Vortex lattices and broken time reversal symmetry in the topological superconductor UPt3

TL;DR

This study investigates vortex lattice configurations in the topological superconductor UPt3, revealing broken time reversal symmetry and chiral order, with field-dependent vortex behavior indicating unconventional superconductivity.

Contribution

It provides direct experimental evidence of vortex lattice asymmetry and non-monotonic rotation in UPt3, demonstrating broken time reversal symmetry and internal vortex degrees of freedom.

Findings

Vortex lattice configuration depends on magnetic field sign.

Discovery of field-driven non-monotonic vortex lattice rotation.

Evidence of broken time reversal symmetry in UPt3 B phase.

Abstract

The topological superconductor UPt3, has three distinct vortex phases, a strong indication of its unconventional character. Using small-angle neutron scattering we have probed the vortex lattice in the UPt3 B phase with the magnetic field along the crystal c-axis. We find a difference in the vortex lattice configuration depending on the sign of the magnetic field relative to the field direction established upon entering the B phase at low temperature in a field sweep, showing that the vortices in this material posses an internal degree of freedom. This observation is facilitated by the discovery of a field driven non-monotonic vortex lattice rotation, driven by competing effects of the superconducting gap distortion and the vortex-core structure. From our bulk measurements we infer that the superconducting order parameter in the UPt3 B phase breaks time reversal symmetry and exhibits chiral symmetry with respect to the c-axis.

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Full text

I I. Introduction

Topological properties of materials are of fundamental as well as practical importance, and are currently the focus of intense research Qi and Zhang (2011). Recently, topological superconductors have received increased attention, as it was realized that vortices in them can host Majorana fermions with potential use in quantum computing Schnyder and Brydon (2015); Beenakker and Kouwenhoven (2016); Kozii et al. (2016). As a result, some well known unconventional superconductors have received renewed attention Mizushima et al. (2016); Sato and Ando (2017), particularly UPt3 Strand et al. (2009); Schemm et al. (2014), Sr2RuO4 Maeno et al. (2012); Steppke et al. (2017) and superfluid 3He Ikegami et al. (2013); Autti et al. (2016). These materials are believed to posses chiral ground states with broken mirror and time-reversal symmetries, as well as chiral fermions associated with the topology of the chiral ground state.

In this report we focus on the heavy-fermion material UPt3 Joynt and Taillefer (2002), which exhibits three distinct vortex phases shown in Fig. 1 and labeled A, B and C, and two zero-field superconducting phases. The latter requires that the superconducting order parameter for UPt3 belong to one of the odd-parity two-dimensional representations of the point group D6h Hess et al. (1989), most likely a $f$-wave pairing states belonging to the $E_{2u}$ irreducible representation, in which the B phase is a chiral ground state that breaks time reversal and mirror symmetries Sauls (1994). The evidence for the latter comes from surface probes, most recently tunneling Strand et al. (2009) and polar Kerr rotation Schemm et al. (2014) in zero applied magnetic field, as well as early $\mu$SR measurements Luke et al. (1993). However there are conflicting reports on the evidence for broken time-reversal symmetry in UPt3 based on $\mu$SR de Réotier et al. (1995). Thus, it is important to have a direct test of broken time-reversal symmetry based on measurements in the bulk superconducting phases.

We report on small-angle neutron scattering (SANS) studies of the vortex lattice (VL) structure in the B phase, extending into the C phase. Measurement were carried out with $\bf{H}\!\parallel\!\bf{c}$ where the VL is especially sensitive to changes in the superconducting state, due to the hexagonal crystal structure and corresponding weak basal plane anisotropy of UPt3 Shivaram et al. (1986); Keller et al. (1994). We observe a difference in the VL diffraction patterns for $\bf{H}$ parallel versus anti-parallel to the original field direction on entering the B phase. From this we infer that time-reversal symmetry is broken in the zero-field B phase, and that the $c$-axis is an axis of chiral symmetry. A similar experiment using rotation in place of magnetic field was used to detect the chirality of the ground state of superfluid 3He-A Walmsley and Golov (2012).

