Evidence for a Spatially-Modulated Superfluid Phase of 3 He under Confinement
Lev V. Levitin, Ben Yager, Laura Sumner, Brian Cowan, Andrew J. Casey, Nikolay Zhelev, Robert G. Bennett, Jeevak M. Parpia, John Saunders
EEvidence for a Spatially-Modulated Superfluid Phase of He under Confinement
Lev V. Levitin, ∗ Ben Yager, † Laura Sumner, ‡ Brian Cowan, Andrew J. Casey, and John Saunders
Department of Physics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK
Nikolay Zhelev, § Robert G. Bennett, ¶ and Jeevak M. Parpia Department of Physics, Cornell University, Ithaca, NY, 14853 USA (Dated: 15 January 2019)In superfluid He-B confined in a slab geometry, domain walls between regions of different orderparameter orientation are predicted to be energetically stable. Formation of the spatially-modulatedsuperfluid stripe phase has been proposed. We confined He in a 1.1 µ m high microfluidic cavityand cooled it into the B phase at low pressure, where the stripe phase is predicted. We measuredthe surface-induced order parameter distortion with NMR, sensitive to the formation of domains.The results rule out the stripe phase, but are consistent with 2D modulated superfluid order. PACS numbers: 67.30.H-, 67.30.hr, 67.30.hj, 74.20.Rp
The pairing of fermions to form a superfluid or super-conductor at sufficiently low temperatures is a relativelyubiquitous phenomenon [1, 2]. Examples include: elec-trically conducting systems from metals to organic mate-rials to metallic oxides [3]; neutral atoms from He [4, 5]to ultracold fermionic gases [6]; and astrophysical ob-jects such as neutron stars and pulsars [7]. In the moststraightforward case the pairs form a macroscopic quan-tum condensate which is spatially uniform. In type-II superconductors a spatially inhomogeneous state, theAbrikosov flux lattice, arises in a magnetic field [8]. Itsorigin is the negative surface energy between normaland superconducting regions. However the realisationand experimental identification of states with spatially-modulated superfluid/superconducting order has provedchallenging.The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state[9, 10], has been predicted to arise in spin-singlet su-perconductors. An imbalance between spin-up and spin-down Fermi momenta, driven by ferromagnetic interac-tions or high magnetic fields, induces pairing with non-zero centre of mass momentum. This results in boththe order parameter and the spin density oscillating inspace with the same wavevector. The FFLO state is pre-dicted to intervene beyond the Pauli limiting field, in-hibiting the destruction of superconductivity [11]. It re-quires orbital effects to be weak, restricting possible ma-terials for its observation. There is evidence of the FFLOstate in the layered organic superconductors κ -(BEDT-TTF) Cu(NCS) and β (cid:48)(cid:48) -(ET) SF CH CF SO [12–16],and in the canonical heavy fermion superconductorCeCu Si [17]. Previously identified as FFLO [18], amore complex state, with intertwined p-wave pair densitywave (PDW) and spin density wave has been proposed inthe heavy fermion d-wave superconductor CeCoIn [19–21]. Elsewhere a PDW commensurate with a chargedensity wave, has been clearly demonstrated in the d-wave cuprate superconductor Bi Sr CaCu O x [22]. Inthe ultracold fermionic gas Li, superfluidity with im- balanced spin populations has been observed [23] withthermodynamic evidence consistent with FFLO [24]. Inaddition to these condensed matter systems it has beenproposed that quantum chromodynamics may providea pathway to inhomogeneous superconductivity, poten-tially realised in astrophysical objects [25].In general the order parameter modulation is expectedto be more complex than the model FFLO state [11, 26].Potential examples are: 1D domain walls of thicknessmuch smaller than the width of domains; 2D modulatedstructures, involving multiple wavevectors [11]. Further-more, nucleation barriers and metastability may inhibitthe formation of periodic states [26].In this paper we report experimental investigation ofa predicted spatially modulated state in the topologicalp-wave, spin-triplet , superfluid He [27]. This requiresthe superfluid to be confined in a thin cavity of uniformthickness. At the heart of this predicted stripe phase isthe stabilisation of a hard domain wall, of thickness com-parable to the superfluid coherence length, in superfluid He-B under confinement in a slab geometry. These B-B domain walls were first classified in Ref. [28], and theanalogy drawn with cosmic domain walls. Their stabilityin the bulk, and possible evidence for their observationis discussed in [29]. Under