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Topology meets time-reversal symmetry breaking in FeSe$_{1-x}$Te$_{x}$ superconductor

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Pith's one-line read Zero-field muon spin relaxation detects spontaneous static internal fields below Tc in bulk FeSe0.36Te0.64, distinct from magnetic order, indicating a time-reversal-symmetry-breaking superconducting state.

desk verdict Bulk muSR evidence for TRSB at the topological composition FeSe0.36Te0.64 is new and mostly convincing, but the homogeneity assumption needs a wTF volume-fraction check before I'd call it airtight. read the letter →

arxiv 2501.02818 v1 pith:RRZU2RPC submitted 2025-01-06 cond-mat.supr-con cond-mat.str-el

M. Roppongi , Y. Cai , K. Ogawa , S. Liu , G. Q. Zhao , M. Oudah , T. Fujii , K. Imamura
show 16 more authors
S. Fang K. Ishihara K. Hashimoto K. Matsuura Y. Mizukami M. Pula C. Young I. Markovic D. A. Bonn T. Watanabe A. Yamashita Y. Mizuguchi G. M. Luke K. M. Kojima Y. J. Uemura T. Shibauchi
This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords zero-fieldmuonspinrelaxationtime-reversalsymmetrybreakingtopologicalsuperconductivityiron-chalcogenidesuperconductorsFeSe1-xTexspontaneousmagneticfieldMajoranazeromodesDiracsurfacestate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to establish that the iron-chalcogenide superconductor FeSe$_{1-x}$Te$_{x}$ at Te composition $x=0.64$, whose surface is already known to host topological states and vortex Majorana zero modes, breaks time-reversal symmetry in its bulk superconducting state. The evidence comes from zero-field muon spin relaxation: below the superconducting transition at 14.5 K a static internal field appears, with an onset near 11.5 K, and the relaxation is suppressed by a 200 G longitudinal field, which is the signature of a static field rather than magnetic order. The same signature, with a smaller field, appears in the nematic composition near 0.35, while higher-Te samples $x=0.75$ and 0.83 show a fast relaxation component that starts above $T_c$ and is not decoupled by a longitudinal field, which the authors attribute to short-range antiferromagnetism. If correct, this makes FeSe$_{1-x}$Te$_{x}$ a bulk superconductor that combines intrinsic time-reversal symmetry breaking with nontrivial topology, a combination predicted to open a Dirac gap in the surface state and to enable chiral edge physics.

What carries the argument

The central probe is zero-field muon spin relaxation, in which spin-polarized muons implanted into the crystal precess and relax according to the local magnetic field distribution. In a time-reversal-symmetry-breaking superconductor, spontaneous currents around defects or domain boundaries generate a small static internal field that raises the muon relaxation rate below $T_c$; applying a longitudinal field of 100-200 G decouples the muons from static fields, which is how the authors distinguish a static spontaneous field from dynamic fluctuations or magnetic order. The argument is carried by a comparative phase-diagram analysis: compositions $x=0.75$ and 0.83 exhibit a fast relaxation component that appears above $T_c$ and survives a longitudinal field, marking short-range antiferromagnetism, while $x=0.64$ and the near-0.35 composition show only the TRSB-type relaxation that decouples under a longitudinal field.

What would settle it

A weak-transverse-field muon measurement on the same $x=0.64$ crystals would settle the point: if the oscillating amplitude begins to drop below $T_c$, that loss would reveal a magnetic volume fraction and contradict the homogeneous TRSB interpretation. Equivalently, a high-statistics two-component analysis of the zero-field spectra at 1.9 K that finds a missing fast component or an initial-asymmetry loss would argue for dilute magnetism instead.