II II. Background

The existence of two zero-field superconducting phases with a small splitting in the transition temperatures ($T_{c},T_{c2}$) is the evidence for the pairing symmetry belonging to a two-dimensional $E$-representation and the presence of a weak symmetry breaking field, e.g. weak in-plane anti-ferromagnetism or strain, which lifts the degeneracy of the multi-dimensional representation leading to two zero-field phases and three VL phases Hess et al. (1989). Identification of the $E_{2u}$ representation proposed in Ref. Sauls (1994), which predicts a time-reversal symmetric A phase and broken time-reversal symmetry (BTRS) in the lower temperature B phase is supported by a a wide range of results. These include the $H$-$T$ phase diagram Adenwalla et al. (1990); Sauls (1994); Shivaram et al. (1986); Choi and Sauls (1991), thermodynamic and transport studies Taillefer et al. (1997); Graf et al. (2000), muon spin rotation experiments Luke et al. (1993), phase-sensitive Josephson tunneling in the B-phase Strand et al. (2009), and the previously mentioned directional tunneling to resolve the nodal structure in the A-phase Strand et al. (2010) as well as polar Kerr rotation that shows evidence for broken time-reversal symmetry in the B phase, but not the A phase Schemm et al. (2014). Furthermore, the linear temperature dependence of the London penetration depth, obtained from both magnetization Signore et al. (1995); Schöttl et al. (1999) and SANS Gannon et al. (2015) measurements, is consistent with quadratic dispersion at the polar nodes of the energy gap structure that is one characteristic of the $E_{2u}$ model. With the exception of the SANS experiment by Gannon et al. Gannon et al. (2015), all of the experiments listed above that support an odd-parity chiral state for the B phase were performed in the Meissner state Luke et al. (1993); Strand et al. (2009, 2010); Schemm et al. (2014). The goal of our SANS studies is to investigate the theoretical prediction of BTRS in the B phase of UPt3 using vortices as the probe.

III III. Methods

The SANS experiments were performed at the GP-SANS beam line at the High Flux Isotope Reactor at Oak Ridge National Laboratory (ORNL) and at the D33 beam line at Institut Laue-Langevin (ILL) Eskildsen et al. (2016). Preliminary, lower-resolution measurements were carried out at the SANS-I and SANS-II beam lines at the Paul Scherrer Institute.

Measurements were performed on VLs prepared using two different field histories illustrated in Fig. 1. For a field reduction (labelled $++$ or $--$) the applied field was first increased into either the time-reversal symmetric C (ILL) or the normal (ORNL) phase, and then reduced to the measurement field ($H_{\text{m}}$) without changing polarity. For a field reversal ($+-$ or $-+$), the field was decreased into the Meissner state with chirality determined by the initial polarity of the field in the C or normal phase, then further decreased through zero such that $H_{\text{m}}$ had the opposite polarity of the initial field in C phase or normal state.

Small-angle neutron scattering studies of the VL rely on the periodic field modulation due to vortices, with a scattered intensity $\propto\lambda^{-4}$ Mühlbauer et al. . For UPt3 the large in-plane penetration depth $\lambda_{ab}\sim 680$ nm Gannon et al. (2015) necessitates a large sample volume. We used the same 15 g, high-quality (RRR $>600$) UPt3 single crystal as in previous experiments Gannon et al. (2015). A long, rod-like crystal was cut into two pieces which were co-aligned and fixed with silver epoxy (EPOTEK E4110) to a copper cold finger. The sample assembly was mounted onto the mixing chamber of a dilution refrigerator and placed inside a superconducting magnet, oriented with the crystalline $a$-axis vertical and the $c$-axis horizontally along the magnetic field and neutron beam. Measurements were performed at base temperature $T\sim 50-65$ mK and with fields between $0.1$ T and 1.7 T. To eliminate effects of eddy current heating, the field changes were paused at 0.1 T from the target field and the sample was allowed to thermalize. Once a stable sample temperature was achieved, the field was driven to the final value. Prior to each SANS measurement a damped field oscillation with an initial amplitude of 20 mT was applied to achieve a homogeneous vortex density and a well ordered VL. Periodically during the SANS measurements a 5 mT field oscillation was performed around the measurement field to maintain an ordered VL. All field oscillations finished by a reduction of the applied field magnitude, corresponding to a decrease of the vortex density.

