Superfluid* Phase: Nonstandard Superfluidity

• Superfluid* phase is a nonstandard superfluid state featuring modified phase coherence and additional internal structure beyond a uniform condensate.
• It arises through mechanisms like emergent U(1)-like degeneracy, confinement-induced dimensional reduction, and phase modulation that alter the classic order parameter.
• Experimental and theoretical studies across helium and lattice systems provide quantifiable benchmarks and novel excitations, deepening our understanding of superfluid variants.

The expression Superfluid* phase is used in several distinct but structurally related ways in the literature. In the works considered here, it denotes superfluid states whose microscopic organization differs from the standard picture of a phase-uniform condensate protected solely by a broken global $U(1)$ symmetry. The deviation may arise from an emergent $U(1)$-like degeneracy, confinement-induced dimensional reduction, unconventional spin-orbital structure, interlayer coherence, phase modulation, or anomalous thermodynamic topology. This suggests that “superfluid*” functions less as a single universality class than as a recurring label for superfluid phases with additional internal structure, unusual order-parameter content, or nonstandard control parameters (Carlstrom et al., 2014, Prisk et al., 2012, Regan et al., 2019, Li et al., 2016).

1. Terminology, baseline structure, and defining motifs

In the conventional formulation, superfluidity is organized by a phase variable. For superconductors the complex order parameter is written as $\Delta=|\Delta|e^{i\phi}$, with superfluid velocity and current given by

$$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$

In confined helium, the relevant phase fingerprint is instead the excitation spectrum encoded in the dynamic structure factor

$$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$

whose phonon–roton, maxon, and layer-mode structure distinguishes microscopically different superfluid states (Holmvall et al., 2019, Prisk et al., 2012).

Within the emergent-superfluidity literature, the low-energy degree of freedom need not originate from an exact microscopic symmetry. A multicomponent system can develop an approximate or exact degeneracy manifold that supports an emergent phase variable $\phi$, with coarse-grained stiffness

$$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$

and helicity modulus

$$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$

In that setting, superfluidity is associated with phase rigidity and vortex-supporting transport even when the underlying symmetry protection is only approximate (Carlstrom et al., 2014).

A recurrent motif across these uses is that the starred phase is identified not merely by dissipationless flow, but by a modified relation between phase, symmetry, and microscopic structure. In some papers the modification is geometric or interfacial; in others it is orbital, topological, or thermodynamic. This suggests a family resemblance centered on nonstandard phase coherence rather than a single canonical order parameter.

2. Emergent phase rigidity and phase-modulated condensates

A direct formulation of a Superfluid* phase appears in the analysis of entropy- and flow-induced superfluidity. There the starting point is a multicomponent Ginzburg–Landau functional with first- and second-order Josephson couplings,

$$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$

or, in a London reduction,

$$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$

The central result is that nontopological phase fluctuations can stabilize superfluidity when the energy landscape has an approximate or exact continuous degeneracy. In that regime the system can move from a state with $U(1)$0 to one with $U(1)$1, while the correlator

$U(1)$2

becomes finite, signaling an induced superfluid channel. The same framework yields a flow-induced stabilization criterion through

$U(1)$3

with lower and upper critical drive scales. Vortices still destroy coherence in the emergent channel, and in one explicit model the $U(1)$4-sector remains superfluid until $U(1)$5, beyond which $U(1)$6 (Carlstrom et al., 2014).

A closely related, though terminologically distinct, development is the phase crystal. There the condensate phase itself crystallizes at finite wave vector,

$U(1)$7

without requiring phase winding or amplitude suppression. The microscopic mechanism is a nonlocal gradient functional

$U(1)$8

whose Fourier-space stiffness $U(1)$9 softens and changes sign at finite $\Delta=|\Delta|e^{i\phi}$0. Near pair-breaking $\Delta=|\Delta|e^{i\phi}$1-wave surfaces this produces a transition with $\Delta=|\Delta|e^{i\phi}$2, $\Delta=|\Delta|e^{i\phi}$3, and $\Delta=|\Delta|e^{i\phi}$4, together with alternating loop currents and spontaneous time-reversal breaking but no topological defects. The same mechanism extends to magnetically active $\Delta=|\Delta|e^{i\phi}$5-wave interfaces, where self-consistent calculations give $\Delta=|\Delta|e^{i\phi}$6 for a 2D annulus (Holmvall et al., 2019).

