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State with spontaneously broken time-reversal symmetry above the superconducting phase transition

Nature Physics volume 17, pages 1254–1259 (2021)Cite this article

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Abstract

The most well-known example of an ordered quantum state—superconductivity—is caused by the formation and condensation of pairs of electrons. Fundamentally, what distinguishes a superconducting state from a normal state is a spontaneously broken symmetry corresponding to the long-range coherence of pairs of electrons, leading to zero resistivity and diamagnetism. Here we report a set of experimental observations in hole-doped Ba1−xKxFe2As2. Our specific-heat measurements indicate the formation of fermionic bound states when the temperature is lowered from the normal state. However, when the doping level is x ≈ 0.8, instead of the characteristic onset of diamagnetic screening and zero resistance expected below the superconducting phase transition, we observe the opposite effect: the generation of self-induced magnetic fields in the resistive state, measured by spontaneous Nernst effect and muon spin rotation experiments. This combined evidence indicates the existence of a bosonic metal state in which Cooper pairs of electrons lack coherence, but the system spontaneously breaks time-reversal symmetry. The observations are consistent with the theory of a state with fermionic quadrupling, in which long-range order exists not between Cooper pairs but only between pairs of pairs.

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Fig. 1: Time-reversal symmetry breaking in a superconducting state.
Fig. 2: Characteristic temperatures.
Fig. 3: Phase diagram.

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Source data are provided with this paper. All other data that support the findings of this paper are available from the corresponding authors upon request.

Code availability

The codes needed to assess the presented results are available from the corresponding authors upon reasonable request.

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Acknowledgements

This work was supported by DFG (GR 4667/1, CA 1931/1-1 (F.C.), GRK 1621 and SFB 1143 (project ID, 247310070)); the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat (EXC 2147, Project ID 390858490); the Swedish Research Council grant nos. 642-2013-7837, 2016-06122, 2016-04516 and 2018-03659; and by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant no. PHY-1607611. The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Center at Linköping, Sweden, partially funded by the Swedish Research Council through grant agreement no. 2018-05973. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas ‘Quantum Liquid Crystals’ (JP19H05823) from JSPS, Japan. This work was further supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 647276-MARS-ERC-2014-CoG) and by the Emmy Noether Program of the German Research Foundation (DFG grant no. 381693882). We also acknowledge support of the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL). We acknowledge fruitful discussions with S.-L. Drechsler, D. Efremov, I. Eremin, E. Herland, C. Hicks, H. Luetkens, Y. Ovchinnikov and P. Volkov. We are thankful to K. Nenkov and C. Klausnitzer for technical support.

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Authors and Affiliations

  1. Institute for Solid State and Materials Physics, Technische Universität Dresden, Dresden, Germany

    Vadim Grinenko, Jochen Wosnitza, Rajib Sarkar, Shanu Dengre, Bernd Büchner & Hans-Henning Klauss

  2. Leibniz Institute for Solid State and Materials Research, Dresden, Germany

    Vadim Grinenko, Federico Caglieris, Christoph Wuttke, Christian Hess, Aliaksei Charnukha, Ruben Hühne, Kornelius Nielsch & Bernd Büchner

  3. Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, China

    Vadim Grinenko

  4. Department of Physics, KTH Royal Institute of Technology, Stockholm, Sweden

    Daniel Weston, Ilaria Maccari & Egor Babaev

  5. Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Wuppertal, Germany

    Christian Hess

  6. Dresden High Magnetic Field Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany

    Tino Gottschall, Denis Gorbunov, Sergei Zherlitsyn & Jochen Wosnitza

  7. Department of Physics, Stockholm University, Stockholm, Sweden

    Andreas Rydh

  8. National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan

    Kunihiro Kihou & Chul-Ho Lee

  9. Institut Denis Poisson UMR CNRS 7013, Université de Tours, Tours, France

    Julien Garaud

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  1. Vadim Grinenko
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Contributions

