We extend the Luttinger-Ward free energy functional to disordered superconductors and superfluids with arbitrary scattering mechanisms, including both impurity disorder and the presence of interfaces. The disorder is taken into account within the self-consistent t-matrix approximation, thus allowing for arbitrary impurity scattering strengths. It is shown that both the interface and the impurity scattering self-energy appear in the functional only implicitly, through self-consistently determined fermionic propagators. The free energy functional is formulated in terms of a generalized integral in the complex energy plane, which encompasses formulations both in terms of retarded/advanced propagators and in terms of Matsubara Green's functions by appropriately choosing the integration path. It can be applied, e.g., to spatially non-uniform and hybrid systems, triplet and other unconventional condensates, and strongly-correlated Fermi liquids. A particularly useful formulation in terms of the quasiclassical propagators is applied to unconventional non-uniform singlet and triplet superconductors and superfluids in an external Zeeman magnetic field.

Anisotropic pair breaking close to surfaces favors the chiral A phase of the superfluid ^{3}He over the time-reversal invariant B phase. Confining the superfluid ^{3}He into a cavity of height D of the order of the Cooper pair size characterized by the coherence length ξ_{0}-ranging between 16 nm (34 bar) and 77 nm (0 bar)-extends the surface effects over the whole sample volume, thus allowing stabilization of the A phase at pressures P and temperatures T where otherwise the B phase would be stable. In this Letter, the surfaces of such a confined sample are covered with a superfluid ^{4}He film to create specular quasiparticle scattering boundary conditions, preventing the suppression of the superfluid order parameter. We show that the chiral A phase is the stable superfluid phase under strong confinement over the full P-T phase diagram down to a quasi-two-dimensional limit D/ξ_{0}=1, where D=80  nm. The planar phase, which is degenerate with the chiral A phase in the weak-coupling limit, is not observed. The gap inferred from measurements over the wide pressure range from 0.2 to 21.0 bar leads to an empirical ansatz for temperature-dependent strong-coupling effects. We discuss how these results pave the way for the realization of the fully gapped two-dimensional p_{x}+ip_{y} superfluid under more extreme confinement.

NMR experiments on liquid $^3$He infused into uniaxially anisotropic silica aerogels show the stabilisation of two equal-spin-pairing chiral phases on cooling from the normal phase. The alignment of the chiral axis relative to the anisotropy axis for these phases is predicted to depend upon temperature. A chiral A-like phase is also stabilized when $^3$He is confined to a slab of thickness $D\sim ξ$, the superfluid coherence length. For both types of confinement, scattering of quasiparticles by the random potential - aerogel or surface - is pair breaking and generates a sub-gap density of quasiparticle states. The random field also conspires with the chiral order parameter to generate skew scattering of quasiparticles in the plane normal to the chiral axis. This scattering mechanism leads to anomalous thermal Hall transport for nonequilibrium quasiparticles driven by a thermal gradient. We report theoretical results for the anomalous thermal Hall conductivity for theoretical models for chiral phases of $^3$He in both anisotropic aerogel and slabs. The anomalous thermal Hall effect (ATHE) provides an important tool to identify signatures of broken time-reversal and mirror symmetries and topology in chiral superconductors/superfluids.

Superfluid 3He, with unconventional spin-triplet p-wave pairing, provides a model system for topological superconductors, which have attracted significant interest through potential applications in topologically protected quantum computing. In topological insulators and quantum Hall systems, the surface/edge states, arising from bulk-surface correspondence and the momentum space topology of the band structure, are robust. Here we demonstrate that in topological superfluids and superconductors the surface Andreev bound states, which depend on the momentum space topology of the emergent order parameter, are fragile with respect to the details of surface scattering. We confine superfluid 3He within a cavity of height D comparable to the Cooper pair diameter ξ0. We precisely determine the superfluid transition temperature Tc and the suppression of the superfluid energy gap, for different scattering conditions tuned in situ, and compare to the predictions of quasiclassical theory. We discover that surface magnetic scattering leads to unexpectedly large suppression of Tc, corresponding to an increased density of low energy bound states.

