Heavy fermion materials can often be tuned by various parameters such as pressure, doping, or magnetic fields. A schematic phase diagram for Ce-based heavy fermion systems under pressure is displayed in Fig. 1(a),^[33] based on the temperature–pressure phase diagram of CeCu₂(Si_1−xGe_x)₂ displayed in Fig. 1(b).^[34] The application of pressure first suppresses magnetic order to a quantum critical point (labelled p_c1), and the system enters a heavy fermion state. In this region, there is a dome of superconductivity mediated by spin fluctuations. This superconductivity is suppressed as pressure further increases the hybridization, but at higher pressures still, there is a second dome of superconductivity in the region of p_c2, well away from magnetic order. It has been proposed that the superconductivity in this second dome is mediated by valence fluctuations,^[33–35] suggesting that the system is in close proximity to a valence transition.

Fig. 1.

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	Fig. 1. (a) A schematic temperature–pressure phase diagram for Ce-based heavy fermion systems under pressure. Reprinted from Yuan et al., Ref. [33]. Copyright 2003 AAAS. (b) Temperature–pressure phase diagram of CeCu2(Si1−xGex)2 reproduced with permission from Yuan et al., Ref. [34]. Copyright 2006 by the American Physical Society.

One of the richest sources of novel physics in heavy fermion compounds has been the Ce_nT_mIn_3n+2m (T = transition metal) series of compounds. CeIn₃ is a cubic compound which orders antiferromagnetically at T_N = 10.2 K.^[36] The magnetic order is suppressed by applying pressure and T_N reaches zero at a critical pressure of around p_c = 2.6 GPa. In this region of the phase diagram, there is a dome of superconductivity with a maximum T_c less than 0.3 K.^[37] Following on from this, the CeTIn₅ (T = Co, Ir, or Rh) family of compounds were discovered, where layers of CeIn₃ are interspersed with TIn₂, leading to an overall much more two-dimensional layered structure.^[38–40] At ambient pressure, both CeCoIn₅ and CeIrIn₅ are superconductors with respective bulk T_c values of 2.2 K and 0.4 K, while CeRhIn₅ is an antiferromagnet with T_N = 3.8 K. The temperature–pressure phase diagram of CeRhIn₅ is displayed in Fig. 2.^[41,42] Upon applying pressure, T_N is suppressed and a superconducting dome emerges. The superconductivity and antiferromagnetism coexist up until p* = 1.75 GPa, where T_N and the superconducting T_c are equal. At higher pressures, T_N is not observed, but measurements in-field reveal the reemergence of magnetism and a quantum critical point at p_c = 2.3 GPa.^[42]

Fig. 2.

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	Fig. 2. Temperature–pressure phase diagram of CeRhIn5 reproduced with permission from Knebel et al., Ref. [41]. Copyright 2006 by the American Physical Society. The antiferromagnetism (AF) is suppressed by pressure until p* where the AF dome meets the superconducting (SC) dome. The quantum critical point pc lies at higher pressure within the SC dome.

Meanwhile the Ce₂TIn₈ (T = Co, Pd or Rh) series are another structural variant which also show either superconductivity at ambient pressure or antiferromagnetism with pressure induced superconductivity.^[43–45]

As a result, the Ce_nT_mIn_3n+2m families of compounds have been very important for studying heavy fermion behavior, magnetism, superconductivity, and the interplay between all of these phenomena. Furthermore, measurements under high applied magnetic fields have been vital for understanding many of the properties displayed in these materials. Below we give an overview of some of the results elucidated from such measurements under field.

A particular interest for studies of the CeTIn₅ materials has been to uncover how the Fermi surface evolves as the various interactions are tuned. CeRhIn₅ is different from both CeCoIn₅ and CeIrIn₅ in that at ambient pressure there is antiferromagnetic order, which indicates the dominance of the RKKY interaction and a weaker Kondo effect in CeRhIn₅. This is consistent with studies of the Fermi surface which show that the 4f electrons are well localized in CeRhIn₅, but in CeCoIn₅ and CeIrIn₅ they are itinerant and contribute to the Fermi surface.^[46–48] It is therefore said that CeRhIn₅ has a small Fermi surface, while the other two materials have large Fermi surfaces. It was found that in CeRhIn₅ at the quantum critical point p_c = 2.3 GPa, where antiferromagnetism is suppressed (Fig. 2), there is a sharp change of the dHvA frequencies and a dramatic increase of the effective quasiparticle mass.^[49] These new frequencies were found to be in good agreement with those of CeCoIn₅, suggesting that at the pressure induced quantum critical point there is a sudden Fermi surface reconstruction from a small Fermi surface to a large Fermi surface. This may correspond to the scenario of a local quantum critical point where there is a breakdown of the Kondo effect at p_c, below which the Kondo singlet does not fully form at the lowest temperatures.^[11–13]

It was therefore desirable to measure the effects of tuning CeRhIn₅ using high magnetic fields. This allows for a comparison between the uses of pressure and field as tuning parameters, and the robustness of the magnetism to applied field allows the change of Fermi surface as a function of field to be probed in the magnetic state.