V.G. designed the experimental study; initiated and supervised the project; co-wrote the paper; performed specific-heat, transport and susceptibility measurements in PPMS (Quantum Design), SQUID (Quantum Design) and NQR; performed ultrasound experiments; and interpreted the experimental data. D.W. wrote independent Monte Carlo code, performed computer simulations and other calculations, analysed the numerical data, and co-wrote the paper. F.C. and C.W. performed the electrical and thermal transport measurements and analysed the data. C.H. supervised the research. T.G. performed the a.c. calorimetry measurements and analysed the data. I.M. wrote independent Monte Carlo code, performed computer simulations and other calculations, analysed the numerical data, and co-wrote the paper. D.G. and S.Z. performed and supervised the ultrasound measurements. J.W. supervised the research. A.R. performed microcalorimetry measurements and analysed the data. K.K. and C.-H.L. prepared the Ba1−xKxFe2As2 single crystals. R.S. and S.D. performed the NQR experiments. J.G. wrote the finite-element energy-minimization code, performed computer simulations and other calculations, and analysed the numerical data. A.C. analysed the data. R.H. supervised research on the structural characterization of the samples. K.N. and B.B. supervised the research. H.-H.K. initiated the project, supervised the research and provided interpretation of the experimental data. E.B. initiated and supervised the project, performed theoretical studies, provided interpretation of the experimental data and numerical results, and co-wrote the paper. All the authors discussed the results and implications and commented on the manuscript.

Corresponding authors

Correspondence to Vadim Grinenko or Egor Babaev.

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Extended data

Extended Data Fig. 1 Extraction of the Spontaneous Nernst effect.

Temperature dependence of the Nernst (left axis) and Seebeck (right axis) signals for x = 0.8 and B = 2 T. The green full and empty symbols represent the Nernst signal for B = 2 T and B = − 2 T, respectively. The red and blue lines represent the evaluated B-antisymmetric and B-symmetric parts of the Nernst signal, respectively. The black dashed line represents the Seebeck signal for B = 2 T. Inset: Spontaneous Nernst effect at B = 2 T, extracted according to Eq. (1).

Extended Data Fig. 2 Comparison of different probes of BTRS state.

(a) Temperature dependence of the zero-field muon spin relaxation rate (left) shown together with the static magnetic susceptibility measured in B∥ab = 0.5 mT (right) for the stack of single crystals with x = 0.78(3)13. The displayed error bars correspond to one standard deviation from the χ2 fit. (b) Temperature dependence of the spontaneous Nernst effect measured in zero magnetic field (left) shown together with the static magnetic susceptibility measured in B∥ab = 0.5 mT (right) for the sample with x = 0.77. The comparison between the μSR data and the spontaneous Nernst signal strongly suggests that the increase of the muon spin relaxation rate above Tc is not an artifact and that the origin of the spontaneous Nernst effect at \({T}_{{{{\rm{c}}}}}^{Z2}\) is spontaneous magnetic fields.

Extended Data Fig. 3 Spontaneous Nernst effect and superconducting fluctuations.

(a-c) Temperature dependence of the spontaneous Nernst effect (left) and electrical resistivity (right) for two Ba1−xKxFe2As2 samples with doping levels x = 0.54 and x = 1 (without quartic phase), compared with the very different behavior of a sample with x = 0.77 that has a quartic phase. A strong spontaneous Nernst signal was observed for x = 0.77, which is clearly seen in the raw data shown in Extended Data Fig. 4b. (d-f) Temperature dependence of the conventional (odd in magnetic field) Nernst effect (Sxy) for the same samples. Dashed lines show a linear behavior in temperature for the quasiparticle contribution in the normal state, Sxy/T. For the sample with x = 0.54, field dependence in the Nernst signal appears a few kelvin above Tc. For the sample with x = 0.77, the Nernst effect has a complex behaviour: it becomes field dependent at \({T}_{{{{\rm{SF}}}}}^{{{{\rm{high}}}}} \sim 2{T}_{{{{\rm{c}}}}}\), has a minimum roughly at \({T}_{{{{\rm{c}}}}}^{{{{\rm{Z}}}}2}\), a maximum at Tc, and goes to zero at \({T}_{{{{\rm{SF}}}}}^{{{{\rm{low}}}}}\) (see also Fig. 2). \({T}_{{{{\rm{SF}}}}}^{{{{\rm{high}}}}}\) is the onset temperature of detectable superconducting fluctuations, and \({T}_{{{{\rm{SF}}}}}^{{{{\rm{low}}}}}\) is the temperature were fluctuations become undetectable. (g-i) Temperature dependence of the longitudinal electrical resistivity in zero field. Solid curves are the experimental data and dashed lines are fits in the normal state (for explanation see the main text). Insets show the temperature dependence of the difference between the fit curves and the experimental data. The resistivity deviates from the normal-state behaviour at \({T}_{{{{\rm{SF}}}}}^{{{{\rm{high}}}}}\).