These two lines of work are conceptually aligned in that they relocate the origin of superfluid order from a rigid, spatially uniform phase to a more delicate object: either an emergent degeneracy coordinate or a finite-$\Delta=|\Delta|e^{i\phi}$7 phase texture. In both cases, the starred or nonstandard phase is defined by how phase rigidity is generated, not merely by the existence of nondissipative transport.

3. Confinement-induced Superfluid* phases in helium systems

In smooth cylindrical nanopores, confined $\Delta=|\Delta|e^{i\phi}$8He exhibits two microscopic superfluid phases distinguished by local density and dimensionality. In FSM-16 pores of diameter $\Delta=|\Delta|e^{i\phi}$9 Å, the thin-film phase at partial filling supports dilute layer modes (DLM) with a modified phonon–roton spectrum and compressed layer modes (CLM), while the saturated phase supports bulk-like 3D modes coexisting with CLM. For the thin-film phase at $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$0 mK and $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$1 mmol/g, the measured parameters are $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$2 m/s, $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$3 Å$$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$4, $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$5 K, and $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$6 amu; for the saturated phase at $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$7 mmol/g they are $$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$8 Å$$\mathbf v_s(\mathbf r)=\frac{\hbar}{2m}\nabla\phi(\mathbf r)-\frac{e}{m}\mathbf A(\mathbf r),\qquad \mathbf j(\mathbf r)=\hat{\rho}_s(\mathbf r)\,\mathbf v_s(\mathbf r).$$9, $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$0 K, and $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$1 amu; the CLM branch is characterized by $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$2 Å$$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$3, $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$4 K, and $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$5 amu. The crossover occurs rapidly around $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$6 mmol/g, where the DLM disappear and a 3D core emerges (Prisk et al., 2012).

Under nanoscale confinement of one spatial direction, $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$7He undergoes an order-parameter reduction from a $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$8 complex matrix to a $$S(q,\omega)=\int dt\,e^{i\omega t}\langle \rho_q(t)\rho_{-q}(0)\rangle,$$9 complex matrix $\phi$0, while the $\phi$1 sector $\phi$2 is suppressed at the transition. The quadratic Landau coefficients satisfy

$\phi$3

so $\phi$4 condenses first. Mean-field theory yields precisely two symmetry-inequivalent minima, the in-plane chiral A-phase and the planar phase, which are accidentally degenerate; strong-coupling corrections favor the A-phase observed experimentally, whereas weak-coupling perturbative RG favors the planar phase. Because $\phi$5, the quasiparticle spectrum is nodal at $\phi$6 regardless of the internal $\phi$7 structure (Sun et al., 2023).

A further confined $\phi$8He realization occurs on the second layer of graphite. Zero-temperature diffusion Monte Carlo finds second-layer promotion at $\phi$9 Å$$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$0, a stable bilayer from $$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$1 Å$$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$2, and a translationally invariant second layer with $$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$3 at $$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$4 and $$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$5 Å$$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$6. This liquid-like phase carries hexatic correlations: the $$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$7 component of the pair distribution exhibits a regular, slowly decaying oscillatory pattern, and that pattern disappears when the first-layer corrugation is smoothed. At $$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$8 Å$$F_{\mathrm{eff}}[\phi]\approx \int d^d x\,\frac{1}{2}\rho_s(\nabla\phi)^2,$$9, a 7/12 registered phase is stable and shows $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$0, whereas the 4/7 commensurate solid is unstable and the triangular incommensurate solids at $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$1 Å$$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$2 are normal (Gordillo et al., 2020).

Taken together, these studies show that confinement can produce Superfluid* behavior through several mechanisms: density-segregated excitation spectra in nanopores, orbital-channel elimination in quasi-2D $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$3He, and corrugation-induced superhexatic or supersolid order on graphite. The common element is that dimensional reduction and substrate structure become part of the phase definition.

4. Vortex-core, anisotropic, and field-stabilized helium phases

In rotating superfluid $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$4He-B, the starred terminology is attached to distinct vortex phases rather than to a single homogeneous condensate. The order parameter is a $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$5 complex matrix $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$6, and a strong-coupling Ginzburg–Landau functional with $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$7 and gradient terms predicts a first-order vortex transition line $$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$8 between a low-$$\Upsilon=\left(\frac{\partial^2 F}{\partial\theta^2}\right)_{\theta=0}.$$9, low-$$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$0 D-core phase and a high-$$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$1, high-$$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$2 A-core phase. The D-core spontaneously breaks axial rotation symmetry, while the A-core hosts both A-phase and $$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$3-phase components in the core. In zero field the calculated triple point is

$$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$4

and in $$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$5 fields with $$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$6 G the A-core stability region expands to lower pressures. A separate supercooling line $$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$7 marks the global instability of the metastable A-core phase (Regan et al., 2019).