The temperature–field phase diagram is displayed in Fig. 3 from Ref. [48], for fields applied along both the a-axis and c-axis. At lower fields, there is a significant anisotropy in the critical fields for the two field directions, as anticipated for a very two-dimensional layered structure. However, at higher fields the two critical field curves converge and the magnetic transition is suppressed to zero temperature at a field induced quantum critical point at B_c0 = 50 T. The more three-dimensional nature of the system at higher fields is reflected in the observation of T_N ∝ (B_c0−B)^2/3, which is the expected behavior for three-dimensional spin fluctuations.

Fig. 3.

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	Fig. 3. (a) Magnetic field dependence of the heat capacity of CeRhIn5 at various temperatures when the field is along the c-axis. (b) Field–temperature phase diagram of CeRhIn5 for fields along the a- and c-axes. AF and P denote antiferromagnetism and paramagnetism, respectively, while the subscripts mark a small (S) or large (L) Fermi surface. Reproduced from Jiao et al., Ref. [48].

Moreover, a clear reconstruction of the Fermi surface is found upon increasing the field, but in this instance it occurs well below the quantum critical point at B* = 30 T, deep inside the antiferromagnetic state. This is clearly illustrated in Fig. 4 for measurements using a piezo-cantilever up to 45 T. The upper inset of Fig. 4(a) shows the oscillations after subtracting the background, where there is a clear change around B* with much more rapid oscillations. To examine the field dependence of the Fermi surface, fast Fourier transforms were performed on the dHvA data over different field windows. As shown in Fig. 4(b), a number of new frequencies emerge at high fields above B*, where the frequencies have much larger values. A more detailed comparison is displayed in Fig. 5 from Ref. [51]. At low fields below B*, it can clearly be seen that the dHvA frequencies of CeRhIn₅ and LaRhIn₅ are very similar, again indicating that they have very similar Fermi surfaces. Since LaRhIn₅ does not have any f-electrons, this is consistent with previous reports that at low fields, the f-electrons of CeRhIn₅ are well localized. However as displayed in Fig. 5(c), the new frequencies above B* are quite close to those of CeCoIn₅, where the f-electrons are itinerant and contribute to the Fermi surface. This analysis indicates that at B* there is a reconstruction of the Fermi surface from a small Fermi surface at low fields to a large Fermi surface at higher fields, where the f-electrons become delocalized. This scenario is further supported by high field Hall resistivity measurements shown in Fig. 6. Upon reducing the temperature, two clear features emerge in the Hall coefficient as a function of field applied parallel to the c-axis. There is an anomaly observed at a field below B*, which is denoted by B_M and likely ascribed to a metamagnetic transition. Meanwhile at B* there is a dramatic drop of the Hall coefficient, which gets more pronounced at lower temperatures and this corresponds to a sudden change of carrier density, indicative of a Fermi surface reconstruction at B*. Following this discovery, a pronounced anisotropy of the in-plane resistivity was found to occur above B*,^[52] which was ascribed to a spin nematic phase, indicating that this change of electronic structure is accompanied by the emergence of novel phases of matter.

Fig. 4.

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Fig. 4. (a) dHvA measurements of CeRhIn5 at high fields parallel to the a-axis performed using a piezo-cantilever in a hybrid magnet up to 45 T. The main panel shows the field dependence of the dHvA amplitude of one of the new frequencies above B*. The inset at the top shows the oscillations after subtracting the background while the 1/B dependence is shown in the lower inset. (b) The Fourier transforms of the dHvA data for a field window below (10 T <B < 30 T) and above (45 T< B< 70 T) B*. Both reproduced from from Jiao et al., Ref. [48].

Fig. 5.

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	Fig. 5. A comparison between the fast Fourier transforms of the dHvA spectra of (a) CeRhIn5 at B >B* and B < B*, (b) CeRhIn5 at B < B* and LaRhIn5 from Ref. [41], and (c) CeRhIn5 at B >B* and CeCoIn5 from Ref. [50]. All reproduced from Jiao et al., Ref. [51].