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Extended reading notes

Core claim

The core discovery is that the tetragonal superconductor FeSe$_{0.36}$Te$_{0.64}$ hosts a bulk superconducting state with broken time-reversal symmetry, identified by the appearance of spontaneous static magnetic fields below $T_c$ that are distinct from any magnetic order. Zero-field muon spin relaxation spectra show an increased relaxation rate in the superconducting state in both non-spin-rotated and spin-rotated geometries; a longitudinal field of 200 G suppresses this relaxation, proving the source is static. The estimated internal field is about 0.76 G in the non-spin-rotated mode and 1.53 G in the spin-rotated mode at the lowest temperature, and the TRSB onset is about 11.5 K, consistent with the earlier Kerr effect. A parallel measurement at a Te fraction near 0.35 finds the same phenomenon with a smaller field. Crucially, the $x=0.64$ samples show no fast relaxing component, no loss of initial muon asymmetry, and no residual density of states in specific heat, so the authors conclude the entire volume becomes a TRSB superconductor without magnetic inclusions.

Load-bearing premise

The claim rests on the assumption that the enhanced zero-field relaxation below $T_c$ in the $x=0.64$ samples comes from a static internal field spread uniformly through the bulk, and not from a small volume fraction of short-range magnetic order that a single-component exponential fit fails to resolve.

Editorial extensions

If this is right

  • At $x=0.64$, the bulk TRSB pairing state coexists with the known topological surface state, so the surface Dirac cone should acquire a gap below the TRSB onset, following the same logic as magnetic topological insulators.
  • The TRSB onset of about 11.5 K being close to the Kerr-effect onset strengthens the case that earlier surface-sensitive signals reflect a bulk broken-time-reversal state rather than a surface-only effect.
  • The phase diagram now places bulk TRSB superconductivity in the nematic and tetragonal regions of both FeSe$_{1-x}$S$_{x}$ and FeSe$_{1-x}$Te$_{x}$, implying that broken time reversal is a generic feature of FeSe-based superconductivity rather than a special property of one composition.
  • For $x=0.75$ and 0.83, any interpretation of topological superconductivity must account for coexisting short-range antiferromagnetic correlations, which appear above $T_c$ and are not suppressed by longitudinal fields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper does not perform is a weak-transverse-field muon measurement on $x=0.64$ below $T_c$; a drop in the oscillating amplitude would signal a magnetic volume fraction and contradict the homogeneous TRSB reading.
  • If the roughly 1 G spontaneous field is intrinsic, one could look for chiral surface currents by scanning SQUID magnetometry on the same crystals, since the predicted two-component order parameter should produce edge currents.
  • Comparing the internal-field magnitude at the near-0.35 and $x=0.64$ compositions with their transition temperatures might reveal whether the spontaneous field scales with $T_c$ or with spin-orbit coupling, a distinction the current data set is too small to settle.
  • The same muon protocol applied to FeSe$_{1-x}$S$_{x}$ at compositions with matching $T_c$ could test whether the TRSB field strength is controlled by the chalcogen mass, which would support the spin-orbit-enhancement scenario.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript reports zero-field (ZF) and longitudinal-field (LF) muon spin relaxation (μSR) measurements on single crystals of FeSe$_{1-x}$Te$_x$ with $x \approx 0.35$, $0.64$, $0.75$, $0.83$, and $0.94$. For $x \approx 0.35$ and $x = 0.64$, the ZF relaxation rate increases below the superconducting transition and is suppressed by a longitudinal field of 100–200 G, which the authors interpret as evidence for static spontaneous fields and thus time-reversal symmetry breaking (TRSB) in the superconducting state. For $x = 0.75$ and $0.83$, a fast, LF-insensitive relaxation component with onset above $T_c$ is attributed to short-range antiferromagnetic correlations, while $x = 0.94$ shows long-range AFM order with a magnetic volume fraction of about 90%. The authors map these findings onto a combined phase diagram and argue that $x = 0.64$ provides a bulk TRSB superconductor with a topological surface state.