confinement the presence ofthe domain wall reduces surface pair-breaking, and canresult in a negative domain wall energy, leading to theformation of the stripe phase [27, 30, 31].In superfluid He the nuclear spins constitute thespin part of the pair wavefunction; thus nuclear mag-netic resonance (NMR) is widely used to provide a di-rect fingerprint of the superfluid order parameter [4, 5].NMR has been predicted to distinguish clearly betweenthe striped and translationally-invariant states of the Bphase [30, 32].To optimise the formation of stripes in this work wechose a slab geometry of height D = 1 . µ m, where the Bphase is stable down to zero pressure [33]. The stripephase was originally predicted in the weak-coupling a r X i v : . [ c ond - m a t . o t h e r] J a n limit of Bardeen-Cooper-Schrieffer theory [27], whilethe strong-coupling corrections to this theory in gen-eral favour the A phase and suppress the stability ofstripes [30]. At present the strong coupling effects arenot fully understood theoretically, leaving the stabilityof the stripe phase an open question [30, 33]. We per-formed the experiment at low pressure to minimise thestrong-coupling effects.Here we show that under our experimental conditionsthe stripe phase is clearly ruled out. However, there isNMR evidence for a spatially modulated superfluid oftwo-dimensional morphology, similar to states discussedin the context of FFLO [11]; we term this polka-dot .The 3 × He allows for multiple superfluid phases with differentbroken symmetries and topological invariants [4, 5]. Inthe bulk, at low pressure and magnetic field the stablestate is the quasi-isotropic B phase with order parametermatrix A = e iφ R ∆, where ∆ is the energy gap, isotropicin the momentum space, φ is the superfluid phase and R = R (ˆ n , θ ) is the matrix of relative spin-orbit rotation,parametrised by angle θ and axis ˆ n . This quasi-isotropicphase is relatively easily distorted by magnetic field orflow. Under confinement the distortion is strong and spa-tially inhomogeneous, induced by surface pair-breaking.In a slab normal to the z -axis, the order parameter ispredicted to take the form A ( z ) = e iφ R ∆ (cid:107) ( z ) ∆ (cid:107) ( z ) ∆ ⊥ ( z ) , (1)with 0 ≤ ∆ ⊥ < ∆ (cid:107) due to stronger surface pair-breakingof Cooper pairs with orbital momentum parallel to theslab surface [34]. This distortion is named planar afterthe planar phase, in which ∆ ⊥ = 0 [5].The order parameter (1) has a large manifold of ori-entations, determined by φ , ˆ n and θ , allowing domainwalls between regions of different orientations [28, 29].Domain walls where ∆ ⊥ changes sign, shown in Fig. 1,are predicted to have negative surface energy in the Bphase confined in a thin slab, close to the A-B tran-sition [27, 35]. As a consequence the slab of He-Bwould spontaneously break into domains, until the do-main walls get close enough that their mutual repulsionbecomes significant. This is predicted to result in the pe-riodic stripe phase with a typical domain size W of order D [27, 30, 31]. Phases with spontaneously broken trans-lational invariance are also predicted to stabilise in Heconfined to narrow pores [36, 37] and in films of d-wavesuperconductors [38].In this experiment we performed pulsed NMR studieson a slab of He confined in a D = 1144 ± He Larmor fre-quency f L = 1 .
02 MHz), using the setup described inRefs. [39, 40]. The measurements were performed at low − − − x /ξ − − − − ∆ / ∆ bu l k ∆ zz ∆ xz ∆ zx FIG. 1. Domain wall at the heart of the stripe phase in He-B [27]. Shown here are the elements of the energy gap matrix ∆ , such that A = e iφ R ∆ . When the domain wall is absent ∆ is diagonal, see Eq. (1). Crossing the domain wall lyingin yz -plane at x = 0, ∆ zz changes from ∆ ⊥ to − ∆ ⊥ , andoff-diagonal elements ∆ xz and ∆ zx emerge, while ∆ xx and∆ yy (not shown) remain close to ∆ (cid:107) . The gap amplitudeswere calculated z = 2 . ξ away from one of the surfaces in a D = 10 ξ thick slab at T = 0 . T bulk c ; ξ is the Cooper pairdiameter, ∆ bulk is the bulk B phase gap. pressure P = 0 .
03 bar, where the bulk superfluid transi-tion temperature T bulk c = 0 .