Fig. 6.

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	Fig. 6. The Hall coefficient of CeRhIn5 as a function of field applied parallel to the c-axis, reproduced from Jiao et al., Ref. [48].

The observation of a field induced reconstruction of the Fermi surface within the antiferromagnetic state is markedly different to that when pressure is applied, where the Fermi surface reconstruction occurs at the quantum critical point. This indicates that the field induced quantum critical point is of the conventional SDW-type, while the pressure induced quantum critical point is the unconventional local type. The possibility of two different types of QCP found from tuning the same material with two different parameters needs to be explored further, in particular to find whether QCPs can be universally classified.

Compared to the CeTIn₅ materials, the Ce₂TIn₈ family has been much less frequently studied, but the presence of superconductivity in two members at ambient pressure makes them an interesting topic for study. Ce₂PdIn₈ is one of these materials, displaying superconductivity at T_c = 0.68 K.^[45] A further interesting feature is that an applied magnetic field is a good tuning parameter, and evidence was found for a field induced quantum critical point.^[53] At zero-field, a superconducting transition is observed in resistivity measurements, which is suppressed as a field is applied.

For fields greater than around 2.4 T, no superconducting transition is found down to 50 mK and the temperature dependence of the resistivity is linear. This indicates the presence of non-Fermi liquid behavior, instead of the ∼ T² dependence of Fermi liquid theory. Meanwhile, at higher fields, the Fermi liquid behavior is again observed and the temperature below which the ∼ T² dependence is found (T_FL) increases with increasing field, as displayed in the phase diagram in Fig. 7. It is also found that the coefficient of the ∼ T² behavior (A) sharply increases approaching the upper critical field. All these results give evidence for the presence of a field-induced quantum critical point coinciding with the suppression of superconductivity at the upper critical field of Ce₂PdIn₈. Similar behavior was suggested to occur in CeCoIn₅,^[54,55] pointing to a degree of universality between the nature of these two families of materials.

Fig. 7.

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	Fig. 7. Field–temperature phase diagram of Ce2PdIn8 obtained from resistivity measurements. The superconducting (SC), non-Fermi liquid (NFL), and Fermi liquid (FL) regions are all labeled. Figure reproduced from Dong et al., Ref. [53].

One particularly promising avenue for exploring the multiple types of QCP is to look for highly frustrated heavy fermion systems, since this has been proposed to promote the unconventional local QCP scenario.^[11–13] In a frustrated system, all contributions to the potential energy cannot be simultaneously minimized and two basic effects that introduce frustration to a system are lattice geometric frustration^[56] and frustration due to irreconcilable exchange interactions.^[57] For example, frustration in magnetic systems arises when the spin configurations cannot be optimized to minimize all the spin interaction energies. In a general heavy fermion phase diagram, besides tuning the hybridization strength between localized moments and the conduction electrons, frustration can add another dimension to the phase diagram that drives the system into intriguing emergent states, as illustrated in Fig. 8.^[58–60] Magnetic fields can tune a system in this two-dimensional phase diagram, but the exact role that an external magnetic field plays needs more theoretical studies and experimental investigations.

Fig. 8.

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	Fig. 8. Generic global phase diagram displaying the combined effects of Kondo coupling (K) and magnetic frustration, or quantum zero-point motion (Q), reproduced with permission from Custers et al., Ref. [60]. Copyright 2010 by the American Physical Society.

A good candidate for studying frustration is the heavy fermion compound CePdAl with a distorted Kagome lattice.^[61] By fitting six inequivalent nuclear Bragg peaks and thirteen inequivalent magnetic Bragg peaks obtained by powder neutron diffraction,^[62] it was concluded that CePdAl is a partially frustrated system in which only two thirds of the Ce moments form long-range antiferromagnetic order with a Neel temperature T_N = 2.7 K, while the other one third of the Ce moments are disordered due to geometric frustration. This magnetic structure is in agreement with a group theoretical analysis.^[62]

For the heavy fermion compound CePdAl, the frustrated moments are screened leading to a heavy fermion state at low temperatures and the intersite correlations between the frustrated moments are likely to be removed, which is supported by neutron scattering^[62] and NMR^[63] measurements. Upon applying a magnetic field, the Neel temperature is smoothly suppressed initially but goes through several metamagnetic transitions to AF₂ and AF₃ phases, in the field range of 3.2–4.0 T as shown in Fig. 9.^[64] A detailed resistivity and susceptibility study on CePdAl by Sun’s group^[65] also shows a similar magnetic field phase diagram, however, crossover behavior is reported to exist between the AF₃ and high field phases shown by the grey shading.