We study thermal transport in a two-dimensional system with coexisting $s$- or $d$-wave superconducting (SC) and spin density wave (SDW) orders. We analyze the nature of coexistence phase in a tight-binding square lattice with $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$ SDW ordering. The electronic thermal conductivity is computed within the framework of the Boltzmann kinetic theory, using Born approximation for the impurity scattering collision integral. We describe the influence of the Fermi surface (FS) topology, the competition between the SC and SDW order parameters, and the presence or absence of zero energy excitations in the coexistence phase, on the low temperature behavior of thermal conductivity of the various pairing states. We present qualitative analytical and fully numerical results that show that the heat transport signatures of various SC states emerging from collinear SDW order are quite distinct and depend on the symmetry properties of the SC order parameter under translation by the SDW nesting vector $\mathbf{Q}$. A combination of $(\ensuremath{\pi},\ensuremath{\pi})$-SDW and the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state results in fully gapped excitations, whereas $(\ensuremath{\pi},\ensuremath{\pi})$-SDW coexisting with either ${d}_{xy}$ or $s$-wave pairing states may always have gapless excitations. There appear special stable Dirac nodal points that are not gapped by the SC order in the coexistence phase, resulting in finite residual heat conductivity.

We predict an inhomogeneous phase of superfluid He films in which translational symmetry is spontaneously broken in the plane of the film. This phase is energetically favored over a range of film thicknesses, Dc2(T ) < D < Dc1(T ), separating distinct homogeneous superfluid phases. The instability at the critical film thickness, Dc2 ≈ 9 ξ(T ), is a single-mode instability generating striped phase order in the film. Numerical calculations of the order parameter and free energy indicate a second-order instability to a periodic lattice of degenerate B-like phases separated by domain walls at Dc1 ≈ 12 ξ(T ). The striped phase should be identifiable in transport and nuclear magnetic resonance experiments.

The recent Comment by Vorontsov [arXiv:2007.13696] claims that surface pair-density-wave superconductivity with critical temperature higher than the bulk FFLO critical temperature is not supported by microscopic theory. The conclusion is reached by using an approximate semi-microscopic quasiclassical approach. Here we show that a fully microscopic approach unambiguously demonstrates the existence of surface pair-density-wave superconductivity.

Superconductivity owes its properties to the phase of the electron pair condensate that breaks the U(1) symmetry. In the most traditional ground state, the phase is uniform and rigid. The normal state can be unstable towards special inhomogeneous superconducting states: the Abrikosov vortex state and the Fulde-Ferrell-Larkin-Ovchinnikov state. Here we show that the phase-uniform superconducting state can go into a fundamentally different and more ordered nonuniform ground state, which we refer to as a phase crystal. This state breaks translational invariance through formation of a spatially periodic modulation of the phase, manifested by unusual superflow patterns and circulating currents, that also break time-reversal symmetry. We list the general conditions needed for realization of phase crystals. Using microscopic theory, we then derive an analytic expression for the superfluid density tensor for the case of a nonuniform environment in a semi-infinite superconductor. We demonstrate how the surface quasiparticle states enter the superfluid density and identify phase crystallization as the main player in several previous numerical observations in unconventional superconductors, and predict the existence of a similar phenomenon in superconductor-ferromagnetic structures. This analytic approach provides a unifying aspect for the exploration of boundary-induced quasiparticles and collective excitations in superconductors. More generally, we trace the origin of phase crystallization to nonlocal properties of the gradient energy, which implies the existence of similar pattern-forming instabilities in many other contexts.

Surfaces of d-wave superconductors may host a substantial density of zero-energy Andreev states. The zero-energy flat band appears due to a topological constraint, but comes with a cost in free energy. We have recently found that an adjustment of the surface states can drive a phase transition into a phase with finite superflow that breaks time-reversal symmetry and translational symmetry along the surface. The associated Doppler shifts of Andreev states to finite energies lower the free energy. Direct experimental verification of such a phase is still technically difficult and controversial, however. To aid further experimental efforts, we use the quasiclassical theory of superconductivity to investigate how the realization and the observability of such a phase are influenced by sample geometry and surface ruggedness. Phase diagrams are produced for relevant geometric parameters. In particular, critical sizes and shapes are identified, providing quantitative guidelines for sample fabrication in the experimental hunt for symmetry-breaking phases.

topological superfluid He P. J. Heikkinen, ∗ A. Casey, L. V. Levitin, X. Rojas, A. Vorontsov, P. Sharma, N. Zhelev, J. M. Parpia, and J. Saunders Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK Department of Physics, Montana State University, Bozeman, Montana 59717, USA Department of Physics, Indian Institute of Science, Bangalore 560 012, India Department of Physics, Cornell University, Ithaca, NY 14853, USA

This paper reviews confinement-driven phase transitions in superconductors and Bardeen–Cooper–Schrieffer superfluids, and the appearance in thin films of new phases that break the time-reversal or translational symmetry. The origins of the new phases are closely tied to the Andreev scattering processes involving particle-hole conversions that create surface quasiparticle states with energies inside the superconducting gap. Restructuring of the low-energy spectrum in the surface region of several coherence lengths ξ 0 results in large spatial variations of the superconducting order parameter. In confined geometry, such as slabs, films, pores or nano-dots, with one or more physical dimensions D ∼10 ξ 0 , the Andreev bound states can dominate properties of a superconductor, leading to modified experimental signatures. They can significantly change the energy landscape, and drive transitions into new superconducting phases. The new phases are expected in a variety of materials, from singlet d -wave superconductors to multi-component triplet superfluid 3 He, but properties of the new phases will depend on the symmetry of the parent state. I will highlight the connection between the Andreev surface states and confinement-stabilized phases with additional broken symmetries, describe recent progress and open questions in the theoretical and experimental investigation of superfluids in confined geometry. This article is part of the theme issue ‘Andreev bound states’.

