Abstract
Mesoscale patterns as observed in, for example, ferromagnets, ferroelectrics, superconductors, monomolecular films or block copolymers1,2 reflect spatial variations of a pertinent order parameter at length scales and time scales that may be described classically. This raises the question for the relevance of mesoscale patterns near zero-temperature phase transitions, also known as quantum phase transitions. Here we report the magnetic susceptibility of LiHoF4—a dipolar Ising ferromagnet—near a well-understood transverse-field quantum critical point (TF-QCP)3,4. When tilting the magnetic field away from the hard axis such that the Ising symmetry is always broken, a line of well-defined phase transitions emerges from the TF-QCP, characteristic of further symmetry breaking, in stark contrast to a crossover expected microscopically. We show that the scenario of a continuous suppression of ferromagnetic domains, representing a breaking of translation symmetry on mesoscopic scales in an environment of broken magnetic Ising symmetry on microscopic scales, is in excellent qualitative and quantitative agreement with the field and temperature dependence of the susceptibility and the magnetic phase diagram of LiHoF4 under tilted field. This identifies a new type of phase transition that may be referred to as mesoscale quantum criticality, which emanates from the textbook example of a microscopic ferromagnetic TF-QCP. Our results establish the surroundings of quantum phase transitions as a regime of mesoscale pattern formation, in which non-analytical quantum dynamics and materials properties without classical analogue may be expected.
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Acknowledgements
We thank P. Böni, C. Castelnovo, T. Enns, M. Garst, P. Jorba Cabre, M. Knap, J. Knolle, M. Lampl, S. Legl, M. Meven, R. Moessner, H. Ronnow, J. Schmalian, S. Säubert, F. Pollmann and W. Zwerger for support and discussions. We also acknowledge support by S. Mayr and the mechanical workshop at Physik-Department, Technische Universität München. Financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Munich Center for Quantum Science and Technology (EXC 2111, project ID 390814868), the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat (EXC 2147, project ID 390858490), SFB 1143 (project ID 247310070) and TRR80 (project ID 107745057) is gratefully acknowledged. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 788031, ExQuiSid).
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A.W., F.R., C.D., M.K. and C.P. conducted the measurements. A.W., F.R. and C.P. analysed the data. C.P. proposed this study. H.E. and M.V. developed the theoretical model. A.W., H.E., M.V. and C.P. conceived the interpretation and wrote the manuscript. All authors discussed the data and commented on the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Miniature ac susceptometer and single-crystal LiHoF4 used in our study.
a, Compact susceptometer for studies under transverse-field geometries at very low temperatures; picture reproduced from ref. 71. A primary coil (P) and a balanced pair of secondaries (S1, S2) are placed on the outside and inside of a sapphire tube (SH), respectively. The sample (S) is mounted on a sapphire rod placed inside the secondaries. The assembly is rigidly mounted inside a body (B) made of high-purity Cu, which provides excellent thermal anchoring. b, Spherical LiHoF4 single crystal as attached to the sapphire rod. c, Photograph of the sapphire tube and primary; picture reproduced from ref. 71. d, Laue X-ray diffraction pattern of the LiHoF4 single crystal along the cylinder axis of the sapphire rod, confirming excellent alignment along the c axis, that is, the easy magnetic axis.
Extended Data Fig. 2 Real and imaginary parts of the transverse susceptibility recorded experimentally.
Data shown in each panel correspond to the value of ϕ stated at the top of each column of the panels. The colour coding of data recorded as a function of magnetic field at fixed temperature changes from blue to red, going from low to high temperatures, respectively. The colour coding of data recorded as a function of temperature at fixed magnetic field changes from blue to red, going from low to high magnetic fields, respectively. Specific values of all parameters are summarized in Extended Data Table 1. a1–f1 (first row), real part of the transverse susceptibility as a function of magnetic field for different temperatures. a2–f2 (second row), imaginary part of the transverse susceptibility as a function of magnetic field for different temperatures. a1–f3 (third row), real part of the transverse susceptibility as a function of temperature for different magnetic fields. Note that a different vertical scale was used for better visibility, in which u = 0.28 for panels a3, b3 and c3 and u = 0 for panels d3, e3 and f3. a4–f4 (bottom row), imaginary part of the transverse susceptibility as a function of temperature for different magnetic fields.