Fig. 9.

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Fig. 9. Magnetic phase diagram of CePdAl for B ∥ c obtained from measurements of different properties. The subscript i for αi and λi denotes the different axes a or c. Open circles indicate the approximate temperature of the broad shoulders of C4f/T below TN, gray squares the position of the maxima of the Schottky anomalies, and red open diamonds the temperature of the C4f maximum. The inset shows an enlarged view of the AF2 and AF3 phases. (b) Entropy S(B) at different fixed temperatures. (c) Frustration parameter fS = TS/TN vs. B. For B >Bc3, C4f/T at T = 0.5 K is plotted against B. All lines are guides to the eye. Figure reproduced with permission from Lucas et al., Ref. [64]. Copyright 2017 by the American Physical Society.

The hybridization strength can be tuned by field and it is possible to liberate the frustrated moments from Kondo screening prior to the suppression of AFM, yielding a metallic spin liquid state in a certain field range. A comprehensive entropy analysis of CePdAl as a function of field^[64] suggests that the frustration level is enhanced when crossing the phase boundary at B_c1 ∼ 3.25 T and reaches its highest value at B_c2 ∼ 3.4 T from AF₂ to AF₃ phase, confirming the boosted frustration and entropy as shown in Fig. 9. It is interpreted that the liberated Ce moments interact with the ordered AF moments and destabilize the AFM order at B_c1. In the AF₃ phase, the unstable frustrated state yields a different magnetic ground state with a lifting of the frustration above B_c3 ∼ 4.1 T. These results strongly suggest that the AF₂ phase in CePdAl is a promising candidate for a metallic spin liquid phase. However, further careful studies are necessary to address this possibility.

Characterizing the order parameter symmetry of the superconducting state is important for understanding the nature of the superconductivity and the microscopic mechanism leading to its formation and an important aspect of this is measuring the structure of the superconducting gap. In the case of the CeTIn₅ materials, the presence of line nodes, that is lines across the Fermi surface where the superconducting gap goes to zero, could be inferred in CeCoIn₅ and CeIrIn₅ from zero-field measurements of the specific heat,^[66] thermal conductivity,^[66] and magnetic penetration depth.^[67,68] However measurements in field allow for not only the presence of nodes to be ascertained but for their positions in k space to be probed.

In Fig. 10, the heat capacity of CeIrIn₅ at 2.05 GPa is displayed from Ref. [69] for fields applied at an azimuthal angle ϕ to the [100] direction. The heat capacity shows a clear four-fold oscillation upon rotating the field through 360°. Besides measurements at the highest field, all the measurements show a minimum along the [100] direction and a maximum along [110]. Although in a simple classical picture this would imply line nodes along [100], this only applies at very low temperatures and fields,^[70] and in fact the results suggest line nodes along [110]. This 45° shift of the positions of the minima and maxima is observed in CeCoIn₅ upon increasing the temperature in an applied field of 1 T.^[71] These results allow for the pairing state to be narrowed further, since although there are several unconventional pairing states with line nodes, only the d_x²−y² state has four-fold symmetry with nodes along [110] and therefore field dependent heat capacity and thermal conductivity measurements indicate the presence of a d_x²−y² state in CeCoIn₅ and CeIrIn₅.^[71–73]

Fig. 10.

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	Fig. 10. Field-angle dependence of the heat capacity of CeIrIn5 at 2.05 GPa, when a field is applied at an angle ϕ to the [100] direction. Figure reproduced with permission from Lu et al., Ref. [69]. Copyright 2002 by the American Physical Society.

Applied magnetic fields can also lead to additional novel behaviors in superconductors. The application of magnetic fields generally leads to the breaking of Cooper pairs by two mechanisms, the orbital effect and paramagnetic limiting. The orbital pair breaking effect arises from the kinetic energy of supercurrents surrounding the vortex flux lines making the superconducting state energetically unfavorable. Meanwhile the paramagnetic limiting effect arises from the Zeeman splitting of the electron bands. When this Zeeman energy becomes sufficiently large compared to the superconducting condensation energy, superconductivity will often be destroyed. Although in most superconducting systems the orbital limiting effect primarily determines the magnitude of the upper critical field, when the paramagnetic limiting mechanism dominates, novel phases have been predicted such as the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state.^[74,75] This state has a spatially modulated order parameter including a periodic arrangement of normal and superconducting regions, and the Cooper pairs have a finite center-of-mass momentum.