All SANS measurements were carried out in a “rocked on” configuration, satisfying the Bragg condition for VL peaks at the top of the two-dimensional position sensitive detector Mühlbauer et al. . Measurements of full rocking curves, where the VL peak intensities are recorded as they are being rotated through the Bragg condition, would greatly exceed the available SANS beam time and were therefore not feasible. Background measurements, obtained either in zero field or above $H_{\text{c2}}$, were subtracted from both the field reduction and field reversal data.

IV IV. Results

Figures 2(a) and (b) show SANS VL diffraction patterns obtained following a field reduction and reversal respectively, illustrating the two central results in this report. First, the scattered intensity was equally divided between two Bragg peaks separated by a splitting angle, $\omega$, for both the field reduced and field reversed case. This indicates the presence of two triangular VL domain orientations, rotated clockwise and counterclockwise about the crystalline $c$-axis. The splitting is also observed with just one of the two crystals illuminated by the neutron beam, indicating that the two VL domain orientations are intrinsic. In real space the splitting of the VL Bragg peaks corresponds to a nearest neighbor direction rotated away from the $\bf{a}^{*}$ direction by $\pm\omega/2$. Previous SANS studies with $\bf{H}\!\parallel\!\bf{c}$ were performed in a magnetic field, $0.19$ T, too low to resolve this splitting Huxley et al. (2000).

Our main result is an observed difference of the Bragg peak splitting for the two field histories used, providing direct evidence for broken time-reversal symmetry of the parent B phase. The field dependence of the VL splitting and the difference in the splitting of the domains ($\Delta\omega=\omega_{+-}-\omega_{++}$ or $\omega_{-+}-\omega_{--}$) are shown in Figs. 3(a) and (b) respectively. The splitting was determined from the angular positions of two-Gaussian fits to the VL diffraction patterns, and error bars represent one standard deviation. The curves in Fig. 3 are fits to an empirical function of magnetic field SM . Below $0.7$ T the field reversed split is slightly smaller than the field reduced one, giving rise to a negative $\Delta\omega$. For fields between $0.8$ T and $1.5$ T the situation is the opposite, where we find $\Delta\omega>0$. While $\Delta\omega$ is small at low fields, the high field results are clear and statistically significant. Finally, we note that there is no difference between measurements where the field reduced or field reversed state originated in the C phase or from the normal state, as is expected if both C and normal phases are time-reversal invariant.

These observations were made possible by extending the field range of previous SANS studies by more than a factor of eight for $\bf{H}\!\parallel\!\bf{c}$ Huxley et al. (2000), which we attribute to crystal quality and magnet homogeneity. This allowed us to explore the high field part of the phase diagram, where the VL is most sensitive to changes in the superconducting state and especially the vortex-core structure. The upper limit to accessible fields results from the reduction of the scattered intensity with increasing field, making SANS count times prohibitively long SM . This is also the reason for the increased error bars for the determination of $\omega$ and $\Delta\omega$ at higher fields.

The location of the B-C phase transition varies between samples apparently owing to crystal quality Adenwalla et al. (1990); Kycia et al. (1998). It is therefore desirable to identify features in the SANS data which allow a determination of this phase boundary. Previous SANS studies with $\bf{H}\!\perp\!\bf{c}$ found a change in the UPt3 VL structure, and in one case a disordering upon entering the C phase Yaron et al. (1997); Gannon et al. (2015). For $\bf{H}\!\parallel\!\bf{c}$ we observe that the VL peak splitting changes with the applied field up to $1.5$ T, as shown in Fig. 3(a). Above this field the splitting stabilizes at a $\omega\sim 8^{\circ}$ (dotted line), deviating from an extrapolation of the lower field data (full and dashed lines).

The radial width of the Bragg peaks in the detector plane ($\Delta Q_{\text{R}}$), shown in Fig. 4(a), also increases abruptly above $1.5$ T, indicating a disordering of the VL. Note that the minimum $\Delta Q_{\text{R}}$, observed at intermediate fields, is smaller than the divergence of the incident neutrons, determined from the undiffracted beam. This corresponds to a highly ordered VL in the B phase, resulting in a diffracted beam that is better collimated than the incident one. The disordering at high fields is reflected in the peak scattered intensity ($I_{0}$), Fig. 4(b). Here the line is a fit of the intermediate field data to an extended London model SM . Above $1.5$ T the measured intensity falls below the extrapolated fit, indicating a broadening of the Bragg peaks in the direction perpendicular to the detector plane (normal to $\Delta Q_{\text{R}}$).