Disorder and anisotropy in aerogel reorganize the helium-3 phase diagram even more strongly. In isotropic silica aerogel, correlated pair breaking suppresses $$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$8 but favors the B phase in zero field; in uniformly stretched silica aerogel an ESP A phase with $$F=\sum_i \frac{|\nabla\psi_i|^2}{2}+\sum_{i,j}\eta_{ij}\psi_i\psi_j^*+\sum_{i,j,k,l}\nu_{ijkl}\psi_i\psi_j\psi_k^*\psi_l^*,$$9 appears at $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$0, followed by a lower-temperature chiral ESP state with $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$1; in uniformly compressed silica aerogel a polar-distorted B phase is stabilized; and in nematically ordered alumina aerogels such as nafen, strong uniaxial anisotropy stabilizes the polar phase at $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$2, with lower-temperature transitions to polar-distorted A phases and the observation of half-quantum vortices. The same body of work emphasizes sharp thermodynamic transitions, with widths as small as $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$3K in high-quality isotropic aerogels (Halperin, 2018).

A particularly explicit field-stabilized phase is the superfluid $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$4 phase of $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$5He in nematic aerogel. Right below the superfluid transition and in high magnetic field, the paper identifies a P1 phase with order parameter

$$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$6

in which only the $$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$7 component condenses, and a lower-temperature distorted-$$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$8 P2 phase

$$H=-\sum_{i,\langle j,k\rangle}\cos(\phi_j^i-\phi_k^i)+\sum_k V(\phi_k^{12},\phi_k^{13}).$$9

which continuously approaches the polar phase as $U(1)$00. The transition temperatures obey

$U(1)$01

and the measured slope ratio is $U(1)$02, close to the bulk-parameter estimate $U(1)$03 at $U(1)$04 bar (Dmitriev et al., 2020).

In these helium-3 examples, the starred phase is encoded in vortex-core topology, anisotropic scattering environments, or spin-split ESP structure. The phase is therefore not exhausted by a scalar condensate amplitude; it requires specification of orbital symmetry, core order, and often the geometry that stabilizes it.

5. Lattice, junction, interlayer, and transport-defined realizations

Optical-lattice work on binary bosonic mixtures identifies a twisted superfluid as a multi-orbital superfluid with a complex order parameter

$U(1)$05

in which the local phase angle varies continuously between neighboring lattice sites. The transition from the normal superfluid to the twisted superfluid is second order, occurs only in binary mixtures, and is visible in time-of-flight as a reduction of first-order Bragg-peak symmetry from $U(1)$06 to $U(1)$07. In the simplified $U(1)$08 case the critical $U(1)$09-orbital admixture is

$U(1)$10

The phase is absent in single-component samples, consistent with the lack of interaction-induced $U(1)$11-$U(1)$12 coupling there (Soltan-Panahi et al., 2011).

Within the quantum phase model for interacting lattice bosons, current-carrying superfluid states are characterized by

$U(1)$13

For the Gaussian Gutzwiller ansatz one finds

$U(1)$14

with an existence boundary

$U(1)$15

and a modulational instability line

$U(1)$16

The maximum carried current occurs exactly at the modulational instability threshold, and in this model energetic and modulational instabilities coincide (Buonsante et al., 2013).

A different interfacial realization is the $U(1)$17-phase of a polarized superfluid Fermi gas. In a real-space BdG treatment of an attractive Hubbard model with a barrier, localized excess majority atoms produce an effective ferromagnetic junction, suppress pairing around the barrier, and stabilize a sign reversal of $U(1)$18 across the junction. The stability is measured by

$U(1)$19

which is negative at low $U(1)$20, approximately $U(1)$21 at $U(1)$22, and crosses zero at $U(1)$23. The $U(1)$24-phase survives at $U(1)$25, but by $U(1)$26 the order parameter no longer changes sign and the 0-phase is favored (Kashimura et al., 2011).

In double bilayer graphene, the phase is defined operationally by interlayer coherence of spatially indirect excitons in the quantum Hall regime. At total filling $U(1)$27, the condensate is diagnosed by

$U(1)$28

For device $U(1)$29 with $U(1)$30 nm, $U(1)$31 T, and $U(1)$32 mK, the upper critical ratio is $U(1)$33, and the minimum counterflow activation gap is $U(1)$34 K near $U(1)$35. Layer imbalance strengthens the condensate and stabilizes it to temperatures in excess of $U(1)$36 K. The condensate appears only when both layers occupy the orbital $U(1)$37 Landau-level states and is favored when the two layers have opposite valley ordering (Li et al., 2016).