CeCoIn₅ had been considered to be a good candidate for the realization of the FFLO state, which has the required dominance of paramagnetic limiting and is in the clean limit.^[66,76] Furthermore it was found that at fields larger than 4.7 T applied along the c-axis (temperatures below 0.31T_c), the transition from the normal to superconducting state changes from second-order to first-order, which is expected for Pauli limited superconductors.^[76] In addition, at higher fields in the region where the superconducting to normal transition is first-order, an additional transition was found in specific heat measurements at lower temperatures in the superconducting state,^[77,78] which was taken as evidence for the FFLO state. In particular, it was found that this transition at T_FFLO disappeared when T_c < T_FFLO, demonstrating an intrinsic link between this transition and the superconducting phase. Meanwhile, while NMR studies all found distinct changes in this high field Q’-phase, conflicting results were also reported.^[79–81] In one study, the appearance of a new higher field resonance line was reported within the Q-phase, where the value of the Knight shift rapidly approached the normal state value with decreasing temperature.^[79] This was taken as evidence for the coexistence of normal and superconducting regions in this high field regime, and therefore indicated that this did indeed correspond to the FFLO phase. On the other hand, different results were reported in another NMR study, where it was found that the spin-susceptibility in the Q-phase was enhanced compared to that in the homogenous superconducting state, yet still less than the normal state value.^[80] This is inconsistent with the simple picture of alternating normal and superconducting regions in the FFLO state. It was also subsequently reported that the broadening of the NMR spectra was different for each of the crystallographic sites.^[81] Such a variation cannot be simply explained by the inhomogeneity arising from the modulation of the order parameter in the FFLO state, since the length scales of these variations are considerably larger than the unit cell, and therefore it would be expected that there should be no difference for the different sites. Instead, these results give evidence for the occurrence of long range magnetic order of localized Ce moments. These unusual results point to a very different relationship between superconducting and magnetism in CeCoIn₅, where instead of an antagonistic relationship between these phases, the occurrence of superconductivity is required for the appearance of long range magnetic order.

Direct evidence for this magnetic order was later reported in single crystal neutron diffraction measurements for fields applied along the [1-10] direction, where magnetic Bragg peaks corresponding to an ordering wavevector of (q, q, 0.5) were reported, with q ∼ 0.44 (Fig. 11).^[82] This magnetic structure has an incommensurate modulation within the ab-plane but is commensurate along the c-axis, and has a small ordered moment of around 0.15 μ_B/Ce aligned along the c-axis. The peak width was limited by the resolution of the instrument, showing that the correlation length of the magnetic order exceeded 60 nm, which is significantly larger than the vortex cores, thereby showing that the magnetism is not purely constrained to the core region. Importantly, the incommensurate modulation q is nearly field independent, and therefore inconsistent with the simple picture of the FFLO state, where the wave-vector of the real space modulation of the order parameter is expected to be proportional to the applied field. Further important findings were revealed by neutron scattering measurements along the [100] direction.^[83] Here a very similar ordering wave vector was found of (q, q, 0.5) with q ∼ 0.45, and therefore contrary to the assumptions of many theoretical proposals at the time, the ordering wave vector is constrained to lie along the nodal directions of the d_x²−y² order parameter, instead of rotating to be aligned with the field.

Fig. 11.

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Fig. 11. Neutron diffraction intensity at a position (h, h, 0.5) as a function of h for CeCoIn5 measured in various high fields close to Hc2. The black circles display the data in the superconducting state, while the grey circles are in the normal state. It can be seen that in the superconducting state the magnetic Bragg peak appears corresponding to the Q-phase, which is absent when the system becomes normal. Figure reproduced with permission from Kenzelmann et al., Ref. [82]. Copyright 2008 AAAS.