We interpret the three features in the SANS data at $1.5$ T (splitting angle, radial width and intensity) as consistent signatures of the B-C transition. Note that disorder in the VL is also seen at low field in both $\Delta Q_{\text{R}}$ and the scattered intensity. This is due to a weakening of the vortex-vortex interactions at low vortex densities Eskildsen et al. (1997), and is not associated with a change in the superconducting order parameter.

To ensure that the SANS results were not affected by a systematic asymmetry of the experimental setup, measurements were carried out using alternating directions of the C phase/normal state field, i.e. $++$/$+-$ and $--$/$-+$. As evident from Fig. 3 this yielded consistent results, eliminating extrinsic effects as the cause for the difference between the splittings of the field reduced and field reversed protocols. Furthermore, results obtained using instruments at two different neutron scattering facilities (ORNL, ILL) are in excellent agreement.

Finally, the measurements are not affected by vortex pinning in any significant manner, evident from a comparison of the vortex density for the two different field histories. This is obtained from the magnitude of the VL scattering vector $Q$, from which is it possible to determine the magnetic induction $B$. As shown in the Supplemental Material, there is perfect agreement between $B$ and the applied field, $H$, for both the field reduced and the field reversed cases, within measurement accuracy SM . This agreement also confirms the triangular symmetry of the VL made up of singly quantized vortices throughout the entire measured field range. Our data does not support theoretical predictions of doubly quantized vortices Tokuyasu and Sauls (1990); Sauls and Eschrig (2009); Ichioka et al. (2012) based on $E_{1u}$ pairing symmetry, or structural distortions/transitions within the B phase or associated with the B-C phase transition Volovik (1988); Tsutsumi et al. (2012); Takamatsu and Yanase (2015) for $\bf{H}\!\parallel\!\bf{c}$.

V. Discussion

The structure of the order parameter in the vortex core depends sensitively on the symmetry of the parent superconducting state in the host material, making vortices an ideal probe of chiral superconductors. The VL symmetry and its orientation relative to the crystalline axes are determined by the anisotropy of the screening current in the plane perpendicular to the applied field, and the effect of this anisotropy on the vortex-vortex interactions. The anisotropy of the screening current can originate from several sources, particularly the Fermi velocity of the parent metallic state Kogan et al. (1997); Champel and Mineev (2001), as well as the order parameter Volovik (1988); Hess et al. (1989); Affleck et al. (1997); Champel and Mineev (2001).

For unconventional superconductors, which spontaneously break the symmetry of the Fermi surface, broken orbital rotational symmetry can lead to an anisotropy of the supercurrents that competes with or dominates the anisotropy arising from the Fermi velocities. This is the situation in UPt3. Here the dominant anisotropy in the A phase arises from the order parameter, which exhibits maximal orbital symmetry breaking with $|\Delta_{\mbox{\tiny A}}(\mathbf{p})|=\Delta_{\mbox{\tiny A}}\,|2\hat{p}_{x}\hat{p}_{y}\hat{p}_{z}|$ for $E_{2u}$ pairing symmetry, and leads to the complex SANS diffraction patterns reported by Huxley et al. Huxley et al. (2000); Champel and Mineev (2001). However, even the B phase exhibits moderate in-plane gap anisotropy resulting from the in-plane symmetry breaking field, with $|\Delta_{\mbox{\tiny B}}(\mathbf{p})|=\Delta_{\mbox{\tiny B}}(T)\,\left(1-\epsilon|\hat{p}_{x}^{2}-\hat{p}_{y}^{2}|\right)|\hat{p}_{z}|$, where $\epsilon\propto\Delta T_{c}/T_{c}\approx 10\%$ is the magnitude of the symmetry breaking field measured in terms of the zero-field splitting of the superconducting transition, $\Delta T_{c}=T_{c}-T_{c_{2}}$ Sauls (1994); Strand et al. (2010). The anisotropy of the B phase gap is tetragonal, which generates a four-fold anisotropy of the screening current around an isolated vortex SM . Notably, the B phase gap anisotropy dominates the hexagonal anisotropy of the in-plane Fermi velocity, as the latter is of the order of $1\,\%$ based on measurements of the in-plane anisotropy of $H_{c_{2}}$ Shivaram et al. (1986); Keller et al. (1994).