93 mK, and close to specularscattering, achieved by preplating the cell walls with a64 µ mol/m ( ∼ He film.We first mapped the phase diagram with small tippingangle, β = 4 ◦ , NMR pulses, Fig. 2a. The A-B transi-tion was observed at T AB = 0 . T bulk c in agreement withtorsional oscillator measurements, with a 1.08 µ m cavity[33]. As we previously observed in the 0.7 µ m cavity, theB phase nucleated stochastically in two spin-orbit ori-entations with distinct NMR signatures: stable B + andmetastable B − [32, 39].The magnitudes of the frequency shifts of translation-ally invariant B + and B − , ∆ f + and ∆ f − , are deter-mined by averages of the gap structure across the cav-ity [32]: (cid:104) ∆ (cid:107) (cid:105) , (cid:104) ∆ (cid:107) ∆ ⊥ (cid:105) and (cid:104) ∆ ⊥ (cid:105) . In case of the pu-tative spatially-modulated phase the averaging is alsoperformed in the plane of the slab. This procedure isvalid when the width of the stripes W is smaller thanthe dipole length ξ D ≈ µ m [32], a condition predictedto hold for this cavity ( W ≈ D √ (cid:28) ξ D ), except veryclose to the stripe-to-B transition [30]. In the stripe phase (cid:104) ∆ (cid:107) ∆ ⊥ (cid:105) = 0 due to ∆ ⊥ having opposite sign in the adja-cent domains [32]. This has clear signatures in the NMRresponse, as a function of tipping angle β .We define the dimensionless gap distortion parameters q = (cid:104) ∆ (cid:107) ∆ ⊥ (cid:105) / (cid:104) ∆ (cid:107) (cid:105) , Q = (cid:113) (cid:104) ∆ ⊥ (cid:105) / (cid:104) ∆ (cid:107) (cid:105) . (2)The tipping angle dependence of the frequency shift ofB + , below the so called “magic angle”, β ∗ > ◦ , scaledby the small-tipping-angle shift of B − is given by [32]∆ f + ( β → f − ( β →
0) = 2 Q − q −
11 + 2 Q , (3a) ∂ ∆ f + ( β ) /∂ cos β ∆ f − ( β →
0) = 2 Q − q Q , (3b) Temperature T / T bulk c − − − − − F r e qu e n c y s h i f t [ k H z ] β = 4 ◦ AB + B − T / T bulk c A ph a s e p e a k s i z e ← − − → β = 20 ◦ β = 40 ◦ β = 60 ◦ Temperature T / T bulk c B + f r e qu e n c y s h i f t [ k H z ] ,,,,, β = 60 ◦ β = 40 ◦ β = 20 ◦ β → β = 4 ◦ (a) (b)(c) FIG. 2. NMR measurements on a D = 1 . µ m slab of super-fluid He. (a) Signatures of the A and B phases observed withsmall tipping pulses. At the second-order normal-superfluidtransition a negative frequency shift develops as the slab en-ters the A phase with the dipole-unlocked orientation. Onfurther cooling a first-order phase transition into the B phasewith a planar distortion occurs, where two spin-orbit orien-tations B + and B − are observed. Inset: sharp A-B transi-tion with small hysteresis. (b) Technique for applying a setof pulses with different tipping angle β but equal heating:all pulses coincide in length and amplitude; β is reduced byapplying the initial section of the pulse with a 180 ◦ phaseshift, which cancels out a similar section that follows (bothlight blue), so only the remainder of the pulse (red) tips thespins. (c) Initial frequency shifts in B + after such pulses yield ∂ ∆ f + /∂ cos β and ∆ f + ( β → f + ( β →
0) obtained here and ∆ f + (4 ◦ ), smoothed from (a),indicates that the heating due to the 20-60 ◦ pulses is neg-ligible. Open/filled symbols show data taken on step-wisewarm-ups after a fast/slow cool-down from the A phase. where β is the tip angle. We further note that: ∆ f − ( β )does not depend on q ; the magic angle at which there isa kink in ∆ f + ( β ) is given by β ∗ = arccos( q − / (2 q + 2)[32]. Thus there is no magic angle expected for the stripephase, for which q = 0.Application of large tipping pulses in our setup resultsin rapid heating of the confined helium via an unidenti-fied mechanism, that previously restricted measurementsof the planar distortion to temperatures well below T AB [32, 40]. Here we focus on moderate pulses, β (cid:46) ◦ ,that allow us to probe the temperature dependence of thedistortion parameters up to T AB , according to Eq. (3).In order to measure the tipping-angle dependence of thefrequency shift at constant temperature, we developed ascheme for applying pulses with different β while induc-ing identical heating, shown in Fig. 2b. Triplets of suchpulses with β = 20 ◦ to 60 ◦ were applied to B + duringstep-wise warm-ups after a fast (at approx. 40 µ K/min
Temperature T / T bulk c P l a n a r d i s t o r t i o n q , Q T AB T BCSAB ,, Qq Temperature T / T bulk c q / Q warm-up 1warm-up 2 A m o un t o f + d o m a i n s a [ % ] (a) (b) FIG. 3. Temperature dependence of planar distortion pa-rameters q , Q (a) and their ratio (b) inferred from NMR mea-surements, Fig. 2. Solid lines show weak-coupling calculationsfor translationally-invariant B phase [44]. In the absence ofstrong coupling the temperature of the AB transition T BCSAB is higher than T AB observed experimentally. While Q is inagreement with the theory, q gets progressively reduced ap-proaching T AB . We interpret this in terms of development ofdomains with opposite sign of ∆ ⊥ on warming. The right-hand vertical axis in (b) estimates the fraction of the slab oc-cupied by the majority domains, taken here to have positive∆ ⊥ [45], under