Significance. If the interpretation for $x = 0.64$ is correct, the work is significant: it would place FeSe$_{0.36}$Te$_{0.64}$ as a bulk superconductor with broken time-reversal symmetry in the same material that hosts a topological surface state and Majorana zero modes, providing a platform to study the predicted Dirac gap and possible chiral edge modes. The paper is well-contextualized and includes useful internal controls: the two-component relaxation observed for $x = 0.75/0.83$ and the $A_0$ loss for $x = 0.94$ demonstrate that the authors can detect magnetic order when it is present, and the LF decoupling and the agreement of $T_{\mathrm{TRSB}} \approx 11.5$ K with Kerr data support the TRSB assignment. Sample characterization (specific heat, magnetization) also supports bulk full superconductivity for $x \le 0.64$. However, the central claim depends on the absence of a small magnetic fraction in the $x = 0.64$ sample, which is not quantified in the present manuscript; this must be addressed before the conclusion can be accepted.

major comments (2)
  1. [Results, Fig. 2b,f and Fig. 3c–e] The central claim for $x = 0.64$ rests on the absence of a magnetic minority phase, but no quantitative bound on the magnetic volume fraction is provided for this composition. The wTF-μSR volume-fraction analysis is shown only for $x = 0.94$ (Fig. 3c–e), and the constancy of the initial asymmetry $A_0$ versus temperature is likewise demonstrated only for $x = 0.94$ (Fig. 3d). For $x = 0.64$, the displayed ZF spectrum (Fig. 2b) is fitted with a single exponential/stretched exponential, and a small volume fraction with a fast-relaxing component could be missed, especially if its relaxation rate exceeds the early-time binning or the local-field distribution is broad. The specific-heat residual-DOS argument (Fig. S2) is not a strong constraint, because a few percent of magnetic clusters would contribute negligibly to the electronic specific heat. Because the sentence “without a sign of inclusion of magnetic order” (p. 7) is the basis for distinguishing TRSB from magnetic order, please perform a wTF-μSR volume-fraction check for $x = 0.64$ (and ideally $x \approx 0.35$), or fit the base-temperature ZF spectra with a two-component function and report an upper bound on the fast fraction. Without such a constraint, the interpretation is not unique.
  2. [Results, Fig. 2f and LF decoupling discussion (p. 6)] The LF decoupling at 200 G is taken as evidence for a static internal field on the ~1 G scale, and this is plausible for the dominant component. However, it does not by itself exclude a minority phase with local fields larger than 200 G or with fast-fluctuating moments, since such a contribution would not be decoupled by 200 G. The early-time asymmetry in Fig. 2b does not resolve a fast component, but a fast component could be hidden in the first ~0.1 μs. Please show the $A_0(T)$ data for $x = 0.64$ (as in Fig. 3d for $x = 0.94$) and report the fit quality or residuals for the single-component fit at base temperature. This is load-bearing because the claim that the spontaneous fields are “distinct from a magnetic order” requires excluding a small magnetic fraction.
minor comments (5)
  1. [p. 5] The typo “relaxatio rate” should be “relaxation rate”.
  2. [Fig. 2 caption] The sentence “the relaxation rate of ZF NSR mode (red circle) and LF (red circles and green squares) correspond to the left axis” is inconsistent; the LF data are plotted as green squares, not red circles. Please correct the caption.
  3. [p. 7] The phrase “we discovered the bulk TRSB superconductivity” is stronger than the presented evidence; suggest “we find evidence for bulk TRSB superconductivity.”
  4. [Results, Fig. 2e] For $x \approx 0.35$, the onset temperature $T_{\mathrm{TRSB}}$ is not stated numerically; please provide the value and its uncertainty, as given for $x = 0.64$.
  5. [Results, p. 6 and Fig. S5] The statement that the ZF relaxation-rate temperature dependence is “similar to that in TF measurements (see Fig. S5)” is an important supporting observation; consider displaying this comparison in the main text for $x = 0.64$.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the muSR inference is externally calibrated and the claimed TRSB detection does not reduce to its own inputs.