In general, for a magnetic ordering wave vector (q, q, 0.5), there are two possible SDW domains, one with Q_h = (q, q,0.5) and the other with Q_v = (q, −q,0.5). Remarkably, it was found from neutron diffraction measurements in the Q-phase with a field rotated in the ab-plane that only one of these domains is found to occur (Fig. 12).^[84] That is when a field is applied along the [1-11] direction, only the SDW domains with an ordering wave-vector perpendicular to the field direction Q_h are populated. Meanwhile as the field is rotated away from [1–10] again only the peaks corresponding to Q_h are observed, until the field is rotated beyond [100] where there is an abrupt change, and only the Q_v domains are populated (Fig. 12). Therefore there is always a single domain structure, even for fields along [100] when both domains should be equivalent. These highly unusual results were explained by theoretical proposals based on the presence of a spatially modulated p-wave pair density wave state, which was proposed to occur from the coupling between d-wave superconductivity and SDW phases.^[85,86] Here the additional p-wave order parameter has a pair of lobes separated by a line node. For a field along [1-10], the lobes of the p-wave order parameter align with the field, gapping the d-wave line node in the field direction, but leaving a nodal direction along [110]. Consequently, only the SDW domains with Q along the nodal direction (Q_h) are allowed to occur. Meanwhile the flipping of the direction of the p-wave order parameter in turn leads to a flipping of the populated SDW domain. Further evidence for this scenario was found from thermal conductivity measurements.^[87] Here distinct differences were seen for measurements when the heat current was parallel and perpendicular to the direction of the ordering vector of the populated domain. Furthermore these results could not be explained simply by the opening of a gap parallel to the SDW wave-vector, since they pointed to a larger gap being opened in the direction perpendicular to that of the populated SDW domain. This is therefore consistent with the pair density wave scenario, where the p-wave order parameter opens a gap along one of the nodal directions of the d-wave order parameter. This state with both d-wave superconductivity and a triplet pair density wave state has a modulation in real space, which arises due to the intricate connection between the superconducting and magnetic orders.^[84]

Fig. 12.

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Fig. 12. (a) The intensity of the neutron diffraction peaks of the two SDW vectors Qh = ( q, q,0.5) and Qv = (q, −q,0.5) as a function of the direction of the applied 11.4 T field in the vicinity of the [100] direction. Here the angle ϕ corresponds to the angle relative to [100]. It can be seen that upon tilting the field beyond [100], there is an abrupt depopulation of one SDW domain and complete population of the other. (b), (c) The hysteresis of this abrupt switch of the domains about the [100] direction, confirming that this is a first order process. All figures are reproduced with permission from Macmillan Publishers Ltd: Nature Physics (Ref. [84]), copyright 2014.

In intermetallic systems, the observation of quantum oscillations in high magnetic fields provides important information about the Fermi surface, while in insulators, the lack of electronic bands crossing the Fermi level should preclude the occurrence of quantum oscillations. However, this seemingly obvious truth was called into question following the discovery of quantum oscillations in the Kondo insulator SmB₆ via high field measurements of the dHvA effect.^[15–17]

Two aspects in particular appear to separate SmB₆ from regular band insulators. Firstly, SmB₆ has generally been considered to be a Kondo insulator where the insulating gap opens at low temperatures due to the strong electronic correlations. As a result of the strong hybridization between the lattice of 4f electrons and the conduction electrons, renormalized heavy bands with very large effective masses form at low temperatures, separated by a hybridization gap. In Kondo insulators, the Fermi level lies within this gap, yielding a low temperature insulating state.^[88] In SmB₆, this is manifested by an extremely large increase in the resistivity at low temperatures, which increases as the temperature is decreased.^[89,90] More recently, it was proposed that some Kondo insulators can also be characterized as topological insulators.^[14] In this case, the resulting insulating electronic structure emerging due to the strong hybridization has a non-trivial Z₂ index. Here the bulk band structure corresponds to a different topological index to the vacuum, which has a trivial topology. As a result, at the boundary between the bulk and vacuum, i.e., the surface of the sample, the insulating gap must vanish, and therefore conducting metallic surface states are anticipated. One of the longstanding mysteries of SmB₆ is that below around 4 K, rather than continuing to decrease with decreasing temperature, the resistivity plateaus and changes little as the material is further cooled. Recent careful transport measurements have suggested that these “in-gap” states in fact correspond to surface conduction, providing experimental evidence for conducting surface states and therefore for SmB₆ being a topological Kondo insulator.^{[18–20,91–93]}

Besides the anomalous low temperature conduction, another unusual observation for an insulator found in SmB₆ is that of quantum oscillations via measurements of the dHvA effect. This phenomenon has been reported by two different research groups using torque magnetometry, on samples grown by two different methods, as displayed in Fig. 13.^[15,16] However, the two different studies came to strikingly different conclusions about the origins of these oscillations. In the case of Li et al.,^[15] the oscillation frequencies were reported to increase strikingly upon rotating the field by 45°, as displayed in Fig. 14. This was taken as a clear indication of the two-dimensionality of the Fermi surface, which is evidence that the quantum oscillations arise from the metallic surface states, rather than the insulating bulk.

Fig. 13.