We make three important observations based on our experimental results. The first is that in the low-field region the gap distortion of the parent B phase leads to the rotation of the hexagonal VL which increases with field as the density of vortices increases. This provides a qualitative explanation for the increase in the domain splitting up to fields of order $H\approx 0.8\,\mbox{T}$.

The second observation is that the non-monotonic field dependence of the VL rotation shown in Fig. 3(a), onsetting at fields of order half the upper critical field, implies a change in the anisotropy of the vortex currents as the vortices become closely packed. This reflects a change in the anisotropy of the order parameter and current density near the vortex-cores. Further evidence that the non-monotonic field dependence is due to the influence of the vortex-core structure is supported by the third observation.

There is asymmetry of the VL rotation for the two different field histories - field reduction compared to field reversal. This is the signature that the parent B phase breaks time reversal symmetry. Specifically, for odd-parity superconductors that break time-reversal symmetry, vortices possess an internal degree of freedom associated with the phase winding of the vortex core, in addition to the global phase winding $n\in\{0,\pm 1,\pm 2,\ldots\}$. The phase winding of the vortex core differs from the global phase winding by the topological winding number of the chiral B phase order parameter, $m=L_{z}^{\text{orb}}/\hbar$. Thus, for a fixed ground state with chirality $\mbox{\sf sgn}(m)=+1$ there are two distinct VLs of singly quantized vortices labelled by $(\mbox{\sf sgn}(m)\,\mbox{\sf sgn}(n))$ corresponding to opposite phase windings embedded in the fixed chiral ground state. With $\mbox{\sf sgn}(m)=+1$: a VL of $(+\,+)$ vortices corresponds to $+{\mathbf{H}}\!\parallel\!L_{z}$ and a VL of $(+\,-)$ vortices corresponds to ${-\mathbf{H}}\!\parallel\!L_{z}$. These time-reversed vortices, embedded in a fixed chiral ground state, have different energies and anisotropic vortex-core structures Tokuyasu et al. (1990). Since chirality is a discrete symmetry in UPt3 it is not possible to deform the $\mbox{\sf sgn}(m)=+1$ chiral ground state into the time-reversed state with $\mbox{\sf sgn}(m)=-1$ by an adiabatic field reversal through the Meissner state. This accounts for the metastable states shown in the Fig. 1 insert.

The vortex-core order parameter is time-reversed relative to the parent B phase order parameter with winding number $p=n+2m$. Strong local anisotropy develops when a $p$-quantized vortex core dissociates, breaking the local rotational symmetry of the parent B phase Tokuyasu et al. (1990); Ichioka and Machida (2002); Sauls and Eschrig (2009); Ichioka et al. (2012). The enhanced asymmetry in the VL rotation at fields of order $H\sim 1\,\mbox{T}$ suggests a broken rotational symmetry of at least one of the vortex cores in the chiral B phase. Quantitative calculations of the VL lattice structure, including the multi-band gap structure of UPt3 Nomoto and Ikeda (2016), can provide a definitive answer to the source of enhanced chiral asymmetry at intermediate fields. Indeed quantitative calculations of the vortex core and current anisotropies in spin-triplet, $p$-wave superfluid 3He have been reported Thuneberg (1986).

V VI. Conclusion

We have used SANS to study the VL in UPt3 with $\bf{H}\!\parallel\!\bf{c}$ in both the B and C phases. We find a non-monotonic field dependence of the VL configuration, attributed to a competition between order parameter anisotropy and vortex-core structure. Depending on the relative direction of the chiral axis and the magnetic field we observe an asymmetry in the VL structure, which becomes more pronounced at higher vortex densities where vortex-core effects become more important. The results provide a direct signature of broken time-reversal symmetry of the bulk parent B phase, in agreement with theoretical predictions.

Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Awards No. DE-SC0005051 (MRE: University of Notre Dame; neutron scattering) and DE-FG02-05ER46248 (WPH: Northwestern University; crystal growth and neutron scattering), and the Center for Applied Physics and Superconducting Technologies (JAS: Northwestern University; theory). The research of JAS is supported by National Science Foundation Grant DMR-1508730. A portion of this research used resources at the High Flux Isotope Reactor, a U.S. DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. Part of this work is based on experiments performed at the Institut Laue-Langevin, Grenoble, France and at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. We are grateful to the technical staff at all of these facilities for their assistance with the measurements.