a qualitative assumption of step-like energygap profile, Eq. (4). Any systematic difference between mea-surements taken while warming after a fast and a slow cool-down through A-B transition (open/filled symbols) is small. rate) and slow (4 µ K/min) cool-down through the A-Btransition. We inferred ∂ ∆ f + /∂ cos β and ∆ f + ( β → f − (4 ◦ ) at the same temperature, Fig. 2a, we deter-mine the planar distortion parameters q and Q throughEq. (3), shown in Fig. 3.In the above analysis the off-diagonal elements of thegap matrix near domain walls (see Fig. 1) have been ne-glected. Incorporating the detailed gap structure at thedomain walls into the NMR model leaves the signatureof the stripe phase, q = 0 , Q >
0, virtually unchanged[30]. We can therefore conclude unambiguously that thestripe phase was not present in our experiment.Nevertheless, while Q matches the weak-coupling cal-culations for the translationally-invariant B phase [44], q is found to be reduced, see Fig. 3. This contrasts withthe good agreement between these calculations and sim-ilar measurements in a D = 0 . µ m slab at higher pres-sure [32, 40], ruling out strong coupling effects as theorigin of this discrepancy. We therefore consider domainstructures in which the amount a and 1 − a of domainswith positive and negative ∆ ⊥ is unequal [45]. For aqualitative estimate we assume that the domain wallsare step-like and ignore gap variation across the slab,∆ (cid:107) = const , | ∆ ⊥ | = const. Then Q = | ∆ ⊥ | ∆ (cid:107) , q = (2 a − | ∆ ⊥ | ∆ (cid:107) , qQ = 2 a − , (4) W WW W ′ W W W (e) q / Q > (d) q / Q ≈ (c) q / Q ≈ (b) q / Q = 0 (a) q / Q = 1 FIG. 4. Possible regular domain configurations in He-B con-fined to a slab, view perpendicular to the slab plane. + / − in-dicates the sign of ∆ ⊥ . (a) Translationally-invariant B phase,(b) predicted stripe phase, (c,d) proposed polka dot phasethat is easier to nucleate than the stripe phase. The values ofthe gap distortion parameter q/Q are derived under a simpli-fied assumption of step-like domain walls, Eq. (4). Pinning ofdomain walls in the experimental cell may introduce disorderto these structures. All features in (c-d) are taken to be ofcharacteristic size similar to the pitch W of the stripe phase.(e) A variant of (d) in which dot diameter W (cid:48) is smaller thandot separation W . demonstrating that while q is sensitive to the presence ofdomains, to the first approximation Q is not, in agree-ment with our observations. The gradual variation of ∆ (cid:107) ,∆ ⊥ and the emergence of the off-diagonal gap matrix el-ements inside the domain walls would lead to correctionsto this model that should be taken into account in futuretheoretical work.Our measurement q/Q = 0 . ± . T AB suggests a+ / − domain proportion of 4 : 1. A likely scenario for animbalance of the domains is a two-dimensional structure.Within possible regular morphologies, Fig. 4, this imbal-ance corresponds to a polka dot phase with hexagonal orsquare symmetry, Fig. 4c,d. Such structures have beensuggested theoretically [27], but the detailed analysis oftheir energetic stability has not been carried out. A pre-liminary Ginzburg-Landau study finds the square latticeof dots, Fig. 4d, to be locally stable, with free energyonly slightly higher than that of the stripe phase [46].Further theoretical work is required to understand thestability of various modulated states in the presence ofstrong coupling effects.Even if less energetically favourable than stripes, thedots may arise because of a lower energy barrier for flip-ping the sign of ∆ ⊥ in a microscopic dot, compared witha stripe, that is macroscopic in one dimension. A latticeof dots would form if the neighbouring dots nucleate closeenough to prevent them from growing beyond a typicalsize W before getting within W of the others.Our experimental protocol is first to cool deep into theB phase, in order to destabilise the domain walls, andthen to take data on warming. The key observation of thedecrease in q/Q with increasing temperature is consistentwith the formation of negative-energy domain walls inthe B phase approaching the transition into the A phase,as predicted theoretically. The observation of a singleNMR line implies that the domain size is shorter than ξ D . The measured temperature dependence of q/Q can be explained by allowing the separation between dots W and their diameter W (cid:48) be unequal, see Fig. 4e. Pinning ofthe domain walls by scratches on the cavity walls [47] mayplay a role in restricting W (cid:48) , and introduce disorder intothe domain morphology. Improved cavities have beendeveloped for future