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	Fig. 13. Quantum oscillations in the topological Kondo insulator SmB6 from measurements of the field dependence of the torque magnetometry. The measurements were performed by two different groups, with panel (a) reprinted with permission from Li et al., Ref. [15]; copyright 2014 AAAS, and panel (b) reprinted with permission from Tan et al., Ref. [16]; copyright 2015 AAAS.

Fig. 14.

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Fig. 14. Angular dependence of the quantum oscillation frequencies of SmB6 from Ref. [15] (a) and Ref. [16] (b). In the case of the former, the angular dependence was reported as being consistent with a 2D Fermi surface, as expected if the quantum oscillations arise from a surface state. On the other hand, the angular dependence of the frequencies in panel (b) was reported to be much more consistent with three-dimensional Fermi surfaces, pointing to a bulk origin. Panel (a) is reprinted with permission from Li et al., Ref. [15]; copyright 2014 AAAS, and panel (b) is reprinted with permission from Tan et al., Ref. [16]; copyright 2015 AAAS.

On the other hand, Tan et al.^[16] also observed quantum oscillations, which were reported over a wider frequency range, up to values larger than 10k_BT. It was suggested by the authors that such large frequencies cannot be accounted for by surface states, due to the need for a correspondingly large mean-free path longer than the depth of the surface layers. Moreover, the angular dependences of the frequencies, which are also displayed in Fig. 14, were reported to be much more three-dimensional, and hence were ascribed to the bulk of the sample. It was also suggested that the Fermi surface is rather similar to other rare-earth hexaborides such as LaB₆, CeB₆, and PrB₆. It was pointed out that due to the robustness of the insulating gap of SmB₆ up to high fields of at least 45 T, this cannot be accounted for by quantum oscillations from a gapless high field state. Consequently, the origin of the quantum oscillations in SmB₆ are hotly contested, and a number of theoretical proposals have been made based on both surface^[94] and bulk^[95,96] origins for this unusual phenomenon.

Topological Kondo insulators are examples of gapped topological materials, where there is a gap in the topologically non-trivial bulk electronic structure, and the Fermi level lies within the gap. Recently, gapless topological semimetals with strong electronic correlations have also been increasingly studied. Examples of gapless topological materials include Dirac and Weyl semimetals, which have bulk band structures hosting linear band crossings, where the nearby excitations are well described by either Dirac or Weyl fermions, respectively.^{[22,23,97–100]} It is of particular interest then to explore examples of such topological materials with strong electronic correlations, to examine the interplay of these two effects.

The R(Sb,Bi) (R = rare earth) family of materials are examples where these effects have been studied, in particular due to the potential for tuning the correlations and electronic structure by substituting with different rare earth elements. Studies of this series of materials were spurred by the finding that LaSb shows a plateau in the resistivity at low temperatures, which was suggested to be similar to that seen in SmB₆,^[101] although angle resolved photoemission spectroscopy (ARPES) studies cannot agree whether there is a non-trivial topology.^[102,103]

High field measurements have been important for probing the non-trivial topology in these materials in two principal ways. Firstly, studies of quantum oscillations either using the dHvA or SdH effects can be utilized to probe the non-trivial Fermi surface, in particular they can be used to look for a non-trivial Berry phase. This can be performed by analyzing the Landau indices, whereby the oscillations are examined as a function of 1/B, the positions of either the oscillation maxima or minima are indexed, and the data are fitted linearly to determine the residual value. Here the value of the residual Landau index provides information about whether or not there is a non-trivial Berry phase, as well as the dimensionality of the Fermi surface.^[24,25] For these measurements, high magnetic fields are particularly important, so as to probe the low lying Landau indices, allowing for an accurate determination of the intercept. The analysis of SdH oscillations in SmSb revealed a residual Landau index corresponding to a non-trivial Berry phase, which could be determined with a low degree of uncertainty due to magnetoresistance measurements being performed in pulsed fields up to 60 T.^[27] Moreover, at low temperatures there was a significant deviation in the temperature dependence of the SdH oscillation amplitudes which could not be accounted for by the Lifshitz–Kosevich expression. A deviation from Lifshitz–Kosevich theory has also been observed in some dHvA measurements of SmB₆,^[16] but in the case of SmSb, anomalous behavior is only seen in SdH oscillations, whereas dHvA measurements show standard behavior. On the other hand, other RSb compounds such as PrSb show evidence for a trivial Berry phase from the Landau index analysis,^[104] suggesting that substituting for different rare earth elements can tune the band topology.^[105,106]