Another transport-defined phase is the sliding Griffiths superfluid in a disordered stack of weakly Josephson-coupled layers. Functional RG generates an interlayer coupling distribution

$U(1)$38

and when $U(1)$39 the c-axis stiffness vanishes while the in-plane stiffness remains finite. The resulting phase has $U(1)$40 but $U(1)$41, together with a strongly anisotropic critical current, and it arises from disorder-induced smearing of the clean 3D XY transition into a Griffiths regime (Pekker et al., 2010).

These realizations show that Superfluid* behavior can be identified by momentum-space symmetry breaking, current-instability structure, phase-biased junction energetics, quantized Hall drag, or anisotropic stiffness. In several cases the phase is most sharply defined by transport rather than by a simple local order parameter.

6. Thermodynamic topology, formal analogies, and unresolved boundaries

Several papers place Superfluid* behavior inside a broader thermodynamic framework. In hypothetical ultra quantum liquids with nuclear mass below $U(1)$42 u, the superfluid order parameter $U(1)$43 is coupled to a liquid–gas order parameter $U(1)$44 through

$U(1)$45

and decreasing mass reorganizes the phase diagram through a sequence in which a tricritical point appears on the superfluid line, the liquid–gas critical point is swallowed beneath a first-order superfluid segment, and the tricritical point disappears when the scattering length crosses zero near $U(1)$46 u (Son et al., 2020). A related mean-field lattice-gas model of $U(1)$47He–$U(1)$48He mixtures gives

$U(1)$49

so the lambda temperature is proportional to the $U(1)$50He density, while vacancies generate a vapor phase and allow tricritical behavior and tricritical Casimir phenomenology (Bafi et al., 2014). A thermodynamically consistent Ginzburg–Landau model for liquid helium instead uses a scalar phase field $U(1)$51 with $U(1)$52 and a control variable $U(1)$53, leading to the equilibrium lambda line

$U(1)$54

and a second-order transition with zero latent heat (Berti et al., 2012).

Other works probe the conceptual boundaries of what counts as superfluid. In the activated-velocity-fluctuation extension of Model F, coupling the order parameter to a stochastic Navier–Stokes field changes the RG structure so strongly that developed turbulence destroys critical fluctuations, while the effective viscosity acquires the scaling exponent $U(1)$55 (Dančo et al., 2015). In a relativistic field-theory vortex solution, the traditional superfluid velocity

$U(1)$56

can become arbitrarily large in the sense $U(1)$57, yet the perturbations remain causal and subluminal; the paper concludes that the superfluid “velocity field” should not be interpreted as the actual velocity of a material constituent (Kourkoulou et al., 2023).

The term also migrates into more formal analogies. A holographic model of a non-relativistic superfluid in an asymptotically Schrödinger black-hole background exhibits a second-order transition at low background mass density, a strongly first-order transition above a multicritical $U(1)$58, and reentrance to the normal phase at high $U(1)$59, with the authors noting that the true $U(1)$60 ground state may lie outside the probe analysis (Adams et al., 2011). In restricted phase-space thermodynamics of a 4D dyonic AdS black hole with Kaniadakis entropy, the mixed $U(1)$61 ensemble displays a “superfluid $U(1)$62-phase transition,” identified by a $U(1)$63-shaped divergence of

$U(1)$64

and by a corresponding $U(1)$65-line in the $U(1)$66–$U(1)$67 plane (Baruah et al., 2024). This suggests that the Superfluid* label can extend beyond microscopic condensates into thermodynamic analogies, although the underlying degrees of freedom are then no longer those of an ordinary many-body superfluid.

A persistent theme across the literature is that nonstandard superfluid phases are easy to diagnose qualitatively but difficult to classify uniquely. Mean-field and weak-coupling RG disagree on order selection in confined $U(1)$68He (Sun et al., 2023); one-loop RG is explicitly insufficient to settle equilibrium fixed-point stability in the activated-velocity model (Dančo et al., 2015); the QPM Gaussian ansatz introduces artifacts cured only by improved trial states (Buonsante et al., 2013); and holographic low-temperature phases may be masked by background thermodynamics (Adams et al., 2011). The broad lesson is that Superfluid* phases are best understood as technically precise, context-dependent deformations of superfluid order rather than as a single universal phase of matter.