Extended Data Fig. 4 Raw thermoelectric data.

Temperature dependence of the zero-field voltage measured with Seebeck (left axis) and Nernst (right axis) contacts when a temperature gradient is applied for (a)x = 0.54 and (b)x = 0.77. A clear difference between the signals is seen in panel (b) only.

Extended Data Fig. 5 In-field properties of the sample with x= 0.8.

(a) Temperature dependence of the specific heat measured by the relaxation method with the relative temperature increase of 2%. (b) Temperature dependence of the longitudinal electrical resistivity measured with the AC current amplitude 1 A/cm2 and the frequency 173 Hz. (c) Temperature dependence of the AC susceptibility measured with AC excitation field B = 0.3 mT and f = 777 Hz in different DC fields applied along the crystallographic c-axis. (d) Temperature dependence of the AC susceptibility measured with AC excitation field B = 0.3 mT and f = 777 Hz in zero DC field (left axis), and the DC susceptibility in B = 0.5 mT applied in the ab-plane (right axis). (e) Experimental magnetic-field phase diagram for the superconducting phase transition defined by zero resistance from the data in panel (b), and AC susceptibility from the data in panel (c), and onset temperature of the dominant specific heat anomaly shown in panel (a) [note that the resolution here is not sufficient to identify \({T}_{{{{\rm{c}}}}}^{Z2}\)]. The relative splitting between the temperatures is increased with magnetic field. (f) The low-field region of the same phase diagram as in panel (e). The displayed error bars correspond to the uncertainty in the determination of Tc.

Extended Data Fig. 6 Ultrasound data.

Temperature dependence of the relative change of the sound velocity (left) for the longitudinal (c11 + c12 + 2c66)/2 and transverse (c11 − c12)/2 acoustic modes with subtracted background (see Fig. S3 in the supplementary information) compared with temperature dependence of the magnetic susceptibility in B∥ab = 0.5 mT (right) measured after cooling in zero magnetic field (ZFC) and cooling in the applied field (FC). The data for the samples with x = 0.71, 0.81, and 1 are shown in panels (a), (b), and (c), respectively. The sample with x = 0.81 exhibits a kink in the velocity of the transverse acoustic mode at a position which agrees with the onset of the broad feature in the specific heat shown in Extended Data Fig. 8 for the same sample. The position of the kink agrees with the position of the Z2 transition indicated by the thermal transport experiments for the sample with x ≈ 0.8 (Fig. 2 in the main text).

Extended Data Fig. 7 Ba1−xKxFe2As2 at different doping levels x.

(a-d) Temperature dependence of the magnetic susceptibility (left axis) measured in B∥ab = 0.5 mT applied after cooling in zero field (ZFC) with consequent measurement while cooling in the same field (FC) and the zero-field specific heat (right axis) measured by the relaxation method with the relative temperature increase of 2% for samples with different K doping level x and mass ms ~ 1 mg. Two specific-heat curves, shown in panel (d), were measured using different techniques: the sample with mass ms ~ 1 μg measured by microcalorimetry was cut from the larger sample with ms = 0.3 mg measured by the relaxation technique. Panels (e-h) present the data for the same sample with ms = 0.3 mg as in (d). (e) Temperature dependence of the ac magnetization (left axis) measured in Bac∥c = 0.3 mT and at fac = 777 Hz and electrical resistivity (right axis) measured in zero magnetic field and at different applied currents. The dashed curve is the electrical resistivity at B∥c = 7 T. (f) Temperature dependence of the ac susceptibility magnified by a factor of 103 (left axis) and electrical resistivity in a log-scale (right axis) measured in zero magnetic field. Both the resistivity and the susceptibility give similar Tc, putting a constraint on inhomogeneity. (g) Temperature dependence of the resistivity in zero magnetic field (solid line) and a T2-fit (dotted line). (h) Temperature dependence of the difference between a T2-fit and the data shown in panel (g). The observed behaviour is very similar to that for the sample with x = 0.77 shown in Extended Data Fig. 3h. (i) Experimental phase diagram (left axis): the superconducting transition is extracted from the DC susceptibility data (closed symbols) and taken from ref. 20 (open symbols); the dome of superconducting s + is state is taken from ref. 20. Green triangles (right axis) show the doping dependence of the residual resistivity ρxx,0 of the Ba1−xKxFe2As2 single crystals. The displayed error bars correspond to the spread of the K concentration within individual samples.