Flat bands of zero-energy states at the edges of quantum materials have a topological origin. However, their presence is energetically unfavorable. If there is a mechanism to shift the band to finite energies, a phase transition can occur. Here we study high-temperature superconductors hosting flat bands of midgap Andreev surface states. In a second-order phase transition at roughly a fifth of the superconducting transition temperature, time-reversal symmetry and continuous translational symmetry along the edge are spontaneously broken. In an external magnetic field, only translational symmetry is broken. We identify the order parameter as the superfluid momentum ps, that forms a planar vector field with defects, including edge sources and sinks. The critical points of the vector field satisfy a generalized Poincaré-Hopf theorem, relating the sum of Poincaré indices to the Euler characteristic of the system.

We study the dynamics of a superconducting condensate in the presence of a domain wall defect in the order parameter. We find that broken translation and reflection symmetries result in collective excitations, bound to the domain wall region. Two additional amplitude/Higgs modes lie below the bulk pair-breaking edge 2 Delta; one of them is a Goldstone mode with vanishing excitation energy. The spectrum of bound collective modes is related to the topological structure and stability of the domain wall. The "unbound" bulk collective modes and transverse gauge field mostly propagate across the domain wall, but the longitudinal component of the gauge field is completely reflected. Softening of the amplitude mode suggests reduced damping and a possible route to its detection in geometrically confined superfluids or in superconductor-ferromagnetic heterostructures.

We calculate electronic energy transport in inhomogeneous superconductors using a fully self-consistent nonequilibrium quasiclassical Keldysh approach. We develop a general theory and apply it to a superconductor with an order parameter that forms domain walls of the type encountered in the Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a domain wall is inherently anisotropic and nonlocal. The bound states in the nonuniform region play a crucial role and control heat transport in several ways: (i) they modify the spectrum of quasiparticle states and result in Andreev reflection processes and (ii) they hybridize with the impurity band and produce a local transport environment with properties very different from those in a uniform superconductor. As a result of this interplay, heat transport becomes highly sensitive to temperature, magnetic field, and disorder. For strongly scattering impurities, we find that the transport across domain walls at low temperatures is considerably more efficient than in the uniform superconducting state.

We calculate the electronic spin susceptibility and spin-lattice relaxation rate in a singlet superconductor near a pair-breaking surface, or in a domain wall of the order parameter. We directly link the presence of high-density Andreev bound states in the inhomogeneous region, combined with coherence factors, to the enhancement of the susceptibility above the normal state's value for certain q vectors. Besides the dominant peak at ferromagnetic vector q = 0, we find significant enhancement of antiferromagnetic correlations at vectors q less than or similar to 2k(f), with q along the domain wall in an S-wave superconductor, and across the domain wall in D-wave (nodes along the wall). These features are destroyed by applying a moderate Zeeman field that splits the zero-energy peak. We solve Bogoliubov-de Gennes equations in momentum space and discuss the deviation of our results from the lattice models investigated previously. Large enhancement of the spin-lattice relaxation rate T-1(-1) at the domain wall provides a clear signature of the quasiparticle bound states, and is in good agreement with recent experiment in organic superconductor kappa-(BEDT-TTF)(2)Cu(NCS)(2).

Spanning a broad range of physical systems, complex symmetry breaking is widely recognized as a hallmark of competing interactions. This is exemplified in superfluid 3He which has multiple thermodynamic phases with spin and orbital quantum numbers S = 1 and L = 1, that emerge on cooling from a nearly ferromagnetic Fermi liquid. The heavy fermion compound UPt3 exhibits similar behavior clearly manifest in its multiple superconducting phases. However, consensus as to its order parameter symmetry has remained elusive. Our small angle neutron scattering measurements indicate a linear temperature dependence of the London penetration depth characteristic of nodal structure of the order parameter. Our theoretical analysis is consistent with assignment of its symmetry to an L = 3 odd parity state for which one of the three thermodynamic phases in non-zero magnetic field is chiral.