Extended Data Fig. 3 Key features in the susceptibility.
Definition of the critical field Bc, the freezing field, Bfr, and the discontinuous change of the susceptibility at the transition, Δχ′, for various tilt angles, ϕ. The critical field, Bc, represents the crossing point between a linear regression of the susceptibility in the paramagnetic state and a linear regression of the fast increase of the susceptibility in the ferromagnetic state. Results reported in our manuscript are insensitive to the precise definition of Bc. The freezing field, Bfr, marks the onset of the decrease of the plateau of χ′ under decreasing field, in which the plateau is only fully developed for ϕ < 5°. The change of the susceptibility, Δχ′, across the transition at Bc represents the difference between the susceptibility at Bc and the susceptibility of the plateau deep in the ordered state. As Bc approaches Bfr for increasing ϕ, the susceptibility of the plateau at low values of ϕ is used to determine Δχ′.
Extended Data Fig. 4 Critical behaviour of the transverse susceptibility and frequency dependence of the freezing behaviour at low temperatures and low magnetic fields.
Data shown were recorded for ϕ = 0, that is, ideal transverse field orientation. a, Typical paramagnetic contribution of the real part of the transverse ac susceptibility, \({\chi }_{{\rm{th}}}^{{\prime} }\), of LiHoF4 for two selected temperatures. Contributions arising from the magnetic domains below Bc, denoted by χ′max, were subtracted from the measured susceptibility, χ′. The magnetic field dependence may be fitted by a power-law dependence, in excellent agreement with the literature3. b, Depiction of \({\chi }_{{\rm{th}}}^{{\prime} }\) as a function of reduced magnetic field on a logarithmic scale at 57 mK. Data points correspond to those shown in ref. 71. The critical exponent γ = −1.02 ± 0.0051 is in excellent agreement with mean-field behaviour and the literature3. c, Critical exponent γ at Bc as a function of temperature. Along the phase boundary up to about 1.2 K, the same mean-field exponent, γ ≈ 1, is observed. d, Real part of the transverse susceptibility, χ′, as a function of magnetic field at T = 139 mK for excitation frequencies between 10 Hz and 5,011 Hz. With increasing frequency, the onset of the freezing shifts to higher magnetic fields characteristic of a slow process. e, Imaginary part of the transverse susceptibility, χ″, as a function of magnetic field at T = 139 mK for various frequencies. χ″ shows a strong peak, as marked by the red triangles characteristic of dissipation at low magnetic fields and for all frequencies measured. With increasing frequency, the peak shifts to higher magnetic fields characteristic of a slow process. Curves have been shifted by a constant for clarity.
Extended Data Fig. 5 Temperature versus magnetic field phase diagram for various field orientations ϕ.
A spherically shaped sample was measured to ensure uniform demagnetizing fields. a, Magnetic phase diagram of single-crystal LiHoF4 for different field angles ϕ. Data are identical to those shown in Fig. 3e–i. Data for ϕ = 0 are in excellent agreement with those reported by Bitko et al.3, as depicted by squares. b, Close-up view of the magnetic phase diagram in the vicinity of the Curie temperature for small fields. The data suggest a tiny regime of re-entrant behaviour as a function of magnetic field for ϕ = 0. Great care was taken to correct the small magnetoresistance of the RuO temperature sensor at low fields72. Small differences of Tc between our data and those reported in ref. 3 may be attributed to small quantitative differences of thermometer calibration.
Extended Data Fig. 6 Misalignment between the transverse field B (ϕ = 0) and the electronic moment J in the plane perpendicular to the easy axis at B = 6 T as calculated from the microscopic model (equation (6) in the Supplementary Information).