experiments [48].As an alternative scenario, we now consider metastabledomain walls with positive energy. Defects are known toform at the A-B transition either due to inhomogeneousnucleation [49, 50] or as relics [51] of defects, presentin the A phase at the start of the transition [52, 53].These may include the domain walls where ∆ ⊥ changessign [29, 51, 54, 55], which, if produced at unusually highdensity, would result in a reduced q/Q ratio; however thisratio would remain constant if the defects are pinned orincrease with time as they decay, contrary to our ob-servation. This does not rule out sparse positive-energydefects with typical separation larger than ξ D , giving riseto small satellite NMR signals, specific to each type of de-fect [51, 53, 56, 57]. Detection of such signals is beyondthe scope of this work. Within errors our observations areindependent of the rate of cooling through the A-B tran-sition, Fig. 3. This supports our proposal that defectsproduced at this transition do not play a major role inthe formation of domains on micron scale. A systematicstudy of the influence of the cooling rate will be subjectof future work.In conclusion our NMR study of superfluid He con-fined in a 1.1 µ m cavity in the vicinity of the AB tran-sition has found neither the predicted stripe phase,nor translationally-invariant planar-distorted B phase.This leads us to propose a superfluid phase with two-dimensional spatial modulation, in a form of a regular ordisordered array of island domains, driven by negative en-ergy of domain walls under confinement. Further system-atic studies of the nucleation of this phase, to determinethe equilibrium morphology, as well as its stability as afunction of pressure, predicted to be influenced by strongcoupling effects, are both desirable. Superfluid He un-der confinement appears to provide a clean model systemfor spatially modulated superconductivity/superfluidity,long sought in a wide variety of physical systems.We thank B. R. Ilic for help with microfluidic cham-ber fabrication and design methodology; A. B. Vorontsovand J. A. Sauls for sharing calculations of the gap profileof confined He; T. Kawakami and T. Mizushima for astimulating discussion and for sharing their preliminaryresult on the stability of the polka dot phase. This workwas supported by EPSRC grants EP/J022004/1 andEP/R04533X/1; NSF grants DMR-1202991 and DMR-1708341, and the European Microkelvin Platform. ∗ [email protected] † Now at Institute for Quantum Computing, University ofWaterloo, Waterloo, Ontario N2L 3G1, Canada ‡ Now at Imperial College London, London SW7 2AZ, UK § Now at Corning Incorporated, USA ¶ Now at Vantage Power Ltd, London, UB6 0FD, UK[1] J. Bardeen, L. N. Cooper and J. R. Schrieffer,Phys. Rev. , 1175 (1957).[2] L. N. Cooper,
BCS: 50 years (World Scientific, 2011).[3] G. R. Stewart, Adv. in Phys. , 75 (2017).[4] A. J. Leggett, Rev. Mod. Phys. , 332 (1975).[5] D. Vollhardt and P. Wolfle, The superfluid phases of He (Taylor and Francis, 1990).[6] S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 1215 (2008).[7] A. B. Migdal, Nucl. Phys. , 655 (1959).[8] A. A. Abrikosov, Sov. Phys.-JETP, , 6, 1174 (1957).[9] A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys.JETP , 762 (1965).[10] P. Fulde and R. A. Ferrell, Phys. Rev. A , 550 (1964).[11] Y. Matsuda and H. Shimahara, J. Phys. Soc. Jpn ,051005 (2007).[12] J. Wosnitza, Ann. Phys. (Berlin) , 1700282 (2018).[13] H. Mayaffre, S. Kr¨amer, M. Horvati´c, C. Berthier,K. Miyagawa, K. Kanoda, and V. F. Mitrovi´c,Nat. Phys. , 928 (2014).[14] C. C. Agosta, N. A. Fortune, S. T. Hannahs,S. Gu, L. Liang, J.-H. Park, and J. A. Schleuter,Phys. Rev. Lett. , 267001 (2017).[15] R. Beyer, B. Bergk, S. Yasin, J. A. Schlueter, and J. Wos-nitza, Phys. Rev. Lett. , 027003 (2012).[16] G. Koutroulakis, H. K¨uhne, J. A. Schlueter, J. Wosnitza,and S. E. Brown, Phys. Rev. Lett. , 067003 (2016).[17] Sh. Kitagawa, G. Nakamine, K. Ishida, H. S. Jeevan,C. Geibel, and F. Steglich, Phys. Rev. Lett. , 157004(2018).[18] A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso,and J. L. Sarrao, Phys. Rev. Lett. , 187004 (2003).[19] G. Koutroulakis, M. D. Stewart, Jr., V. F. Mitrovi´c,M. Horvati´c, C. Berthier, G. Lapertot, and J. Flouquet,Phys. Rev. Lett. , 087001 (2010).[20] D. Y. Kim, S.-Z. Lin, F. Weickert, M. Kenzelmann,E. D. Bauer, F. Ronning, J. D. Thompson, andR. Movshovich, Phys. Rev. X , 041059 (2016).[21] D. Y. Kim, S.-Z. Lin, F. Weickert, E. D. Bauer,F. Ronning, J. D. Thompson, and R. Movshovich,Phys. Rev. Lett. , 197001 (2017).[22] M. H. Hamidian, S. D. Edkins, S. H. Joo, A. Kostin,H. Eisaki, S. Uchida, M.J. Lawler, E.