47 SC sequence (Avraham et al., 2 Sep 2025). The interpretation explicitly invokes the competition between $_5$48 and $_5$49, and the authors argue that the many-body effect cannot be described within a single-junction framework.

The same theme of field-stabilized or finite-momentum superconductivity appears in more strongly momentum-space-driven theories. In twisted trilayer graphene, the high-field superconducting phase is proposed to emerge when the in-plane field drives the superfluid stiffness through zero while leaving the pairing amplitude finite, producing a quantum Lifshitz transition in the phase mode and a high-field finite-momentum state (Lake et al., 2021). In altermagnets with dispersive-phonon-mediated attraction, altermagnetism suppresses singlet pairing at low temperature, but thermal phonon scattering suppresses the effective altermagnetic pair breaking, leading to re-entrant superconductivity at finite temperature and stabilization of an FFLO-like finite-momentum superconducting state at low temperature (Iorsh, 24 Mar 2025). A unifying suggestion is that RSC often marks a regime where the instability of the uniform condensate opens a nearby channel—triplet, finite-$_5$50, or geometry-selected—that becomes dominant only after the original one has been weakened.

6. Conceptual boundaries, recurring misconceptions, and open directions

A recurrent misconception is that RSC always means superconductivity destroyed by magnetism and restored only by magnetic compensation. The record summarized here is broader. In some systems the mechanism is indeed magnetic, as in dirty superconducting alloys with magnetic impurities and in uranium heavy fermions (Borycki et al., 2010). In others the dominant ingredient is proximity interference in an exchange field (Zdravkov et al., 2013), field-driven Fermi-surface reshaping under strong spin–orbit coupling (Maryenko et al., 2 Oct 2025), geometric phase winding in Josephson structures (Lesser et al., 20 Jan 2026), or competition between different superconducting instabilities in multicomponent triplet order parameters (Goryo, 30 Dec 2025). This suggests that “re-entrant” is a phenomenological classification rather than a microscopic one.

A second misconception is that the re-entered superconducting phase must be identical to the low-field or high-temperature phase. The bilayer with a conical ferromagnet is strictly reentrant in the sense that the high- and low-temperature phases are the same normal state (Wu et al., 2011). By contrast, the high-field phases in uranium compounds are commonly discussed as distinct non-unitary or fluctuation-stabilized triplet states (Feng et al., 2020), and the high-field phase in twisted trilayer graphene is proposed to be finite-momentum paired (Lake et al., 2021). In spin-triplet GL theory, the field reorganizes the hierarchy of instabilities so that the leading mode crosses over from spinful to spin-polarized along the same instability line (Goryo, 30 Dec 2025). A plausible implication is that RSC often signals an internal reconstruction of the order parameter rather than simple recovery of a pre-existing condensate.

A third misconception is that RSC is always a bulk thermodynamic phase transition. Corbino Josephson junctions show a fluxoid-quantized reappearance of dissipationless Josephson current governed by geometry and topology rather than by bulk restoration of the superconducting order parameter (Lesser et al., 20 Jan 2026). Granular aluminum shows RSC in a phase-fluctuating array where charge screening and Josephson coupling are the decisive scales (Avraham et al., 2 Sep 2025). These examples place RSC at the boundary between superconducting thermodynamics and superconducting coherence.

Across the surveyed systems, several technical themes recur. One is the existence of at least two competing energy scales that cross as the control parameter is varied: $_5$51 versus $_5$52 in HoNi$_5$53–NbN–HoNi$_5$54, $_5$55 versus $_5$56 in grAl, orbital stiffness versus quartic-gradient stabilization in twisted trilayer graphene, and spinful versus spin-polarized channels in triplet GL theory (Singh et al., 2013, Avraham et al., 2 Sep 2025, Lake et al., 2021, Goryo, 30 Dec 2025). Another is the special role of anisotropy: HoNi$_5$57 easy-plane magnetization, orthorhombic uranium compounds, (110) oxide interfaces, and polygonal Corbino boundaries all use anisotropy to prevent simple monotonic suppression and to open a re-entrant channel (Singh et al., 2013, Zambra et al., 10 Jun 2025, Maryenko et al., 2 Oct 2025, Lesser et al., 20 Jan 2026).

Open questions are system dependent but show strong overlap. In heavy fermions, a more microscopic theory linking order-parameter structure, metamagnetic criticality, and field-angle dependence remains unresolved (Zambra et al., 10 Jun 2025). In S/F hybrids, direct probes of interfacial magnetization and superconducting gap profiles are needed to determine how exchange fields reshape the effective superconducting thickness through the re-entrant window (Singh et al., 2013). In low-dimensional platforms, the precise character of the intervening resistive state—vortex liquid, phase-fluctuating superconductor, or distinct paired state—remains open in oxide interfaces and granular arrays (Maryenko et al., 2 Oct 2025, Avraham et al., 2 Sep 2025). In multicomponent and topological settings, the direct experimental identification of the high-field or geometry-selected order parameter—spin-polarized triplet, finite-$_5$58, or Majorana-mediated even-harmonic state—is the natural next step (Goryo, 30 Dec 2025, Lesser et al., 20 Jan 2026).

Re-entrant superconductivity therefore occupies a distinctive place in superconductivity research: it is a robust diagnostic of competing scales, frustrated coherence, and order-parameter reorganization. Rather than contradicting superconductivity, the re-entrant regime exposes the conditions under which superconductivity survives by changing its character, its spatial structure, or the route by which phase coherence is established.