full rationale

The central claim—spontaneous static fields below Tc in x=0.64 (and x~0.35)—is inferred directly from zero-field muSR data: the ZF relaxation rate increases below Tc, the increase is suppressed by a 200 G (100 G) longitudinal field, and the temperature dependence tracks the superfluid density (Fig. S5). These are external experimental observables analyzed with standard muSR methods, not outputs of the same pairing theory the paper discusses. The x=0.75/0.83/0.94 samples provide internal controls: their fast, LF-insensitive relaxation begins above Tc and is attributed to (short-range) AFM, whereas x=0.64 shows no resolvable fast component and near-zero residual specific heat, supporting a homogeneous bulk origin. No equation reduces to its own input: Bint is a standard transformation of the fitted relaxation rate, and the LF data independently distinguish static fields from dynamic fluctuations. Self-citations such as ref. 22 (prior FeSe1-xSx muSR) supply context and methodology but are not load-bearing for the new x=0.64 result, which is measured in this paper. The paper itself notes that 'additional investigations are essential to fully elucidate the nature and detail of the TRSB pairing state,' a limitation about pairing symmetry, not about the TRSB detection. A wTF volume-fraction check on x=0.64 is not shown (only x=0.94), which is a robustness gap for excluding dilute magnetic fractions but not a circular reduction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new theoretical entities are introduced. The paper uses standard muSR analysis and prior theory. The main load-bearing ingredients are the fit parameters of the muSR spectra and the interpretive assumption that LF-suppressed ZF relaxation in a single-component signal indicates bulk TRSB rather than a hidden magnetic fraction.

free parameters (1)
  • ZF relaxation rates lambda_ZF, lambda_fast, lambda_slow = ~0.03-0.05 us^-1 for x=0.64; fast component >10 us^-1 for x>=0.75
    Fit parameters from exponential fits to muSR asymmetry spectra. Their temperature dependence is the observed quantity; they are standard data-analysis parameters, not theoretical constants, but the TRSB claim depends on the fitted increase below Tc.
assumptions (4)
  • domain assumption Zero-field muSR relaxation enhancement that is suppressed by a longitudinal field is a valid indicator of static spontaneous internal fields, i.e. TRSB.
    Standard muSR interpretation invoked in refs 22 and 40; used to infer TRSB from the ZF vs LF comparison in Fig. 2e,f.
  • domain assumption The x=0.64 crystals are free of excess iron and short-range magnetic correlations, so the muSR signal represents the bulk superconducting state.
    Relies on Te-annealing removing excess Fe and on the residual specific heat going to zero, as stated in the sample characterization and Fig. S2.
  • domain assumption A single exponential or stretched exponential fit adequately captures the x=0.64 ZF spectra, with no hidden fast component.
    If a small magnetic fraction existed, a single-component fit could miss it; this is the weakest point in the TRSB-vs-magnetism discrimination.
  • domain assumption Prior ARPES and STM evidence for a topological surface state and Majorana vortex modes at nearby compositions transfers to the x=0.64 samples measured here.
    The convergence claim assumes the same electronic structure in these crystals; cited refs 9, 11, 18 and 30.

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Pith. "Pith review of Topology meets time-reversal symmetry breaking in FeSe$_{1-x}$Te$_{x}$ superconductor." pith.science (2026). https://pith.science/paper/RRZU2RPC

@misc{pith2026250102818,
  author       = {Pith},
  title        = {Pith review of: Topology meets time-reversal symmetry breaking in FeSe$_1-x$Te$_x$ superconductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRZU2RPC}},
  note         = {Machine review of arXiv:2501.02818}
}
abstract

Time-reversal symmetry breaking (TRSB) in magnetic topological insulators induces a Dirac gap in the topological surface state (TSS), leading to exotic phenomena such as the quantum anomalous Hall effect. Yet, the interplay between TRSB and topology in superconductors remains underexplored due to limited suitable materials. Here we employ zero-field muon spin relaxation ($\mu$SR) as a sensitive probe of TRSB to map out the electronic phase diagrams of iron-chalcogenide superconductors FeSe$_{1-x}$Te$_{x}$. For the Te composition $x=0.64$ with the highest superconducting transition temperature $T_{\rm c}=14.5$ K, which is known to host a TSS and Majorana zero modes within vortices, we detect spontaneous magnetic fields below $T_{\rm c}$ distinct from a magnetic order. This signifies a TRSB superconducting state in the bulk, revealing the convergence of unconventional TRSB superconductivity with topologically nontrivial electronic structures in FeSe$_{1-x}$Te$_{x}$. Given the relatively high $T_{\rm c}$ and the tunability of the Fermi level through chemical substitution, iron-chalcogenide superconductors offer an intriguing platform for investigating the synergy between topological superconductivity and TRSB.

Figures

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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

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