-A. Kim,A.P. Mackenzie, K. Fujita, J. Lee, and J. C. S´eamusDavis, Nature , 343 (2016).[23] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, andW. Ketterle, Science, , 492 (2006).[24] M. C. Revelle, J. A. Fry, B. A. Olsen, and R. G. Hulet,Phys. Rev. Lett. , 235301 (2016).[25] R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. , 263(2004).[26] S. Dutta and E. J. Mueller, Phys. Rev. A , 023612(2017).[27] A. B. Vorontsov and J. A. Sauls, Phys. Rev. Lett. ,045301 (2007), J. Low Temp. Phys. , 283 (2005).[28] M.M. Salomaa and G.E. Volovik, Phys. Rev. B , 9298(1988).[29] M. Silveri, T. Turunen, and E. Thuneberg,Phys. Rev. B , 184513 (2014). [30] J. J. Wiman and J. A. Sauls, J. Low Temp. Phys. ,1054 (2016).[31] K. Aoyama, J. Phys. Soc. Jpn. , 094604 (2016).[32] L. V. Levitin, R. G. Bennett, E. V. Surovtsev,J. M. Parpia, B. Cowan, A. J. Casey, J. Saunders,Phys. Rev. Lett. , 235304 (2013).[33] N. Zhelev, T. S. Abhilash, E. N. Smith, R. G. Bennett,X. Rojas, L. Levitin, J. Saunders, and J. M. Parpia,Nat. Comm. , 15963 (2017).[34] Y. Nagato and K. Nagai, Physica B , 269 (2000).[35] Note that the state with ∆ ⊥ < ⊥ > π phase shift and acombined rotation R (ˆ n (cid:48) , θ (cid:48) ) = R (ˆ n , θ ) R (ˆ z , π ): e iφ R (ˆ n , θ ) ∆ (cid:107) ∆ (cid:107) − ∆ ⊥ = − e iφ R (ˆ n , θ ) − ∆ (cid:107) − ∆ (cid:107) ∆ ⊥ = e i ( φ + π ) R (ˆ n , θ ) R (ˆ z , π ) ∆ (cid:107) ∆ (cid:107) ∆ ⊥ . [36] K. Aoyama, Phys. Rev. B , 140502R (2014).[37] J. J. Wiman and J. A. Sauls, arXiv:1802.08719 (2018).[38] A. B. Vorontsov, Phys. Rev. Lett. , 177001 (2009).[39] L. V. Levitin, R. G. Bennett, A. Casey, B. Cowan,J. Saunders, D. Drung, Th. Schurig, and J. M. Parpia,Science , 841 (2013).[40] See Supplemental Material at [URL will be inserted bypublisher] for details of the experimental setup, NMR,thermometry, comparison of the measured q and Q toRef. [32], and qualitative comparison of domain configu-rations. This includes Refs. [41–43].[41] L. .V. Levitin, R. G. Bennett, A. Casey, B. Cowan,J. Saunders, D. Drung, Th. Schurig, J. M. Parpia, B. Ilic,and N. Zhelev, J. Low Temp. Phys. , 667 (2014).[42] V. Dotsenko and N. Mulders. J. Low Temp. Phys. ,443 (2004).[43] P. J. Heikkinen, A. Casey, L. V. Levitin, X. Rojas,A. Vorontsov, N. Zhelev, J. M. Parpia, and J. Saunders, Tuning pair-breaking at the surface of topological super-fuid He (unpublished).[44] A. B. Vorontsov and J. A. Sauls, Phys. Rev. B , 064508(2003) and private communication.[45] Since the sign of ∆ ⊥ can be flipped throughout the sam-ple with a uniform transformation [35], we only need toconsider the case of ∆ ⊥ being positive in the larger partof the slab, a ≥ .
5, so q ≥
0, as required in Eq. (3) [32].[46] T. Kawakami and T. Mizushima, private communication(2018).[47] Analogous to weak pinning of the A-B boundary observedin the similarly fabricated D = 0 . µ m cavity [39].[48] N. Zhelev, T. S. Abhilash, R. G. Bennett, E. N. Smith,B. Ilic, J. M. Parpia, L. V. Levitin, X. Rojas, A. Casey,and J. Saunders, Rev. Sci. Instr. , 073902 (2018).[49] D. I. Bradley, S. N. Fisher, A. M. Gu´enault, R. P. Haley,J. Kopu, H. Martin, G. R. Pickett, J. E. Roberts, andV. Tsepelin, Nat. Phys. , 46 (2008).[50] P. J. Heikkinen, S. Autti, V. B. Eltsov, R. P. Haley, andV. V. Zavjalov, J. Low Temp. Phys. , 681 (2014).[51] J. T. M¨akinen, , V. V. Dmitriev, J. Nissinen, J. Rysti,G. E. Volovik, A. N. Yudin, K. Zhang, V. B. Eltsov,arXiv:1807.04328v3. [52] P. M. Walmsley and A. I. Golov, Phys. Rev. Lett. ,215301 (2012).[53] J. Kasai, Y. Okamoto, K. Nishioka, T. Takagi, andY. Sasaki, Phys. Rev. Lett. , 205301 (2018).[54] Yu. Mukharsky, O. Avenel, and E. Varoquaux,Phys. Rev. Lett. , 210402 (2004).[55] C. B. Winkelmann, J. Elbs, Yu. M. Bunkov, and H. God-frin, Phys. Rev. Lett. , 205301 (2006). [56] V. B. Eltsov, R. Blaauwgeers, N. B. Kopnin, M. Kru-sius, J. J. Ruohio, R. Schanen, and E. V. Thuneberg,Phys. Rev. Lett. , 065301 (2002).[57] S. Autti, V. V. Dmitriev, J. T. M¨akinen, A. A. Solda-tov, G. E. Volovik, A. N. Yudin, V. V. Zavjalov, andV. B. Eltsov, Phys. Rev. Lett. , 255301 (2016). Supplemental Material
EXPERIMENTAL SETUP
The apparatus of this experiment is in most aspectsidentical to that used in Ref. [39] and described in itssupplementary material. The main difference is the im-proved uniformity of the cavity, see Fig. S1, achievedthrough optimised nanofabrication, introduction of a par-tition wall and restricting the measurements to low pres-sure [41]. In addition the cell fill line was interrupted witha superfluid-leaktight cryogenic valve [41, 42] at 6 mK, toavoid gradual depletion of the He film due to the foun-tain effect.