Extended Data Fig. 8 Sample with x = 0.81, ms = 0.1mg.

(a) Magnetic susceptibility in B∥ab = 0.5 mT (left) and zero-field specific heat (right) measured by the AC technique. (b) Temperature dependence of the electrical resistivity in zero magnetic field. Both the resistivity and the susceptibility give a similar Tc. An anomaly in the specific heat is observed above Tc. (c) Temperature dependence of the resistivity in zero magnetic field. (d) Temperature dependence of the difference between a T2 fit and the data. The observed T2 behaviour is very similar to the sample with x = 0.77 shown in Extended Data Fig. 3h.

Extended Data Fig. 9 Spontaneous magnetic field.

Magnetic field induced by a domain wall in the presence of thermal gradients in the BTRS quartic metal state. The surface elevation together with the colouring, represent the magnitude of the spontaneous magnetic field B(6), normalized to the maximal value \({B}_{\max }\). The domain wall itself is illustrated on the near face of the domain: the color scheme represents two different BTRS states, schematically depicted by arrows. The near and far faces are maintained at different temperatures, resulting in a thermal gradient along the domain wall. Parameters are given in the text.

Extended Data Fig. 10 Fluctuation-induced phases in multiband models.

Panels (a)-(d) show results for the two-component approximation of a three-band model in zero external magnetic field in the extreme type-II limit for various values of the mixed gradient coupling ν. (a) Phase diagram. The U(1) and Z2 transitions are clearly split apart for sufficiently large ν, giving rise to the quartic metal phase. The gray dashed line indicates the value ν = 0.6. (b) Binder cumulant U at ν = 0.6 for different system sizes as function of the inverse temperature β. (c) Helicity modulus for the phase sum, ϒ+, at ν = 0.6 for different system sizes versus β. (d) Illustrative example of the meaning of the two-component Ising order parameter m. Panels (e) -(i) Show results for a three-component Ginzburg-Landau model. (e) Heat capacity L−3 d〈E〉/dT versus β for the system with applied field that we consider. The heat capacity shows a signature of the Z2 transition to a non-superconducting state associated with the breaking of time-reversal symmetry. (f) Histogram of the Ising order parameter m for β = 4.1. For this inverse temperature the Z2 symmetry is clearly broken. (g) Illustration of the order parameter m for the three-component case. Structure factors for (h) the vorticity of ψ1 and (i) the magnetic field, at the same inverse temperature β = 4.1 as for the above histogram. Snapshots are shown to the left and thermal averages to the right. In the presence of a vortex lattice, the structure factors will have pronounced peaks. The absence of such peaks indicates that the system is in a resistive vortex-liquid state which spontaneously breaks Z2 symmetry due to nontrivial phase locking. (We remove the trivial zero-wave-vector components of the structure factors for clarity, and normalize the remaining components to the zero-wave-vector component.)

Supplementary information

Supplementary Information

Supplementary Figs. 1–4, data and discussion.

Source data

Source Data Fig. 2

Source susceptibility, specific-heat, transport and thermal transport data for all the panels.

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Grinenko, V., Weston, D., Caglieris, F. et al. State with spontaneously broken time-reversal symmetry above the superconducting phase transition. Nat. Phys. 17, 1254–1259 (2021). https://doi.org/10.1038/s41567-021-01350-9

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