The symmetry of LiHoF4 allows two configurations of the F− ions in the hard plane, which lead to different signs of the crystal field coefficient \({B}_{6}^{4}(S)\)18. Owing to this small symmetry-breaking crystal field, the projection of the electronic moment onto the hard plane is not perfectly oriented along the applied magnetic field but instead shows a small misalignment θ − θ′. Thus, even if the field is applied along a high-symmetry direction, such as θ = 0, B = (B, 0, 0)T, the electronic moment, has a finite component perpendicular to the field axis θ′ ≠ 0, Jy ≠ 0. Therefore, the only spin symmetry that is not already broken in the disordered state at B > Bc is the Ising symmetry Jz → −Jz. a, Qualitative sketch of the two equivalent configurations of the F− ions (circles) projected onto the hard magnetic plane as depicted in red and blue shading, resulting in opposite sign of the crystal field parameter \({B}_{6}^{4}(S)\). b, Misalignment angle θ − θ′ between the electronic moment J and the magnetic field direction, θ, within the hard magnetic plane as a function of the orientation of the field component in the hard plane.
Extended Data Fig. 7 Energy due to domains, Edom, as a function of the number of domains, N, at various magnetic fields.
The behaviour observed here implies that the theoretical analysis does not depend on the precise choice of the domain structure. Moreover, it allows to motivate the choice of parameters selected for the numerical evaluation. For details, see Supplementary Note S8. a, Individual contributions to Edom as a function of N at zero field, namely, stray-field energy and energy of the domain walls. The ideal number of domains is determined by the competition between these two terms, scaling roughly as 1/N and N at B = 0. b, Total Edom/M as a function of N. With increasing field strength, the stray-field contribution strongly increases, whereas the contributions by the domain walls decrease. The full-domain energy near the QCP is, therefore, dominated by stray-field contributions and only weakly dependent on N. The minimal energy and, thereby, the optimal N is reached at N > 1,000. c, Total Edom/M as a function of N when hypothetically assuming a domain wall energy that is 100 times larger than realistic. This shifts the optimal N to 80. The minimum becomes shallow at large fields and, therefore, deviations from the optimal N only lead to minor quantitative changes.
Extended Data Fig. 8 Evolution of magnetic domains under magnetic field and various field directions ϕ.
a, Domain structure assumed in our model, in which planar sheets are stacked along the y axis. The spontaneous and the field-induced magnetization are oriented along the z and x axes, respectively. The magnetic field is applied in the z–x plane. D1 and D2 denote the widths of the up and the down domains, respectively. The up and the down states represent the majority and minority domains, respectively. b, Free energy F in units K as a function of domain ratio v = D2/(D1 + D2). The dashed vertical lines mark values of v, for which v* denotes the stable configuration. c, Optimal domain ratio v* of the minimum of F as a function of magnetic field for different tilt angles ϕ.
Extended Data Fig. 9 Interplay of hyperfine coupling with the non-Kramers ground state at zero temperature.
Treatment of the Ising anisotropy in terms of the full non-Kramers crystal-field terms is denoted CF. Treatment of the Ising anisotropy in terms of a simple single-ion anisotropy acting on a Kramers moment is denoted SIA. a, Calculated critical field, Bc, as a function of field orientation, ϕ, with and without hyperfine coupling A. The full set of crystal-field terms, VCF, is taken into account. Bc strongly increases owing to the hyperfine coupling because the electronic moment of the Ho ion, \(| \,J| =\sqrt{\langle \,{{J}^{x}}^{2}+{{J}^{y}}^{2}+{{J}^{z}}^{2}\,\rangle }\), increases substantially in the ordered phase (b). In turn, the Ho moment benefits more strongly from the hyperfine coupling to anti-aligned nuclear spins. As ϕ increases, this effect is rapidly suppressed owing to the crystal field terms. This accounts for the rapid decrease of Bc as a function of increasing ϕ and the suppression of the increase of Bc towards zero temperature observed for ϕ = 0. c, Calculated critical field, Bc, as a function of field orientation, ϕ, with and without hyperfine coupling A, in which the Ising character is accounted for by a SIA acting on a Kramers moment. The critical field does not exhibit a substantial dependence on the hyperfine coupling. The sensitivity of Bc to changes of ϕ is greatly reduced, because ∣J∣ is essentially field independent (d). The tiny variation of ∣J∣ under these conditions, highlighted in the inset, reflects the hyperfine-induced entanglement of the electronic moment with the nuclear spin.
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Wendl, A., Eisenlohr, H., Rucker, F. et al. Emergence of mesoscale quantum phase transitions in a ferromagnet. Nature 609, 65–70 (2022). https://doi.org/10.1038/s41586-022-04995-5
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DOI: https://doi.org/10.1038/s41586-022-04995-5
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