Another important probe for looking for Weyl fermions is from the presence of the chiral anomaly, which can be inferred from a negative longitudinal magnetoresistance. Weyl semimetals host Weyl points with different chiralities, and the chiral anomaly arises due to the parallel magnetic and electric fields pumping charges from one Weyl point to another of opposite chirality, which in turn leads to a population imbalance.^[28,29] Evidence for Weyl fermions was found in the magnetic Kondo system CeSb from the observation of signatures of the chiral anomaly from the measurements of the longitudinal magnetoresistance. CeSb orders antiferromagnetically at low temperatures, but as shown in Fig. 15(a), in the field-induced ferromagnetic state a negative longitudinal magnetoresistance is observed, where a slight rotation of the field away from the current direction destroys this effect.^[31] This scenario is accounted for by electronic structure calculations, which show that in zero-field there is a non-trivial band structure with band inversions.^[31] In the field-induced ferromagnetic state, the Zeeman field splits the degeneracy of the bands, except at non-trivial band crossing points corresponding to Weyl nodes, which is illustrated in Fig. 15(b).

Fig. 15.

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Fig. 15. (a) Resistivity of CeSb as a function of field, when the applied field is parallel to the current direction. At low temperatures, a negative longitudinal magnetoresistance is observed after the transition from the antiferro-ferromagnetic (AFF) state to the field-induced ferromagnetic (FM) state. (b) Schematic diagram of how Weyl nodes occur in the field induced FM state of CeSb. The spin–orbit coupling gaps out the degenerate band crossing points, giving rise to a band inversion. The breaking of time reversal symmetry in the field-induced FM state splits the degenerate bands, giving rise to doubly degenerate band crossing points, which correspond to Weyl nodes. Both panels are reproduced from Guo et al., Ref. [31].

Meanwhile evidence for Weyl fermions from magnetotransport measurements was also observed in the heavy fermion semimetal YbPtBi,^[32] where the extremely large Sommerfeld coefficient of around 8 J⋅mol⁻¹⋅K⁻² indicates that the electronic bands are strongly renormalized at low temperatures due to the Kondo interaction.^[107] At elevated temperatures above 20 K, the magnetotransport shows clear signatures of the chiral anomaly, exhibiting the expected temperature, field, field-angle, and doping dependences.^[32] Furthermore, the presence of Weyl fermions is supported by the electronic structure calculations and ARPES measurements, showing evidence for triply degenerate fermion points in zero field, which are each split into a Weyl point and a trivial crossing in applied fields. At low temperatures, the evidence for the chiral anomaly disappears, yet as displayed in Fig. 16, evidence for Weyl fermions is found from a pronounced topological Hall effect, as well as a T³ dependence of the specific heat, which can be accounted for by the strong renormalization of the electronic bands dramatically reducing the Fermi velocity v*. This evolution of the Weyl points as the charge carriers pick up a large effective mass at low temperatures, is illustrated in Fig. 17,^[32] where the greatly reduced v* due to the Kondo effect leads to a negligible chiral anomaly contribution to the conductivity c_a, but a sizeable specific heat term C_e/T³ ∼ (1/v*)³. As such, YbPtBi offers the prospect of studying the interplay of topology, strong correlations, and quantum criticality.

Fig. 16.

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Fig. 16. Evidence for Weyl fermions at low temperatures in the heavy fermion state of YbPtBi. (a) Anomalous contribution to the Hall resistivity, after subtracting the normal contribution from the charge carriers. (b) Topological Hall angle at different temperatures after subtracting a contribution from the Hall resistivity proportional to the magnetization. The significant value at low temperatures arises from the Berry curvature generated by the Weyl nodes. (c) Temperature dependence of the specific heat as Cp/T in various applied fields. The solid line shows a fit to a Kondo model, which cannot describe the low temperature data. (d) Cp/T vs. T2 in various applied fields. The linear relationship reveals a cubic temperature dependence of the specific heat, which is a signature of the linear dispersion of heavy fermion bands expected in the vicinity of Weyl points. Figure reproduced from Guo et al., Ref. [32].

Fig. 17.

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Fig. 17. Schematic diagram to illustrate the evolution of Weyl fermions in YbPtBi, as the temperature is reduced and the Kondo interaction becomes stronger. The dramatic decrease of the Fermi velocity v* at low temperatures leads to the disappearance of the chiral anomaly contribution ca, but the large entropy associated with the renormalized bands leads to a sizeable cubic term in the temperature dependence of the specific heat Ce. Meanwhile a sizeable topological Hall effect contribution can be detected in both the low and intermediate temperature ranges, arising from the Berry curvature generated by the Weyl points. Figure reproduced from Guo et al., Ref. [32].