TEMPERATURE CORRECTION
The thermometry in our experimental setup is basedon monitoring the temperature of the silver sinter heat
Slab thickness D [ µ m] D i s t r . d e n s i t y [ % / n m ] D = 728 ±
28 nmat P = 5.5 bar mm D [ µ m ] D = 1144 ± P = 0.0 bar mm D [ µ m ] (a) (b)(c) FIG. S1. Spectroscopic measurement of the thickness D ofsilicon-glass microfluidic cavities. The 1.1 µ m cavity used inthe experiment described here is compared to the 0.7 µ m cav-ity from Refs. [32, 39], inflated under pressure necessary tostabilise the B phase. In the present work the slab uniformityis improved, one of the reasons being the partition wall in themiddle of the cell [41]. exchanger with a Pt NMR thermometer, calibratedagainst the He melting curve. In recent work on a D = 0 . µ m slab [43] the temperature gradient betweensilver sinter (at T Ag ) and helium (at T He ) in the heat ex-changer was carefully determined for various surface Hecoverages. With He plating similar to the one used inthis experiment, the correction was found to be T . − T . = C, (S1)corresponding to thermal boundary resistance R K ( T ) ∝ /T . . The constant C in (S1), determined by heat leak Heat exchanger temperature T Ag [mK] H e li u m t e m p e r a t u r e T H e [ m K ] T He = T Ag T − T = 0.09mK Temperature [mK] − − − − A ph a s e f r e qu e n c y s h i f t [ k H z ] T He = T Ag T − T = 0.09mK bulk A phase (a)(b) FIG. S2. Temperature correction. (a) Relationship betweentemperatures T He and T Ag of helium and silver sinter in theheat exchanger. (b) Frequency shift in the A phase is ex-pected to have the bulk value, since the energy gap in the Aphase is not suppressed in a slab with specular walls. This isfound to be the case after the temperature correction. Herethe bulk frequency shift is modelled based on the initial slope2 f L ∂ (cid:12)(cid:12) ∆ f ( T ) (cid:12)(cid:12) /∂ (1 − T /T c ) | T → T c = 3 . × Hz measuredin the D = 0 . µ m cavity with 98%-specularly scatteringwalls and the calculated temperature dependence of the weak-coupling A phase energy gap. Here the initial slope is definedas the slope of a straight line fitted between 0 . T c and 1 . T c . Time after heating pulse [min] H e a t i n g [ µ K ] D = 0.7 µ m P = 5.5 bar T = 1.4 mK24 µ T × slabbulk marker Time after heating pulse [msec] S l a bh e a t i n g [ µ K ] D = 1.1 µ m P = 0.0 bar T = 0.8 mK18 µ T × Heating pulse length τ [msec] S l a bh e a t i n g [ µ K ] D = 1.1 µ m P = 0.0 bar T = 0.8 mK B = 18 µ T pulses τ [msec ] S l a bh e a t i n g [ µ K ] D = 1.1 µ m P = 0.0 bar T = 0.8 mK B = 18 µ T pulses
Heating pulse RF field B [ µ T] S l a bh e a t i n g [ µ K ] D = 1.1 µ m P = 0.0 bar T = 0.8 mK τ = 1.5 msec pulses B [10 µ T ] S l a bh e a t i n g [ µ K ] D = 1.1 µ m P = 0.0 bar T = 0.8 mK τ = 1.5 msec pulses (a)(b) (c)(d) (e)(f) FIG. S3. Characterisation of heating of the confined He by NMR pulses. Helium temperature is inferred from frequency shiftprobed regularly with small NMR pulses, while the heating is induced by large pulses, applied sufficiently far from Larmorfrequency not to tip He spins and rarely enough (once every 20 minutes) to let the sample cool down before it is heatedagain. (a) Heating in the slab and the “bulk marker” volume at the mouth of the fill line. Only the former rapidly warms updue to local heating we are concerned with; in addition both slowly respond to Joule heating of the metallic elements of theexperimental setup, transmitted via the heat exchanger. Data from the D = 0 . µ m slab with a large clearly-visible bulk marker[39]. The rest of the measurements were performed on the D = 1 . µ m slab presented in this paper. (b) Time evolution of theslab temperature shortly after a large pulse. (c-f) Dependence of heating on the pulse duration τ and power of radio-frequencyfield B measured 20 msec after pulses, when the slab reaches its highest temperature. to the helium sample, is obtained from the suppressionof bulk T c , registered with NMR in the bulk He markerat the mouth of the fill line. Fig. S2 demonstrates thatsuch correction with a slightly different C is adequatefor the present experiment, which utilised the same heatexchanger. In this work the NMR pulses were appliedrarely enough to have no effect on C . HEATING BY NMR PULSES
NMR experiments on superfluid He under regu-lar confinement in silicon-glass and fully-silicon cavities(Refs. [32, 39, 43] and this work) have manifested heatingof an unidentified origin that couples to the confined liq-uid directly, Fig. S3a. Most heating occurs within 1 msecafter the pulse (the duration of the probe NMR pulse plusthe dead time of the NMR spectrometer), and the slabtemperature is nearly constant for many msec afterwards,Fig. S3b. The dependence of the heating on duration τ and amplitude B of the sine-wave pulses, Fig. S3c-f, issteeper than expected for linear dissipation ( ˙ Q ∝ τ B ),especially when taking into account that specific heat of He in the slab increases with temperature. This pointstowards an exotic origin of this parasitic effect.In addition to the signatures shown in Fig. S3 we found that the heating is not resonant near the He Larmorfrequency, and that heating due to pulses with the initial‘antipulse’, Fig. 2b, only depends on the total durationand amplitude, but not on the length of the ‘antipulse’part.As shown in Fig. S3b, the free induction decay aftera large pulse with characteristic T ∗ ≈ − − cos β − ∆ f A ( β ) [ k H z ] The A phase T = 0.76 mK∆ T = 0.05 mK − − cos β ∆ f + ( β ) [ k H z ] B + T = 0.57 mK∆ T ≈ FIG. S4. NMR measurements on the A and B phase in D = 1 . µ m slab with 18 µ T × T above the temperature T of thehelium in the heat exchanger. Different tipping angles areachieved by applying the initial part of the pulse with a 180 ◦ phase shift, see Fig. 2b. In the dipole-unlocked A phase∆ f A ( β ) ∝ cos β , see [32] for ∆ f + ( β ). Temperature T / T bulk c P l a n a r d i s t o r t i o n Q D /ξ = 13 D /ξ = 20 D /ξ = 40 Reduced thickness D /ξ AB P l a n a r d i s t o r t i o n Q D /ξ = 13 D /ξ = 20 D /ξ = 40 Reduced thickness D /ξ AB P l a n a r d i s t o r t i o n Q D = 1.1 µ m P = 0.0 bar D = 0.7 µ m P = 5.5 bar Temperature T / T bulk c P l a n a r d i s t o r t i o n q D /ξ = 13 D /ξ = 20 D /ξ = 40 Reduced thickness D /ξ AB P l a n a r d i s t o r t i o n q D /ξ = 13 D /ξ = 20 D /ξ = 40 Reduced thickness D /ξ AB P l a n a r d i s t o r t i o n q D = 1.1 µ m P = 0.0 bar D = 0.7 µ m P = 5.5 bar (a)(b) (c)(d) (e)(f) FIG. S5. Universal scaling of planar distortion parameters q and Q with reduced slab thickness D/ξ AB . (a-d) Calculationsof planar distortion of spatially-invariant planar-distorted B phase at various D/ξ [44]. See Eq. (S3) for the definition of thecoherence length. (e-f) Comparison of these calculations to measurements on 0.7 µ m slab (filled symbols) [32] and 1.1 µ m slab(open symbols, showing together both warm-ups presented in Fig. 3). The horizontal bars represent the range of D/ξ AB thatcan be probed in at given D and P limited by T = T AB and T = 0. temperature T of helium in the heat exchanger and fillline. The use of ‘antipulses’, Fig. 2b, allows us to mea-sure the NMR response as a function of tipping angle ata constant elevated temperature. This is illustrated inFig. S4. We note that the strong frequency dependenceof NMR tipping by ‘antipulses’, renders them unusableaway from their carrier frequency. For this reason B − ismissing from Fig. S4, due to relatively large frequencyrange spanned by ∆ f − ( β ) ≈ − × cos( β ). ProbingB + above the magic angle β ∗ is equally problematic andwas not studied here in detail.In contrast the 0.7 µ m slab was probed with groups oflarge pulses that caused different heating, restricting suchmeasurements to the T (cid:46) . T c limit where the temper-ature dependence of the frequency shifts is weak [32].In this work we demonstrated that tipping angles upto 60 ◦ could be reached with 8 µ T × T (cid:46) µ K, small compared to the tem-perature range over which we probed the B phase gapdistortion.
UNIVERSAL SCALING OF PLANARDISTORTION
Within the Ginzburg-Landau regime, T − T c (cid:28) T c ,the effects of confinement on properties of the super-fluid are determined by a single control parameter, thereduced thickness D/ξ ( T, P ), where ξ is the coherence length. This universality breaks down at lower tempera-tures, i.e. see the supplementary of Ref. [39]. To study theA-B transition outside of the Ginzburg-Landau regimethe coherence length has been defined as ξ ∆ ( T, P ) = (cid:126) v F ( P )∆ B ( T, P ) √ , (S2)were v F is the Fermi velocity and ∆ B is the bulk B phasegap. We observe that the weak-coupling quasiclassicalcalculations [44] of q ( T ) and Q ( T ) at different D collapse,see Fig. S5a-d, expressed as a function of D/ξ adoptinga slightly different coherence length ξ AB ( T, P ) = D AB ( T, P ) π √ , (S3)where D AB ( T, P ) is the thickness of the slab at whichthe A to B transition occurs at temperature T and pres-sure P (the inverse of the T AB ( D/ξ ( P )) function [34]).Here we restrict the discussion to the calculations fora slab with specular boundaries and recognise that thepressure P only enters the weak-coupling calculations viathe pressure dependence of the bulk transition temper-ature T bulk c ( P ) and the Cooper pair diameter ξ ( P ) = (cid:126) v F ( P ) / πk B T bulk c ( P ). We find that ξ AB /ξ ∆ → . T → T c and ξ AB /ξ ∆ → . T → PLANAR DISTORTION: THEORY VSEXPERIMENTS
We compare the calculations discussed in the previ-ous section with the NMR measurements of the planardistortion of the B phase in 0.7 and 1.1 µ m slabs inFig. S5e,f. The former, obtained at P = 5 . T = 0 . . T bulk c are in good agreement with thetheory in terms of both q and Q . This confirms that thestrong coupling effects, known to increase with pressure,are not responsible for the reduced q presented in thispaper, Fig. 3 and Fig. S5f. QUALITATIVE DISCUSSION OF DOMAINCONFIGURATIONS
In this section we compare possible structures of do-mains using the length of domain walls per unit area ofthe slab
L/A as a figure of merit of the free energy gaindue to formation of domain walls. We consider a two-dimensional problem of energetic stability of thin domainwalls with hard-core repulsion at distance W . Dots areassumed to be circular. We find L/A = 1 /W for thestripe phase, L/A = π/ W √ ≈ . /W for hexagonallattice and L/A = π/ W ≈ . /W/W